![4 1 Introduction to mathematical epidemiology
1.1 A brief history of infectious diseases
In the year 2020, everybody–at home on their couch–became an infectious disease
expert. To calibrate all this newly gained knowledge, it seems a good idea to briefly
reiterate the basic concepts and nomenclature of infectious diseases. Infectious dis-
eases are spread by either bacterial or viral agents and are ever-present in society
[8]. Every once in a while, this may result in outbreaks that have a significant impact
on a local or global level. Depending on their spatial and temporal spread, we can
classify outbreaks as endemic, epidemic, or pandemic [28]. Endemic outbreaks, for
example chickenpox, are permanently present in a region or a population. Epidemic
outbreaks, for example the seasonal flu, affects a lot of people in a short period of
time, spread across several communities, and then disappear. Pandemic outbreaks,
for example the Spanish flu, are epidemic outbreaks that affects a lot of people in a
short period of time and spread across the entire world. Mathematical modeling of
infectious diseases is important to understand their outbreak dynamics and inform
political decision making to manage their spread [4].
Table 1.1 History of recent infectious disease outbreaks. Time period, type of disease, number
of deaths, and location.
period disease deaths location
1346 - 1350 Black Death 100,000,000 1/3 of Europe
1665 - 1666 Great Plague 100,000 1/4 of London
1918 - 1920 Spanish flu 50,000,000 worldwide
1980 measles 2,600,000 worldwide / year
2003 SARS 774 worldwide
2009 - 2010 H1N1 18,500 worldwide
2011 tuberculosis 1,400,000 worldwide / year
HIV/AIDS 1,200,000 worldwide / year
malaria 627,000 worldwide / year
2011 measles 160,000 worldwide (-94%)
2012 - 2020 MERS 866 worldwide
seasonal flu 35,000 United States / year
2014 - 2016 ebola 11,000 Africa
2018 ebola 2,280 Congo
2019 -2021 COVID-19 3,300,000 worldwide
The COVID-19 pandemic. On March 11, 2020, Tedros Adhanom Ghebreyesus, the
Director-General of the World Health Organization, declared the COVID-19 outbreak
a global pandemic [49]. On that day, it had affected 126,702 people worldwide. What
nobody could have foreseen is that within the following year, by March 11, 2021,
this number had increased by three orders of magnitude, to 118.57 million [12].
Naturally, we often think of the COVID-19 pandemic as the deadliest and most
devastating infectious disease in modern history.](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-21-320.jpg)
![6 1 Introduction to mathematical epidemiology
Table 1.1 summarizes recent infectious disease outbreaks ranging from the Black
Death in the 14th century with an estimated 100 million deaths across Europe to the
current COVID-19 pandemic with 3.3 million deaths worldwide to date. Notably, in
1980, the annual death toll of the measles with 2.6 million was of comparable size.
Today, after massive vaccination campaigns, this number has dropped significantly,
by 94% to 160,000 [5].
Fig. 1.1 The Lessons of the Pandemic. In this famous Science publication, George A. Soper
reflects on the Spanish flu that resulted in 50 million deaths worldwide in 1918 and 1919.
Figure 1.1 shows the cover page of the Science publication The Lessons from the
Pandemic from May 1919, a reflection on the scientific understanding of the Spanish
flu [29]. While some of it is specific to the nature of the 1918 influenza and the time
during which it occurred, much of it still applies to the COVID-19 pandemic today:
The Spanish flu lasted from 1918 to 1920 and, similar to COVID-19, occurred in
multiple waves. Both, the Spanish flu and COVID-19, are contagious respiratory
illnesses that spread from person to person, mainly by droplets, through cough,
sneeze, or talk. Naturally, increased hygiene, mask wearing, and physical distancing
in addition to strict isolation and quarantine are successful strategies to manage both
conditions [10]. While COVID-19 is caused by a coronavirus, SARS-CoV-2, the
1918 influenza was caused by the H1N1 influenza A virus, a virus of avian origin.
Within 25 months, it infected 500 million people, one third of the world’s population,
and resulted in more than 50 million deaths [5]. The major differences between the
Spanish flu and COVID-19 are their high risk populations and the mechanisms of
death: In contrast to the Spanish flu, which affected mainly healthy adults between
25 and 40 years of age, COVID-19 affects mainly individuals of 65 years and older
with comorbidities. Victims of the 1918 influenza mainly died from secondary
bacterial pneumonia, whereas victims of COVID-19 die from an overactive immune
response that results in organ failure [32]. Nevertheless, comparing the COVID-
19 outbreak to previous pandemics can provide insight and guidance to manage
the current COVID-19 pandemic. An important element of this comparison are
mathematical and computational tools that have been designed to quantify and
explain the spreading mechanisms of infectious diseases. This is the objective of
mathematical and computational epidemiology [3].](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-23-320.jpg)
![1.2 Introduction to epidemiology 7
1.2 Introduction to epidemiology
Epidemiology is the study of distributions, patterns, and determinants of health-
related events in human populations [3]. It is a cornerstone of public health , and
shapes policy decisions by identifying risk factors for outbreaks and targets for
prevention [28].
Epidemiology literally means the study of what is upon the people. It is
derived from the greek words epi, meaning upon demos, meaning people, and
logos meaning study, suggesting that it applies only to human populations. By
this definition, epidemiology is the scientific, systematic, data-driven study of
the distribution, i.e., where, who, and when, and determinants, i.e., causes and
risk factors, of health-related patterns and events in specified populations, i.e.,
the world, a country, state, county, city, school, neighborhood. Epidemiology
also includes the study of outbreak dynamics, outbreak control, and informing
political decision making.
Agent, host, and environment. A key premise in epidemiology is that health-related
events are not evenly distributed in a population; rather, they affect some individuals
more than others [3]. An important goal of epidemiology is to identify the causes that
put these individuals at a higher risk [2]. A simple but popular model to analyze and
explain disease causation is the epidemiologic triangle. The epidemiologic triangle
summarizes the interplay of the three components that contribute to the spread of an
infectious disease: an external agent, a susceptible host, and an environment in which
agent and host interact [9]. Descriptive epidemiologists characterize this interaction
as the seed, the soil, and the climate [42]. Effective public health measures assess
all three components and their interactions to control or prevent the spreading of a
disease. Figure 1.2 illustrates the epidemiologic triangle of COVID-19 with agent,
host, and environment. For COVID-19, the agent is the SARS-CoV-2 virus, the
hosts are people, and the environment are droplets. Interventions between any two
of these three components can help reduce the spread of the disease [20]. For exam-
ple, reduced exposure, vaccination, and antiviral treatment can modulate agent-host
interactions and reduce the number of new infections.
From descriptive to mathematical epidemiology. Mathematical models of infec-
tious diseases date back to Daniel Bernoulli’s model for smallpox in 1760 [5], and
they have been developed and improved extensively since the 1920s [14]. In the
middle of the 19th century, the English physician John Snow conducted a famous
series of experiments of the cholera outbreak in London to discover the cause of the
disease and to prevent its recurrence [46]. Because his research illustrates the classic
sequence–from descriptive epidemiology and hypothesis generation, to mathemat-
ical epidemiology and hypothesis testing–John Snow is considered the father of
modern epidemiology. Figure 1.4 summarizes the experiments of John Snow that
ended the cholera outbreak in London by removing the handle of a public water](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-24-320.jpg)
![8 1 Introduction to mathematical epidemiology
Fig. 1.2 Epidemiologic triangle of COVID-19 with agent, host, and environment. The epidemi-
ologic triangle illustrates the interplay of the three components that contribute to the spread of a
disease: an external agent, a susceptible host, and an environment in which agent and host interact.
For COVID-19, the agent is the SARS-CoV-2 virus, the hosts are people, and the environment are
droplets. Interventions between any of these three components can reduce the spread of the disease.
Fig. 1.3 Daniel Bernoulli is considered the most famous
mathematician of the Bernoulli family, although he actually
studied medicine. He was born on February 8, 1700 in
Groningen, the Netherlands, and is best known for his
applications of mathematics to mechanics. To prove the
efficacy of vaccination against smallpox, he proposed the
first compartment model in 1760 [5]. It only had two
compartments, but demonstrated the potential of modeling
to understand the mechanisms of transmission, predict future
spread, and control the outbreak through vaccination. Daniel
Bernoulli died on March 27, 1782 in Basel, Switzerland.
pump. Today the field of epidemiology is considered a quantitative discipline that
uses rigorous mathematical tools of probability and statistics to develop and test
hypotheses to understand and explain health-related events [3]. In simple terms,
today’s epidemiologists count, scale, and compare. They count the number of cases;
scale this count by a characteristic population to define fractions; and compare the
evolution of these fractions over time [9]. Mathematical epidemiology has advanced
significantly throughout the past decades [11], and models have become more so-
phisticated and complex [39]. This implies that in epidemiology today, is often no
longer possible to solve epidemiological models analytically and in closed form [3].
From mathematical to computational epidemiology. Computational epidemiol-
ogy is a multidisciplinary field that integrates mathematics, computer science, and
public health to better understand central questions in epidemiology such as the](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-25-320.jpg)
![1.2 Introduction to epidemiology 9
Fig. 1.4 The English physician John Snow is considered one
of the founders of modern epidemiology, in part because of
his research during the cholera outbreak in London in 1954
[46]. John Snow lived from March 15, 1813 to June 16, 1858.
He questioned the widely accepted paradigm that cholera
would spread by polluted bad air. He postulated that water
would be the cause of the cholera outbreak in Soho, London
in 1854. By talking to local residents, he identified the source
of the outbreak as the public water pump on Broad Street.
Although his chemical and microscopic observations were
inconclusive, his spreading patterns of the cholera outbreak
were convincing enough to persuade the local council to
disable a street pump on Broad Street by removing its handle.
There is a common belief that this action marked the ending
of the outbreak.
spread of diseases or the effectiveness of public health interventions [22]. Real-time
epidemiology is a rapidly developing area within computational epidemiology that
seeks to support policy makers–in real time–as an outbreak is unfolding. An impor-
tant application of real-time epidemiology is disease surveillance, the data-driven
collection, analysis, and interpretation of large volumes of disease data from a va-
riety of sources. This information can help to evaluate the effectiveness of control
and preventative health measures [12]. Central to computational epidemiology is
the accurate knowledge of people who are affected by an infectious disease at any
given point in time. During the COVID-19 pandemic, for the first time in history,
this information has been collected conscientiously, shared publicly, updated daily,
and made freely available in real time [13].
Sensitivity and specificity. A diagnostic tests, for example, the nasal swab
test, can be inaccurate in two ways: a false positive result erroneously labels
a healthy person as infected, resulting in unnecessary quarantine and contact
tracing; a false negative result oversees an infected person, resulting in the
risk of infecting others.
sensitivity = true positives / all sick people
Sensitivity measures the proportion of positives, diseased people, that are
correctly identified and not overlooked.
specificity = true negatives / all healthy people
Specificity measures the proportion of negatives, healthy people, that are
correctly identified and not classified as diseased. A perfect test has 100%
sensitivity and 100% specificity.](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-26-320.jpg)
![1.4 The basic reproduction number 11
Fig. 1.5 Testing for COVID-19. The objective of COVID-19
testing is to probe whether an individual is currently or
has previously been infected with SARS-CoV-2. A nasal or
throat swab test is a diagnostic test that provides information
about an acute infection. Diagnostic testing is critical to
provide early treatment, quarantine individuals, and trace
and isolate their contacts to reduce the spread of the virus. A
blood test is an antibody test that provides information about
a previous infection. Antibody tests are critical to estimate
the overall dimension of an outbreak. Throughout this book,
we use COVID-19 case data from public dashboards based
on confirmed positive diagnostic tests.
An antibody test can detect antibodies that are made by your immune system in
response to a previous infection with SARS-CoV-2. In contrast to genetic material
from the virus or specific proteins made by the virus, antibodies can take several
days or even weeks to develop, but they can remain present for several weeks or
months after recovery. A serology test is an antibody test that looks for antibodies in
blood samples. These venous blood samples are typically collected at the doctor’s
office or in the clinic. Antibody tests should not be used to diagnose an active infec-
tion. Instead, antibody testing is important to understand the kinetics of the immune
response to infection, clarify whether an infection protects from future infection,
characterize how long immunity will last, and estimate the overall dimension of an
outbreak.
Table 1.2 contrasts the three most common types of tests for COVID-19. Throughout
this book, we use COVID-19 case data from public dashboards based on confirmed
positive diagnostic tests. Only in Chapters 8 and 13, where we explore the effects of
asymptomatic transmission, we use specialized models that combine case data from
diagnostic tests with seroprevalence data from antibody tests.
1.4 The basic reproduction number
The basic reproduction number is a powerful but simple concept to explain the
contagiousness and transmissibility of an infectious disease [6]. For decades, epi-
demiologists have successfully used the basic reproduction number to quantify how
many new infections a single infectious individual creates in an otherwise com-
pletely susceptible population [6]. During the COVID-19 pandemic, the public me-
dia, scientists, and political decision makers across the globe have adopted the basic
reproduction number as an illustrative metric to explain and justify the need for dif-
ferent outbreak control strategies [10]: An outbreak will continue for reproduction
numbers larger than one, R0 > 1, and come to an end for reproduction numbers
smaller than one, R0 < 1 [8]. However, especially in the midst of a global pandemic,
it is difficult–if not impossible–to measure R0 directly [48].](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-28-320.jpg)
![12 1 Introduction to mathematical epidemiology
Table 1.3 Basic reproduction numbers and herd immunity thresholds for common infectious
diseases. Herd immunity is the indirect protection from an infectious disease that occurs when a
large fraction of the population has become immune. The herd immunity threshold, H = 1 − 1/R0,
beyond which this protection occurs is a function of the basic reproduction number R0.
disease R0 H disease R0 H
measles 12 - 18 92 - 95% mumps 4.0 - 7.0 75 - 86%
pertussis 12 - 17 92 - 94% COVID-19 2.0 - 6.0 50 - 83%
rubella 6 - 7 83 - 86% SARS 2.0 - 5.0 50 - 80%
smallpox 6 - 7 83 - 86% ebola 1.5 - 2.5 33 - 60%
polio 5 - 7 80 - 86% influenza 1.5 - 1.8 33 - 44%
Table 1.3 summarizes the basic reproduction numbers for common infectious dis-
eases. It varies from R0 = 1.5 − 1.8, for less contagious diseases like influenza, to
R0 = 12 − 18 for the measles and pertussis. Knowing the precise value of R0 is
important, but challenging, because of limited testing, inconsistent reporting, and
incomplete data [6]. Throughout this book, instead of measuring the basic repro-
duction number directly, we estimate it using mathematical modeling and reported
case data [17]. Mathematical models interpret the basic reproduction number R0
as the ratio between the infectious period C, the period during which an infectious
individual can infect others, and the contact period B, the average time it takes to
come into contact with another individual [8],
R0 = C/B . (1.1)
The longer a person is infectious, and the more contacts the person has during this
time, the larger the reproduction number [6]. While we cannot control the infectious
period C, we can change our behavior to increase the contact period B [17]. This is
precisely what community mitigation strategies and political interventions seek to
attempt.
The reproduction number of COVID-19. Since the beginning of the coronavirus
pandemic, no other number has been discussed more controversially than the re-
production number of COVID-19 [22]. The earliest COVID-19 study that followed
the first 425 cases of the Wuhan outbreak via direct contact tracing reported a basic
reproduction number of R0 = 2.2 [15]. However, especially during the early stages
of the outbreak, information was limited because of insufficient testing, changes
in case definitions, and overwhelmed healthcare systems. While the concept of R0
seems fairly simple, the reported basic reproduction numbers for COVID-19 vary
hugely with country, culture, calculation, and time [17]. Most basic reproduction
numbers of COVID-19 we see in the public media today are estimates of mathemat-
ical models. These estimates depend critically on the choice of the model, its initial
conditions, and many other modeling assumptions [6]. To no surprise, the mathe-
matically predicted basic reproduction numbers cover a wide range, from R0 = 2−4
for exponential growth models to R0 = 4 − 7 for more sophisticated compartment
models [22].](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-29-320.jpg)
![1.4 The basic reproduction number 13
Throughout this book, we identify or infer the reproduction number of COVID-
19–and the uncertainty associated with it–using computational epidemiology [22],
Bayesian analysis [10], and reported case data [12]: In the first example for the very
early COVID-19 outbreak in China in Section 7.8, we identify a basic reproduction
number of R0 = 12.58±3.17 across 30 Chinese provinces. In the second example for
the early outbreak in the United States in Section 10.2, we find a basic reproduction
number of R0 = 5.30 ± 0.95 across all 54 states and territories. This value suggests
that COVID-19 is less contagious than the measles and pertussis with R0 = 12 − 18,
as infectious as rubella, smallpox, polio, and mumps with R0 = 4 − 7, slightly more
infectious than SARS with R0 = 2 − 5, and more infectious than an influenza with
R0 = 1.5 − 1.8. Our basic reproduction number for the United States is significantly
lower than our basic reproduction number for China, which could be caused by
an increased awareness of COVID-19 transmission a few weeks into the global
pandemic. In our third example for the early outbreak in Europe in Section 12.2, the
basic reproduction number takes similar values of R0 = 4.62 ± 1.32 across all 27
countries.
During these early stages of exponential growth, with new case numbers doubling
within two or three days, the most urgent question amongst health care providers
and political decision makers was: Can we reduce the reproduction number? For the
broad public, this question became famously and illustratively rephrased as: Can we
flatten the curve [15]? For the modeling community, the quest for a lower reproduc-
tion number all of a sudden meant that traditional epidemiology models were no
longer suitable because of changes in the disease dynamics [12]. While traditional
models with static parameters were well-suited to model the outbreak dynamics
of unconstrained, freely evolving infectious diseases with fixed basic reproduction
numbers in the early 20th century, they fail capture how behavioral changes and po-
litical interventions can modulate the reproduction number to manage the COVID-19
pandemic in the 21st century [17]. In fact, static reproduction numbers are probably
the single most common cause of model failure in COVID-19 modeling [12].
Fortunately, several months into the pandemic, most countries have successfully
managed to flatten the new-case-number curves and the reproduction numbers have
dropped to values closer to or below one. To model these changes in disease dynamics
and reproduction, in Section 7.7, we introduce a dynamic reproduction number,
R(t), that accounts for time-varying contact periods. In our fourth example for the
European Union in Section 12.2, we show that the initial basic reproduction number
of R0 = 4.22±1.69 dropped to an effective reproduction number of R(t) = 0.67±0.18
by mid May 2020. Using machine learning, we correlate mobility and reproduction
and identify the responsiveness between the drop in air traffic, driving, walking,
and transit mobility and the drop in reproduction to ∆t = 17.24 ± 2.00 days. In
the final examples in Sections 13.2 and 13.3 of nine locations across the world,
we systematically infer the dynamic reproduction number R(t) throughout a time
window of 100 days while accounting for both symptomatic and asymptomatic
transmission.
From the failure of traditional static epidemiology models [15], we have now
learned that we need to introduce dynamic time-varying model parameters if we](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-30-320.jpg)
![14 1 Introduction to mathematical epidemiology
Fig. 1.6 Basic reproduction numbers and herd immunity thresholds for common infectious
diseases. Herd immunity is the indirect protection from an infectious disease that occurs when a
large fraction of the population has become immune, either through previous infection or through
vaccination. The black line highlights the herd immunity threshold, H = 1 − 1/R0, beyond which
this protection occurs, as a function of the basic reproduction number R0. The herd immunity
threshold varies from 33-44% for influenza to 92-95% for the measles.
want to correctly model behavioral and political changes and reproduce the reported
case numbers [17]. This naturally introduces a lot of freedom, a large number of
unknowns, and a high level of uncertainty. However, in stark contrast to the epidemic
outbreaks in the early 20th century, we now have thoroughly-reported case data and
the appropriate tools [23] to address this challenge. The massive amount of COVID-
19 case data, well documented and freely available, has induced a clear paradigm
shift from traditional mathematical epidemiology towards data-driven, physics-based
modeling of infectious disease [1]. This new technology naturally learns the most
probable model parameters–in real time–from the continuously emerging case data,
allows us to make projections into the future, and quantifies the uncertainty on the
estimated parameters and predictions [26].
1.5 Concept of herd immunity
An important consequence of the basic reproduction number R0 is the condition
for herd immunity [9]. Herd immunity describes the indirect protection from an
infectious disease that occurs when a large fraction of the population has become
immune, either through previous infection or through vaccination [3]. The critical
threshold at which the disease reaches this endemic steady state is called the herd
immunity threshold,
H = 1 − 1/R0 . (1.2)
The larger the basic reproduction number R0, the higher the herd immunity threshold
H. Table 1.3 and Figure 1.6 summarize the basic reproduction numbers R0 for](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-31-320.jpg)
![1.6 Concept of immunization 15
several common infectious diseases along with the estimates of their herd immunity
thresholds H. The herd immunity threshold varies from 33-44% for influenza to 92-
95% for the measles. For the reported basic reproduction numbers of R0 = 2.0 − 6.0
of COVID-19, the estimated herd immunity threshold would range from 50-83%.
Recent studies that account for the emerging new and more infectious B.1.351 and
B.1.1.7 variants of COVID-19 estimate these values to 75-95% [20].
1.6 Concept of immunization
To prevent or revert an epidemic outbreak, we need to ensure that, on average,
every infectious individual infects less than one new individual. The concept of herd
immunity describes the natural path towards ending an outbreak. However, if the
basic reproduction number R0 is large, the herd immunity threshold H = 1 − 1/R0 is
high, and waiting for herd immunity through infection alone can be quite devastating.
Vaccination is a powerful strategy to accelerate the path towards herd immunity [1].
Fig. 1.7 Vaccination against COVID-19. The objective of
COVID-19 vaccination is to provide immunity against severe
acute respiratory syndrome coronavirus 2 or SARS-CoV-2,
the virus that causes COVID-19 [30]. By May, 2021,
thirteen vaccines were authorized for public use: two RNA
vaccines (Pfizer–BioNTech and Moderna), five conventional
inactivated vaccines (BBIBP-CorV, CoronaVac, Covaxin,
WIBP-CorV and CoviVac), four viral vector vaccines
(Sputnik V, Oxford–AstraZeneca, Convidecia, and Johnson
& Johnson), and two protein subunit vaccines (EpiVacCorona
and RBD-Dimer); and more than a billion doses of COVID-
19 vaccines were administered worldwide.
Immunization threshold. Vaccination effectively reduces the susceptible population
S. Successfully immunizing a fraction I of the population, reduces the susceptible
population from S to S∗
0 = [ 1 − I ] S0 and, with it, the reproduction number from R0
to R∗
0 = [ 1 − I ] R0. The critical immunization threshold I, below which the effective
basic reproduction number is smaller than one, R∗
0 = [ 1 − I ] R0 < 1, defines the
fraction of the population that needs to be immunized to prevent or revert the outbreak
of an epidemic,
I > 1 − 1/R0 . (1.3)
From Table 1.3, we conclude that the required immunization fraction varies signif-
icantly, from I > 92 − 95% for the measles with a basic reproduction number of
R0 = 12 − 18 to I > 33 − 44% for the common influenza with a basic reproduction
number of R0 = 1.5−1.8. This explains, at least in part, why some infectious diseases
are a lot more difficult to control through immunization than others.](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-32-320.jpg)
![16 1 Introduction to mathematical epidemiology
Eradication through vaccination. Once enough individuals are immunized–either
through infection or through vaccination–the outbreak stops. A disease that stops
circulating in a specific region is considered eliminated in that region. Polio, for
example, was eliminated in the United States by 1979 after widespread vaccination
efforts. A disease that is eliminated worldwide is considered eradicated. Eradicating
a disease through vaccination is a desirable but elusive goal [1]. Malaria has been
a candidate for eradication, but although its incidence has been drastically reduced
through vaccination, completely eradicating it remains challenging because infection
does not result in life long immunity. Polio has been eliminated in most countries
through massive vaccination efforts, but still remains present in some regions be-
cause its early symptoms often remain unnoticed and infected individuals continue to
infect others. Measles have been the target of widespread vaccination, but although
the disease is highly recognizable through its characteristic rash, a long latent pe-
riod from exposure to the first onset of symptoms complicates outbreak control. To
this day, smallpox is the only human infectious disease that has been successfully
eradicated through vaccination. The eradication of smallpox is the result of focused
surveillance, rapid identification, and ring vaccination [8]. In a massive vaccina-
tion campaign launched in 1967, anyone who could have possibly been exposed to
smallpox was quickly identified and vaccinated to prevent its further spread. The
last known case of smallpox occurred in Somalia in 1977. In 1980 World Health
Organization declared smallpox eradicated. The eradication of smallpox remains one
of the most notable and profound public health successes in history.
Efficacy and risk ratio. The efficacy e of a vaccine is the relative reduction in the
disease attack rate between the unvaccinated placebo group npla and the vaccinated
group nvac. For a randomized trial with an equal allocation, meaning equally sized
placebo and vaccinated groups, npla = nvac, the efficacy is
e =
npla − nvac
npla
· 100% =
1 −
nvac
npla
· 100% = 1 − r . (1.4)
The risk ratio r is the ratio between the attack rate of the vaccinated group nvac and
the placebo group npla,
r =
nvac
npla
· 100% = 1 − e . (1.5)
The efficacy is an important measure to characterize the success of a vaccine and
define critical thresholds below which a vaccination trial should stop.
Table 1.4 Contingency table to quantify the significance of a vaccine. The table compares the
total number of vaccinated and placebo individuals that developed and did not develop the disease.
positive negative total
vaccine a b a+b
placebo c d c+d
total a+c b+d n](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-33-320.jpg)
![1.6 Concept of immunization 17
Example: Efficacy of the first COVID-19 vaccine. On November 9, 2020, the
Pfizer and BioNTech trial reported a number or COVID-19 cases of ncovid =
npla + nvac = 94 and an efficacy of e emin with emin = 90%. The efficacy
e of a vaccine is the relative reduction in the disease attack rate between the
unvaccinated placebo group npla and the vaccinated group nvac,
e =
1 −
nvac
npla
· 100% = 1 − r with r =
nvac
npla
· 100% = 1 − e .
where r is the risk ratio. The Pfizer and BioNTech trial was a randomized trial
with an equal allocation [30]. Although it did not report detailed numbers, we
can estimate the number of placebo and vaccinated cases npla and nvac and the
risk ratio r from the reported efficacy emin = 90% with nvac = ncovid − npla,
e =
1 −
ncovid − npla
npla
· 100% =
2 −
ncovid
npla
· 100% emin .
Solving for the number of placebo cases yields the general equation,
npla
ncovid
2 − emin
,
and, for the Pfizer and BioNTech case, npla ncovid/1.1 = 85.45. This implies
that for a total of ncovid = 94 COVID-19 positive cases, at a number of placebo
cases npla = 86 and vaccinated cases nvac = 8, the efficacy is larger than 90%.
Back-calculating the efficacy for these populations,
e =
1 −
nvac
npla
· 100% =
1 −
8
86
· 100% = 90.7% 90% = emin ,
confirms the simulation. According to the protocol, Pfizer and BioNTech
planned to take a look at the data at five stages with ncovid = 32,64,92,120,164
reported positive cases and only continue the trial if the efficacy was above
e ecrit with ecrit = 62.7%. From this information, we can calculate the
critical numbers npla and nvac below which the trial would have stopped at any
of the five stages. Solving for the number of placebo cases yields the general
equation,
npla
ncovid
2 − ecrit
,
and, for the Pfizer and BioNTech case, npla ncovid/1.373. This implies that for
ncovid = 32,64,92,120,164, the minimum number of placebo cases to continue
the trial was npla ≥ 24,47,68,88,120, for which the resulting efficacies of
e = 66.7%,63.8%,64.7%,63.6%,63.3% would all have been slightly above
the critical threshold of ecrit = 62.7%.](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-34-320.jpg)
![1.7 Mathematical modeling in epidemiology 19
positive negative total
vaccine 8 21,761 21,769
placebo 86 21,683 21,769
total 94 43,444 43,538
We can use the contingency table to estimate the efficiency of the Pfizer BioN-
Tech vaccine. Fisher’s exact test calculates the significance of the deviation
from a null hypothesis,
p =
21769! 21769! 94! 43444!
8! 21761! 86! 21683! 43538!
,
and confirms, for the Pfizer and BioNTech case with p 0.00001, that indi-
viduals in the vaccinated and placebo groups will not equally likely contract
COVID-19.
1.7 Mathematical modeling in epidemiology
During the early onset of the COVID-19 pandemic, all eyes were on mathematical
modeling with the general expectation that mathematical models could precisely
predict the trajectory of the pandemic. Mathematical modeling rapidly became front
and center to understanding the exponential increase of infections, the shortage of
ventilators, and the limited capacity of hospital beds; too rapidly as we now know.
Bold and catastrophic predictions not only initiated a massive press coverage, but
also a broad anxiety in the general population [15]. However, within only a few
weeks, the vastly different predictions and conflicting conclusions began to create
the impression that all mathematical models are generally unreliable and inherently
wrong [12]. While the failure of COVID-19 modeling–often by an order of mag-
nitude and more–was devastating for policymakers and public health practitioners,
initial mistakes are not new to the modeling community where an iterative cycle of
prediction, failure, and redesign is common standard and best practice [26]. However,
the successful use of mathematical models implies to set the expectations right [45].
Understanding what models can and cannot predict is critical to the Art of Mod-
eling. Epidemiologists distinguish two kinds of models to understand the outbreak
dynamics of an infectious disease: statistical models and mechanistic models [12].
Depending on the degree of complexity, the most popular mechanistic models are
compartment models and agent-based models.
Statistical models, or more precisely, purely statistical models, use machine learn-
ing or regression to analyze massive amounts of data and project the number of
infections into the future. The essential idea is to select a function D(t), use statisti-
cal tools to fit its coefficients to reported case data D̂(t), and make projections into
the future. The function can be quadratic, cubic, logistic, power-law, or exponential,](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-36-320.jpg)
![20 1 Introduction to mathematical epidemiology
Table 1.5 Mathematical modeling of COVID-19. Summary of the three most common models.
Statistical or forecasting models fit nonlinear functions to case data over time, whereas mechanistic
models, including compartment and agent-based models, simulate outbreak and contact dynamics.
Throughout this book, we use compartment models to simulate the outbreak dynamics of infectious
diseases including COVID-19.
statistical models mechanistic models
forecasting models compartment models agent-based models
use machine learning, statis-
tics, regression, or method of
least squares
use physics-based modeling
based on nonlinear reaction-
diffusion equations
use rule-based approaches to
study the interaction of au-
tonomous systems
model case numbers through
a nonlinear function
model population through
compartments
model every individual as an
independent agent
formulate number of cases as
a function of time
formulate rules by which in-
dividuals pass through the
compartments
formulate simple rules by
which individual agents in-
teract
fit coefficients that are purely
phenomenological
infer parameters that have a
mechanistic interpretation
identify parameters that
summarize human behavior
predicts case numbers from
fitting a function
predict outbreak dynam-
ics, characterize sensitivi-
ties, quantify uncertainties
predict outbreak dynamics
as emergent collective be-
havior of individual agents
no feedback mechanisms nonlinear feedback discrete contact networks
predictions are inexpensive,
but very unreliable
predictions are reliable only
for a small time window
predictions are detailed, but
computationally expensive
for example, D(ϑ,t) = exp(c0 + c1 t + c2 t2 + c3 t3), where D(ϑ,t) are the modeled
cumulative cases per day, ϑ = {c0,c1,c2,c3} are the model parameters, and t is
the time. One of the simplest statistical tools to compare the model D(ϑ,t) to the
data D̂(t) and identify values for its parameters ϑ is the method of least squares.
It is important to understand that these parameters are purely phenomenological,
they are derived purely by fitting a curve, and typically do not have a mechanistic
interpretation. Early in an outbreak, when little is known about disease transmission,
epidemiologists often use statistical models because they do not rely on any prior
knowledge of the disease. An example for a statistical model of COVID-19 is the ini-
tial IHME model [23]. Early in the pandemic, the IHME model used case data from
China and Italy to create similar curves, forecast case numbers in the United States,
and inform the White House’s response to the pandemic. Carefully constructed sta-
tistical frameworks can be used for short-time forecasting using machine learning or
regression. This could potentially be useful to understand how to allocate resources
or make rapid short-term recommendations. However, purely statistical models can
neither capture the dynamics of disease transmission nor the effects of mitigation
strategies. This explains, at least in part, why the early COVID-19 predictions based
on purely statistical models were off by an order of magnitude or more. To address
these serious limitations, several COVID-19 models have now been adjusted to com-
bine both statistical modeling and mechanistic modeling.
Mechanistic models simulate the outbreak through interacting disease mechanisms
by using local nonlinear population dynamics and global mixing of populations](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-37-320.jpg)
![1.7 Mathematical modeling in epidemiology 21
[4]. The underlying idea is to identify fundamental mechanisms that drive disease
dynamics, for example the duration of the infectious period or the number of con-
tacts an infectious individual has during this time. Unlike purely statistical models,
mechanistic models include important nonlinear feedback: The more people be-
come infected, the faster the disease spreads. By their very nature, the parameters
of mechanistic models are not just fitting parameters, they usually have a clear epi-
demiological interpretation. This makes mechanistic modeling a powerful strategy
to explore different outbreak scenarios or study how an outbreak would change
under various assumptions and political interventions [12]. Another advantage of
mechanistic models is that we can adjust and improve them dynamically as more
information becomes available. Throughout this book, we gradually improve a class
of mechanistic models by adding new information. For example, we introduce time-
varying dynamic contact rates that vary in different lockdown levels and add the
effect of asymptomatic transmission [26]. Even if we do not precisely know the
dimension of asymptomatic disease spread, we can use mechanistic models to study
what-if scenarios: What would the disease landscape look like if two third of all
infectious were asymptomatic? Mechanistic modeling naturally extends into sensi-
tivity analysis and uncertainty quantification. As such, it not only provides valuable
information about the robustness of the model, but also about the most effective
parameters to modulate a disease outbreak [19]. Rather than studying a single one
disease trajectory, we could explore a range of trajectories around the mean and
characterize the best- and worst-case scenarios. The two most popular mechanistic
models in epidemiology are compartment models and agent-based models, and both
have been used to understand the outbreak dynamics of COVID-19. When choosing
between compartment models and agent-based, it is important to understand the
major strengths and weaknesses of each model.
Compartment models are the most common approach to model the epidemiology
of an infectious disease [14]. Compartment models simulate the collective behavior
of subgroups of the population through a number of compartments with labels, for
example, SEIR for susceptible, exposed, infectious, and recovered. Individuals move
between compartments and the order of the labels indicates the successive motion,
for example, SEIS means susceptible, exposed, infectious, then susceptible again [4].
The underlying principle is to model the time evolution of these groups through a set
of coupled ordinary differential equations, identify rate constants that characterize
their interaction using reported case data, and vary these rate constants to probe
different outbreak scenarios.
Figure 1.8 illustrates a compartment model that represents the characteristic time-
line of COVID-19 through six compartments, the susceptible, exposed, infectious,
recovered, hospitalized, and dead groups. This SEIRHD model is defined through a
set of six ordinary differential equations that simulate how many individuals reside
in each compartment throughout the duration of the outbreak. The model parameters
of a compartment model define the transition rates between the individual compart-
ments and, for more complex models, the fraction of individuals that transition into
a particular path of the disease. For this example, the parameters ϑ = {β,α,γ,νh,νd}](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-38-320.jpg)
![22 1 Introduction to mathematical epidemiology
Fig. 1.8 Characteristic timeline of COVID-19. On day 0, a fraction of all susceptible individuals
is exposed to the virus. After a latent period of A = 1/α = 3 days, the exposed individuals become
infectious. After an infectious period of C = 1/γ = 10 days, a fraction (1 − νh) transition to
the recovered group, whereas a fraction νh develops severe symptoms and is hospitalized. Of the
hospitalized individuals, a fraction (1 − νd) recovers, whereas a fraction νd becomes dead. We can
simulate this behavior through an SEIRHD model with six compartments.
are the contact rate β, latent rate α, and infectious rate γ, and the hospitalized and
dead fractions, νh and νh · νd. From reported case numbers, hospitalizations, and
deaths, we can identify or infer the set of model parameters ϑ, the rate constants
and fractions, that best explain the model output using statistical tools. Importantly,
in contrast to purely statistical models, compartment models are based on model
parameters that have a clear physical interpretation. The most important parameters
of any compartment model are the contact rate β and the infectious rate γ. Together,
they define an important nonlinear feedback that is not present in purely statistical
models: The more people become infected, the faster the spread of the disease. The
basic reproduction number R0 = β/γ, the ratio of the contact rate β and the infec-
tious rate γ, characterizes the magnitude of this feedback. It is an easy-to-understand
disease metric that explains how quickly susceptible individuals become infected
and how fast a disease spreads across a population [6]. Some epidemiologists ar-
gue that, because of their mechanistic nature, compartment models are better suited
for long-term predictions than purely statistical models. While this might be true
for infectious diseases that develop freely, without any political intervention, the
COVID-19 pandemic has taught us that long-term predictions of outbreak dynamics
are challenging, even with the most sophisticated mechanistic models [15]. Under-
standing the potential and the limitations of compartment modeling is one of the
main objectives of this book.
In Chapters 2 and 3, we introduce two simple classical compartment models be-
fore we introduce the most common compartment model for COVID-19 in Chapter
4. We show how we can use these models to estimate the reproduction number
from reported COVID-19 case data. Knowing the precise reproduction number has
important consequences for estimating the dimensions of herd immunity and im-
munization. Compartment models capture the fundamental dynamics of disease
transmission and the effects of public health interventions; however, classical com-
partment models ignore the dynamics of the contact rate and its variation across
a population. In Section 7.7 we introduce a compartment model that explicitly ac-
counts for a time-varying dynamic contact rate, β(t), and captures a varying contact
behavior in different subgroups of the population. The dynamic nature of the contact](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-39-320.jpg)
![1.8 Data-driven modeling in epidemiology 23
rate naturally introduces a dynamic reproduction number, R(t) = β(t)/γ and allows
us to quantify the effectiveness of policy measures as we discuss in Section 12.2.
Agent-based models simulate individuals or agents interacting in various social
settings and estimate the spread of a disease as these agents come into contact with
one another. The underlying idea is to represent each agent individually, formulate
relatively simple rules by which individual agents interact, and interpret the collec-
tive behavior across all agents as the emergent dynamics of an outbreak. A strength
of agent-based models is that they simulate human behavior very granularly: They
can assign different parameters or behavior patterns to each individual agent instead
of simulating the collective behavior of entire populations. As such, agent-based
models offer a lot more freedom than compartment models, but also require a lot
more detail. For example, to formulate rules of interaction, agent-based models draw
on social connectivity networks, from activity surveys, cell phone locations, public
transportation, or airlines statistics. By their very nature, agent-based approaches are
computationally expensive, especially for large populations. For small populations,
agent-based modeling is a powerful strategy to predict how individual behavior, for
example the violation of quarantine, leads to a collective behavior and modulates
disease spread. For large populations, agent-based modeling can help rationalize
collective model parameters, for example contact rates or reproduction numbers,
that feed into more abstract, population level models. Above a certain population
size, agent-based models simply become computationally unfeasible and most epi-
demiologists would turn to a more macroscopic approach that represents groups of
individuals collectively as subgroups of the population. Since our objective is to
design and discuss data-driven models for the COVID-19 pandemic, throughout the
remainder of this book, we focus exclusively on compartment models.
When choosing between purely statistical, compartment, and agent-based models,
it is important to know upfront which questions the model should address [45]. Ta-
ble 1.5 compares the three models and summarizes their strengths and weaknesses.
Throughout this book, we focus on mechanistic compartment modeling.
1.8 Data-driven modeling in epidemiology
One year after the World Health Organization had declared the COVID-19 outbreak a
global pandemic, SARS-CoV-2 has resulted in more than 118 million reported cases
across more than 180 countries and over 2.6 million deaths worldwide. Unlike any
other disease in history, the COVID-19 pandemic has generated an unprecedented
volume of data, well documented, continuously updated, and broadly available to
the general public. There is a critical need for time- and cost-efficient strategies
to analyze and interpret these data to systematically manage the pandemic on a
global level. Yet, the precise role of physics-based modeling and machine learning
in providing quantitative insight into the dynamics of COVID-19 remains a topic of
ongoing debate.](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-40-320.jpg)
![24 1 Introduction to mathematical epidemiology
Physics-based modeling is a successful strategy to integrate multiscale, multi-
physics data and uncover mechanisms that explain the dynamics of specific outbreak
characteristics. However, physics-based modeling alone often fails to efficiently
combine large data sets from different sources and different levels of resolution [26].
Machine learning is as a powerful technique to integrate multimodality, multifidelity
data, from cities, counties, states, and countries across the world, and reveal cor-
relations between different disease phenomena. However, machine learning alone
ignores the fundamental laws of physics and can result in ill-posed problems or non-
physical solutions [1]. Throughout this book, we illustrate how data-driven modeling
can integrate classical physics-based modeling and machine learning to infer critical
disease parameters–in real time–from reported case data to make informed predic-
tions and guide political decision making. As a valuable by product, this approach
naturally lends itself in sensitivity analysis and uncertainty quantification. From the
COVID-19 pandemic, we have learnt that even small inaccuracies in the model can
trigger large changes in the number of cases. To understand the vulnerability of the
model to these small changes, especially in view of the varying reporting practices
of the COVID-19 case data, sensitivity analysis and quantifying uncertainty have
become critical elements of robust predictive modeling.
Epidemiology data. In data-driven modeling, we need data to fit or infer our model
parameters. Unlike earlier pandemics for which case data are often sparse, irregular,
or incomplete, the COVID-19 pandemic is amazingly well documented [13]. On
hundreds of public COVID-19 dashboards, we can find and download a vast variety
of case numbers: daily new cases, active cases, recovered cases, seriously critical
cases, cumulative cases, and deaths, at the level of cities, counties, states, countries,
or the entire world [9, 12, 25]. Local dashboards often also share the number of hos-
pitalizations and intensive care units, which were of great concern especially at the
early onset of the pandemic. More recently, these dashboards have also included the
number of tests and vaccines. Seroprevalence data with information about the history
of the disease, both asymptomatic and symptomatic, are rare and often only avail-
able from scientific publications rather than governmental databases. These data are
typically updated on a daily basis, and contain notable weekday-weekend alterations.
When using the data to infer model parameters and learn about the outbreak behav-
ior, we usually smoothen these alteration using seven-day moving averages. Finally,
to compare data from different locations, we typically scale the reported case data
by the population. A common metric for comparison is the seven-day-per-100,000
incidence, the number of new cases per 100,000 individuals across a seven-day win-
dow [38]. Policy makers across the globe use this incidence value to characterize the
severity of the outbreak and justify the need for political interventions.
Figure 1.9 illustrates a typical data set for the COVID-19 outbreak that we use to
infer our epidemiological model parameters. The orange lines summarize the out-
break dynamics worldwide throughout the year after the World Health Organization
had declared COVID-19 a global pandemic, from March 11, 2020 to March 11,
2021. The light orange lines represent the reported daily new cases, which display
notable weekday-weekend fluctuations associated with testing and reporting irreg-](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-41-320.jpg)
![1.8 Data-driven modeling in epidemiology 25
Fig. 1.9 Daily new cases of COVID-19 and their seven-day moving average. Daily new COVID-
19 cases are reported on public dashboards worldwide. The light orange curve represents the raw
data, the dark orange curve is the seven-day moving average, both reported throughout the year
after the WHO had declared COVID-19 a global pandemic on March 11, 2020.
ularities. The dark orange lines are the associated seven-day moving average ˆ
I(t),
which smoothens the fluctuations and displays a much clearer trend in the disease
dynamics. Within this one-year window, the absolute daily case numbers peaked
on January 8, 2021 with 841,304 new cases. The moving seven-day average peaked
on January 11, 2021 with ∆Imax = 745,404 cases. If we assume an infectious pe-
riod of C = 1/γ = 7 days, this would result in a maximum infectious population
of Imax = 7 · ∆Imax = 7 · 745,404 = 5,217,828, meaning that mid January 2021,
more than 5 million people were sick with COVID-19. For a total population of
N = 7.8 billion people, this corresponds to a peak seven-day-per-100,000 incidence
of Imax · 100,000/N = 67. There are many different ways to use the reported case
data. The simplest way is to select a function, use statistical tools to fit its coefficients
to the reported case data, and make projections into the future. However, not only the
daily new cases, but also the seven-day average in Figure 1.9 display substantial fluc-
tuation and it seems difficult to find a function that could explain the orange curves.
For data-driven modeling, it is often easier to use the total cumulative case numbers,
which always increase monotonically and tend to be more smooth in general.
Figure 1.10 shows the total cumulative cases of COVID-19 D̂(t) and a simple
statistical model D(t) to fit the data throughout the first year of the pandemic.
During the very early stages of an outbreak, an exponential growth model, D(t) =
D0 exp(G t), with a growth rate G often provides a good approximation of the total
number of cases D̂(t) and is easy to fit. Indeed, this, or similar exponential models,
is what many early approaches used. While the initial phase of the outbreak is
well represented by exponential growth models, they soon tend to overestimate the
outbreak. This is why, during the early COVID-19 pandemic, there was a broad
overestimate of the number of cases and of the number of ventilators and hospital
beds needed to tread diseased individuals [15]. In this book, instead of using purely
statistical models, we use mechanistic models like the compartment model in Figure](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-42-320.jpg)
![26 1 Introduction to mathematical epidemiology
Fig. 1.10 Total cumulative cases of COVID-19 and a statistical model to fit the data. Cumulative
COVID-19 cases, the sum of all daily new cases to that date, are reported on public dashboards
worldwide. The solid orange curve represents the raw data, the dashed orange curve is an example
of a statistical model, both reported throughout the year after the WHO had declared COVID-19 a
global pandemic on March 11, 2020.
1.8. For this SEIRHD compartment model, the number of total cases is the sum
of all infectious, hospitalized, recovered and dead individuals, I(t) + R(t) + H(t) +
D(t). Throughout this book, we use different types of compartment models, infer
their model parameters using reported COVID-19 case data from public databases
[9, 12, 25], seroprevalence data from scientific publications [26], and mobility data
[2, 10, 10], and vary the parameters to probe different outbreak scenarios.
Problems
1.1 Testing, testing, testing. On June 6, 2020, Donald J. Trump, the President of
the United States, famously said “Remember this, when you test more, you have
more cases.” What is wrong with this statement? What did he really mean to say?
Discuss the implications of the testing frequency on reported case numbers, and,
ultimately, on policy making.
1.2 Herd immunity. On September 15, 2020, Donald J. Trump, the President of the
United States, claimed that COVID-19 would go away, without a vaccine. “You’ll
develop herd. Like a herd mentality.” What did he really mean to say? Discuss the
effects of vaccination on herd immunity.
1.3 Herd immunity. Assume the basic reproduction number of the swine flu, caused
by the H1N1 virus, was on the order of R0 = 1.4 − 1.6. Calculate its herd immunity
threshold H and compare it against the herd immunity thresholds of other infectious
diseases in Table 1.3 and Figure 1.6. Comment on whether this makes the swine flu
a good candidate for eradication through vaccination.](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-43-320.jpg)
![1.8 Data-driven modeling in epidemiology 27
1.4 COVID-19 variants B.1.351 and B.1.1.7. Assume the basic reproduction num-
ber of COVID-19 in December 2020 was R0 = 2.0. Calculate the herd immunity
threshold H. Now, assume that the variant B.1.351 with a 50% increased basic re-
production number has been introduced into the population [20]. How does the herd
immunity threshold change? How would the variant B.1.1.7 with a 56% increased
basic reproduction number change the herd immunity threshold? Comment on your
results.
Fig. 1.11 The Lessons of the Pandemic. Twelve condensed rules to the the avoidance of unneces-
sary personal risks and to the promotion of better personal health by George A. Soper in reflection
of the Spanish flu in 1919.
1.5 The Lessons of the Pandemic. In his Science publication The Lessons of the
Pandemic, George A. Soper lists twelve public health measures to manage the Spanish
flu in 1919. Read the twelve recommendations in Figure 1.11 and comment on which
of them are still valid for the COVID-19 pandemic today.
1.6 Efficacy of the AstraZeneca COVID-19 vaccine. The randomized, equal allo-
cation AstraZeneca trial enrolled 11636 participants in its interim efficacy analysis,
5807 in the vaccinated group and 5829 in the unvaccinated placebo group. Of the
vaccinated group, 30 developed COVID-19, and of the placebo group 101. Calculate
the efficacy and risk ratio of the AstraZeneca vaccine.
1.7 Case rates during the AstraZeneca COVID-19 trial. The AstraZeneca trial en-
rolled 11636 participants in its interim efficacy analysis, 5807 in the vaccinated
group and 5829 in the unvaccinated placebo group. Of the vaccinated group, 30
developed COVID-19, and of the placebo group 101. Create a contingency table for
the interim efficacy analysis. Calculate the overall case rate of the trial and the case
rates in the vaccinated and placebo groups.
1.8 Efficacy of the Janssen COVID-19 vaccine. The randomized, equal allocation
Janssen COVID-19 trial enrolled 39,321 participants, 19,630 received the vaccine](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-44-320.jpg)
![Chapter 2
The classical SIS model
Abstract The SIS model is the simplest compartment model with only two pop-
ulations, the susceptible and infectious groups S and I. It characterizes infectious
diseases like the common cold or influenza that do not provide immunity upon
infection. While the SIS model is too simplistic to explain the outbreak dynamics
of complex infectious diseases, it is the only compartment model with an explicit
analytical solution for the time course of its populations. This makes it a popular
model to explain and illustrate the basic principles of compartment modeling. The
learning objectives of this chapter on classical SIS modeling are to
• explain concepts of mass action incidence and constant rate recovery
• interpret contact and infectious rates and periods
• solve the analytical solutions for the susceptible and infectious populations
• distinguish disease free and endemic equilibria
• analyze the final size relation
• demonstrate how the infectious rate, reproduction number, and initial conditions
modulate outbreak dynamics
• discuss limitations of classical SIS modeling
By the end of the chapter, you will be able to analyze, simulate, and predict the
outbreak dynamics of simple infectious diseases like the common cold or influenza
that do not provide immunity to reinfection.
2.1 Introduction of the SIS model
Some infectious diseases, for example from the common cold or influenza, do not
provide lifelong immunity upon recovery from infection [2, 2], and previously in-
fected individuals become susceptible again [10]. We can model their behavior
through the simplest of all compartment models, the SIS model [5]. Figure 11.4
shows that the SIS model consists of only two populations, the susceptible group S
and the infectious group I [3]. As such, it provides valuable insight into the concept
33
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
E. Kuhl, Computational Epidemiology, https://doi.org/10.1007/978-3-030-82890-5_2](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-49-320.jpg)
![34 2 The classical SIS model
Fig. 2.1 Classical SIS model. The classical SIS model contains two compartments for the sus-
ceptible and infectious populations, S and I. The transition rates between the compartments, the
contact and infectious rates, β and γ, are inverses of the contact and infectious periods, B = 1/β
and C = 1/γ.
of compartment modeling [8]. The SIS model is often used to illustrate the basic
features of compartment models because we can solve the dynamics of its two pop-
ulations, S(t) and I(t), analytically at any point t in time [7]. The SIS model also has
simple analytical solutions for the converged final sizes, S∞ and I∞, as t → ∞ [13].
For the transition from the susceptible to the infectious group, we assume a mass
action incidence [4], which implies that the rate of new infections is proportional
to the size of the susceptible and infectious groups S and I weighted by the contact
rate β, Û
I = β SI. For the transition from the infectious to the susceptible group, we
Fig. 2.2 The transition from the susceptible to the infectious
group is based on the assumption of mass action incidence
for which the rate of new infections is proportional to the size
of the susceptible and infectious groups S and I, weighted
by the contact rate β, Û
I = β SI. The transition from the
infectious to the susceptible group is based on the assumption
of constant rate recovery for which the rate of recovered
infections is proportional to the size of the infectious group
I, weighted by the infectious rate γ, Û
I = −γI.
assume a constant rate recovery, which implies that the rate of recovered infections
is proportional to the size of the infectious group I weighted by the infectious rate
γ, Û
I = γI. Figure 2.2 illustrates the dynamics of these two assumptions that result in
the following system of two coupled ordinary differential equations,
Û
S = − β SI + γ I
Û
I = + β SI − γ I .
(2.1)
The transition rates between both compartments are the contact rate β and the
infectious rate γ in units [1/days], which are the inverses of the contact period
B = 1/β and the infectious period C = 1/γ in units [days]. The ratio between the
contact and infectious rates, or similarly, between the infectious and contact periods,
defines the basic reproduction number R0,
R0 =
β
γ
=
C
B
. (2.2)
In this simple format, the SIS model (2.1) neglects all vital dynamics, Û
S + Û
I 0 and
S + I = const. = 1. It does not account for births, natural deaths, or death from the
disease.](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-50-320.jpg)
![2.2 Analytical solution of the SIS model 35
2.2 Analytical solution of the SIS model
From the condition of non-vital dynamics, S = 1 − I, we can rephrase the system of
equations of the SIS model (2.1) in terms of only one independent variable, the size
of the infectious group I, governed by the following nonlinear ordinary differential
equation,
Û
I = β [ 1 − I ] I − γ I = [ β − γ ] I
1 −
I
1 − γ/β
. (2.3)
Equation (2.3) is a logistic differential equation of the form,
Û
I = r I [ 1 − I/K ] with r = β − γ and K = 1 − γ/β, (2.4)
where K is the carrying capacity. This type of equation has an explicit analytical
solution,
I(t) =
K I0
I0 + [ K − I0 ] exp (−r t)
, (2.5)
where I0 = I(0) is the initial infectious population [10]. It proves convenient to
reparameterize the equation for the infectious population (2.5) in terms of the basic
reproduction number R0 = β/γ and the infectious period C = 1/γ with r = [R0 −
1]/C and K = 1 − 1/R0. This provides the analytical solution for the SIS model in
terms of the infectious period C, the basic reproduction number R0, and the initial
infectious population I0,
S(t) = 1−
[1 − 1/R0] I0
I0 + [1 − 1/R0 − I0] exp ([1 − R0] t/C)
I(t) =
[1 − 1/R0] I0
I0 + [1 − 1/R0 − I0] exp ([1 − R0] t/C)
.
(2.6)
Figures 2.3, 2.4, and 2.5 highlight the outbreak dynamics of the SIS model, solved
analytically using equations (2.6), for the time period of one year. The three figures
demonstrate the sensitivity of the SIS model for varying infectious periods C, and
varying basic reproduction numbers R0, and initial infectious populations I0. Increas-
ing the infectious period C delays convergence to the endemic equilibrium, but the
final sizes S∞ and I∞ remain unchanged. Increasing the basic reproduction number
R0 accelerates convergence to the endemic equilibrium, decreases S∞, and increases
I∞. Increasing the initial infectious population I0 accelerates the onset of the out-
break, but the final sizes S∞ and I∞ remain unchanged. Interestingly, increasing the
initial exposed population I0 by an order of magnitude shifts the population dynam-
ics by a constant time increment, for the current parameterization by 50 days. This
highlights the exponential nature of the SIS model, which causes a constant acceler-
ation of the outbreak for a logarithmic increase of the initial infectious population,
while the overall outbreak dynamics remain the same.](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-51-320.jpg)
![36 2 The classical SIS model
Fig. 2.3 Classical SIS model. Sensitivity with respect to the infectious period C. Increasing
the infectious period C delays convergence to the endemic equilibrium, but the final sizes S∞ and
I∞ remain unchanged. Basic reproduction number R0 = 2.0, initial infectious population I0 = 0.01,
and infectious period C = 5, 10, 15, 20, 25, 30 days.
Fig. 2.4 Classical SIS model. Sensitivity with respect to the basic reproduction number R0.
Increasing the basic reproduction number R0 accelerates convergence to the endemic equilibrium,
decreases S∞, and increases I∞. Infectious period C = 20 days, initial infectious population
I0 = 0.01, and basic reproduction number R0 = 1.5, 1.7, 2.0, 2.4, 3.0, 5.0, 10.0.
2.3 Final size relation of the SIS model
For practical purposes, it is interesting to estimate the final susceptible and infectious
populations S∞ and I∞ [15]. From the analytical solution (2.5),
I∞ =
K I0
I0 + [ K − I0 ] exp (−r t∞)
with r = β − γ and K = 1 − γ/β, (2.7)](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-52-320.jpg)
![2.3 Final size relation of the SIS model 37
Fig. 2.5 Classical SIS model. Sensitivity with respect to the initial infectious population I0.
Increasing the initial infectious population I0 accelerates the onset of the outbreak, but the final
sizes S∞ and I∞ remain unchanged. Infectious period C = 20 days, basic reproduction number
R0 = 2.0, and initial infectious population I0 = 10−1, 10−2, 10−3, 10−4, 10−5, 10−6.
it is easy to show that we can distinguish two cases. If there is a finite initial infectious
population, I0 0, the infectious population will converge to either zero, for β γ,
r 0, and exp (−r t∞) → ∞, or to K for β γ, r 0, and exp (−r t∞) → 0 [2, 9],
thus
I∞ =
0 if β γ
K if β γ .
(2.8)
We can rephrase these limit conditions (2.8) in terms of the basic reproduction
number R0. In classical epidemiology, the two converged states are known as the
disease-free equilibrium and the endemic equilibrium [3]. The equations that define
the final converged populations of the endemic equilibrium state are the final size
relation.
R0 1 disease-free equilibrium: S∞ = 1 and I∞ = 0
R0 1 endemic equilibrium: S∞ = 1/R0 and I∞ = 1 − 1/R0 .
(2.9)
Figure 2.6 illustrates the final size relation and emphasizes the role of the basic
reproduction number R0 as the distinguishing outbreak characteristic between the
disease-free and endemic equilibrium. For reproduction numbers smaller than one,
R0 1, the SIS model converges to a disease-free equilibrium with S∞ = 1 and
I∞ = 0. For reproduction numbers larger than one, R0 1, the SIS model converges
to an endemic equilibrium with S∞ = 1/R0 and I∞ = 1 − 1/R0. The larger the
reproduction number R0, the smaller the final susceptible population S∞ and the
larger the final infectious population I∞ [3].](https://image.slidesharecdn.com/23116792-250602081652-27d136a5/85/Computational-Epidemiology-Datadriven-Modeling-Of-Covid19-Ellen-Kuhl-53-320.jpg)
Computational Epidemiology Datadriven Modeling Of Covid19 Ellen Kuhl
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- 5. Computational Epidemiology Ellen Kuhl Data-Driven Modeling of COVID-19
- 7. Ellen Kuhl Computational Epidemiology Data-Driven Modeling of COVID-19
- 8. Ellen Kuhl Mechanical Engineering Stanford University Stanford, CA, USA ISBN 978-3-030-82889-9 ISBN 978-3-030-82890-5 (eBook) https://doi.org/10.1007/978-3-030-82890-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
- 9. To my mom, with whom I had the most scientific discussions about COVID-19.
- 10. Foreword Nothing will ever be quite the same. The Great COVID-19 Pandemic that started in December 2019 will be remem- bered as the main event of the first part of the 21st Century. It affected all parts of the world, it laid bare social inequalities, revealed the excesses of modern life, shook all strata of society, and changed profoundly how individuals, groups, and nations interact with each other. Early in the crisis, it was realized that theoretical epidemiology and modeling are essential tools to confront the problem. As Daniel Bernoulli stated in the first-ever epidemiological studies of 1760: "I simply wish that, in a matter which so closely concerns the wellbeing of the human race, no decision shall be made without all the knowledge which a little analysis and calculation can provide". By January 2020, the first ‘little analysis and calculation’ provided by epidemiological studies revealed the seriousness of the situation and scientists around the world rang the alarm bell. As the crisis escalated, the attention of the media, governments, and the public turned towards modelers with burning questions: how bad is it? how will it evolve? how many cases should we anticipate? will the hospitals be overwhelmed? how many deaths? what can we do? Like many people in academia, Ellen Kuhl found herself naturally interested in the problem. As an extreme athlete, a marathoner, a triathlete and an iron-(wo)man, Ellen is fearless. Early on, she realized that the methods she had been developing to understand neurodegenerative diseases could be readily adapted to the evolving crisis. As a scientist, Ellen combines a unique ability for modeling, great technical skills, and a wonderful intuition for good problems. She quickly built elegant data- driven models for how the disease spreads around different parts of the world through the airline network that soon became landmark studies. As an educator, she also realized that students in Stanford would gain from learning more about the science behind the disease. Rapidly, she put together a new course that was taught in the 2020 winter term, as the second wave was gathering strength. In her course that I attended and enjoyed tremendously, she taught both the basics of epidemiology and her own research as well as many topics related to the wider social and political vii
- 11. viii Foreword impacts of the crisis. This book is the combined result of her course, state-of-the-art research, and general reflections. The first part of the book is a fast-paced self-contained review of the fundamental ideas of epidemiology. It is full of wonderful stories, biographies, and anecdotes that brings life to the topic. It introduces and defines important concepts such as the reproduction number, herd immunity, and vaccine efficacy. On the mathematical side, it is centered around the development of the the famous SIR model that captures the evolution of three homogeneous populations of people susceptible (S ), Infected (Is ) and Recovered (R) going back to the work of Kermack and McKendrick in 1927. The model will tell you that for a reproductive number R larger than one, after an initial exponential growth, there will be saturation when enough of the population has been infected before the disease eventually runs its course. This classic material is updated for our computational world in Part II where Ellen shows how to implement efficiently the classic theory and how it can be generalized to problems that are directly relevant to the COVID crisis. In particular, in Chapter 7 and 8 two extra populations, the exposed (E) and asymptomatic (Ia), are introduced to capture the propagation of the disease through populations that do not display yet signs of infections, a crucial feature of this disease. As Ellen shows, this simple system of four or five differential equations is already enough to capture many features of the disease and can be used to understand early disease outbreaks. The third part of the book brings the topic of epidemiology to current research level. One of the key features of a pandemic is the propagation of the disease from one region to another. In our hyper-connected world there are many ways diseases can travel. At the global level, the airline network is a natural way for the disease to spread especially through people who show no symptoms. Ellen shows how the SEIR model can then be extended to a network and efficiently implemented. Remarkably, this simple idea is sufficient to understand the early spread and different phases of the disease. Yet, a key question remains. How do we use daily data about the number of cases to validate and infer parameters for the various models? Part IV of the book addresses directly this question and brings the topic to the forefront of current data research by presenting Bayesian inference methods and how it can be efficiently applied to the COVID crisis. This data-driven approach combined with the network models bring state-of-the-art techniques to the study of the disease dynamics and shows how secondary data, such as mobility data from cell phones, can be used as a barometer to predict the development of the disease. Through the fog of war, governments have made many mistakes that have cost countless lives. As Anne Frank wrote: “What is done cannot be undone, but at least one can keep it from happening again". How do we keep it from happening? We learn from our mistakes, sharpen our tools and models and confront the current and next crises with the full power of science. Ellen’s wonderful book gives us the ideas, methods, tools, knowledge, and concepts to understand the current pandemic and be prepared for the next one. Oxford, May 2021 Alain Goriely
- 12. Preface The objective of this book is to understand the outbreak dynamics of the COVID-19 pandemic through the lens of computational modeling. Computational modeling can provide valuable insights into the dynamics and control of a global pandemic to guide public health decisions. So... why didn’t it? Why did dozens of computational models make predictions that were orders of magnitude off? This book seeks to an- swer this question by integrating innovative concepts of mathematical epidemiology, computational modeling, physics-based simulation, and probabilistic programming. We illustrate how we can infer critical disease parameters–in real time–from reported case data to make informed predictions and guide political decision making. We crit- ically discuss questions that COVID-19 models can and cannot answer and showcase controversial decisions around the early outbreak dynamics, outbreak control, and gradual return to normal. As scientists, it is our ethic responsibility to educate the public to ask the right questions and to communicate the limitations of our answers. Throughout this book, we will create data-driven models for COVID-19 to do so. Who is this book for? If you are a student, educator, basic scientist, or medical researcher in the natural or social sciences, or someone passionate about big data and human health: This book is for you! Don’t worry, this book is introductory and doesn’t require a deep knowledge in epidemiology. A fascination for numbers and a general excitement for physics-based modeling, data science, and public health are a lot more important. And this is why this book is both, a textbook for undergraduates and graduate students, and a monograph for researchers and scientists. As a textbook, this book can be used in the mathematical life sciences suitable for courses in applied mathematics, biomedical engineering, biostatistics, computer science, data science, epidemiology, health sciences, machine learning, mathematical biology, numerical methods, and probabilistic programming. As a monograph, this book integrates the basic fundamentals of mathematical and computational epidemiology with modern concepts of data-driven modeling and probabilistic programming. It serves researchers in epidemiology and public health, with timely examples of computational modeling, and scientists in the data science and machine learning, with applications to COVID-19 and human health. ix
- 13. x Preface What does this book cover? This book consists of four main parts that gradually build from mathematical epidemiology via computational and network epidemiology to data-driven epidemiology. Part I. Mathematical Epidemiology introduces the basic concepts of epidemiology in view of the COVID-19 pandemic with the objectives to understand the cause of an infectious disease, predict its outbreak dynamics, and design strategies to control it. We introduce the paradigm of compartment modeling and revisit analytical solutions for the classical SIS, SIR, and SEIR models using outbreak data of the COVID-19 pandemic. Typical problems include estimating the herd immunity threshold and efficacy of different COVID-19 vaccines and the growth rate, basic reproduction number, contact period of the COVID-19 outbreak at different locations. Part II. Computational Epidemiology introduces numerical methods for ordinary differential equations and applies these methods to discretize, linearize, and solve the governing equations of SIS, SIR, SEIR, and SEIIR models to interpret the case data of COVID-19. Typical problems include simulating the first wave of the COVID-19 outbreak using reported case data and understanding the effects of early community spreading and outbreak control. We discuss why classical epidemiology models have failed to predict the outbreak dynamics of COVID-19 and introduce new concepts of data-driven dynamic contact rates and serology-informed asymptomatic transmission to address these shortcomings. Part III. Network Epidemiology discusses numerical methods for partial differ- ential equations and applies these methods to discretize, linearize, and solve the spreading of infectious diseases using discrete mobility networks and finite element models. Instead of solving the outbreak dynamics of COVID-19 locally for each region, state, or country, we now allow individuals to travel and populations to mix globally, informed by cell phone mobility data and air travel statistics. Typi- cal problems include understanding the early outbreak patterns, the effect of travel restrictions, and the risk of reopening after lockdown. Part IV. Data-driven Epidemiology covers the most timely methods and applica- tions of this book. It focuses on probabilistic programming with the objectives to understand, predict, and control the outbreak dynamics of the COVID-19 pandemic. We integrate computational epidemiology and data-driven modeling to explore dis- ease data in view of different compartment models using a probabilistic approach and quantify the uncertainties of our analysis. Typical problems include inferring the reproduction dynamics, visualizing the effects of asymptomatic transmission, and correlating case data and mobility. This book is by no means complete. It does not cover agent-based modeling, age- dependent modeling, population mixing, purely statistical, stochastic, or probabilistic modeling, forecasting, and many other key aspects of computational epidemiology. Instead, it is a personal reflection on the role of data-driven modeling during the COVID-19 pandemic, motivated by the curiosity to understand it. Because Science. Stanford, May 2021 Ellen Kuhl
- 14. Acknowledgements Now, that’s a wrap! I started writing this book on January 14, 2021, during the United States peak of the COVID-19 pandemic and finished on May 31, 2021, at the lowest incidence in 14 months. Dozens of people have contributed to this book, directly or indirectly, through endless discussions around the pandemic. I was fortunate to collaborate with an amazing international team, Alain Goriely from the UK, Francisco Sahli Costabal from Chile, Henry van den Bedem from the Netherlands, Kevin Linka from Germany, Mathias Peirlinck from Belgium, Paris Perdikaris from Greece, and Proton Rahman from Canada. The daily discussions of our local outbreak dynamics–while never meeting in person–were an essential part of my pandemic life made possible by Zoom. Thank you for sharing this unique experience! I enjoyed regular COVID-19 meetings with my scientific friends around the world, Silvia Budday, Krishna Garikipati, Tom Hughes, Tinsley Oden, Paul Steinmann, Tarek Zohdi, and the IMAG/MSM Working Group for Multiscale Modeling and Viral Pandemics. But the true motivation for this book came from the students of the new course Data-driven modeling of COVID-19 that we created at Stanford University in the Fall of 2020 to make online learning a bit more tangible. The enrollment ranged from undergraduates in biology, classics, computer science, engineering, ethics, human biology, mechanical engineering, and Spanish, to master and PhD students in aeronautics, astronautics, computer science, environment and resources, management science and engineering, and mechanical engineering. Together, we designed this course as the pandemic unfolded–in real time–and this book is a collection of class notes and feedback from students, guest speakers, and lecturers. We received support from the Stanford Bio-X Program and the School of Engineering COVID-19 Research and Assistance Fund. Massive thanks to all students in the class, to our amazing course assistants, Amelie Schäfer, Oguz Tikenogullari, Mathias Peirlinck, and Kevin Linka, and to my favorite guest speaker, Alain Goriely. Last but not least, I thank the true heroes of the pandemic, Jasper and Syb, and all the kids who not only had to bear the uncertainties of a global pandemic, but also endure their parents working from home. Thank you, Henry, for getting through this together, with 354 miles of swimming, 3611 miles of biking, and 3279 miles of running. This has been a truly memorable time. I’m glad it’s over–at least for now. xi
- 15. Contents Part I Mathematical epidemiology 1 Introduction to mathematical epidemiology . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 A brief history of infectious diseases . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Introduction to epidemiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Testing, testing, testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 The basic reproduction number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Concept of herd immunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Concept of immunization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.7 Mathematical modeling in epidemiology . . . . . . . . . . . . . . . . . . . . . . . 19 1.8 Data-driven modeling in epidemiology. . . . . . . . . . . . . . . . . . . . . . . . . 23 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 The classical SIS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1 Introduction of the SIS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Analytical solution of the SIS model. . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 Final size relation of the SIS model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 The classical SIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1 Introduction of the SIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Growth rate of the SIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Maximum infection of the SIR model . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Final size relation of the SIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5 The Kermack-McKendrick theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 xiii
- 16. xiv Contents 4 The classical SEIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Introduction of the SEIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Growth rate of the SEIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3 Maximum exposure and infection of the SEIR model . . . . . . . . . . . . . 66 4.4 Final size relation of the SEIR model . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Part II Computational epidemiology 5 Introduction to computational epidemiology . . . . . . . . . . . . . . . . . . . . . . 81 5.1 Numerical methods for ordinary differential equations . . . . . . . . . . . . 81 5.2 Explicit time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3 Implicit time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.4 Comparison of explicit and implicit time integration . . . . . . . . . . . . . 92 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6 The computational SIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.1 Explicit time integration of the SIR model . . . . . . . . . . . . . . . . . . . . . . 99 6.2 Implicit time integration of the SIR model . . . . . . . . . . . . . . . . . . . . . . 101 6.3 Comparison of explicit and implicit SIR models . . . . . . . . . . . . . . . . . 102 6.4 Comparison of SIR and SIS models . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.5 Sensitivity analysis of the SIR model . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.6 Example: Early COVID-19 outbreak in Austria . . . . . . . . . . . . . . . . . . 106 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7 The computational SEIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.1 Explicit time integration of the SEIR model . . . . . . . . . . . . . . . . . . . . 117 7.2 Implicit time integration of the SEIR model . . . . . . . . . . . . . . . . . . . . 119 7.3 Comparison of explicit and implicit SEIR models. . . . . . . . . . . . . . . . 121 7.4 Comparison of SEIR and SIR models . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.5 Sensitivity analysis of the SEIR model . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.6 Maximum exposure and infection of the SEIR model . . . . . . . . . . . . . 127 7.7 Dynamic SEIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.8 Example: Early COVID-19 outbreak dynamics in China . . . . . . . . . . 131 7.9 Example: Early COVID-19 outbreak control in Germany . . . . . . . . . 136 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8 The computational SEIIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.1 Introduction of the SEIIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.2 Discrete SEIIR model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8.3 Similar symptomatic and asymptomatic disease dynamics. . . . . . . . . 153 8.4 Different symptomatic and asymptomatic disease dynamics . . . . . . . 155
- 17. Contents xv 8.5 Example: Early COVID-19 outbreak in the Netherlands. . . . . . . . . . . 158 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Part III Network epidemiology 9 Introduction to network epidemiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9.1 Numerical methods for partial differential equations. . . . . . . . . . . . . . 169 9.2 Network diffusion method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.3 Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 9.4 Comparison of network diffusion and finite element methods . . . . . . 191 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 10 The network SEIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 10.1 Network diffusion method of the SEIR model . . . . . . . . . . . . . . . . . . . 199 10.2 Example: COVID-19 spreading across the United States . . . . . . . . . . 202 10.3 Example: Travel restrictions across the European Union . . . . . . . . . . 207 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Part IV Data-driven epidemiology 11 Introduction to data-driven epidemiology . . . . . . . . . . . . . . . . . . . . . . . . . 221 11.1 Introduction to data-driven modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 222 11.2 Bayes’ theorem for discrete parameter values . . . . . . . . . . . . . . . . . . . 225 11.3 Bayes’ theorem for continuous parameter values . . . . . . . . . . . . . . . . . 228 11.4 Data-driven SIS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 12 Data-driven dynamic SEIR model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 12.1 Introduction of the data-driven dynamic SEIR model . . . . . . . . . . . . . 250 12.2 Example: Inferring reproduction of COVID-19 in Europe . . . . . . . . . . 256 12.3 Example: Correlating mobility and reproduction of COVID-19 . . . . . 259 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 13 Data-driven dynamic SEIIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 13.1 Introduction of the data-driven dynamic SEIIR model . . . . . . . . . . . . 270 13.2 Example: Visualizing asymptomatic COVID-19 transmission . . . . . . 277 13.3 Example: Asymptomatic COVID-19 transmission worldwide. . . . . . . 280 13.4 Example: Inferring the outbreak date of COVID-19 . . . . . . . . . . . . . . 282 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
- 18. xvi Contents 14 Data-driven network SEIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 14.1 Introduction of the data-driven network SEIR model . . . . . . . . . . . . . 289 14.2 Example: The Newfoundland story . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
- 20. Chapter 1 Introduction to mathematical epidemiology Abstract Mathematical epidemiology is the science of understanding the cause of a disease, predicting its outbreak dynamics, and developing strategies to control it. Throughout the past century, mathematical epidemiology has become the corner- stone of public health. This success story is a result of the rapid advancement of mathematical models and the broad availability of massive amounts of data. Now, two fundamental things have changed that may forever affect our understanding of mod- ern epidemiology: the outbreak of the COVID-19 pandemic, in the age of machine learning. Here, to embrace this opportunity, we introduce the concept of mathe- matical epidemiology in light of data-driven modeling and physics-based learning. We briefly review the traditional concepts of epidemiology, and discuss how our classical understanding of testing, reproduction, herd immunity, and immunization has changed in view of the COVID-19 pandemic. We compare the strengths and weaknesses of purely statistical and mechanistic models and illustrate how we can integrate the large volume of COVID-19 data into mechanistic compartment models to infer model parameters, learn correlations, and identify causation. The learning objectives of this chapter on mathematical epidemiology are to • recognize the ever-present risk of infectious diseases, now and beyond COVID-19 • interpret epidemiological data from molecular, antigen, and antibody tests • explain the basic reproduction number and its impact on public health decisions • estimate herd immunity thresholds for given basic reproduction numbers • demonstrate how vaccination can accelerate the path towards herd immunity • calculate the efficacy and risk ratio of a vaccine • understand the strengths and weaknesses of statistical and mechanistic models • explain the importance of nonlinear feedback in compartment modeling • find and prepare COVID-19 case data towards creating data-driven models • discuss limitations of testing, reported data, and modeling By the end of the chapter, you will be able to think in terms of data-driven modeling, know where to find and how to interpret your data, judge how to select your model, and understand why so many models have failed to explain and predict the outbreak dynamics of the COVID-19 pandemic. 3 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Kuhl, Computational Epidemiology, https://doi.org/10.1007/978-3-030-82890-5_1
- 21. 4 1 Introduction to mathematical epidemiology 1.1 A brief history of infectious diseases In the year 2020, everybody–at home on their couch–became an infectious disease expert. To calibrate all this newly gained knowledge, it seems a good idea to briefly reiterate the basic concepts and nomenclature of infectious diseases. Infectious dis- eases are spread by either bacterial or viral agents and are ever-present in society [8]. Every once in a while, this may result in outbreaks that have a significant impact on a local or global level. Depending on their spatial and temporal spread, we can classify outbreaks as endemic, epidemic, or pandemic [28]. Endemic outbreaks, for example chickenpox, are permanently present in a region or a population. Epidemic outbreaks, for example the seasonal flu, affects a lot of people in a short period of time, spread across several communities, and then disappear. Pandemic outbreaks, for example the Spanish flu, are epidemic outbreaks that affects a lot of people in a short period of time and spread across the entire world. Mathematical modeling of infectious diseases is important to understand their outbreak dynamics and inform political decision making to manage their spread [4]. Table 1.1 History of recent infectious disease outbreaks. Time period, type of disease, number of deaths, and location. period disease deaths location 1346 - 1350 Black Death 100,000,000 1/3 of Europe 1665 - 1666 Great Plague 100,000 1/4 of London 1918 - 1920 Spanish flu 50,000,000 worldwide 1980 measles 2,600,000 worldwide / year 2003 SARS 774 worldwide 2009 - 2010 H1N1 18,500 worldwide 2011 tuberculosis 1,400,000 worldwide / year HIV/AIDS 1,200,000 worldwide / year malaria 627,000 worldwide / year 2011 measles 160,000 worldwide (-94%) 2012 - 2020 MERS 866 worldwide seasonal flu 35,000 United States / year 2014 - 2016 ebola 11,000 Africa 2018 ebola 2,280 Congo 2019 -2021 COVID-19 3,300,000 worldwide The COVID-19 pandemic. On March 11, 2020, Tedros Adhanom Ghebreyesus, the Director-General of the World Health Organization, declared the COVID-19 outbreak a global pandemic [49]. On that day, it had affected 126,702 people worldwide. What nobody could have foreseen is that within the following year, by March 11, 2021, this number had increased by three orders of magnitude, to 118.57 million [12]. Naturally, we often think of the COVID-19 pandemic as the deadliest and most devastating infectious disease in modern history.
- 22. 1.1 A brief history of infectious diseases 5 Endemic. The word endemic is derived from the greek words en meaning in and demos meaning people. An infectious disease is endemic when it is constantly maintained at a baseline level in a geographic region without external inputs. For example, chicken pox is endemic in the United Kingdom. A person-to-person transmitted disease is endemic if each infected person passes the disease to one other person on average. The infection neither dies out nor increases exponentially, it is in an endemic steady state. Infectious disease experts ask: • How many people are infectious at any give time? – What is I(t)? • How fast do new infections arise? – What is dI/dt? – Stability analysis • What are the effects of quarantine or vaccination? – What is R0? • Can we eradicate the disease? – What is H = 1 − 1/R0? – Limit analysis Epidemic. The word epidemic is derived from the greek words epi meaning upon and demos meaning people. An infectious disease is epidemic when it spreads rapidly across a large number of people in a short time period. For example, the seasonal flu is epidemic. Epidemics often come in waves with several recurring outbreaks, which can sometimes be seasonal. There is usually no increase in susceptibles and the disease dies out as the number of infectives decreases because a large enough fraction of the population has become immune. Infectious disease experts ask: • How severe will the epidemic be? – What are Imax? • When will it reach its peak? – What is t(Imax)? • How long will it last? – What are S∞ and R∞? What is t(S∞)? • What are the effects of vaccination? – What are R0 and H = 1−1/R0 and I? Pandemic. The word pandemic is derived from the greek words pan meaning all and demos meaning people. An infectious disease is pandemic when it spreads rapidly across a large region or worldwide in a short time period. A pandemic is a global outbreak. For example, smallpox, tuberculosis, the black death, the Spanish flu were pandemics, and COVID-19 is now. Infectious disease experts ask: • How severe is the pandemic, when will it peak? – What are Imax and t(Imax)? • What are effective measures to manage the outbreak? – What is R(t)? • What are the effects of quarantine or lockdown? – What is β(t)? • What are the effects of travel restrictions? – What are κ and Lij? • How do we prioritize vaccination? – What are R0 and H = 1 − 1/R0 and I?
- 23. 6 1 Introduction to mathematical epidemiology Table 1.1 summarizes recent infectious disease outbreaks ranging from the Black Death in the 14th century with an estimated 100 million deaths across Europe to the current COVID-19 pandemic with 3.3 million deaths worldwide to date. Notably, in 1980, the annual death toll of the measles with 2.6 million was of comparable size. Today, after massive vaccination campaigns, this number has dropped significantly, by 94% to 160,000 [5]. Fig. 1.1 The Lessons of the Pandemic. In this famous Science publication, George A. Soper reflects on the Spanish flu that resulted in 50 million deaths worldwide in 1918 and 1919. Figure 1.1 shows the cover page of the Science publication The Lessons from the Pandemic from May 1919, a reflection on the scientific understanding of the Spanish flu [29]. While some of it is specific to the nature of the 1918 influenza and the time during which it occurred, much of it still applies to the COVID-19 pandemic today: The Spanish flu lasted from 1918 to 1920 and, similar to COVID-19, occurred in multiple waves. Both, the Spanish flu and COVID-19, are contagious respiratory illnesses that spread from person to person, mainly by droplets, through cough, sneeze, or talk. Naturally, increased hygiene, mask wearing, and physical distancing in addition to strict isolation and quarantine are successful strategies to manage both conditions [10]. While COVID-19 is caused by a coronavirus, SARS-CoV-2, the 1918 influenza was caused by the H1N1 influenza A virus, a virus of avian origin. Within 25 months, it infected 500 million people, one third of the world’s population, and resulted in more than 50 million deaths [5]. The major differences between the Spanish flu and COVID-19 are their high risk populations and the mechanisms of death: In contrast to the Spanish flu, which affected mainly healthy adults between 25 and 40 years of age, COVID-19 affects mainly individuals of 65 years and older with comorbidities. Victims of the 1918 influenza mainly died from secondary bacterial pneumonia, whereas victims of COVID-19 die from an overactive immune response that results in organ failure [32]. Nevertheless, comparing the COVID- 19 outbreak to previous pandemics can provide insight and guidance to manage the current COVID-19 pandemic. An important element of this comparison are mathematical and computational tools that have been designed to quantify and explain the spreading mechanisms of infectious diseases. This is the objective of mathematical and computational epidemiology [3].
- 24. 1.2 Introduction to epidemiology 7 1.2 Introduction to epidemiology Epidemiology is the study of distributions, patterns, and determinants of health- related events in human populations [3]. It is a cornerstone of public health , and shapes policy decisions by identifying risk factors for outbreaks and targets for prevention [28]. Epidemiology literally means the study of what is upon the people. It is derived from the greek words epi, meaning upon demos, meaning people, and logos meaning study, suggesting that it applies only to human populations. By this definition, epidemiology is the scientific, systematic, data-driven study of the distribution, i.e., where, who, and when, and determinants, i.e., causes and risk factors, of health-related patterns and events in specified populations, i.e., the world, a country, state, county, city, school, neighborhood. Epidemiology also includes the study of outbreak dynamics, outbreak control, and informing political decision making. Agent, host, and environment. A key premise in epidemiology is that health-related events are not evenly distributed in a population; rather, they affect some individuals more than others [3]. An important goal of epidemiology is to identify the causes that put these individuals at a higher risk [2]. A simple but popular model to analyze and explain disease causation is the epidemiologic triangle. The epidemiologic triangle summarizes the interplay of the three components that contribute to the spread of an infectious disease: an external agent, a susceptible host, and an environment in which agent and host interact [9]. Descriptive epidemiologists characterize this interaction as the seed, the soil, and the climate [42]. Effective public health measures assess all three components and their interactions to control or prevent the spreading of a disease. Figure 1.2 illustrates the epidemiologic triangle of COVID-19 with agent, host, and environment. For COVID-19, the agent is the SARS-CoV-2 virus, the hosts are people, and the environment are droplets. Interventions between any two of these three components can help reduce the spread of the disease [20]. For exam- ple, reduced exposure, vaccination, and antiviral treatment can modulate agent-host interactions and reduce the number of new infections. From descriptive to mathematical epidemiology. Mathematical models of infec- tious diseases date back to Daniel Bernoulli’s model for smallpox in 1760 [5], and they have been developed and improved extensively since the 1920s [14]. In the middle of the 19th century, the English physician John Snow conducted a famous series of experiments of the cholera outbreak in London to discover the cause of the disease and to prevent its recurrence [46]. Because his research illustrates the classic sequence–from descriptive epidemiology and hypothesis generation, to mathemat- ical epidemiology and hypothesis testing–John Snow is considered the father of modern epidemiology. Figure 1.4 summarizes the experiments of John Snow that ended the cholera outbreak in London by removing the handle of a public water
- 25. 8 1 Introduction to mathematical epidemiology Fig. 1.2 Epidemiologic triangle of COVID-19 with agent, host, and environment. The epidemi- ologic triangle illustrates the interplay of the three components that contribute to the spread of a disease: an external agent, a susceptible host, and an environment in which agent and host interact. For COVID-19, the agent is the SARS-CoV-2 virus, the hosts are people, and the environment are droplets. Interventions between any of these three components can reduce the spread of the disease. Fig. 1.3 Daniel Bernoulli is considered the most famous mathematician of the Bernoulli family, although he actually studied medicine. He was born on February 8, 1700 in Groningen, the Netherlands, and is best known for his applications of mathematics to mechanics. To prove the efficacy of vaccination against smallpox, he proposed the first compartment model in 1760 [5]. It only had two compartments, but demonstrated the potential of modeling to understand the mechanisms of transmission, predict future spread, and control the outbreak through vaccination. Daniel Bernoulli died on March 27, 1782 in Basel, Switzerland. pump. Today the field of epidemiology is considered a quantitative discipline that uses rigorous mathematical tools of probability and statistics to develop and test hypotheses to understand and explain health-related events [3]. In simple terms, today’s epidemiologists count, scale, and compare. They count the number of cases; scale this count by a characteristic population to define fractions; and compare the evolution of these fractions over time [9]. Mathematical epidemiology has advanced significantly throughout the past decades [11], and models have become more so- phisticated and complex [39]. This implies that in epidemiology today, is often no longer possible to solve epidemiological models analytically and in closed form [3]. From mathematical to computational epidemiology. Computational epidemiol- ogy is a multidisciplinary field that integrates mathematics, computer science, and public health to better understand central questions in epidemiology such as the
- 26. 1.2 Introduction to epidemiology 9 Fig. 1.4 The English physician John Snow is considered one of the founders of modern epidemiology, in part because of his research during the cholera outbreak in London in 1954 [46]. John Snow lived from March 15, 1813 to June 16, 1858. He questioned the widely accepted paradigm that cholera would spread by polluted bad air. He postulated that water would be the cause of the cholera outbreak in Soho, London in 1854. By talking to local residents, he identified the source of the outbreak as the public water pump on Broad Street. Although his chemical and microscopic observations were inconclusive, his spreading patterns of the cholera outbreak were convincing enough to persuade the local council to disable a street pump on Broad Street by removing its handle. There is a common belief that this action marked the ending of the outbreak. spread of diseases or the effectiveness of public health interventions [22]. Real-time epidemiology is a rapidly developing area within computational epidemiology that seeks to support policy makers–in real time–as an outbreak is unfolding. An impor- tant application of real-time epidemiology is disease surveillance, the data-driven collection, analysis, and interpretation of large volumes of disease data from a va- riety of sources. This information can help to evaluate the effectiveness of control and preventative health measures [12]. Central to computational epidemiology is the accurate knowledge of people who are affected by an infectious disease at any given point in time. During the COVID-19 pandemic, for the first time in history, this information has been collected conscientiously, shared publicly, updated daily, and made freely available in real time [13]. Sensitivity and specificity. A diagnostic tests, for example, the nasal swab test, can be inaccurate in two ways: a false positive result erroneously labels a healthy person as infected, resulting in unnecessary quarantine and contact tracing; a false negative result oversees an infected person, resulting in the risk of infecting others. sensitivity = true positives / all sick people Sensitivity measures the proportion of positives, diseased people, that are correctly identified and not overlooked. specificity = true negatives / all healthy people Specificity measures the proportion of negatives, healthy people, that are correctly identified and not classified as diseased. A perfect test has 100% sensitivity and 100% specificity.
- 27. 10 1 Introduction to mathematical epidemiology 1.3 Testing, testing, testing During a global pandemic, knowing how many people currently have the disease or have previously had it provides crucial information for healthcare providers and policy makers, both on the individual and population levels. Unfortunately, there is no method to provide this information with absolute accuracy. We usually use two measures to characterize the degree of accuracy of a test: sensitivity, the fraction of correctly identified positive, diseased individuals; and specificity, the fraction of correctly identified negative, healthy individuals. Understanding these limitations is important when modeling, simulating, and predicting the outbreak dynamics of a pandemic, especially in view of disease management and political decision making. Table 1.2 Testing for COVID-19. Summary of the three most common tests for SARS-CoV-2. Diagnostic tests, including molecular and antigen tests, provide information about acute infection, whereas antibody or serology tests provide information about previous infection. Publicly available case data are based on diagnostic testing; seroprevalence reports are based on antibody testing. diagnostic testing antibody testing molecular testing antigen testing serology testing provides information about acute COVID-19 infection provides information about acute COVID-19 infection does not provide information about acute infection does not provide information about past infection does not provide information about past infection tests for antibody presence, does not guarantee immunity can take hours to days takes minutes to hours takes minutes to hours relatively accurate early in the infection faster and cheaper than molecular tests quick results accuracy drops later in infec- tion less accurate than molecular tests tests can vary in accuracy point of care testing; collect swabs at home and mail in point of care testing; collect swabs at home and mail in blood test that can be done at doctor’s office or at home Testing for COVID-19. Since its outbreak in late 2019, the COVID-19 pandemic has generated an exponentially growing demand for testing, and diagnostic assays that enable mass screening have been developed at an unprecedented pace. To success- fully use these tests to inform public health strategies, it is critical to understand their individual strengths and limitations. There are two different types of assays for SARS-CoV-2, the virus that causes COVID-19: diagnostic tests and antibody tests. A diagnostic test can show if you have an active COVID-19 infection and need to take steps to quarantine or isolate yourself from others. Molecular tests and antigen tests fall under this category. A molecular test is a diagnostic test that detects genetic material from the virus using, for example, reverse transcription polymerase chain reaction or nucleic acid amplification. An antigen test is a diagnostic test that detects specific proteins made by the virus. Samples for diagnostic tests are typically col- lected with a nasal or throat swab or with saliva from spitting into a tube. Diagnostic testing is critical to provide early treatment, quarantine individuals sooner, and trace and isolate their contacts to reduce the spread of the virus.
- 28. 1.4 The basic reproduction number 11 Fig. 1.5 Testing for COVID-19. The objective of COVID-19 testing is to probe whether an individual is currently or has previously been infected with SARS-CoV-2. A nasal or throat swab test is a diagnostic test that provides information about an acute infection. Diagnostic testing is critical to provide early treatment, quarantine individuals, and trace and isolate their contacts to reduce the spread of the virus. A blood test is an antibody test that provides information about a previous infection. Antibody tests are critical to estimate the overall dimension of an outbreak. Throughout this book, we use COVID-19 case data from public dashboards based on confirmed positive diagnostic tests. An antibody test can detect antibodies that are made by your immune system in response to a previous infection with SARS-CoV-2. In contrast to genetic material from the virus or specific proteins made by the virus, antibodies can take several days or even weeks to develop, but they can remain present for several weeks or months after recovery. A serology test is an antibody test that looks for antibodies in blood samples. These venous blood samples are typically collected at the doctor’s office or in the clinic. Antibody tests should not be used to diagnose an active infec- tion. Instead, antibody testing is important to understand the kinetics of the immune response to infection, clarify whether an infection protects from future infection, characterize how long immunity will last, and estimate the overall dimension of an outbreak. Table 1.2 contrasts the three most common types of tests for COVID-19. Throughout this book, we use COVID-19 case data from public dashboards based on confirmed positive diagnostic tests. Only in Chapters 8 and 13, where we explore the effects of asymptomatic transmission, we use specialized models that combine case data from diagnostic tests with seroprevalence data from antibody tests. 1.4 The basic reproduction number The basic reproduction number is a powerful but simple concept to explain the contagiousness and transmissibility of an infectious disease [6]. For decades, epi- demiologists have successfully used the basic reproduction number to quantify how many new infections a single infectious individual creates in an otherwise com- pletely susceptible population [6]. During the COVID-19 pandemic, the public me- dia, scientists, and political decision makers across the globe have adopted the basic reproduction number as an illustrative metric to explain and justify the need for dif- ferent outbreak control strategies [10]: An outbreak will continue for reproduction numbers larger than one, R0 > 1, and come to an end for reproduction numbers smaller than one, R0 < 1 [8]. However, especially in the midst of a global pandemic, it is difficult–if not impossible–to measure R0 directly [48].
- 29. 12 1 Introduction to mathematical epidemiology Table 1.3 Basic reproduction numbers and herd immunity thresholds for common infectious diseases. Herd immunity is the indirect protection from an infectious disease that occurs when a large fraction of the population has become immune. The herd immunity threshold, H = 1 − 1/R0, beyond which this protection occurs is a function of the basic reproduction number R0. disease R0 H disease R0 H measles 12 - 18 92 - 95% mumps 4.0 - 7.0 75 - 86% pertussis 12 - 17 92 - 94% COVID-19 2.0 - 6.0 50 - 83% rubella 6 - 7 83 - 86% SARS 2.0 - 5.0 50 - 80% smallpox 6 - 7 83 - 86% ebola 1.5 - 2.5 33 - 60% polio 5 - 7 80 - 86% influenza 1.5 - 1.8 33 - 44% Table 1.3 summarizes the basic reproduction numbers for common infectious dis- eases. It varies from R0 = 1.5 − 1.8, for less contagious diseases like influenza, to R0 = 12 − 18 for the measles and pertussis. Knowing the precise value of R0 is important, but challenging, because of limited testing, inconsistent reporting, and incomplete data [6]. Throughout this book, instead of measuring the basic repro- duction number directly, we estimate it using mathematical modeling and reported case data [17]. Mathematical models interpret the basic reproduction number R0 as the ratio between the infectious period C, the period during which an infectious individual can infect others, and the contact period B, the average time it takes to come into contact with another individual [8], R0 = C/B . (1.1) The longer a person is infectious, and the more contacts the person has during this time, the larger the reproduction number [6]. While we cannot control the infectious period C, we can change our behavior to increase the contact period B [17]. This is precisely what community mitigation strategies and political interventions seek to attempt. The reproduction number of COVID-19. Since the beginning of the coronavirus pandemic, no other number has been discussed more controversially than the re- production number of COVID-19 [22]. The earliest COVID-19 study that followed the first 425 cases of the Wuhan outbreak via direct contact tracing reported a basic reproduction number of R0 = 2.2 [15]. However, especially during the early stages of the outbreak, information was limited because of insufficient testing, changes in case definitions, and overwhelmed healthcare systems. While the concept of R0 seems fairly simple, the reported basic reproduction numbers for COVID-19 vary hugely with country, culture, calculation, and time [17]. Most basic reproduction numbers of COVID-19 we see in the public media today are estimates of mathemat- ical models. These estimates depend critically on the choice of the model, its initial conditions, and many other modeling assumptions [6]. To no surprise, the mathe- matically predicted basic reproduction numbers cover a wide range, from R0 = 2−4 for exponential growth models to R0 = 4 − 7 for more sophisticated compartment models [22].
- 30. 1.4 The basic reproduction number 13 Throughout this book, we identify or infer the reproduction number of COVID- 19–and the uncertainty associated with it–using computational epidemiology [22], Bayesian analysis [10], and reported case data [12]: In the first example for the very early COVID-19 outbreak in China in Section 7.8, we identify a basic reproduction number of R0 = 12.58±3.17 across 30 Chinese provinces. In the second example for the early outbreak in the United States in Section 10.2, we find a basic reproduction number of R0 = 5.30 ± 0.95 across all 54 states and territories. This value suggests that COVID-19 is less contagious than the measles and pertussis with R0 = 12 − 18, as infectious as rubella, smallpox, polio, and mumps with R0 = 4 − 7, slightly more infectious than SARS with R0 = 2 − 5, and more infectious than an influenza with R0 = 1.5 − 1.8. Our basic reproduction number for the United States is significantly lower than our basic reproduction number for China, which could be caused by an increased awareness of COVID-19 transmission a few weeks into the global pandemic. In our third example for the early outbreak in Europe in Section 12.2, the basic reproduction number takes similar values of R0 = 4.62 ± 1.32 across all 27 countries. During these early stages of exponential growth, with new case numbers doubling within two or three days, the most urgent question amongst health care providers and political decision makers was: Can we reduce the reproduction number? For the broad public, this question became famously and illustratively rephrased as: Can we flatten the curve [15]? For the modeling community, the quest for a lower reproduc- tion number all of a sudden meant that traditional epidemiology models were no longer suitable because of changes in the disease dynamics [12]. While traditional models with static parameters were well-suited to model the outbreak dynamics of unconstrained, freely evolving infectious diseases with fixed basic reproduction numbers in the early 20th century, they fail capture how behavioral changes and po- litical interventions can modulate the reproduction number to manage the COVID-19 pandemic in the 21st century [17]. In fact, static reproduction numbers are probably the single most common cause of model failure in COVID-19 modeling [12]. Fortunately, several months into the pandemic, most countries have successfully managed to flatten the new-case-number curves and the reproduction numbers have dropped to values closer to or below one. To model these changes in disease dynamics and reproduction, in Section 7.7, we introduce a dynamic reproduction number, R(t), that accounts for time-varying contact periods. In our fourth example for the European Union in Section 12.2, we show that the initial basic reproduction number of R0 = 4.22±1.69 dropped to an effective reproduction number of R(t) = 0.67±0.18 by mid May 2020. Using machine learning, we correlate mobility and reproduction and identify the responsiveness between the drop in air traffic, driving, walking, and transit mobility and the drop in reproduction to ∆t = 17.24 ± 2.00 days. In the final examples in Sections 13.2 and 13.3 of nine locations across the world, we systematically infer the dynamic reproduction number R(t) throughout a time window of 100 days while accounting for both symptomatic and asymptomatic transmission. From the failure of traditional static epidemiology models [15], we have now learned that we need to introduce dynamic time-varying model parameters if we
- 31. 14 1 Introduction to mathematical epidemiology Fig. 1.6 Basic reproduction numbers and herd immunity thresholds for common infectious diseases. Herd immunity is the indirect protection from an infectious disease that occurs when a large fraction of the population has become immune, either through previous infection or through vaccination. The black line highlights the herd immunity threshold, H = 1 − 1/R0, beyond which this protection occurs, as a function of the basic reproduction number R0. The herd immunity threshold varies from 33-44% for influenza to 92-95% for the measles. want to correctly model behavioral and political changes and reproduce the reported case numbers [17]. This naturally introduces a lot of freedom, a large number of unknowns, and a high level of uncertainty. However, in stark contrast to the epidemic outbreaks in the early 20th century, we now have thoroughly-reported case data and the appropriate tools [23] to address this challenge. The massive amount of COVID- 19 case data, well documented and freely available, has induced a clear paradigm shift from traditional mathematical epidemiology towards data-driven, physics-based modeling of infectious disease [1]. This new technology naturally learns the most probable model parameters–in real time–from the continuously emerging case data, allows us to make projections into the future, and quantifies the uncertainty on the estimated parameters and predictions [26]. 1.5 Concept of herd immunity An important consequence of the basic reproduction number R0 is the condition for herd immunity [9]. Herd immunity describes the indirect protection from an infectious disease that occurs when a large fraction of the population has become immune, either through previous infection or through vaccination [3]. The critical threshold at which the disease reaches this endemic steady state is called the herd immunity threshold, H = 1 − 1/R0 . (1.2) The larger the basic reproduction number R0, the higher the herd immunity threshold H. Table 1.3 and Figure 1.6 summarize the basic reproduction numbers R0 for
- 32. 1.6 Concept of immunization 15 several common infectious diseases along with the estimates of their herd immunity thresholds H. The herd immunity threshold varies from 33-44% for influenza to 92- 95% for the measles. For the reported basic reproduction numbers of R0 = 2.0 − 6.0 of COVID-19, the estimated herd immunity threshold would range from 50-83%. Recent studies that account for the emerging new and more infectious B.1.351 and B.1.1.7 variants of COVID-19 estimate these values to 75-95% [20]. 1.6 Concept of immunization To prevent or revert an epidemic outbreak, we need to ensure that, on average, every infectious individual infects less than one new individual. The concept of herd immunity describes the natural path towards ending an outbreak. However, if the basic reproduction number R0 is large, the herd immunity threshold H = 1 − 1/R0 is high, and waiting for herd immunity through infection alone can be quite devastating. Vaccination is a powerful strategy to accelerate the path towards herd immunity [1]. Fig. 1.7 Vaccination against COVID-19. The objective of COVID-19 vaccination is to provide immunity against severe acute respiratory syndrome coronavirus 2 or SARS-CoV-2, the virus that causes COVID-19 [30]. By May, 2021, thirteen vaccines were authorized for public use: two RNA vaccines (Pfizer–BioNTech and Moderna), five conventional inactivated vaccines (BBIBP-CorV, CoronaVac, Covaxin, WIBP-CorV and CoviVac), four viral vector vaccines (Sputnik V, Oxford–AstraZeneca, Convidecia, and Johnson & Johnson), and two protein subunit vaccines (EpiVacCorona and RBD-Dimer); and more than a billion doses of COVID- 19 vaccines were administered worldwide. Immunization threshold. Vaccination effectively reduces the susceptible population S. Successfully immunizing a fraction I of the population, reduces the susceptible population from S to S∗ 0 = [ 1 − I ] S0 and, with it, the reproduction number from R0 to R∗ 0 = [ 1 − I ] R0. The critical immunization threshold I, below which the effective basic reproduction number is smaller than one, R∗ 0 = [ 1 − I ] R0 < 1, defines the fraction of the population that needs to be immunized to prevent or revert the outbreak of an epidemic, I > 1 − 1/R0 . (1.3) From Table 1.3, we conclude that the required immunization fraction varies signif- icantly, from I > 92 − 95% for the measles with a basic reproduction number of R0 = 12 − 18 to I > 33 − 44% for the common influenza with a basic reproduction number of R0 = 1.5−1.8. This explains, at least in part, why some infectious diseases are a lot more difficult to control through immunization than others.
- 33. 16 1 Introduction to mathematical epidemiology Eradication through vaccination. Once enough individuals are immunized–either through infection or through vaccination–the outbreak stops. A disease that stops circulating in a specific region is considered eliminated in that region. Polio, for example, was eliminated in the United States by 1979 after widespread vaccination efforts. A disease that is eliminated worldwide is considered eradicated. Eradicating a disease through vaccination is a desirable but elusive goal [1]. Malaria has been a candidate for eradication, but although its incidence has been drastically reduced through vaccination, completely eradicating it remains challenging because infection does not result in life long immunity. Polio has been eliminated in most countries through massive vaccination efforts, but still remains present in some regions be- cause its early symptoms often remain unnoticed and infected individuals continue to infect others. Measles have been the target of widespread vaccination, but although the disease is highly recognizable through its characteristic rash, a long latent pe- riod from exposure to the first onset of symptoms complicates outbreak control. To this day, smallpox is the only human infectious disease that has been successfully eradicated through vaccination. The eradication of smallpox is the result of focused surveillance, rapid identification, and ring vaccination [8]. In a massive vaccina- tion campaign launched in 1967, anyone who could have possibly been exposed to smallpox was quickly identified and vaccinated to prevent its further spread. The last known case of smallpox occurred in Somalia in 1977. In 1980 World Health Organization declared smallpox eradicated. The eradication of smallpox remains one of the most notable and profound public health successes in history. Efficacy and risk ratio. The efficacy e of a vaccine is the relative reduction in the disease attack rate between the unvaccinated placebo group npla and the vaccinated group nvac. For a randomized trial with an equal allocation, meaning equally sized placebo and vaccinated groups, npla = nvac, the efficacy is e = npla − nvac npla · 100% = 1 − nvac npla · 100% = 1 − r . (1.4) The risk ratio r is the ratio between the attack rate of the vaccinated group nvac and the placebo group npla, r = nvac npla · 100% = 1 − e . (1.5) The efficacy is an important measure to characterize the success of a vaccine and define critical thresholds below which a vaccination trial should stop. Table 1.4 Contingency table to quantify the significance of a vaccine. The table compares the total number of vaccinated and placebo individuals that developed and did not develop the disease. positive negative total vaccine a b a+b placebo c d c+d total a+c b+d n
- 34. 1.6 Concept of immunization 17 Example: Efficacy of the first COVID-19 vaccine. On November 9, 2020, the Pfizer and BioNTech trial reported a number or COVID-19 cases of ncovid = npla + nvac = 94 and an efficacy of e emin with emin = 90%. The efficacy e of a vaccine is the relative reduction in the disease attack rate between the unvaccinated placebo group npla and the vaccinated group nvac, e = 1 − nvac npla · 100% = 1 − r with r = nvac npla · 100% = 1 − e . where r is the risk ratio. The Pfizer and BioNTech trial was a randomized trial with an equal allocation [30]. Although it did not report detailed numbers, we can estimate the number of placebo and vaccinated cases npla and nvac and the risk ratio r from the reported efficacy emin = 90% with nvac = ncovid − npla, e = 1 − ncovid − npla npla · 100% = 2 − ncovid npla · 100% emin . Solving for the number of placebo cases yields the general equation, npla ncovid 2 − emin , and, for the Pfizer and BioNTech case, npla ncovid/1.1 = 85.45. This implies that for a total of ncovid = 94 COVID-19 positive cases, at a number of placebo cases npla = 86 and vaccinated cases nvac = 8, the efficacy is larger than 90%. Back-calculating the efficacy for these populations, e = 1 − nvac npla · 100% = 1 − 8 86 · 100% = 90.7% 90% = emin , confirms the simulation. According to the protocol, Pfizer and BioNTech planned to take a look at the data at five stages with ncovid = 32,64,92,120,164 reported positive cases and only continue the trial if the efficacy was above e ecrit with ecrit = 62.7%. From this information, we can calculate the critical numbers npla and nvac below which the trial would have stopped at any of the five stages. Solving for the number of placebo cases yields the general equation, npla ncovid 2 − ecrit , and, for the Pfizer and BioNTech case, npla ncovid/1.373. This implies that for ncovid = 32,64,92,120,164, the minimum number of placebo cases to continue the trial was npla ≥ 24,47,68,88,120, for which the resulting efficacies of e = 66.7%,63.8%,64.7%,63.6%,63.3% would all have been slightly above the critical threshold of ecrit = 62.7%.
- 35. 18 1 Introduction to mathematical epidemiology Fisher’s exact test and contingency tables. Fisher’s exact test is a statistical sig- nificance test to analyze contingency tables that quantify the effects of a vaccine compared to a placebo. Table 1.4 illustrates a generic contingency table. From it, we can calculate the case rate across the entire trial as the ratio between the total number of positive cases and the total number of enrolled individuals, νdisease = a + b a + b + c + d with a + b + c + d = ntot . (1.6) The case rates across the vaccinated and placebo groups are, νvac = a a + b with a = nvac and νpla = c c + d with c = npla . (1.7) Most vaccination trials are designed as a randomized trial, meaning they assign participants at random to a vaccinated or placebo group, with equal allocation, meaning they target an equal enrollment into both groups, a + b ≈ c + d ≈ ntot/2. We can use the contingency table to calculate the statistical significance p of the deviation from a null hypothesis, p = a+b a c+d c n a+c = a+b b c+d d n b+d = (a + b)!(c + d)!(a + c)!(b + d)! a! b! c! d! n! , (1.8) in terms of binominal coefficients or factorial operators using a hypergeometric distribution. For a vaccination trial, the lower the p-value, the larger the effect on the vaccinated group compared to the placebo group. Example: Statistical significance of the first COVID-19 vaccine. OnNovem- ber 9, 2020, the Pfizer and BioNTech trial reported ntot = 43,538 enrolled par- ticipants. From this number, we can calculate the case rates across the entire trial, in the vaccinated group, and in the placebo group assuming a random- ization at 1:1 between the vaccinated and placebo groups. The case rate of the entire trial is the ratio between the total number of COVID-19 positive cases and the total number of enrolled individuals, νcovid = ncovid ntot = 94 43,538 = 0.190% . The case rates of the two groups are the ratios between the vaccinated and placebo COVID-19 positive cases and half of the enrolled cases, νvac = nvac ntot/2 = 8 43,538/2 = 0.037% νpla = npla ntot/2 = 86 43,538/2 = 0.395% . Using Fisher’s exact test, we can test the null hypothesis that vaccinated and placebo participants will equally likely contract COVID-19.
- 36. 1.7 Mathematical modeling in epidemiology 19 positive negative total vaccine 8 21,761 21,769 placebo 86 21,683 21,769 total 94 43,444 43,538 We can use the contingency table to estimate the efficiency of the Pfizer BioN- Tech vaccine. Fisher’s exact test calculates the significance of the deviation from a null hypothesis, p = 21769! 21769! 94! 43444! 8! 21761! 86! 21683! 43538! , and confirms, for the Pfizer and BioNTech case with p 0.00001, that indi- viduals in the vaccinated and placebo groups will not equally likely contract COVID-19. 1.7 Mathematical modeling in epidemiology During the early onset of the COVID-19 pandemic, all eyes were on mathematical modeling with the general expectation that mathematical models could precisely predict the trajectory of the pandemic. Mathematical modeling rapidly became front and center to understanding the exponential increase of infections, the shortage of ventilators, and the limited capacity of hospital beds; too rapidly as we now know. Bold and catastrophic predictions not only initiated a massive press coverage, but also a broad anxiety in the general population [15]. However, within only a few weeks, the vastly different predictions and conflicting conclusions began to create the impression that all mathematical models are generally unreliable and inherently wrong [12]. While the failure of COVID-19 modeling–often by an order of mag- nitude and more–was devastating for policymakers and public health practitioners, initial mistakes are not new to the modeling community where an iterative cycle of prediction, failure, and redesign is common standard and best practice [26]. However, the successful use of mathematical models implies to set the expectations right [45]. Understanding what models can and cannot predict is critical to the Art of Mod- eling. Epidemiologists distinguish two kinds of models to understand the outbreak dynamics of an infectious disease: statistical models and mechanistic models [12]. Depending on the degree of complexity, the most popular mechanistic models are compartment models and agent-based models. Statistical models, or more precisely, purely statistical models, use machine learn- ing or regression to analyze massive amounts of data and project the number of infections into the future. The essential idea is to select a function D(t), use statisti- cal tools to fit its coefficients to reported case data D̂(t), and make projections into the future. The function can be quadratic, cubic, logistic, power-law, or exponential,
- 37. 20 1 Introduction to mathematical epidemiology Table 1.5 Mathematical modeling of COVID-19. Summary of the three most common models. Statistical or forecasting models fit nonlinear functions to case data over time, whereas mechanistic models, including compartment and agent-based models, simulate outbreak and contact dynamics. Throughout this book, we use compartment models to simulate the outbreak dynamics of infectious diseases including COVID-19. statistical models mechanistic models forecasting models compartment models agent-based models use machine learning, statis- tics, regression, or method of least squares use physics-based modeling based on nonlinear reaction- diffusion equations use rule-based approaches to study the interaction of au- tonomous systems model case numbers through a nonlinear function model population through compartments model every individual as an independent agent formulate number of cases as a function of time formulate rules by which in- dividuals pass through the compartments formulate simple rules by which individual agents in- teract fit coefficients that are purely phenomenological infer parameters that have a mechanistic interpretation identify parameters that summarize human behavior predicts case numbers from fitting a function predict outbreak dynam- ics, characterize sensitivi- ties, quantify uncertainties predict outbreak dynamics as emergent collective be- havior of individual agents no feedback mechanisms nonlinear feedback discrete contact networks predictions are inexpensive, but very unreliable predictions are reliable only for a small time window predictions are detailed, but computationally expensive for example, D(ϑ,t) = exp(c0 + c1 t + c2 t2 + c3 t3), where D(ϑ,t) are the modeled cumulative cases per day, ϑ = {c0,c1,c2,c3} are the model parameters, and t is the time. One of the simplest statistical tools to compare the model D(ϑ,t) to the data D̂(t) and identify values for its parameters ϑ is the method of least squares. It is important to understand that these parameters are purely phenomenological, they are derived purely by fitting a curve, and typically do not have a mechanistic interpretation. Early in an outbreak, when little is known about disease transmission, epidemiologists often use statistical models because they do not rely on any prior knowledge of the disease. An example for a statistical model of COVID-19 is the ini- tial IHME model [23]. Early in the pandemic, the IHME model used case data from China and Italy to create similar curves, forecast case numbers in the United States, and inform the White House’s response to the pandemic. Carefully constructed sta- tistical frameworks can be used for short-time forecasting using machine learning or regression. This could potentially be useful to understand how to allocate resources or make rapid short-term recommendations. However, purely statistical models can neither capture the dynamics of disease transmission nor the effects of mitigation strategies. This explains, at least in part, why the early COVID-19 predictions based on purely statistical models were off by an order of magnitude or more. To address these serious limitations, several COVID-19 models have now been adjusted to com- bine both statistical modeling and mechanistic modeling. Mechanistic models simulate the outbreak through interacting disease mechanisms by using local nonlinear population dynamics and global mixing of populations
- 38. 1.7 Mathematical modeling in epidemiology 21 [4]. The underlying idea is to identify fundamental mechanisms that drive disease dynamics, for example the duration of the infectious period or the number of con- tacts an infectious individual has during this time. Unlike purely statistical models, mechanistic models include important nonlinear feedback: The more people be- come infected, the faster the disease spreads. By their very nature, the parameters of mechanistic models are not just fitting parameters, they usually have a clear epi- demiological interpretation. This makes mechanistic modeling a powerful strategy to explore different outbreak scenarios or study how an outbreak would change under various assumptions and political interventions [12]. Another advantage of mechanistic models is that we can adjust and improve them dynamically as more information becomes available. Throughout this book, we gradually improve a class of mechanistic models by adding new information. For example, we introduce time- varying dynamic contact rates that vary in different lockdown levels and add the effect of asymptomatic transmission [26]. Even if we do not precisely know the dimension of asymptomatic disease spread, we can use mechanistic models to study what-if scenarios: What would the disease landscape look like if two third of all infectious were asymptomatic? Mechanistic modeling naturally extends into sensi- tivity analysis and uncertainty quantification. As such, it not only provides valuable information about the robustness of the model, but also about the most effective parameters to modulate a disease outbreak [19]. Rather than studying a single one disease trajectory, we could explore a range of trajectories around the mean and characterize the best- and worst-case scenarios. The two most popular mechanistic models in epidemiology are compartment models and agent-based models, and both have been used to understand the outbreak dynamics of COVID-19. When choosing between compartment models and agent-based, it is important to understand the major strengths and weaknesses of each model. Compartment models are the most common approach to model the epidemiology of an infectious disease [14]. Compartment models simulate the collective behavior of subgroups of the population through a number of compartments with labels, for example, SEIR for susceptible, exposed, infectious, and recovered. Individuals move between compartments and the order of the labels indicates the successive motion, for example, SEIS means susceptible, exposed, infectious, then susceptible again [4]. The underlying principle is to model the time evolution of these groups through a set of coupled ordinary differential equations, identify rate constants that characterize their interaction using reported case data, and vary these rate constants to probe different outbreak scenarios. Figure 1.8 illustrates a compartment model that represents the characteristic time- line of COVID-19 through six compartments, the susceptible, exposed, infectious, recovered, hospitalized, and dead groups. This SEIRHD model is defined through a set of six ordinary differential equations that simulate how many individuals reside in each compartment throughout the duration of the outbreak. The model parameters of a compartment model define the transition rates between the individual compart- ments and, for more complex models, the fraction of individuals that transition into a particular path of the disease. For this example, the parameters ϑ = {β,α,γ,νh,νd}
- 39. 22 1 Introduction to mathematical epidemiology Fig. 1.8 Characteristic timeline of COVID-19. On day 0, a fraction of all susceptible individuals is exposed to the virus. After a latent period of A = 1/α = 3 days, the exposed individuals become infectious. After an infectious period of C = 1/γ = 10 days, a fraction (1 − νh) transition to the recovered group, whereas a fraction νh develops severe symptoms and is hospitalized. Of the hospitalized individuals, a fraction (1 − νd) recovers, whereas a fraction νd becomes dead. We can simulate this behavior through an SEIRHD model with six compartments. are the contact rate β, latent rate α, and infectious rate γ, and the hospitalized and dead fractions, νh and νh · νd. From reported case numbers, hospitalizations, and deaths, we can identify or infer the set of model parameters ϑ, the rate constants and fractions, that best explain the model output using statistical tools. Importantly, in contrast to purely statistical models, compartment models are based on model parameters that have a clear physical interpretation. The most important parameters of any compartment model are the contact rate β and the infectious rate γ. Together, they define an important nonlinear feedback that is not present in purely statistical models: The more people become infected, the faster the spread of the disease. The basic reproduction number R0 = β/γ, the ratio of the contact rate β and the infec- tious rate γ, characterizes the magnitude of this feedback. It is an easy-to-understand disease metric that explains how quickly susceptible individuals become infected and how fast a disease spreads across a population [6]. Some epidemiologists ar- gue that, because of their mechanistic nature, compartment models are better suited for long-term predictions than purely statistical models. While this might be true for infectious diseases that develop freely, without any political intervention, the COVID-19 pandemic has taught us that long-term predictions of outbreak dynamics are challenging, even with the most sophisticated mechanistic models [15]. Under- standing the potential and the limitations of compartment modeling is one of the main objectives of this book. In Chapters 2 and 3, we introduce two simple classical compartment models be- fore we introduce the most common compartment model for COVID-19 in Chapter 4. We show how we can use these models to estimate the reproduction number from reported COVID-19 case data. Knowing the precise reproduction number has important consequences for estimating the dimensions of herd immunity and im- munization. Compartment models capture the fundamental dynamics of disease transmission and the effects of public health interventions; however, classical com- partment models ignore the dynamics of the contact rate and its variation across a population. In Section 7.7 we introduce a compartment model that explicitly ac- counts for a time-varying dynamic contact rate, β(t), and captures a varying contact behavior in different subgroups of the population. The dynamic nature of the contact
- 40. 1.8 Data-driven modeling in epidemiology 23 rate naturally introduces a dynamic reproduction number, R(t) = β(t)/γ and allows us to quantify the effectiveness of policy measures as we discuss in Section 12.2. Agent-based models simulate individuals or agents interacting in various social settings and estimate the spread of a disease as these agents come into contact with one another. The underlying idea is to represent each agent individually, formulate relatively simple rules by which individual agents interact, and interpret the collec- tive behavior across all agents as the emergent dynamics of an outbreak. A strength of agent-based models is that they simulate human behavior very granularly: They can assign different parameters or behavior patterns to each individual agent instead of simulating the collective behavior of entire populations. As such, agent-based models offer a lot more freedom than compartment models, but also require a lot more detail. For example, to formulate rules of interaction, agent-based models draw on social connectivity networks, from activity surveys, cell phone locations, public transportation, or airlines statistics. By their very nature, agent-based approaches are computationally expensive, especially for large populations. For small populations, agent-based modeling is a powerful strategy to predict how individual behavior, for example the violation of quarantine, leads to a collective behavior and modulates disease spread. For large populations, agent-based modeling can help rationalize collective model parameters, for example contact rates or reproduction numbers, that feed into more abstract, population level models. Above a certain population size, agent-based models simply become computationally unfeasible and most epi- demiologists would turn to a more macroscopic approach that represents groups of individuals collectively as subgroups of the population. Since our objective is to design and discuss data-driven models for the COVID-19 pandemic, throughout the remainder of this book, we focus exclusively on compartment models. When choosing between purely statistical, compartment, and agent-based models, it is important to know upfront which questions the model should address [45]. Ta- ble 1.5 compares the three models and summarizes their strengths and weaknesses. Throughout this book, we focus on mechanistic compartment modeling. 1.8 Data-driven modeling in epidemiology One year after the World Health Organization had declared the COVID-19 outbreak a global pandemic, SARS-CoV-2 has resulted in more than 118 million reported cases across more than 180 countries and over 2.6 million deaths worldwide. Unlike any other disease in history, the COVID-19 pandemic has generated an unprecedented volume of data, well documented, continuously updated, and broadly available to the general public. There is a critical need for time- and cost-efficient strategies to analyze and interpret these data to systematically manage the pandemic on a global level. Yet, the precise role of physics-based modeling and machine learning in providing quantitative insight into the dynamics of COVID-19 remains a topic of ongoing debate.
- 41. 24 1 Introduction to mathematical epidemiology Physics-based modeling is a successful strategy to integrate multiscale, multi- physics data and uncover mechanisms that explain the dynamics of specific outbreak characteristics. However, physics-based modeling alone often fails to efficiently combine large data sets from different sources and different levels of resolution [26]. Machine learning is as a powerful technique to integrate multimodality, multifidelity data, from cities, counties, states, and countries across the world, and reveal cor- relations between different disease phenomena. However, machine learning alone ignores the fundamental laws of physics and can result in ill-posed problems or non- physical solutions [1]. Throughout this book, we illustrate how data-driven modeling can integrate classical physics-based modeling and machine learning to infer critical disease parameters–in real time–from reported case data to make informed predic- tions and guide political decision making. As a valuable by product, this approach naturally lends itself in sensitivity analysis and uncertainty quantification. From the COVID-19 pandemic, we have learnt that even small inaccuracies in the model can trigger large changes in the number of cases. To understand the vulnerability of the model to these small changes, especially in view of the varying reporting practices of the COVID-19 case data, sensitivity analysis and quantifying uncertainty have become critical elements of robust predictive modeling. Epidemiology data. In data-driven modeling, we need data to fit or infer our model parameters. Unlike earlier pandemics for which case data are often sparse, irregular, or incomplete, the COVID-19 pandemic is amazingly well documented [13]. On hundreds of public COVID-19 dashboards, we can find and download a vast variety of case numbers: daily new cases, active cases, recovered cases, seriously critical cases, cumulative cases, and deaths, at the level of cities, counties, states, countries, or the entire world [9, 12, 25]. Local dashboards often also share the number of hos- pitalizations and intensive care units, which were of great concern especially at the early onset of the pandemic. More recently, these dashboards have also included the number of tests and vaccines. Seroprevalence data with information about the history of the disease, both asymptomatic and symptomatic, are rare and often only avail- able from scientific publications rather than governmental databases. These data are typically updated on a daily basis, and contain notable weekday-weekend alterations. When using the data to infer model parameters and learn about the outbreak behav- ior, we usually smoothen these alteration using seven-day moving averages. Finally, to compare data from different locations, we typically scale the reported case data by the population. A common metric for comparison is the seven-day-per-100,000 incidence, the number of new cases per 100,000 individuals across a seven-day win- dow [38]. Policy makers across the globe use this incidence value to characterize the severity of the outbreak and justify the need for political interventions. Figure 1.9 illustrates a typical data set for the COVID-19 outbreak that we use to infer our epidemiological model parameters. The orange lines summarize the out- break dynamics worldwide throughout the year after the World Health Organization had declared COVID-19 a global pandemic, from March 11, 2020 to March 11, 2021. The light orange lines represent the reported daily new cases, which display notable weekday-weekend fluctuations associated with testing and reporting irreg-
- 42. 1.8 Data-driven modeling in epidemiology 25 Fig. 1.9 Daily new cases of COVID-19 and their seven-day moving average. Daily new COVID- 19 cases are reported on public dashboards worldwide. The light orange curve represents the raw data, the dark orange curve is the seven-day moving average, both reported throughout the year after the WHO had declared COVID-19 a global pandemic on March 11, 2020. ularities. The dark orange lines are the associated seven-day moving average ˆ I(t), which smoothens the fluctuations and displays a much clearer trend in the disease dynamics. Within this one-year window, the absolute daily case numbers peaked on January 8, 2021 with 841,304 new cases. The moving seven-day average peaked on January 11, 2021 with ∆Imax = 745,404 cases. If we assume an infectious pe- riod of C = 1/γ = 7 days, this would result in a maximum infectious population of Imax = 7 · ∆Imax = 7 · 745,404 = 5,217,828, meaning that mid January 2021, more than 5 million people were sick with COVID-19. For a total population of N = 7.8 billion people, this corresponds to a peak seven-day-per-100,000 incidence of Imax · 100,000/N = 67. There are many different ways to use the reported case data. The simplest way is to select a function, use statistical tools to fit its coefficients to the reported case data, and make projections into the future. However, not only the daily new cases, but also the seven-day average in Figure 1.9 display substantial fluc- tuation and it seems difficult to find a function that could explain the orange curves. For data-driven modeling, it is often easier to use the total cumulative case numbers, which always increase monotonically and tend to be more smooth in general. Figure 1.10 shows the total cumulative cases of COVID-19 D̂(t) and a simple statistical model D(t) to fit the data throughout the first year of the pandemic. During the very early stages of an outbreak, an exponential growth model, D(t) = D0 exp(G t), with a growth rate G often provides a good approximation of the total number of cases D̂(t) and is easy to fit. Indeed, this, or similar exponential models, is what many early approaches used. While the initial phase of the outbreak is well represented by exponential growth models, they soon tend to overestimate the outbreak. This is why, during the early COVID-19 pandemic, there was a broad overestimate of the number of cases and of the number of ventilators and hospital beds needed to tread diseased individuals [15]. In this book, instead of using purely statistical models, we use mechanistic models like the compartment model in Figure
- 43. 26 1 Introduction to mathematical epidemiology Fig. 1.10 Total cumulative cases of COVID-19 and a statistical model to fit the data. Cumulative COVID-19 cases, the sum of all daily new cases to that date, are reported on public dashboards worldwide. The solid orange curve represents the raw data, the dashed orange curve is an example of a statistical model, both reported throughout the year after the WHO had declared COVID-19 a global pandemic on March 11, 2020. 1.8. For this SEIRHD compartment model, the number of total cases is the sum of all infectious, hospitalized, recovered and dead individuals, I(t) + R(t) + H(t) + D(t). Throughout this book, we use different types of compartment models, infer their model parameters using reported COVID-19 case data from public databases [9, 12, 25], seroprevalence data from scientific publications [26], and mobility data [2, 10, 10], and vary the parameters to probe different outbreak scenarios. Problems 1.1 Testing, testing, testing. On June 6, 2020, Donald J. Trump, the President of the United States, famously said “Remember this, when you test more, you have more cases.” What is wrong with this statement? What did he really mean to say? Discuss the implications of the testing frequency on reported case numbers, and, ultimately, on policy making. 1.2 Herd immunity. On September 15, 2020, Donald J. Trump, the President of the United States, claimed that COVID-19 would go away, without a vaccine. “You’ll develop herd. Like a herd mentality.” What did he really mean to say? Discuss the effects of vaccination on herd immunity. 1.3 Herd immunity. Assume the basic reproduction number of the swine flu, caused by the H1N1 virus, was on the order of R0 = 1.4 − 1.6. Calculate its herd immunity threshold H and compare it against the herd immunity thresholds of other infectious diseases in Table 1.3 and Figure 1.6. Comment on whether this makes the swine flu a good candidate for eradication through vaccination.
- 44. 1.8 Data-driven modeling in epidemiology 27 1.4 COVID-19 variants B.1.351 and B.1.1.7. Assume the basic reproduction num- ber of COVID-19 in December 2020 was R0 = 2.0. Calculate the herd immunity threshold H. Now, assume that the variant B.1.351 with a 50% increased basic re- production number has been introduced into the population [20]. How does the herd immunity threshold change? How would the variant B.1.1.7 with a 56% increased basic reproduction number change the herd immunity threshold? Comment on your results. Fig. 1.11 The Lessons of the Pandemic. Twelve condensed rules to the the avoidance of unneces- sary personal risks and to the promotion of better personal health by George A. Soper in reflection of the Spanish flu in 1919. 1.5 The Lessons of the Pandemic. In his Science publication The Lessons of the Pandemic, George A. Soper lists twelve public health measures to manage the Spanish flu in 1919. Read the twelve recommendations in Figure 1.11 and comment on which of them are still valid for the COVID-19 pandemic today. 1.6 Efficacy of the AstraZeneca COVID-19 vaccine. The randomized, equal allo- cation AstraZeneca trial enrolled 11636 participants in its interim efficacy analysis, 5807 in the vaccinated group and 5829 in the unvaccinated placebo group. Of the vaccinated group, 30 developed COVID-19, and of the placebo group 101. Calculate the efficacy and risk ratio of the AstraZeneca vaccine. 1.7 Case rates during the AstraZeneca COVID-19 trial. The AstraZeneca trial en- rolled 11636 participants in its interim efficacy analysis, 5807 in the vaccinated group and 5829 in the unvaccinated placebo group. Of the vaccinated group, 30 developed COVID-19, and of the placebo group 101. Create a contingency table for the interim efficacy analysis. Calculate the overall case rate of the trial and the case rates in the vaccinated and placebo groups. 1.8 Efficacy of the Janssen COVID-19 vaccine. The randomized, equal allocation Janssen COVID-19 trial enrolled 39,321 participants, 19,630 received the vaccine
- 45. 28 1 Introduction to mathematical epidemiology and 19,691 received placebo. At least 14 days after vaccination, the trial recorded 116 COVID-19 cases in the vaccine group and 348 cases in the placebo group. Calculate the efficacy of the Janssen COVID-19 vaccine and compare it to the efficacy of the Pfizer BioNTech and AstraZeneca vaccines. How would you expect the efficacy to change after another 14 days? 1.9 Efficacy of the Janssen COVID-19 vaccine. The randomized, equal allocation Janssen COVID-19 trial enrolled 39,321 participants, 19,630 received the vaccine and 19,691 received placebo. At least 28 days after vaccination, the trial recorded 66 COVID-19 cases in the vaccine group and 193 cases in the placebo group. Calculate the efficacy of the Janssen COVID-19 vaccine after 28 days and compare it to the efficacy after 14 days. Comment on why you would or would not have expected this result. 1.10 Statistical models. Find and download the total cumulative COVID-19 case data D̂(t) from your own city, county, state, country, or the world. Plot the case data similar to Figure 1.10. Try to fit a linear function D(t) = D0 + c1 t and a quadratic function D(t) = D0 + c1 t + c2 t2 to the very early stages of the outbreak. Which function is easier to fit? What does that tell you about the early outbreak? 1.11 Statistical models. Find and download the total cumulative COVID-19 case data D̂(t) from your own city, county, state, country, or the world. Plot the case data similar to Figure 1.10. Fit an exponential function D(t) = D0 exp(G t) to the early stages of the outbreak. What is your growth rate G? When does the exponential function fail to describe the case data D̂(t)? 1.12 Epidemiology data. Find and download the daily new COVID-19 case data from your own city, county, state, country, or the world. Plot the raw data similar to Figure 1.9. Explain the local fluctuations in the reported case data that occur on the order of days. 1.13 Epidemiology data. Find and download the daily new COVID-19 case data from your own city, county, state, country, or the world. Calculate and plot the seven- day moving average similar to Figure 1.9. How many waves can you identify? Explain the global fluctuations that occur on the order of weeks or months. Interpret the growth or decay in case numbers in view of specific events or political interventions. 1.14 Epidemiology data. Find the COVID-19 case data from your own state or country and your furthest away vacation destination. Compare the different outbreak dynamics. Do you think the reporting between both locations is consistent? Identify at least four potential sources of error in reporting daily COVID-19 case data. 1.15 Incidence. Find and download the daily new COVID-19 case data from your own city, county, state, country, or the world. Calculate the seven-day moving average. Identify the peak seven-day moving average ∆Imax within your simulation window. Assume an infectious period of C = 1/γ = 7 days. Calculate the maximum infectious population Imax = 7·∆Imax. Find the total population N of your location and calculate its peak seven-day-per-100,000 incidence, Imax · 100,000/N.
- 46. References 29 1.16 Incidence and the effect of scale. Studying an outbreak at a more global scale tends to smoothen fluctuations and local peaks. Find and download the daily new COVID-19 case data from the next smaller or larger scale compared to the previous problem. If you have studied your state, now study your city, county, or country. Identify the peak seven-day moving average ∆Imax within your simulation window and calculate the maximum infectious population Imax = 7 · ∆Imax. Find the total population N of your location and calculate its peak seven-day-per-100,000 incidence, Imax ·100,000/N. Compare you results against the seven-day-per-100,000 incidence at the smaller or larger scale. Interpret your results. 1.17 Epidemiology data and compartment models. The compartment model in Figure 1.8 introduces three possible disease paths from infection: direct recovery at a fraction (1 − νh), hospitalization and recovery at a fraction νh (1 − νd), and hospitalization and death at a fraction νh · νd. The fraction νh · νd is called the case fatality rate and is about 2% worldwide for COVID-19. Find extreme values for the case fatality rate. Discuss which factors influence the case fatality rate both globally and locally. 1.18 Epidemiology data and compartment models. Assume you want to learn parameters for your compartment model, for example the one in Figure 1.8, from reported case data, hospitalizations, and deaths. Early in the pandemic, when testing was slow and reporting was delayed, epidemiologists suggested to use deaths rather than daily new cases for model calibration. This initiated a controversial and still ongoing discussion how to count COVID-19 deaths. Discuss the difference between death from and death with COVID-19 in terms of absolute numbers, case fatality ratios, and model parameters. References 1. Alber M, Buganza Tepole A, Cannon W, De S, Dura-Bernal S, Garikipati K, Karniadakis G, Lytton WW, Perdikaris P, Petzold L, Kuhl E (2019) Integrating machine learning and multiscale modeling: Perspectives, challenges, and opportunities in the biological, biomedical, and behavioral sciences. npj Digital Medicine 2:115. 2. Anderson RM, May RM (1982) Directly transmitted infectious diseases: control by vaccination. Science 215:1053-1060. 3. Anderson RM, May RM (1991) Infectious Diseases of Humans. Oxford University Press, Oxford. 4. Apple Mobility Trends. https://www.apple.com/covid19/mobility. accessed: June 1, 2021. 5. Bernoulli D (1760) Essay d’une nouvelle analyse de la mortalite causee par la petite verole et des avantages de l’inoculation pour la prevenir. Mémoires de Mathématiques et de Physique, Académie Royale des Sciences, Paris 1-45. 6. Brauer F, Castillo-Chavez C (2001) Mathematical Models in Population Biology and Epidemi- ology. Springer-Verlag New York. 7. Brauer F, van den Dreissche P, Wu J (2008) Mathematical Epidemiology. Springer-Verlag Berlin Heidelberg.
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- 48. References 31 33. Linka K, Peirlinck M, Kuhl E (2020) The reproduction number of COVID-19 and its correlation with public heath interventions. Computational Mechanics 66:1035-1050. 34. Linka K, Goriely A, Kuhl E (2021) Global and local mobility as a barometer for COVID-19 dynamics. Biomechanics and Modeling in Mechanobiology 20:651–669. 35. Linka K, Peirlinck M, Schafer A, Ziya Tikenogullari O, Goriely A, Kuhl E (2021) Effects of B.1.1.7 and B.1.351 on COVID-19 dynamics. A campus reopening study. Archives of Computational Methods in Engineering. doi:10.1007/s11831-021-09638-y. 36. Liu J, Shang, X (2020) Computational Epidemiology. Springer International Publishing. 37. Liu Y, Gayle AA, Wilder-Smith A, Rocklöv J (2020) The reproductive number of COVID-19 is higher compared to SARS coronavirus. Journal of Travel Medicine (2020) 27:taaa021. 38. Lu H, Weintz C, Pace J, Indana D, Linka K, Kuhl E (2021) Are college campuses superspread- ers? A data-driven modeling study. Computer Methods in Biomechanics and Biomedical Engineering doi:10.1080/10255842.2020.1869221. 39. Martcheva M (2015) An Introduction to Mathematical Epidemiology. Springer Science + Business Media New York. 40. New York Times (2020) Coronavirus COVID-19 Data in the United States. https://github .com/nytimes/covid-19-data/blob/master/us-states.csv assessed: June 1, 2021. 41. Osvaldo M (2018) Bayesian Analysis with Python: Introduction to Statistical Modeling and Probabilistic Programming Using PyMC3 and ArviZ. Packt Publishing, 2nd edition. 42. Paul JR (1966) Clinical Epidemiology. University of Chicago Press, Chicago. 43. Peirlinck M, Linka K, Sahli Costabal F, Bendavid E, Bhattacharya J, Ioannidis J, Kuhl E (2020) Visualizing the invisible: The effect of asymptomatic transmission on the outbreak dynamics of COVID-19. Computer Methods in Applied Mechanics and Engineering 372:113410. 44. Peng GCY, Alber M, Buganza Tepole A, Cannon W, De S, Dura-Bernal S, Garikipati K, Karniadakis G, Lytton WW, Perdikaris P, Petzold L, Kuhl E (2021) Multiscale modeling meets machine learning: What can we learn? Archive of Computational Methods in Engineering 28:1017-1037. 45. Siegenfeld AF, Taleb NN, Bar-Yam Y (2020) What models can and cannot tell us about COVID-19. Proceedings of the National Academy of Sciences 117:16092-16095. 46. Snow J (1855) On the Mode of Communication of Cholera (2nd edition). London, John Churchill. 47. Soper GA (1919) The lessons of the pandemic. Science XLIX 501-506. 48. Viceconte G, Petrosillo N (2020) COVID-19 R0: Magic number or conundrum? Infectious Disease Reports 12:8516. 49. World Health Organization. WHO Virtual Press Conference on COVID-19, March 11, 2020 https://www.who.int/docs/default-source/coronaviruse/transcripts/who-a udio-emergencies-coronavirus-press-conference-full-and-final-11mar2020 .pdf?sfvrsn=cb432bb32 accessed: June 1, 2021.
- 49. Chapter 2 The classical SIS model Abstract The SIS model is the simplest compartment model with only two pop- ulations, the susceptible and infectious groups S and I. It characterizes infectious diseases like the common cold or influenza that do not provide immunity upon infection. While the SIS model is too simplistic to explain the outbreak dynamics of complex infectious diseases, it is the only compartment model with an explicit analytical solution for the time course of its populations. This makes it a popular model to explain and illustrate the basic principles of compartment modeling. The learning objectives of this chapter on classical SIS modeling are to • explain concepts of mass action incidence and constant rate recovery • interpret contact and infectious rates and periods • solve the analytical solutions for the susceptible and infectious populations • distinguish disease free and endemic equilibria • analyze the final size relation • demonstrate how the infectious rate, reproduction number, and initial conditions modulate outbreak dynamics • discuss limitations of classical SIS modeling By the end of the chapter, you will be able to analyze, simulate, and predict the outbreak dynamics of simple infectious diseases like the common cold or influenza that do not provide immunity to reinfection. 2.1 Introduction of the SIS model Some infectious diseases, for example from the common cold or influenza, do not provide lifelong immunity upon recovery from infection [2, 2], and previously in- fected individuals become susceptible again [10]. We can model their behavior through the simplest of all compartment models, the SIS model [5]. Figure 11.4 shows that the SIS model consists of only two populations, the susceptible group S and the infectious group I [3]. As such, it provides valuable insight into the concept 33 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Kuhl, Computational Epidemiology, https://doi.org/10.1007/978-3-030-82890-5_2
- 50. 34 2 The classical SIS model Fig. 2.1 Classical SIS model. The classical SIS model contains two compartments for the sus- ceptible and infectious populations, S and I. The transition rates between the compartments, the contact and infectious rates, β and γ, are inverses of the contact and infectious periods, B = 1/β and C = 1/γ. of compartment modeling [8]. The SIS model is often used to illustrate the basic features of compartment models because we can solve the dynamics of its two pop- ulations, S(t) and I(t), analytically at any point t in time [7]. The SIS model also has simple analytical solutions for the converged final sizes, S∞ and I∞, as t → ∞ [13]. For the transition from the susceptible to the infectious group, we assume a mass action incidence [4], which implies that the rate of new infections is proportional to the size of the susceptible and infectious groups S and I weighted by the contact rate β, Û I = β SI. For the transition from the infectious to the susceptible group, we Fig. 2.2 The transition from the susceptible to the infectious group is based on the assumption of mass action incidence for which the rate of new infections is proportional to the size of the susceptible and infectious groups S and I, weighted by the contact rate β, Û I = β SI. The transition from the infectious to the susceptible group is based on the assumption of constant rate recovery for which the rate of recovered infections is proportional to the size of the infectious group I, weighted by the infectious rate γ, Û I = −γI. assume a constant rate recovery, which implies that the rate of recovered infections is proportional to the size of the infectious group I weighted by the infectious rate γ, Û I = γI. Figure 2.2 illustrates the dynamics of these two assumptions that result in the following system of two coupled ordinary differential equations, Û S = − β SI + γ I Û I = + β SI − γ I . (2.1) The transition rates between both compartments are the contact rate β and the infectious rate γ in units [1/days], which are the inverses of the contact period B = 1/β and the infectious period C = 1/γ in units [days]. The ratio between the contact and infectious rates, or similarly, between the infectious and contact periods, defines the basic reproduction number R0, R0 = β γ = C B . (2.2) In this simple format, the SIS model (2.1) neglects all vital dynamics, Û S + Û I 0 and S + I = const. = 1. It does not account for births, natural deaths, or death from the disease.
- 51. 2.2 Analytical solution of the SIS model 35 2.2 Analytical solution of the SIS model From the condition of non-vital dynamics, S = 1 − I, we can rephrase the system of equations of the SIS model (2.1) in terms of only one independent variable, the size of the infectious group I, governed by the following nonlinear ordinary differential equation, Û I = β [ 1 − I ] I − γ I = [ β − γ ] I 1 − I 1 − γ/β . (2.3) Equation (2.3) is a logistic differential equation of the form, Û I = r I [ 1 − I/K ] with r = β − γ and K = 1 − γ/β, (2.4) where K is the carrying capacity. This type of equation has an explicit analytical solution, I(t) = K I0 I0 + [ K − I0 ] exp (−r t) , (2.5) where I0 = I(0) is the initial infectious population [10]. It proves convenient to reparameterize the equation for the infectious population (2.5) in terms of the basic reproduction number R0 = β/γ and the infectious period C = 1/γ with r = [R0 − 1]/C and K = 1 − 1/R0. This provides the analytical solution for the SIS model in terms of the infectious period C, the basic reproduction number R0, and the initial infectious population I0, S(t) = 1− [1 − 1/R0] I0 I0 + [1 − 1/R0 − I0] exp ([1 − R0] t/C) I(t) = [1 − 1/R0] I0 I0 + [1 − 1/R0 − I0] exp ([1 − R0] t/C) . (2.6) Figures 2.3, 2.4, and 2.5 highlight the outbreak dynamics of the SIS model, solved analytically using equations (2.6), for the time period of one year. The three figures demonstrate the sensitivity of the SIS model for varying infectious periods C, and varying basic reproduction numbers R0, and initial infectious populations I0. Increas- ing the infectious period C delays convergence to the endemic equilibrium, but the final sizes S∞ and I∞ remain unchanged. Increasing the basic reproduction number R0 accelerates convergence to the endemic equilibrium, decreases S∞, and increases I∞. Increasing the initial infectious population I0 accelerates the onset of the out- break, but the final sizes S∞ and I∞ remain unchanged. Interestingly, increasing the initial exposed population I0 by an order of magnitude shifts the population dynam- ics by a constant time increment, for the current parameterization by 50 days. This highlights the exponential nature of the SIS model, which causes a constant acceler- ation of the outbreak for a logarithmic increase of the initial infectious population, while the overall outbreak dynamics remain the same.
- 52. 36 2 The classical SIS model Fig. 2.3 Classical SIS model. Sensitivity with respect to the infectious period C. Increasing the infectious period C delays convergence to the endemic equilibrium, but the final sizes S∞ and I∞ remain unchanged. Basic reproduction number R0 = 2.0, initial infectious population I0 = 0.01, and infectious period C = 5, 10, 15, 20, 25, 30 days. Fig. 2.4 Classical SIS model. Sensitivity with respect to the basic reproduction number R0. Increasing the basic reproduction number R0 accelerates convergence to the endemic equilibrium, decreases S∞, and increases I∞. Infectious period C = 20 days, initial infectious population I0 = 0.01, and basic reproduction number R0 = 1.5, 1.7, 2.0, 2.4, 3.0, 5.0, 10.0. 2.3 Final size relation of the SIS model For practical purposes, it is interesting to estimate the final susceptible and infectious populations S∞ and I∞ [15]. From the analytical solution (2.5), I∞ = K I0 I0 + [ K − I0 ] exp (−r t∞) with r = β − γ and K = 1 − γ/β, (2.7)
- 53. 2.3 Final size relation of the SIS model 37 Fig. 2.5 Classical SIS model. Sensitivity with respect to the initial infectious population I0. Increasing the initial infectious population I0 accelerates the onset of the outbreak, but the final sizes S∞ and I∞ remain unchanged. Infectious period C = 20 days, basic reproduction number R0 = 2.0, and initial infectious population I0 = 10−1, 10−2, 10−3, 10−4, 10−5, 10−6. it is easy to show that we can distinguish two cases. If there is a finite initial infectious population, I0 0, the infectious population will converge to either zero, for β γ, r 0, and exp (−r t∞) → ∞, or to K for β γ, r 0, and exp (−r t∞) → 0 [2, 9], thus I∞ = 0 if β γ K if β γ . (2.8) We can rephrase these limit conditions (2.8) in terms of the basic reproduction number R0. In classical epidemiology, the two converged states are known as the disease-free equilibrium and the endemic equilibrium [3]. The equations that define the final converged populations of the endemic equilibrium state are the final size relation. R0 1 disease-free equilibrium: S∞ = 1 and I∞ = 0 R0 1 endemic equilibrium: S∞ = 1/R0 and I∞ = 1 − 1/R0 . (2.9) Figure 2.6 illustrates the final size relation and emphasizes the role of the basic reproduction number R0 as the distinguishing outbreak characteristic between the disease-free and endemic equilibrium. For reproduction numbers smaller than one, R0 1, the SIS model converges to a disease-free equilibrium with S∞ = 1 and I∞ = 0. For reproduction numbers larger than one, R0 1, the SIS model converges to an endemic equilibrium with S∞ = 1/R0 and I∞ = 1 − 1/R0. The larger the reproduction number R0, the smaller the final susceptible population S∞ and the larger the final infectious population I∞ [3].
- 54. 38 2 The classical SIS model Fig. 2.6 Classical SIS model. Final size relation as a function of the basic reproduction num- ber R0. For R0 1, the SIS model converges to a disease-free equilibrium with S∞ = 1 and I∞ = 0. For R0 1, the SIS model converges to an endemic equilibrium with S∞ = 1/R0 and I∞ = 1 − 1/R0. Increasing R0 reduces the susceptible population S∞ and increases the infectious population I∞. Problems 2.1 Basic reproduction number. Throughout the winter of 1980, a Scottish board- ing school reported an influenza that, on any day, affected on average 408 of its 1632 students. Estimate the basic reproduction number. 2.2 Basic reproduction number. Throughout the winter of 1980, a Scottish board- ing school reported an influenza with an infectious period of C = 4 days and a contact period of B = 2.8 days. How many of its 1632 students are infectious at endemic equilibrium? 2.3 Basic reproduction number. Assume the common flu has an infectious period of C = 7 days and a contact rate of β = 0.2/ days. Determine the basic reproduction number R0 and the final sizes of the susceptible and infectious populations S∞ and I∞. 2.4 Contact rate. Assume the common flu has an infectious period of C = 7 days. Determine the contact rate β for which the infectious population I never increases beyond 20% of the population. What is the basic reproduction number under these conditions? 2.5 Contact rate. Assume the common flu has an infectious period of C = 7 days. Determine the contact rate β for which the infectious population I never increases beyond 30% of the population. Comment on how the basic reproduction number for a maximum infectious population of 30% differs to the basic reproduction number for a maximum infectious population of 20%.
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- 56. “Nothing of the kind,” said Thorndyke; “it’s unmitigated pleasure; the pleasure of the voyage and your high well-born society.” “I didn’t mean that,” grumbled the captain, “but, if you are coming as guests, send your man for your nightgear and let us bring you back tomorrow evening.” “We won’t disturb Polton,” said my colleague; “we can take the train from Blackwall and fetch our things ourselves. Eleven o’clock, you said?” “Thereabouts,” said Captain Grumpass; “but don’t put yourselves out.” The means of communication in London have reached an almost undesirable state of perfection. With the aid of the snorting train and the tinkling, two-wheeled “gondola,” we crossed and re-crossed the town with such celerity that it was barely eleven when we reappeared on Trinity Wharf with a joint Gladstone and Thorndyke’s little green case. The tender had hauled out of Bow Creek, and now lay alongside the wharf with a great striped can buoy dangling from her derrick, and Captain Grumpass stood at the gang way, his jolly, red face beaming with pleasure. The buoy was safely stowed forward, the derrick hauled up to the mast, the loose shrouds rehooked to the screw- lanyards, and the steamer, with four jubilant hoots, swung round and shoved her sharp nose against the incoming tide. For near upon four hours the ever-widening stream of the “London River” unfolded its moving panorama. The smoke and smell of Woolwich Reach gave place to lucid air made soft by the summer haze; the grey huddle of factories fell away and green levels of cattle-spotted marsh stretched away to the high land bordering the river valley. Venerable training ships displayed their chequered hulls by the wooded shore, and whispered of the days of oak and hemp, when the tall three-decker, comely and majestic, with her soaring heights of canvas, like towers of ivory, had not yet given place to the mud-coloured saucepans that fly the white ensign now-a-days and
- 57. devour the substance of the British taxpayer: when a sailor was a sailor and not a mere seafaring mechanic. Sturdily breasting the flood tide, the tender threaded her way through the endless procession of shipping; barges, billy-boys, schooners, brigs; lumpish Black-seamen, blue-funnelled China tramps, rickety Baltic barques with twirling windmills, gigantic liners, staggering under a mountain of top-hamper. Erith, Purfleet, Greenhithe, Grays greeted us and passed astern. The chimneys of Northfleet, the clustering roofs of Gravesend, the populous anchorage and the lurking batteries, were left behind, and, as we swung out of the Lower Hope, the wide expanse of sea reach spread out before us like a great sheet of blue- shot satin. About half-past twelve the ebb overtook us and helped us on our way, as we could see by the speed with which the distant land slid past, and the freshening of the air as we passed through it. But sky and sea were hushed in a summer calm. Balls of fleecy cloud hung aloft, motionless in the soft blue; the barges drifted on the tide with drooping sails, and a big, striped bell buoy—surmounted by a staff and cage and labelled “Shivering Sand”—sat dreaming in the sun above its motionless reflection, to rouse for a moment as it met our wash, nod its cage drowsily, utter a Solemn ding-dong, and fall asleep again. It was shortly after passing the buoy that the gaunt shape of a screw-pile lighthouse began to loom up ahead, its dull red paint turned to vermilion by the early afternoon sun. As we drew nearer, the name Girdler, painted in huge, white letters, became visible, and two men could be seen in the gallery around the lantern, inspecting us through a telescope. “Shall you be long at the lighthouse, sir?” the master of the tender inquired of Captain Grumpass; “because we’re going down to the North-East Pan Sand to fix this new buoy and take up the old one.” “Then you’d better put us off at the lighthouse and come back for us when you’ve finished the job,” was the reply. “I don’t know how long
- 58. we shall be.” The tender was brought to, a boat lowered, and a couple of hands pulled us across the intervening space of water. “It will be a dirty climb for you in your shore-going clothes,” the captain remarked—he was as spruce as a new pin himself, “but the stuff will all wipe off.” We looked up at the skeleton shape. The falling tide had exposed some fifteen feet of the piles, and piles and ladder alike were swathed in sea-grass and encrusted with barnacles and worm tubes. But we were not such town-sparrows as the captain seemed to think, for we both followed his lead without difficulty up the slippery ladder, Thorndyke clinging tenaciously to his little green case, from which he refused to be separated even for an instant. “These gentlemen and I,” said the captain, as we stepped on the stage at the head of the ladder, “have come to make inquiries about the missing man, James Brown. Which of you is Jeffreys?” “I am, sir,” replied a tall, powerful, square-jawed, beetle-browed man, whose left hand was tied up in a rough bandage. “What have you been doing to your hand?” asked the captain. “I cut it while I was peeling some potatoes,” was the reply. “It isn’t much of a cut, sir.” “Well, Jeffreys,” said the captain, “Brown’s body has been picked up and I want particulars for the inquest. You’ll be summoned as a witness, I suppose, so come in and tell us all you know.” We entered the living-room and seated ourselves at the table. The captain opened a massive pocket-book, while Thorndyke, in his attentive, inquisitive fashion, looked about the odd, cabin-like room as if making a mental inventory of its contents. Jeffreys’ statement added nothing to what we already knew. He had seen a boat with one man in it making for the lighthouse. Then the fog had drifted up and he had lost sight of the boat. He started the fog-horn and kept a bright look-out, but the boat never arrived. And
- 59. that was all he knew. He supposed that the man must have missed the lighthouse and been carried away on the ebb-tide, which was running strongly at the time. “What time was it when you last saw the boat?” Thorndyke asked. “About half-past eleven,” replied Jeffreys. “What was the man like?” asked the captain. “I don’t know, sir; he was rowing, and his back was towards me.” “Had he any kit-bag or chest with him?” asked Thorndyke. “He’d got his chest with him,” said Jeffreys. “What sort of chest was it?” inquired Thorndyke. “A small chest, painted green, with rope beckets.” “Was it corded?” “It had a single cord round, to hold the lid down.” “Where was it stowed?” “In the stern-sheets, sir.” “How far off was the boat when you last saw it?” “About half-a-mile.” “Half-a-mile!” exclaimed the captain. “Why, how the deuce could you see that chest half-a-mile away?” The man reddened and cast a look of angry suspicion at Thorndyke. “I was watching the boat through the glass, sir,” he replied sulkily. “I see,” said Captain Grumpass. “Well, that will do, Jeffreys. We shall have to arrange for you to attend the inquest. Tell Smith I want to see him.” The examination concluded, Thorndyke and I moved our chairs to the window, which looked out over the sea to the east. But it was not the sea or the passing ships that engaged my colleague’s attention. On the wall, beside the window, hung a rudely-carved
- 60. pipe-rack containing five pipes. Thorndyke had noted it when we entered the room, and now, as we talked, I observed him regarding it from time to time with speculative interest. “You men seem to be inveterate smokers,” he remarked to the keeper, Smith, when the captain had concluded the arrangements for the “shift.” “Well, we do like our bit of ‘baccy, sir, and that’s a fact,” answered Smith. “You see, sir,” he continued, “it’s a lonely life, and tobacco’s cheap out here.” “How is that?” asked Thorndyke. “Why, we get it given to us. The small craft from foreign, especially the Dutchmen, generally heave us a cake or two when they pass close. We’re not ashore, you see, so there’s no duty to pay.” “So you don’t trouble the tobacconists much? Don’t go in for cut tobacco?” “No, sir; we’d have to buy it, and then the cut stuff wouldn’t keep. No, it’s hard-tack to eat out here and hard tobacco to smoke.” “I see you’ve got a pipe-rack, too, quite a stylish affair.” “Yes,” said Smith, “I made it in my off-time. Keeps the place tidy and looks more ship-shape than letting the pipes lay about anywhere.” “Some one seems to have neglected his pipe,” said Thorndyke, pointing to one at the end of the rack which was coated with green mildew. “Yes; that’s Parsons, my mate. He must have left it when he went off near a month ago. Pipes do go mouldy in the damp air out here.” “How soon does a pipe go mouldy if it is left untouched?” Thorndyke asked. “It’s according to the weather,” said Smith. “When it’s warm and damp they’ll begin to go in about a week. Now here’s Barnett’s pipe that he’s left behind—the man that broke his leg, you know, sir—it’s
- 61. just beginning to spot a little. He couldn’t have used it for a day or two before he went.” “And are all these other pipes yours?” “No, sir. This here one is mine. The end one is Jeffreys’, and I suppose the middle one is his too, but I don’t know it.” “You’re a demon for pipes, doctor,” said the captain, strolling up at this moment; “you seem to make a special study of them.” “‘The proper study of mankind is man,’” replied Thorndyke, as the keeper retired, “and ‘man’ includes those objects on which his personality is impressed. Now a pipe is a very personal thing. Look at that row in the rack. Each has its own physiognomy which, in a measure, reflects the peculiarities of the owner. There is Jeffreys’ pipe at the end, for instance. The mouthpiece is nearly bitten through, the bowl scraped to a shell and scored inside and the brim battered and chipped. The whole thing speaks of rude strength and rough handling. He chews the stem as he smokes, he scrapes the bowl violently, and he bangs the ashes out with unnecessary force. And the man fits the pipe exactly: powerful, square-jawed and, I should say, violent on occasion.” “Yes, he looks a tough customer, does Jeffreys,” agreed the captain. “Then,” continued Thorndyke, “there is Smith’s pipe, next to it; ‘coked’ up until the cavity is nearly filled and burnt all round the edge; a talker’s pipe, constantly going out and being relit. But the one that interests me most is the middle one.” “Didn’t Smith say that was Jeffreys’ too?” I said. “Yes,” replied Thorndyke, “but he must be mistaken. It is the very opposite of Jeffreys’ pipe in every respect. To begin with, although it is an old pipe, there is not a sign of any tooth-mark on the mouthpiece. It is the only one in the rack that is quite unmarked. Then the brim is quite uninjured: it has been handled gently, and the silver band is jet-black, whereas the band on Jeffreys’ pipe is quite bright.”
- 62. “I hadn’t noticed that it had a band,” said the captain. “What has made it so black?” Thorndyke lifted the pipe out of the rack and looked at it closely. “Silver sulphide,” said he, “the sulphur no doubt derived from something carried in the pocket.” “I see,” said Captain Grumpass, smothering a yawn and gazing out of the window at the distant tender. “Incidentally it’s full of tobacco. What moral do you draw from that?” Thorndyke turned the pipe over and looked closely at the mouthpiece. “The moral is,” he replied, “that you should see that your pipe is clear before you fill it.” He pointed to the mouthpiece, the bore of which was completely stopped up with fine fluff. “An excellent moral too,” said the captain, rising with an other yawn. “If you’ll excuse me a minute I’ll just go and see what the tender is up to. She seems to be crossing to the East Girdler.” He reached the telescope down from its brackets and went out onto the gallery. As the captain retreated, Thorndyke opened his pocket knife, and, sticking the blade into the bowl of the pipe, turned the tobacco out into his hand. “Shag, by Jove!” I exclaimed. “Yes,” he answered, poking it back into the bowl. “Didn’t you expect it to be shag?” “I don’t know that I expected anything,” I admitted. “The silver band was occupying my attention.” “Yes, that is an interesting point,” said Thorndyke, “but let us see what the obstruction consists of.” He opened the green case, and, taking out a dissecting needle, neatly extracted a little ball of fluff from the bore of the pipe. Laying this on a glass slide, he teased it out in a drop of glycerine and put on a cover-glass while I set up the microscope.
- 63. “Better put the pipe back in the rack,” he said, as he laid the slide on the stage of the instrument. I did so and then turned, with no little excitement, to watch him as he examined the specimen. After a brief inspection he rose and waved his hand towards the microscope. “Take a look at it, Jervis,” he said. I applied my eye to the instrument, and, moving the slide about, identified the constituents of the little mass of fluff. The ubiquitous cotton fibre was, of course, in evidence, and a few fibres of wool, but the most remarkable objects were two or three hairs—very minute hairs of a definite zigzag shape and having a flat expansion near the free end like the blade of a paddle. “These are the hairs of some small animal,” I said; “not a mouse or rat or any rodent, I should say. Some small insectivorous animal, I fancy. Yes! Of course! They are the hairs of a mole.” I stood up, and, as the importance of the discovery flashed on me, I looked at my colleague in silence. “Yes,” he said, “they are unmistakable; and they furnish the keystone of the argument.” “You think that this is really the dead man’s pipe, then?” I said. “According to the law of multiple evidence,” he replied, “it is practically a certainty. Consider the facts in sequence. Since there is no sign of mildew on it, this pipe can have been here only a short time, and must belong either to Barnett, Smith, Jeffreys or Brown. It is an old pipe, but it has no tooth-marks on it. Therefore it has been used by a man who has no teeth. But Barnett, Smith and Jeffreys all have teeth and mark their pipes, whereas Brown has no teeth. The tobacco in it is shag. But these three men do not smoke shag, whereas Brown had shag in his pouch. The silver band is encrusted with sulphide; and Brown carried sulphur-tipped matches loose in his pocket with his pipe. We find hairs of a mole in the bore of the pipe; and Brown carried a mole skin pouch in the pocket in which he appears to have carried his pipe. Finally, Brown’s pocket contained a pipe which was obviously not his and which closely resembled that
- 64. of Jeffreys; it contained tobacco similar to that which Jeffreys smokes and different from that in Brown’s pouch. It appears to me quite conclusive, especially when we add to this evidence the other items that are in our possession.” “What items are they?” I asked. “First there is the fact that the dead man had knocked his head heavily against some periodically submerged body covered with acorn barnacles and serpulæ. Now the piles of this lighthouse answer to the description exactly, and there are no other bodies in the neighbourhood that do: for even the beacons are too large to have produced that kind of wound. Then the dead man’s sheath- knife is missing, and Jeffreys has a knife-wound on his hand. You must admit that the circumstantial evidence is overwhelming.” At this moment the captain bustled into the room with the telescope in his hand. “The tender is coming up towing a strange boat,” he said. “I expect it’s the missing one, and, if it is, we may learn something. You’d better pack up your traps and get ready to go on board.” We packed the green case and went out into the gallery, where the two keepers were watching the approaching tender; Smith frankly curious and interested, Jeffreys restless, fidgety and noticeably pale. As the steamer came opposite the lighthouse, three men dropped into the boat and pulled across, and one of them—the mate of the tender—came climbing up the ladder. “Is that the missing boat?” the captain sang out. “Yes, sir,” answered the officer, stepping onto the staging and wiping his hands on the reverse aspect of his trousers, “we saw her lying on the dry patch of the East Girdler. There’s been some hanky-panky in this job, sir.” “Foul play, you think, hey?” “Not a doubt of it, sir. The plug was out and lying loose in the bottom, and we found a sheath-knife sticking into the kelson forward
- 65. among the coils of the painter. It was stuck in hard as if it had dropped from a height.” “That’s odd,” said the captain. “As to the plug, it might have got out by accident.” “But it hadn’t sir,” said the mate. “The ballast-bags had been shifted along to get the bottom boards up. Besides, sir, a seaman wouldn’t let the boat fill; he’d have put the plug back and baled out.” “That’s true,” replied Captain Grumpass; “and certainly the presence of the knife looks fishy. But where the deuce could it have dropped from, out in the open sea? Knives don’t drop from the clouds— fortunately. What do you say, doctor?” “I should say that it is Brown’s own knife, and that it probably fell from this staging.” Jeffreys turned swiftly, crimson with wrath. “What d’ye mean?” he demanded. “Haven’t I said that the boat never came here?” “You have,” replied Thorndyke; “but if that is so, how do you explain the fact that your pipe was found in the dead man’s pocket and that the dead man’s pipe is at this moment in your pipe-rack?” The crimson flush on Jeffreys’ face faded as quickly as it had come. “I don’t know what you’re talking about,” he faltered. “I’ll tell you,” said Thorndyke. “I will relate what happened and you shall check my statements. Brown brought his boat alongside and came up into the living-room, bringing his chest with him. He filled his pipe and tried to light it, but it was stopped and wouldn’t draw. Then you lent him a pipe of yours and filled it for him. Soon afterwards you came out on this staging and quarrelled. Brown defended himself with his knife, which dropped from his hand into the boat. You pushed him off the staging and he fell, knocking his head on one of the piles. Then you took the plug out of the boat and sent her adrift to sink, and you flung the chest into the sea. This happened about ten minutes past twelve. Am I right?”
- 66. Jeffreys stood staring at Thorndyke, the picture of amazement and consternation; but he uttered no word in reply. “Am I right?” Thorndyke repeated. “Strike me blind!” muttered Jeffreys. “Was you here, then? You talk as if you had been. Anyhow,” he continued, recovering somewhat, “you seem to know all about it. But you’re wrong about one thing. There was no quarrel. This chap, Brown, didn’t take to me and he didn’t mean to stay out here. He was going to put off and go ashore again and I wouldn’t let him. Then he hit out at me with his knife and I knocked it out of his hand and he staggered backwards and went overboard.” “And did you try to pick him up?” asked the captain. “How could I,” demanded Jeffreys, “with the tide racing down and me alone on the station? I’d never have got back.” “But what about the boat, Jeffreys? Why did you scuttle her?” “The fact is,” replied Jeffreys, “I got in a funk, and I thought the simplest plan was to send her to the cellar and know nothing about it. But I never shoved him over. It was an accident, sir; I swear it!” “Well, that sounds a reasonable explanation,” said the captain. “What do you say, doctor?” “Perfectly reasonable,” replied Thorndyke, “and, as to its truth, that is no affair of ours.” “No. But I shall have to take you off, Jeffreys, and hand you over to the police. You understand that?” “Yes, sir, I understand,” answered Jeffreys. “That was a queer case, that affair on the Girdler,” remarked Captain Grumpass, when he was spending an evening with us some six months later. “A pretty easy let off for Jeffreys, too—eighteen months, wasn’t it?” “Yes, it was a very queer case indeed,” said Thorndyke. “There was something behind that ‘accident,’ I should say. Those men had
- 67. probably met before.” “So I thought,” agreed the captain. “But the queerest part of it to me was the way you nosed it all out. I’ve had a deep respect for briar pipes since then. It was a remarkable case,” he continued. “The way in which you made that pipe tell the story of the murder seems to me like sheer enchantment.” “Yes,” said I, “it spoke like the magic pipe—only that wasn’t a tobacco-pipe—in the German folk-story of the ‘Singing Bone.’ Do you remember it? A peasant found the bone of a murdered man and fashioned it into a pipe. But when he tried to play on it, it burst into a song of its own— ‘My brother slew me and buried my bones Beneath the sand and under the stones.’” “A pretty story,” said Thorndyke, “and one with an excellent moral. The inanimate things around us have each of them a song to sing to us if we are but ready with attentive ears.”
- 69. PART I THE SPINSTERS’ GUEST The lingering summer twilight was fast merging into night as a solitary cyclist, whose evening-dress suit was thinly disguised by an overcoat, rode slowly along a pleasant country road. From time to time he had been overtaken and passed by a carriage, a car or a closed cab from the adjacent town, and from the festive garb of the occupants he had made shrewd guesses at their destination. His own objective was a large house, standing in somewhat extensive grounds just off the road, and the peculiar circumstances that surrounded his visit to it caused him to ride more and more slowly as he approached his goal. Willowdale—such was the name of the house—was, to-night, witnessing a temporary revival of its past glories. For many months it had been empty and a notice-board by the gate-keeper’s lodge had silently announced its forlorn state; but to-night, its rooms, their bare walls clothed in flags and draperies, their floors waxed or carpeted, would once more echo the sound of music and cheerful voices and vibrate to the tread of many feet. For on this night the spinsters of Raynesford were giving a dance; and chief amongst the spinsters was Miss Halliwell, the owner of Willowdale. It was a great occasion. The house was large and imposing; the spinsters were many and their purses were long. The guests were numerous and distinguished, and included no less a person than Mrs. Jehu B. Chater. This was the crowning triumph of the function, for the beautiful American widow was the lion (or should we say lioness?) of the season. Her wealth was, if not beyond the dreams of avarice, at least beyond the powers of common British arithmetic,
- 70. and her diamonds were, at once, the glory and the terror of her hostesses. All these attractions notwithstanding, the cyclist approached the vicinity of Willowdale with a slowness almost hinting at reluctance; and when, at length, a curve of the road brought the gates into view, he dismounted and halted irresolutely. He was about to do a rather risky thing, and, though by no means a man of weak nerve, he hesitated to make the plunge. The fact is, he had not been invited. Why, then, was he going? And how was he to gain admittance? To which questions the answer involves a painful explanation. Augustus Bailey lived by his wits. That is the common phrase, and a stupid phrase it is. For do we not all live by our wits, if we have any? And does it need any specially brilliant wits to be a common rogue? However, such as his wits were, Augustus Bailey lived by them, and he had not hitherto made a fortune. The present venture arose out of a conversation overheard at a restaurant table and an invitation-card carelessly laid down and adroitly covered with the menu. Augustus had accepted the invitation that he had not received (on a sheet of Hotel Cecil notepaper that he had among his collection of stationery) in the name of Geoffrey Harrington-Baillie; and the question that exercised his mind at the moment was, would he or would he not be spotted? He had trusted to the number of guests and the probable inexperience of the hostesses. He knew that the cards need not be shown, though there was the awkward ceremony of announcement. But perhaps it wouldn’t get as far as that. Probably not, if his acceptance had been detected as emanating from an uninvited stranger. He walked slowly towards the gates with growing discomfort. Added to his nervousness as to the present were certain twinges of reminiscence. He had once held a commission in a line regiment— not for long, indeed; his “wits” had been too much for his brother
- 71. officers—but there had been a time when he would have come to such a gathering as this an invited guest. Now, a common thief, he was sneaking in under a false name, with a fair prospect of being ignominiously thrown out by the servants. As he stood hesitating, the sound of hoofs on the road was followed by the aggressive bellow of a motor-horn. The modest twinkle of carriage lamps appeared round the curve and then the glare of acetylene headlights. A man came out of the lodge and drew open the gates; and Mr. Bailey, taking his courage in both hands, boldly trundled his machine up the drive. Half-way up—it was quite a steep incline—the car whizzed by; a large Napier filled with a bevy of young men who economized space by sitting on the backs of the seats and on one another’s knees. Bailey looked at them and decided that this was his chance, and, pushing forward, he saw his bicycle safely bestowed in the empty coach-house and then hurried on to the cloak-room. The young men had arrived there before him and, as he entered, were gaily peeling off their over coats and flinging them down on a table. Bailey followed their example, and, in his eagerness to enter the reception room with the crowd, let his attention wander from the business of the moment, and, as he pocketed the ticket and hurried away, he failed to notice that the bewildered attendant had put his hat with another man’s coat and affixed his duplicate to them both. “Major Podbury, Captain Barker-Jones, Captain Sparker, Mr. Watson, Mr. Goldsmith, Mr. Smart, Mr. Harrington Baillie!’ As Augustus swaggered up the room, hugging the party of officers and quaking inwardly, he was conscious that his hostesses glanced from one man to another with more than common interest. But at that moment the footman’s voice rang out, sonorous and clear— “Mrs. Chater, Colonel Crumpler!” and, as all eyes were turned towards the new arrivals, Augustus made his bow and passed into the throng. His little game of bluff had “come off,” after all.
- 72. He withdrew modestly into the more crowded portion of the room, and there took up a position where he would be shielded from the gaze of his hostesses. Presently, he reflected, they would forget him, if they had really thought about him at all, and then he would see what could be done in the way of business. He was still rather shaky, and wondered how soon it would be decent to steady his nerves with a “refresher.” Meanwhile he kept a sharp look-out over the shoulders of neighbouring guests, until a movement in the crowd of guests disclosed Mrs. Chater shaking hands with the presiding spinster. Then Augustus got a most uncommon surprise. He knew her at the first glance. He had a good memory for faces, and Mrs. Chater’s face was one to remember. Well did he recall the frank and lovely American girl with whom he had danced at the regimental ball years ago. That was in the old days when he was a subaltern, and before that little affair of the pricked court-cards that brought his military career to an end. They had taken a mutual liking, he remembered, that sweet-faced Yankee maid and he; had danced many dances and had sat out others, to talk mystical nonsense which, in their innocence, they had believed to be philosophy. He had never seen her since. She had come into his life and gone out of it again, and he had forgotten her name, if he had ever known it. But here she was, middle aged now, it was true, but still beautiful and a great personage withal. And, ye gods! what diamonds! And here was he, too, a common rogue, lurking in the crowd that he might, perchance, snatch a pendant or “pinch” a loose brooch. Perhaps she might recognize him. Why not? He had recognized her. But that would never do. And thus reflecting, Mr. Bailey slipped out to stroll on the lawn and smoke a cigarette. Another man, somewhat older than himself, was pacing to and fro thoughtfully, glancing from time to time through the open windows into the brilliantly-lighted rooms. When they had passed once or twice, the stranger halted and addressed him.
- 73. “This is the best place on a night like this,” he remarked; “it’s getting hot inside already. But perhaps you’re keen on dancing.” “Not so keen as I used to be,” replied Bailey; and then, observing the hungry look that the other man was bestowing on his cigarette, he produced his case and offered it. “Thanks awfully!” exclaimed the stranger, pouncing with avidity on the open case. “Good Samaritan, by Jove. Left my case in my overcoat. Hadn’t the cheek to ask, though I was starving for a smoke.” He inhaled luxuriously, and, blowing out a cloud of smoke, resumed: “These chits seem to be running the show pretty well, h’m? Wouldn’t take it for an empty house to look at it, would you?” “I have hardly seen it,” said Bailey; “only just come, you know.” “We’ll have a look round, if you like,” said the genial stranger, “when we’ve finished our smoke, that is. Have a drink too; may cool us a bit. Know many people here?” “Not a soul,” replied Bailey. “My hostess doesn’t seem to have turned up.” “Well, that’s easily remedied,” said the stranger. “My daughter’s one of the spinsters—Granby, my name; when we’ve had a drink, I’ll make her find you a partner—that is, if you care for the light fantastic.” “I should like a dance or two,” said Bailey, “though I’m getting a bit past it now, I suppose. Still, it doesn’t do to chuck up the sponge prematurely.” “Certainly not,” Granby agreed jovially; “a man’s as young as he feels. Well, come and have a drink and then we’ll hunt up my little girl.” The two men flung away the stumps of their cigarettes and headed for the refreshments. The spinsters’ champagne was light, but it was well enough if taken in sufficient quantity; a point to which Augustus—and Granby too— paid judicious attention; and when he had supplemented the wine with a few sandwiches, Mr. Bailey felt in notably better spirits. For, to
- 74. tell the truth, his diet, of late, had been somewhat meagre. Miss Granby, when found, proved to be a blonde and guileless “flapper” of some seventeen summers, childishly eager to play her part of hostess with due dignity; and presently Bailey found himself gyrating through the eddying crowd in company with a comely matron of thirty or thereabouts. The sensations that this novel experience aroused rather took him by surprise. For years past he had been living a precarious life of mean and sordid shifts that oscillated between mere shabby trickery and downright crime; now conducting a paltry swindle just inside the pale of the law, and now, when hard pressed, descending to actual theft; consorting with shady characters, swindlers and knaves and scurvy rogues like himself; gambling, borrowing, cadging and, if need be, stealing, and always slinking abroad with an apprehensive eye upon “the man in blue.” And now, amidst the half-forgotten surroundings, once so familiar; the gaily-decorated rooms, the rhythmic music, the twinkle of jewels, the murmur of gliding feet and the rustle of costly gowns, the moving vision of honest gentlemen and fair ladies; the shameful years seemed to drop away and leave him to take up the thread of his life where it had snapped so disastrously. After all, these were his own people. The seedy knaves in whose steps he had walked of late were but aliens met by the way. He surrendered his partner, in due course, with regret—which was mutual—to an inarticulate subaltern, and was meditating another pilgrimage to the refreshment-room, when he felt a light touch upon his arm. He turned swiftly. A touch on the arm meant more to him than to some men. But it was no wooden-faced plain-clothes man that he confronted; it was only a lady. In short, it was Mrs. Chater, smiling nervously and a little abashed by her own boldness. “I expect you’ve forgotten me,” she began apologetically, but Augustus interrupted her with an eager disclaimer.
- 75. “Of course I haven’t,” he said; “though I have forgotten your name, but I remember that Portsmouth dance as well as if it were yesterday; at least one incident in it—the only one that was worth remembering. I’ve often hoped that I might meet you again, and now, at last, it has happened.” “It’s nice of you to remember,” she rejoined. “I’ve often and often thought of that evening and all the wonderful things that we talked about. You were a nice boy then; I wonder what you are like now. What a long time ago it is!” “Yes,” Augustus agreed gravely, “it is a long time. I know it myself; but when I look at you, it seems as if it could only have been last season.” “Oh, fie!” she exclaimed. “You are not simple as you used to be. You didn’t flatter then; but perhaps there wasn’t the need.” She spoke with gentle reproach, but her pretty face flushed with pleasure nevertheless, and there was a certain wistfulness in the tone of her concluding sentence. “I wasn’t flattering,” Augustus replied, quite sincerely; “I knew you directly you entered the room and marvelled that Time had been so gentle with you. He hasn’t been as kind to me.” “No. You have gotten a few grey hairs, I see, but after all, what are grey hairs to a man? Just the badges of rank, like the crown on your collar or the lace on your cuffs, to mark the steps of your promotion —for I guess you’ll be a colonel by now.” “No,” Augustus answered quickly, with a faint flush, “I left the army some years ago.” “My! what a pity!” exclaimed Mrs. Chater. “You must tell me all about it—but not now. My partner will be looking for me. We will sit out a dance and have a real gossip. But I’ve forgotten your name—never could recall it, in fact, though that didn’t prevent me from remembering you; but, as our dear W. S. remarks, ‘What’s in a name?’”
- 76. “Ah, indeed,” said Mr. Harrington-Baillie; and apropos of that sentiment, he added: “Mine is Rowland—Captain Rowland. You may remember it now.” Mrs. Chater did not, however, and said so. “Will number six do?” she asked, opening her programme; and, when Augustus had assented, she entered his provisional name, remarking complacently: “We’ll sit out and have a right-down good talk, and you shall tell me all about yourself and if you still think the same about free-will and personal responsibility. You had very lofty ideals, I remember, in those days, and I hope you have still. But one’s ideals get rubbed down rather faint in the friction of life. Don’t you think so?” “Yes, I am afraid you’re right,” Augustus assented gloomily. “The wear and tear of life soon fetches the gilt off the gingerbread. Middle age is apt to find us a bit patchy, not to say naked.” “Oh, don’t be pessimistic,” said Mrs. Chater; “that is the attitude of the disappointed idealist, and I am sure you have no reason, really, to be disappointed in yourself. But I must run away now. Think over all the things you have to tell me, and don’t forget that it is number six.” With a bright smile and a friendly nod she sailed away, a vision of glittering splendour, compared with which Solomon in all his glory was a mere matter of commonplace bullion. The interview, evidently friendly and familiar, between the unknown guest and the famous American widow had by no means passed unnoticed; and in other circumstances, Bailey might have endeavoured to profit by the reflected glory that enveloped him. But he was not in search of notoriety; and the same evasive instinct that had led him to sink Mr. Harrington-Baillie in Captain Rowland, now advised him to withdraw his dual personality from the vulgar gaze. He had come here on very definite business. For the hundredth time he was “stony-broke,” and it was the hope of picking up some “unconsidered trifles” that had brought him. But, somehow, the atmosphere of the place had proved unfavourable. Either opportunities were lacking or he failed to seize them. In any case, the game pocket that formed an unconventional feature of his dress-
- 77. coat was still empty, and it looked as if a pleasant evening and a good supper were all that he was likely to get. Nevertheless, be his conduct never so blameless, the fact remained that he was an uninvited guest, liable at any moment to be ejected as an impostor, and his recognition by the widow had not rendered this possibility any the more remote. He strayed out onto the lawn, whence the grounds fell away on all sides. But there were other guests there, cooling themselves after the last dance, and the light from the rooms streamed through the windows, illuminating their figures, and among them, that of the too-companionable Granby. Augustus quickly drew away from the lighted area, and, chancing upon a narrow path, strolled away along it in the direction of a copse or shrubbery that he saw ahead. Presently he came to an ivy-covered arch, lighted by one or two fairy lamps, and, passing through this, he entered a winding path, bordered by trees and shrubs and but faintly lighted by an occasional coloured lamp suspended from a branch. Already he was quite clear of the crowd; indeed, the deserted condition of the pleasant retreat rather surprised him, until he reflected that to couples desiring seclusion there were whole ranges of untenanted rooms and galleries available in the empty house. The path sloped gently downwards for some distance; then came a long flight of rustic steps and, at the bottom, a seat between two trees. In front of the seat the path extended in a straight line, forming a narrow terrace; on the right the ground sloped up steeply towards the lawn; on the left it fell away still more steeply towards the encompassing wall of the grounds; and on both sides it was covered with trees and shrubs. Bailey sat down on the seat to think over the account of himself that he should present to Mrs. Chater. It was a comfortable seat, built into the trunk of an elm, which formed one end and part of the back. He leaned against the tree, and, taking out his silver case, selected a cigarette. But it remained unlighted between his fingers as he sat and meditated upon his unsatisfactory past and the
- 78. melancholy tale of what might have been. Fresh from the atmosphere of refined opulence that pervaded the dancing-rooms, the throng of well-groomed men and dainty women, his mind travelled back to his sordid little flat in Bermondsey, encompassed by poverty and squalor, jostled by lofty factories, grimy with the smoke of the river and the reek from the great chimneys. It was a hideous contrast. Verily the way of the transgressor was not strewn with flowers. At that point in his meditations he caught the sound of voices and footsteps on the path above and rose to walk on along the path. He did not wish to be seen wandering alone in the shrubbery. But now a woman’s laugh sounded from somewhere down the path. There were people approaching that way too. He put the cigarette back in the case and stepped round behind the seat, intending to retreat in that direction, but here the path ended, and beyond was nothing but a rugged slope down to the wall thickly covered with bushes. And while he was hesitating, the sound of feet descending the steps and the rustle of a woman’s dress left him to choose between staying where he was or coming out to confront the newcomers. He chose the former, drawing up close behind the tree to wait until they should have passed on. But they were not going to pass on. One of them—a woman—sat down on the seat, and then a familiar voice smote on his ear. “I guess I’ll rest here quietly for a while; this tooth of mine is aching terribly; and, see here, I want you to go and fetch me something. Take this ticket to the cloak-room and tell the woman to give you my little velvet bag. You’ll find in it a bottle of chloroform and a packet of cotton-wool.” “But I can’t leave you here all alone, Mrs. Chater,” her partner expostulated. “I’m not hankering for society just now,” said Mrs. Chater. “I want that chloroform. Just you hustle off and fetch it, like a good boy. Here’s the ticket.”
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