Sir model cat 1 (1)
- 1. SIR MODEL FOR SPREAD OF DISEASES IN EPIDEMIOLOGY GROUP 11
- 2. APPLIED EPIDEMIOLOGY MPH 5202 LECTURER: DR. JOHN KARIUKI MEMBERS SHAFICI ISMAIL JAMA- MPH/2022/48923 KAVEKE MULANDI-MPH/2021/86630
- 3. Introduction • The SIR model refers to an epidemiological model that computes the theoretical number of people infected with an infectious disease within a closed population (without migration) over a given period of time (Nasution et al., 2020). • Historically, this model has been used to explain the rapid rise and fall in number of infected people observed during epidemics.
- 4. • The SIR model aims to predict the number of individuals who are susceptible to infection, also the number of individual are actively infected, or have recovered from infection at any given time. • This model was introduced in 1927, less than a decade after the 1918 influenza pandemic,4 and its popularity may be due in part to its simplicity, which allows modelers to approximate disease behavior by estimating a small number of parameters.
- 5. Population Classes in SIR Model • The SIR model is a compartmental model hence it categorises individuals within a closed population into mutually exclusive compartments termed as population classes (Lourenco et al., 2020). • This therefore implies that an individual in the population can only fall into one compartment or population class at a given time period.
- 6. Population Classes in SIR Model Cont’ The term SIR is an abbreviation for the following three population classes (Lourenco et al., 2020): S: Susceptible people (individuals capable of becoming infected). I: Infected or infective people (individuals capable of causing infection) – individuals in this compartment may be symptomatic or asymptomatic.
- 7. Population Classes in SIR Model Cont’ R: People that have recovered or are no longer contagious – this entails individuals that: o Had the disease and recovered, o Died of the disease during the given time period, o Gained immunity against the disease upon recovery, and o Those that are isolated until recovery
- 8. Progression in SIR Model • The assumed progression in this model entails: a) A susceptible individual becomes infected of the disease through contact with an already infected individual, b) As an infected individual, the contact becomes infectious over a certain period of time, • Then the infected individual advances to the non- contagious state which is termed as recovery and entails recovering and getting immune from the disease of dying from it (Zlojutro • & Gardner, 2019).
- 9. Progression in SIR Model Cont’ • During most epidemics, all population begins in the susceptible compartment hence they are not ill but have the capability of getting infected of the disease (Nasution et al., 2020). • This therefore implies that no one is immune against the disease at the start of the outbreak. • Upon spread of the infectious agent, the susceptible individuals start getting infected hence making them contagious.
- 10. Progression in SIR Model Cont’ • The rate at which the susceptible individuals become infected and contagious of the disease during an epidemic depends on the existing number of both susceptible and infected individuals. • Therefore, at the start of an epidemic the disease spreads slowly owing to the few infected individuals. • As more individuals become infected, they contribute to the spread and increase the rate of infection.
- 11. Assumptions of the SIR Model: • The SIR model assumes that: a) The population size is fixed (closed population) hence no migration, births or deaths due to causes other than the specific disease of interest during the given time period. b) The latent period is short enough to be negligible hence a susceptible who contracts the disease becomes infective right away. c) After recovery, the infected gain immunity. d) Individuals in the closed population freely mix and come into contact with one another.
- 12. Assumptions of the SIR Model: • A fixed fraction k of the infected group will recover during any given day.
- 13. Parameters of the SIR Model • The SIR model is defined by two parameters which entail: I. Effective contact rate (β) – it accounts for transmissibility of the disease and the mean number of contacts per individual. Thus it affects the transition from the susceptible compartment to the infected compartment. For example, some COVID-19 mitigation strategies such as social distancing and quarantining infected individuals have helped lower the value of β thus reducing the spread of the disease (Lourenco et al., 2020).
- 14. Parameters of the SIR Model Cont’ II. Rate of recovery or mortality (γ) – this affects transition from the infected compartment to the recovered compartment (Chen et al., 2020). • If the rate at which individuals become infected exceeds the rate at which infected individuals recover, then there will be accumulation of individuals in the infected compartment. • The ratio between β and γ gives the basic reproduction number (RO) which refers to the mean number of new infection caused by a single individual over the course of their illness.
- 15. Parameters of the SIR Model Cont’ • Community mitigation strategies against an epidemic usually decrease the effective contact rate (β) which in turn reduces the basic reproduction number (RO) (Chen et al., 2020). • This ultimately delays and lowers the peak infection rate that occurs in the epidemic. In this regard, to maintain the decrease in total infections, the decrease in RO must be maintained through the community mitigation strategies.
- 16. Differential Equations for SIR Model • Identification of independent and dependent variables a) The independent variable is time (t) measured in days b) Dependent variables- There are two related sets i. Counts people in each of the groups, each as a function of time. • S= S (t) is the number of susceptible individuals • I = I(t) is the number of infected individuals • R = R(t) is the number of recovered individuals.
- 17. Differential Equations for SIR Model Cont’ ii. Represents the fraction of the total population in each of the three categories. If N is the total population, we have; • s(t) = S(t)/N the susceptible fraction of the population • i(t) = I(t)/N the infected fraction of the population, • r(t) = R(t)/N the recovered fraction of the population.
- 18. Differential Equations for SIR Model Cont’ • The SIR model comprises of the following differential equations: • dS/dt = - βSI • dI/dt = βSI – γI • dR/dt = γI ; where; • S is the number of the susceptible people, I is the number of infected people, and R is the number of people who have recovered from the illness and gained immunity. β is the infection rate while γ is the recovery rate (Cooper et al., 2020).
- 19. Differential Equations for SIR Model Cont’ • RO = βS/ γ where RO is the number of secondary infections caused by a single primary infected individual (Cooper et al., 2020).
- 21. Limitations of the SIR Model i. The model leaves out the concept of latent period between exposure and time at which an infected individual is capable of transmitting the disease. ii. The model also assumes homogeneous mixing of the population such that all individuals in the population have an equal probability of coming into contact with each other. This is not the case in the general population since we live in a society marked by social structures and groups. iii. It also assumes a closed population without births, migrations, or even deaths from causes other than the disease of interest (Lourenco et al., 2020).
- 22. References • Chen, Y. C., Lu, P. E., Chang, C. S., & Liu, T. H. (2020). A time- dependent SIR model for COVID-19 with undetectable infected persons. IEEE Transactions on Network Science and Engineering, 7(4), 3279-3294. • Cooper, I., Mondal, A., & Antonopoulos, C. G. (2020). A SIR model assumption for the spread of COVID-19 in different communities. Chaos, Solitons & Fractals, 139, 110057. • Lourenco, J., Paton, R., Ghafari, M., Kraemer, M., Thompson, C., Simmonds, P., ... & Gupta, S. (2020). Fundamental principles of epidemic spread highlight the immediate need for large-scale serological surveys to assess the stage of the SARS-CoV-2 epidemic. MedRxiv.
- 23. References Cont’ • Nasution, H., Jusuf, H., Ramadhani, E., & Husein, I. (2020). Model of Spread of Infectious Disease. Systematic Reviews in Pharmacy, 11(2). • Zlojutro, A., Rey, D., & Gardner, L. (2019). A decision-support framework to optimize border control for global outbreak mitigation. Scientific reports, 9(1), 1-14.