![Estimation of Parameters
• How does one estimate 𝛽 and 𝜌 from
1
𝑏
= 𝛽(1 − 𝜖)(1 − 𝑐) and
1
𝑒
=
𝜖𝜌(1 − 𝑐)?
• Define function 𝑓: 0,1 × [−1,1] as:
• On input (𝑟, 𝑠), set 𝜌 = 𝑟 and 𝑐 = 𝑠, compute 𝛽 and 𝜖, and use it to compute
trajectory of 𝑈 and 𝑅𝑈 for current phase. Compare with 𝑇 and 𝑅𝑇 to estimate
𝑇 + 𝑅𝑇 =
1
𝑎
𝑈 + 𝑅𝑈 + 𝑠′. Output (
𝑎+1
𝑒𝑎 1−𝑠′ , 𝑠′).
• Value (𝜌, 𝑐) is a fix-point of 𝑓.
Experimentally, it is found that 𝑓 has unique fix-point that can be
found quickly by iterating 𝑓 fifteen times from a random point.](https://image.slidesharecdn.com/sutramodel-ibm-220213133319/85/Sutra-model-ibm-52-320.jpg)
Sutra model ibm
- 1. The COVID SUTRA Manindra Agrawal IIT Kanpur
- 2. Modelling of Pandemics • Pandemics such as plague, flu, cholera exhibit sharp rise and fall: Spanish flu deaths in UK (Source: https://doi.org/10.3201/eid1201.050979, CC)
- 3. Modelling of Pandemics • To explain this phenomenon, Kermack-McKendrik (1927) proposed a mathematical model called SIR model. • Susceptible: population not yet infected • Infected: population with infection • Removed: population no longer infected (includes fatalities) Susceptible Infected Removed
- 4. SIR Model • Let 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡) represent fraction of population in each of three groups at time 𝑡. • Susceptible get infected in proportion to both 𝑆(𝑡) and 𝐼(𝑡): • Suppose an infected person meets k persons per day and transfers infection to each with probability p. • Then one infected person newly infects 𝑝𝑘𝑆 = 𝛽𝑆 persons in one day. • Therefore, fraction of infected persons in one day is: 𝑆(𝑡) + 𝐼(𝑡) + 𝑅(𝑡) = 1 𝑑𝑆 𝑑𝑡 = −𝛽𝑆𝐼
- 5. SIR Model • Infected get removed in proportion to 𝐼(𝑡): • 𝐼(𝑡) changes with new infections coming in and earlier infected removed: 𝑑𝐼 𝑑𝑡 = 𝛽𝑆𝐼 − 𝛾𝐼 𝑑𝑅 𝑑𝑡 = 𝛾𝐼
- 6. SIR Model • Fatalities 𝐷(𝑡) are a subset of 𝑅(𝑡) and change in 𝐷(𝑡) is also proportional to 𝐼(𝑡): • Constants 𝛽, 𝛾, and 𝜂 determine the trajectory of the pandemic. 𝑑𝐷 𝑑𝑡 = 𝜂𝐼 Traditionally estimated by studying virus properties, population dynamics, and healthcare infrastructure.
- 7. A Property of SIR Model • Let 𝑁(𝑡) denote fraction of new infections at time t. • Then: • Alternately: 𝑁 = 𝛽𝑆𝐼 = 𝛽 1 − 𝐼 − 𝑅 𝐼 = 𝛽𝐼 − 𝛽 𝐼 + 𝑅 𝐼 𝐼 = 1 𝛽 𝑁 + 𝐼 + 𝑅 𝐼
- 8. A Property of SIR Model • Let 𝑁 𝑡 = 𝑃0𝑁 𝑡 , 𝐼 𝑡 = 𝑃0𝐼 𝑡 , 𝑅 𝑡 = 𝑃0𝑅 𝑡 denote respective actual numbers at time t. • Then: 𝐼 = 1 𝛽 𝑁 + 1 𝑃0 𝐼 + 𝑅 𝐼
- 9. A Property of SIR Model • This demonstrates a linear relationship between 𝐼, 𝑁, and 𝐼 + 𝑅 𝐼. • If above three quantities can be measured, then parameter 𝛽 can be estimated. The problem is that reported values of infections may differ greatly from actual values. That is why epidemiologists need to use other methods to estimate parameter values.
- 10. SEIR Model • In some diseases, e.g. measles, an infected person starts infecting after a gestation period. This was captured by a variant called SEIR model. • Exposed: infected population that is not yet spreading to others • Infected: infected population that is spreading to others Susceptible Exposed Infected Removed
- 11. SEIR Model • Let 𝑆(𝑡), 𝐸(𝑡), 𝐼(𝑡), and 𝑅(𝑡) represent fraction of population in each of four groups at time 𝑡. • Dynamics: 𝑆 𝑡 + 𝐸 𝑡 + 𝐼(𝑡) + 𝑅(𝑡) = 1 𝑑𝑆 𝑑𝑡 = −𝛽𝑆𝐼 𝑑𝐸 𝑑𝑡 = 𝛽𝑆𝐼 − 𝛼𝐸 𝑑𝐼 𝑑𝑡 = 𝛼𝐸 − 𝛾𝐼 𝑑𝑅 𝑑𝑡 = 𝛾𝐼 Adding 𝐸 and 𝐼 reduces to SIR model.
- 12. SAIR Model • In some diseases, e.g. Covid-19, an infected person can remain asymptomatic but still infect others. • This is captured by a variant called SAIR model. • Asymptomatic: infected asymptomatic population • Infected: infected population with symptoms • Removed: splits into two – Removed Asymptomatic and Removed Infected Susceptible Asymptomatic Removed Infected Removed
- 13. SAIR Model • Let 𝑆(𝑡), 𝐴(𝑡), 𝐼(𝑡), 𝑅𝐴(𝑡) and 𝑅𝐼(𝑡) represent fraction of population in each of five groups at time 𝑡. • Dynamics: 𝑆 𝑡 + 𝐴 𝑡 + 𝐼 𝑡 + 𝑅𝐴 𝑡 + 𝑅𝐼(𝑡) = 1 𝑑𝑆 𝑑𝑡 = −𝛽𝑆(𝐴 + 𝐼) 𝑑𝐴 𝑑𝑡 = 𝛽𝑆 𝐴 + 𝐼 − 𝛿𝐴 − 𝛾𝐴 𝑑𝐼 𝑑𝑡 = 𝛿𝐴 − 𝛾𝐼 𝑑𝑅𝐴 𝑑𝑡 = 𝛾𝐴 𝑑𝑅𝐼 𝑑𝑡 = 𝛾𝐼 Adding 𝐴 and 𝐼, 𝑅𝐴 and 𝑅𝐼 reduces to SIR model
- 14. COVID-19 Pandemic • Different than earlier pandemics: • Most of these asymptomatic cases are not detected, and continue passing infection to others. • Nearly all cases with severe symptoms get detected. Has a large number of asymptomatic cases Without detecting, how does one estimate asymptomatic cases?
- 15. Modelling Spread of Pandemic • Due to this, estimating becomes more difficult since even denominator is not known. • No model has studied this so far. The spread of pandemic has been controlled in most countries How does one estimate spread of the pandemic at different time?
- 16. Modelling Spread of Pandemic • Let 𝑃(𝑡) denote the population of region within reach of the pandemic at time 𝑡. • Define 𝜌 = 𝑃/𝑃0, a new parameter that increases over time from 0 to 1, where 𝑃0 is the total population of the region. • Parameter 𝜌 is called reach of the pandemic.
- 17. COVID-19 Pandemic • On the positive side, extensive data is available for the first time about pandemic progression in different regions. • Under-reporting and testing limitations take reported data even further away from actual numbers. Can one use reported data to estimate parameter values?
- 18. A Question • Let • 𝑁𝑇 𝑡 denote fraction of daily reported infections, • 𝑇(𝑡) fraction of reported active infections, and • 𝑅𝑇(𝑡) fraction of cumulative removed among reported infections at time t. • Data available is 𝑇 𝑡 = 𝜌𝑃0𝑇, 𝑅𝑇(𝑡) = 𝜌𝑃0𝑅𝑇, and 𝑁𝑇 𝑡 = 𝜌𝑃0𝑁𝑇 as daily time series. Is there any relationship between these three measurable quantities?
- 19. A Hypothesis 𝑇, 𝑁𝑇 and 𝑇 + 𝑅𝑇 𝑇 satisfy linear relation: 𝑇 = 𝑏𝑁𝑇 + 𝑒 𝑃0 𝑇 + 𝑅𝑇 𝑇 To verify, we plot 𝑇 − 𝑏𝑁𝑇 against 𝑇 + 𝑅𝑇 𝑇 for suitably chosen 𝑏 Recall that we have: 𝐼 = 1 𝛽 𝑁 + 1 𝑃0 𝐼 + 𝑅 𝐼
- 20. India Data -5000.0 0.0 5000.0 10000.0 15000.0 20000.0 25000.0 0.0 100000000.0 200000000.0 300000000.0 400000000.0 500000000.0 600000000.0 700000000.0 800000000.0 900000000.0 T – b * N T * (T + RT) March 23 – April 23, 2020 b = 3.86 e = 39164 R2 = 0.998 India data is taken from www.covid19india.org
- 21. India Data b = 6.38 e = 917 R2 = 0.999 -20000.0 0.0 20000.0 40000.0 60000.0 80000.0 100000.0 120000.0 140000.0 160000.0 0.0 50000000000.0 100000000000.0 150000000000.0 200000000000.0 250000000000.0 T – b* N T * (T + RT) April 29 – June 20, 2020
- 22. India Data b = 6.29 e = 165 R2 = 0.999 0.0 200000.0 400000.0 600000.0 800000.0 1000000.0 1200000.0 1400000.0 0.0 2000000000000.0 4000000000000.0 6000000000000.0 8000000000000.0 10000000000000.0 12000000000000.0 T – b * N T * (T + RT) July 21 – August 21, 2020
- 23. India Data b = 6.68 e = 82 R2 = 0.999 0.0 500000.0 1000000.0 1500000.0 2000000.0 2500000.0 0.0 5000000000000.0 10000000000000.0 15000000000000.0 20000000000000.0 25000000000000.0 30000000000000.0 35000000000000.0 40000000000000.0 T – b * N T * (T + RT) September 21 – November 1, 2020
- 24. India Data b = 4.93 e = 83.6 R2 = 0.999 0.0 500000.0 1000000.0 1500000.0 2000000.0 2500000.0 0.0 5000000000000.0 10000000000000.0 15000000000000.0 20000000000000.0 25000000000000.0 30000000000000.0 35000000000000.0 T – b * N T * (T + RT) November 12 – December 31, 2020
- 25. India Data b = 4.29 e = 83.1 R2 = 0.999 0.0 100000.0 200000.0 300000.0 400000.0 500000.0 600000.0 700000.0 800000.0 900000.0 11000000000000.0 11500000000000.0 12000000000000.0 12500000000000.0 13000000000000.0 13500000000000.0 T – b * N T * (T + RT) January 22 – January 31, 2021
- 26. India Data b = 2.64 e = 68.3 R2 = 0.999 0.0 200000.0 400000.0 600000.0 800000.0 1000000.0 1200000.0 1400000.0 0.0 5000000000000.0 10000000000000.0 15000000000000.0 20000000000000.0 25000000000000.0 30000000000000.0 T – b * N T * (T + RT) March 22 – March 28, 2021
- 27. India Data b = 3.52 e = 38.6 R2 = 0.999 0.0 2000000.0 4000000.0 6000000.0 8000000.0 10000000.0 12000000.0 14000000.0 16000000.0 18000000.0 0.0 100000000000000.0 200000000000000.0 300000000000000.0 400000000000000.0 500000000000000.0 600000000000000.0 700000000000000.0 T – b * N T * (T + RT) April 24 – May 17, 2021
- 28. Observations • There are nine different phases with different values of b and e • Is this unique to India? The equation holds for ~62% days in the entire timeline Simulations of 26 countries, 35 states and UTs, and 200+ districts of India show same phenomenon!
- 29. US Data b = 3.14 e = 483.0 R2 = 0.999 US data is taken from https://datahub.io/core/covid-19#resource-time-series-19-covid-combined -100000.0 0.0 100000.0 200000.0 300000.0 400000.0 500000.0 600000.0 700000.0 800000.0 0.0 100000000000.0 200000000000.0 300000000000.0 400000000000.0 500000000000.0 600000000000.0 700000000000.0 T – b * N T * (T + RT) March 18 – April 12, 2020
- 30. US Data b = 6.75 e = 78.8 R2 = 0.985 0.0 100000.0 200000.0 300000.0 400000.0 500000.0 600000.0 700000.0 800000.0 2300000000000.0 2400000000000.0 2500000000000.0 2600000000000.0 2700000000000.0 2800000000000.0 2900000000000.0 T – b * N T * (T + RT) May 15 – June 10, 2020
- 31. US Data b = 5.80 e = 29.8 R2 = 0.998 0.0 500000.0 1000000.0 1500000.0 2000000.0 2500000.0 0.0 5000000000000.0 10000000000000.0 15000000000000.0 20000000000000.0 25000000000000.0 T – b * N T * (T + RT) June 21 – September 09, 2020
- 32. US Data b = 4.27 e = 14.2 R2 = 0.990 0.0 500000.0 1000000.0 1500000.0 2000000.0 2500000.0 3000000.0 3500000.0 4000000.0 4500000.0 0.0 20000000000000.0 40000000000000.0 60000000000000.0 80000000000000.0 100000000000000.0 T – b * N T * (T + RT) October 28 – November 30, 2020
- 33. US Data b = 4.34 e = 10.6 R2 = 0.999 0.0 1000000.0 2000000.0 3000000.0 4000000.0 5000000.0 6000000.0 7000000.0 8000000.0 9000000.0 0.0 50000000000000.0 100000000000000.0 150000000000000.0 200000000000000.0 250000000000000.0 300000000000000.0 T – b * N T * (T + RT) December 11 – December 29, 2020
- 34. US Data b = 3.67 e = 9.2 R2 = 0.999 0.0 2000000.0 4000000.0 6000000.0 8000000.0 10000000.0 12000000.0 0.0 50000000000000.0 100000000000000.0 150000000000000.0 200000000000000.0 250000000000000.0 300000000000000.0 350000000000000.0 400000000000000.0 T – b * N T * (T + RT) January 4 – February 18, 2021
- 35. US Data b = 4.39 e = 7.7 R2 = 0.998 0.0 500000.0 1000000.0 1500000.0 2000000.0 2500000.0 3000000.0 3500000.0 4000000.0 4500000.0 5000000.0 150000000000000.0 155000000000000.0 160000000000000.0 165000000000000.0 170000000000000.0 175000000000000.0 180000000000000.0 185000000000000.0 T – b * N T * (T + RT) February 26 – March 7, 2021
- 36. US Data b = 1.98 e = 8.6 R2 = 0.999 0.0 500000.0 1000000.0 1500000.0 2000000.0 2500000.0 3000000.0 3500000.0 4000000.0 4500000.0 0.0 20000000000000.0 40000000000000.0 60000000000000.0 80000000000000.0 100000000000000.0 120000000000000.0 140000000000000.0 160000000000000.0 T – b * N T * (T + RT) March 24 – May 22, 2021
- 37. Observations • There are eight different phases with different values of b and e • For both India and US, value of e starts high and reduces rapidly The equation holds for ~70% days in the entire timeline e Phase 1 Phase 2 Phase 3 Phase 4 Phase 5 Phase 6 Phase 7 Phase 8 Phase 9 India 5759253.2 39164.2 917.1 165.0 81.9 83.6 83.1 68.3 43.3 US 483.0 78.8 29.8 14.2 10.6 9.2 7.7 8.6 -
- 38. Questions Why does equation hold for majority of days? Why does equation not hold for some days? What is meaning of rapidly decreasing value of 𝑒?
- 39. The SUTRA Model Authors: M Agrawal (IITK), M Kanitkar (Int Defence Staff), M Vidyasagar (IITH)
- 40. SUTRA Model • Group 𝑈: infected but undetected population • It will mostly consist of asymptomatic cases • Group 𝑇: infected and tested positive population • Most of symptomatic cases will be in 𝑇 Susceptible Undetected Removed Tested +ve Removed A at the end stands for Approach
- 41. SUTRA Model • Let 𝑆(𝑡), 𝑈(𝑡), 𝑇(𝑡), 𝑅𝑈(𝑡) and 𝑅𝑇(𝑡) represent fraction of population in each of five groups at time 𝑡. • Dynamics: 𝑆 𝑡 + 𝑈 𝑡 + 𝑇 𝑡 + 𝑅𝑈 𝑡 + 𝑅𝑇(𝑡) = 1 𝑑𝑆 𝑑𝑡 = −𝛽𝑆𝑈 𝑑𝑈 𝑑𝑡 = 𝛽𝑆𝑈 − 𝑁𝑇 − 𝛾𝑈 𝑑𝑇 𝑑𝑡 = 𝑁𝑇 − 𝛾𝑇 𝑑𝑅𝑈 𝑑𝑡 = 𝛾𝑈 𝑑𝑅𝑇 𝑑𝑡 = 𝛾𝑇 Adding 𝑈 and 𝑇, 𝑅𝑈 and 𝑅𝑇 reduces to SIR model
- 42. SUTRA Model: Transition from U to T • As in SAIR model, we can choose 𝑁𝑇 = 𝛿𝑈 • However, this is not a good choice since only recently infected move from 𝑈 to 𝑇 • Due to contact tracing protocol • Also, analysis is hard • Idea: 𝑘𝛽𝑆𝑈 is a better approximation of size of recently infected population for constant 𝑘. • Hence, set 𝑁𝑇 = 𝛿𝑘𝛽𝑆𝑈 = 𝜖𝛽𝑆𝑈 • This also makes the analysis very neat!
- 43. SUTRA Model: Analysis • Compare equations for 𝑈 and 𝑇: • This gives: 𝑑𝑈 𝑑𝑡 + 𝛾𝑈 = 𝛽(1 − 𝜖)𝑆𝑈 𝑑𝑇 𝑑𝑡 + 𝛾𝑇 = 𝛽𝜖𝑆𝑈 𝑑(𝑇 − 𝜖𝑈) 𝑑𝑡 = −𝛾 𝑇 − 𝜖𝑈 , 𝜖 = 𝜖/(1 − 𝜖)
- 44. SUTRA Model: Analysis • Therefore: • Thus 𝑈 quickly converges to 𝑇/𝜖. • We also get, for a constant 𝑐: • Therefore, 𝑅𝑈 quickly converges to 𝑅𝑇/𝜖 + 𝑐. 𝑇 = 𝜖𝑈 + 𝑎𝑒−𝛾𝑡 𝑈 + 𝑅𝑈 = 1/𝜖(𝑇 + 𝑅𝑇) + 𝑐
- 45. SUTRA Model: Analysis • We have: 𝑁𝑇 = 𝜖𝛽𝑆𝑈 = 𝛽 1 − 𝜖 𝑆𝑇 = 𝛽 1 − 𝜖 1 − 𝑈 − 𝑇 − 𝑅𝑈 − 𝑅𝑇 𝑇 = 𝛽 1 − 𝜖 1 − 𝑐 − 𝑇 + 𝑅𝑇 𝜖 𝑇 = 𝛽 1 − 𝜖 1 − 𝑐 𝑇 − 𝛽(1 − 𝜖)/𝜖 𝑇 + 𝑅𝑇 𝑇
- 46. SUTRA Model: Analysis • Resulting in: • With 𝑏 = 1 𝛽 1−𝜖 1−𝑐 and 𝑒 = 1 𝜖𝜌(1−𝑐) . 𝑇 = 1 𝛽 1 − 𝜖 1 − 𝑐 𝑁𝑇 + 1 𝜖𝜌 1 − 𝑐 𝑃0 𝑇 + 𝑅𝑇 𝑇 = 𝑏𝑁𝑇 + 𝑒 𝑃0 𝑇 + 𝑅𝑇 𝑇 Fundamental sutra of the model
- 47. SUTRA Model: Parameters • 𝛽 : Contact rate, governs speed at which people get infected • 𝛾 : Removal rate, governs speed at which infected people get removed • 𝜂 : Mortality rate • 𝜖 : Ratio of detected to total infections • 𝑐 : Constant connecting 𝑅𝑇 to 𝑅𝑈. • 𝜌 : reach of the pandemic
- 48. Estimation of Parameters • Also available is 𝐷 𝑡 = 𝜌𝑃0𝐷 as daily time series • Using equations values of 𝛾 and 𝜂 can be calculated. 𝑑𝑅𝑇 𝑑𝑡 = 𝛾𝑇, 𝑑𝐷 𝑑𝑡 = 𝜂𝑇 Standard least square error method is used in estimation
- 49. Estimation of Parameters • From the fundamental sutra: values of 𝑏 = 1 𝛽(1−𝜖)(1−𝑐) and 𝑒 = 1 𝜖𝜌(1−𝑐) can be calculated. • These values change over time, as observed in example data. • Change of these values is called a phase change. 𝑇 = 𝑏𝑁𝑇 + 𝑒 𝑃0 (𝑇 + 𝑅𝑇)𝑇
- 50. Causes of Phase Change • Lockdowns, personal protection measures reduce 𝛽 • Crowding and mutants increase 𝛽 • Testing policies change 𝜖 • Spread of infection to new areas increases 𝜌 It is reasonable to assume that parameter values drift for some time after phase change and then stabilize.
- 51. Answers to Questions SUTRA model shows why equation holds for majority of days Equation does not hold for some days due to drift in parameter values Value of 𝑒 is large at beginning due to small value of 𝜌 and it reduces as 𝜌 increases
- 52. Estimation of Parameters • How does one estimate 𝛽 and 𝜌 from 1 𝑏 = 𝛽(1 − 𝜖)(1 − 𝑐) and 1 𝑒 = 𝜖𝜌(1 − 𝑐)? • Define function 𝑓: 0,1 × [−1,1] as: • On input (𝑟, 𝑠), set 𝜌 = 𝑟 and 𝑐 = 𝑠, compute 𝛽 and 𝜖, and use it to compute trajectory of 𝑈 and 𝑅𝑈 for current phase. Compare with 𝑇 and 𝑅𝑇 to estimate 𝑇 + 𝑅𝑇 = 1 𝑎 𝑈 + 𝑅𝑈 + 𝑠′. Output ( 𝑎+1 𝑒𝑎 1−𝑠′ , 𝑠′). • Value (𝜌, 𝑐) is a fix-point of 𝑓. Experimentally, it is found that 𝑓 has unique fix-point that can be found quickly by iterating 𝑓 fifteen times from a random point.
- 53. India: Parameter Values Start Date Drift Period β η 1/ϵ ρ (in %) Phase 1 02-03-2020 5 0.33 ±0.03 0.002 ±0.0005 37 0 ±0 Phase 2 20-03-2020 0 0.26 ±0.01 0.0063 ±0.0004 37 ±0 0.1 ±0 Phase 3 24-04-2020 5 0.16 ±0 0.0041 ±0.0002 37 ±0 4 ±0.4 Phase 4 21-06-2020 30 0.16 ±0 0.0019 ±0.0001 37 ±0 22.4 ±1.5 Phase 5 22-08-2020 10 0.15 ±0 0.0012 ±0 37 ±0 45.2 ±1.2 Phase 6 02-11-2020 10 0.21 ±0.04 0.0011 ±0 37 ±0 44.3 ±5.9 Phase 7 01-01-2021 10 0.22 ±0.01 0.0009 ±0 37 ±0 44.5 ±1.1 Phase 8 10-02-2021 40 0.39 ±0.01 0.0008 ±0 37 ±0 54.2 ±1.3 Phase 9 29-03-2021 26 0.33 ±0.02 0.0011 ±0 37 ±0 85.3 ±4.9 We fix 𝛾 = 0.1 These values should be taken with a pinch of salt, as they depend on calibration chosen. However, percentage change is independent of calibration.
- 54. India: Pandemic Spread Simulation 0 50 100 150 200 250 300 350 400 450 3/1/2020 4/20/2020 6/9/2020 7/29/2020 9/17/2020 11/6/2020 12/26/2020 2/14/2021 4/5/2021 5/25/2021 Infections Thousands Date Detected New Infections (7 day average) Actual Data Model Computed on 29th April
- 55. US: Parameter Values Start Date Drift Period β η 1/ϵ ρ (in %) Phase 1 15-03-2020 3 0.4 ±0.02 0.0091 ±0.0002 5 1 ±0.1 Phase 2 13-04-2020 32 0.19 ±0 0.0049 ±0.0003 5 ±0 6.3 ±0.2 Phase 3 11-06-2020 10 0.21 ±0.01 0.0017 ±0 5.1 ±0.1 17.1 ±1.2 Phase 4 13-09-2020 45 0.29 ±0.01 0.0012 ±0 5.1 ±0 36.1 ±1.3 Phase 5 01-12-2020 10 0.29 ±0.02 0.0013 ±0 5.2 ±0 48.8 ±3.1 Phase 6 30-12-2020 5 0.34 ±0.02 0.0016 ±0 5.2 ±0 56.5 ±1.5 Phase 7 19-02-2021 7 0.28 ±0.03 0.0023 ±0.0001 5.2 ±0 67.3 ±3.2 Phase 8 08-03-2021 16 0.61 ±0.1 0.0013 ±0.0001 5.6 ±0.4 65.3 ±6.7 We fix 𝛾 = 0.1 These values should be taken with a pinch of salt, as they depend on calibration chosen. However, percentage change is independent of calibration.
- 56. US: Pandemic Spread Simulation 0 50 100 150 200 250 300 3/10/2020 6/8/2020 9/6/2020 12/5/2020 3/5/2021 6/3/2021 Infections Thousands Date Detected New Infections (7 day average) Actual Data Model Computed Data
- 57. UP: Pandemic Spread Simulation 0 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 5/1/2020 7/30/2020 10/28/2020 1/26/2021 4/26/2021 Infections Date Detected New Infections Actual Data Model Computed on 9th May
- 58. Kanpur: Pandemic Spread Simulation 0 500 1000 1500 2000 2500 5/1/2020 7/30/2020 10/28/2020 1/26/2021 4/26/2021 Infections Date Detected New Infections (7 day average) Actual Data Model Computed on 6th May More simulations at www.sutra-india.in
- 59. Strengths and Weaknesses of the Model • Strengths: • Probably the first model that can estimate values of all parameters only from daily reported infections and deaths data • Can provide an excellent understanding of the past • Can provide future projections up to medium term, assuming that parameters do not change significantly • Can provide what-if analysis through setting parameters to different values • Weaknesses: • During drift period, estimating parameter values is difficult and so predictions are likely to be wrong • Cannot predict future values of parameters
- 60. Future Work • Incorporate loss of immunity over time • Needs regular serosurvey data for validation • Incorporate immunity induced by vaccination • Prove that, under reasonable conditions, function f has a unique fixed point • Find a way of estimating stable parameter values during initial drift period of a phase
- 61. Thank You