Stochastic Modeling of an Infectious Disease, Part I: Understand the Negative Binomial Distribution and Predict an Epidemic More Reliably
Abstract
Why are the epidemic patterns of COVID19 so different among different cities or countries which are similar in their populations, medical infrastructures, and people's behavior? Why are forecasts or predictions made by socalled experts often grossly wrong, concerning the numbers of people who get infected or die? The purpose of this study is to better understand the stochastic nature of an epidemic disease, and answer the above questions. Much of the work on infectious diseases has been based on "SIR deterministic models," (Kermack and McKendrick:1927.) We will explore stochastic models that can capture the essence of the seemingly erratic behavior of an infectious disease. A stochastic model, in its formulation, takes into account the random nature of an infectious disease. The stochastic model we study here is based on the "birthanddeath process with immigration" (BDI for short), which was proposed in the study of population growth or extinction of some biological species. The BDI process model ,however, has not been investigated by the epidemiology community. The BDI process is one of a few birthanddeath processes, which we can solve analytically. Its timedependent probability distribution function is a "negative binomial distribution" with its parameter $r$ less than $1$. The "coefficient of variation" of the process is larger than $\sqrt{1/r} > 1$. Furthermore, it has a long tail like the zeta distribution. These properties explain why infection patterns exhibit enormously large variations. The number of infected predicted by a deterministic model is much greater than the median of the distribution. This explains why any forecast based on a deterministic model will fail more often than not.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 DOI:
 10.48550/arXiv.2006.01586
 arXiv:
 arXiv:2006.01586
 Bibcode:
 2020arXiv200601586K
 Keywords:

 Quantitative Biology  Populations and Evolution;
 Statistics  Methodology;
 00;
 G.3;
 I.6;
 J.3
 EPrint:
 28 pages, 14 figures