Stochastic Modeling of an Infectious Disease, Part I: Understand the Negative Binomial Distribution and Predict an Epidemic More Reliably

TL;DR

This paper explores stochastic models, specifically the birth-and-death process with immigration, to better understand and predict the erratic patterns of infectious diseases like COVID-19, highlighting the limitations of deterministic models.

Contribution

It introduces the BDI process as a novel stochastic model in epidemiology, deriving its negative binomial distribution and explaining variability in epidemic patterns.

Findings

The BDI process has a negative binomial distribution with parameter r<1.

The model explains large variations and long tails in infection data.

Deterministic models often overestimate infection numbers.

Abstract

Why are the epidemic patterns of COVID-19 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 so-called 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 "birth-and-death 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 birth-and-death processes, which we can solve analytically. Its time-dependent 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.