**Spring 2022 ECE Distinguished Seminar Series **

**A Stochastic Model of an Infectious Disease, **

** Based on the Birth-and-Death-with-Immigration Process **

Hisashi Kobayashi

The Sherman Fairchild University Professor, Emeritus

Princeton University

Friday, March 4, 2022, 2:00 pm

Zoom meeting link: https://gmu.zoom.us/j/95160126923

**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 a new stochastic model that can capture the essence of the seemingly erratic behavior of an infectious disease.

The stochastic model we propose is based on the "birth-and-death process with immigration" (BDI for short), which was originally proposed in the study of population growth or extinction of some biological species. The BDI process
model, however, does not seem to have been investigated by the epidemiology community. The BDI process is one of a few birth-and-death processes, which we can solve analytically. We revisit the partial differential equation (PDE) for the probability generating
function (PGF) of the time nonhomogeneous BDI process and derive a closed form solution

We clarify the relationships among the basic reproduction number, effective reproduction number, the exponential growth/decay parameter, the proportion of vaccinated
population, vaccine effectiveness, and the behavioral factor of the public.