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: 



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. 


We also derive maximum-likelihood estimation of the model parameters, and present surprisingly simple formulas for estimating our model parameters, and the exponential growth/decay parameter. We identify some challenges in parameter estimation arising from two important characteristics of COVID-19 and its variants. First, a significant proportion of infections are asymptomatic. Second, each infection has its incubation period, which is a random variable.