Stochastic Modeling of an Infectious Disease, Part III-B: Analysis of the Time-Nonhomogeneous BDI Process and Simulation Experiments of both BD and BDI Processes

TL;DR

This paper derives a new closed-form solution for the probability generating function of the time-nonhomogeneous BDI process, extends known distributions to this case, and confirms results through extensive simulation experiments.

Contribution

It provides the first closed-form PDE solution for the time-nonhomogeneous BDI process and extends the negative binomial distribution to this broader context.

Findings

Derived a new PDE solution for the PGF of the BDI process.

Confirmed analytic results with extensive simulation experiments.

Extended the negative binomial distribution to the time-nonhomogeneous case.

Abstract

In Section 1, we revisit the partial differential equation (PDE) for the probability generating function (PGF) of the time-nonhomogeneous BDI (birth-and-death-with-immigration) process and derive a closed form solution. To the best of our knowledge, this is a new mathematical result. We state this result as Proposition 1. We state as Corollary 1 that the negative binomial distribution of the time-homogeneous BDI process discussed in Part I extends to the general time-nonhomogeneous case, provided that the ratio of the immigration rate to the birth rate is a constant. In section 1.2, we take up the heuristic approach discussed by Bartlett and Bailey (1964), and carry it out to completion by arriving at the solution obtained above,. In Section 2, we present the results of our extensive simulation experiments of the time-nonhomogeneous BD process that was analyzed in Part III-A and confirm our analytic results. In Section 3, we undertake similar simulation experiments for the BDI process that is analyzed in Section 1. As we discuss in Section 4, our stochastic model now seems more promising and powerful than has been heretofore expected. In Appendix B, a closed form solution for the M(t)/M(t)/infinity queue is obtained, as a special case of this BDI process model.