Limiting Behaviors of Stochastic Spread Models Using Branching Processes

In this paper, we introduce a spread model using multi-type branching processes to investigate the evolution of the population during a pandemic in which individuals are classified into different types. We study some limiting behaviors of the population including the growth rate of the population and the spread rate of each type. In particular, the work in this paper focuses on the cases where the offspring mean matrices are non-primitive but can be decomposed into two primitive components, A and B, with maximal eigenvalues ${\rho }_A$ and ${\rho }_B$, respectively. It is shown that the growth rate and the spread rate heavily depend on the conditions of these two maximal eigenvalues and are related to the corresponding eigenvectors. In particular, we find the spread rates for the case with ${\rho }_B>{\rho }_A>1$ and the case with ${\rho }_A>{\rho }_B>1$. In addition, some numerical examples and simulations are also provided to support the theoretical results.