The Effect of Demographic Variability and Periodic Fluctuations on Disease Outbreaks in a Vector–Host Epidemic Model

Abstract

Seasonality and contact patterns affect the dynamics of disease outbreaks. Recent studies applied to deterministic and stochastic epidemic models with periodic environments have shown that the average basic reproduction number is not sufficient to predict an outbreak. We extend these studies to a time-nonhomogeneous stochastic vector–host model with demographic variability and periodic fluctuations to better understand the combined effects of variability and periodicity on the risk of a disease outbreak. A multitype branching process approximation is used to calculate the probability of a disease outbreak. The approximation follows from the solution of a system of differential equations which is derived from the backward Kolmogorov differential equations. This approximation shows that the risk of a disease outbreak is also periodic and depends on the particular time at which either an infected vector or an infected host is introduced into the susceptible vector and susceptible host populations. Numerical examples with periodic transmission rates for vector and host illustrate the times at which there is the greatest probability of an outbreak and also demonstrate how these times are related to the peak transmission rates for vector or host.