Effects of demographic, environmental and seasonal variability on disease outbreaks in stochastic vector-host, multi-patch and dengue epidemic models

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, a multi-patch epidemic model, and a dengue model, all of which include 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 equation. This approximation shows that the risk of a disease outbreak is also periodic and depends on the particular time, location, and the number of the initial infected individuals. Numerical examples with periodic transmission rates for vector and/or host or periodic dispersal and for various types of connectivity relations between patches illustrate that seasonality has a large impact on the probability of an outbreak. Similar techniques are applied to a dengue model of stochastic differential equations for which the adult vectors emerge from the larval stage either vary periodically or randomly.