A West Nile Virus Model with Vertical Transmission and Periodic Time Delays

Abstract

Seasonal change has played a critical role in the evolution dynamics of West Nile virus transmission. In this paper, we formulate and analyze a novel delay differential equation model, which incorporates seasonality, the vertical transmission of the virus, the temperature-dependent maturation delay and the temperature-dependent extrinsic incubation period in mosquitoes. We first introduce the basic reproduction ratio $$R_0$$ for this model and then show that the disease is uniformly persistent if $$R_0>1$$. It is also shown that the disease-free periodic solution is attractive if $$R_0<1$$, provided that there is only a small invasion. In the case where all coefficients are constants and the disease-induced death rate of birds is zero, we establish a threshold result on the global attractivity in terms of $$R_0$$. Numerically, we study the West Nile virus transmission in Orange County, California, USA.