Existence, Persistence and Stability of Periodic Solutions of a Seasonally Forced West Nile Virus Modelwith Lateral Hosts and Vertical Vector Transmission

Abstract

West Nile virus persists in tropical settings where temperature drives vector development, birds transmit laterally through predation, scavenging and a contaminated environment, and mosquitoes transmit vertically to their progeny. We analyse a twenty-two compartment, seasonally forced model of such transmission in which corvid and non-corvid reservoirs, the full mosquito life cycle, three human infection outcomes and an environmental viral reservoir are coupled through temperature-dependent vital and transmission rates of period one year. Existence of a positive periodic solution is established by Mawhin’s continuation theorem after uniform a priori bounds are derived for every compartment; the non-negative orthant is positively invariant, the periodic solution strictly positive, and all trajectories ultimately bounded within a compact absorbing set on which the periodic solution is unique. Setting the avian introduction terms to zero yields a unique disease-free periodic solution, and the periodic reproduction ratio is obtained as the spectral radius of the next-infection operator. Reduction of the thirteen-dimensional next-generation kernel to a five-dimensional block-triangular matrix separates an autonomous human block from a temperature-driven vector-reservoir block. Numerical evaluation gives a threshold of 1.494471, which seasonal forcing depresses well below the constant-temperature value, while a positive bird seeding index sustains transmission below threshold. The endemic periodic orbit is globally asymptotically stable.