  Wolbachia is possibly the most studied reproductive parasite of arthropod
species. It appears to be a promising candidate for biocontrol of some mosquito
borne diseases. We begin by developing a sex-structured model for a Wolbachia
infected mosquito population. Our model incorporates the key effects of
Wolbachia infection including cytoplasmic incompatibility and male killing. We
also allow the possibility of reduced reproductive output, incomplete maternal
transmission, and different mortality rates for uninfected/infected male/female
individuals. We study the existence and local stability of equilibria,
including the biologically relevant and interesting boundary equilibria. For
some biologically relevant parameter regimes there may be multiple coexistence
steady states including, very importantly, a coexistence steady state in which
Wolbachia infected individuals dominate. We also extend the model to
incorporate West Nile virus (WNv) dynamics, using an SEI modelling approach.
Recent evidence suggests that a particular strain of Wolbachia infection
significantly reduces WNv replication in Aedes aegypti. We model this via
increased time spent in the WNv-exposed compartment for Wolbachia infected
female mosquitoes. A basic reproduction number $R_0$ is computed for the WNv
infection. Our results suggest that, if the mosquito population consists mainly
of Wolbachia infected individuals, WNv eradication is likely if WNv replication
in Wolbachia infected individuals is sufficiently reduced.
