# Two-patch model for the spread of West Nile virus

## Abstract

A two-patch model for the spread of West Nile virus between two discrete geographic regions is established to incorporate a mobility process which describes how contact transmission occurs between individuals from and between two regions. In the mobility process, we assume that the host birds can migrate between regions, but not the mosquitoes. The basic reproduction number \(R_{0}\) is computed by the next generation matrix method. We prove that if \(R_{0}<1\), then the disease-free equilibrium is globally asymptotically stable. If \(R_{0}>1\), the endemic equilibrium is globally asymptotically stable for any nonnegative nontrivial initial data. Using the perturbation theory, we obtain the concrete expression of the endemic equilibrium of the model with a mild restriction of the birds movement rate between patches. Finally, numerical simulations demonstrate that the disease becomes endemic in both patches when birds move back and forth between the two regions. Some numerical simulations for \(R_{0}\) in terms of the birds movement rate are performed which show that the impacts could be very complicated.

## Keywords

West Nile virus Patch model Mosquitoes Birds migration Basic reproduction number Stability## References

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