Models of impulsive culling of mosquitoes to interrupt transmission of West Nile virus to birds

A mathematical model describing the transmission of West Nile virus (WNV) between vector mosquitoes and birds, incorporating a control strategy of culling mosquitoes and defined by impulsive differential equations is presented and its properties investigated. First, we consider a strategy of periodic impulsive culling of the mosquitoes. Theoretical results indicate that if the threshold R₀ is greater than unity the disease uniformly persists, but, if not, the disease does not necessarily become extinct. The explicit conditions determining the backward or forward bifurcation were obtained. The culling rate has a major effect on the occurrence of backward bifurcation. Analysis shows that the disease is most sensitive to mosquito-bird contacts, mosquito-culling rate and intervals between culls. The dependence of the outcomes of the culling strategy on mosquito biting rate is discussed. When the complete elimination of disease is impossible, mosquito culls are implemented once the infected birds reach a predefined but adjustable threshold value. Numerical analysis shows that the period of mosquito culling finally stabilizes at a fixed value. In addition, variations of mean prevalence of WNV in birds and the culling period are simulated.