Mathematical Analysis of West Nile Virus Model with Discrete Delays

Abstract

The paper presents the basic model for the transmission dynamics of West Nile virus (WNV). The model, which consists of seven mutually-exclusive compartments representing the birds and vector dynamics, has a locally-asymptotically stable disease-free equilibrium whenever the associated reproduction number (R 0) is less than unity. As reveal in [3, 20], the analyses of the model show the existence of the phenomenon of backward bifurcation (where the stable diseasefree equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity). It is shown, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. Analysis of the reproduction number of the model shows that, the disease will persist, whenever R 0 > 1, and increase in the length of incubation period can help reduce WNV burden in the community if a certain threshold quantities, denoted by ∆ b and ∆ v are negative. On the other hand, increasing the length of the incubation period increases disease burden if ∆ b > 0 and ∆ v > 0. Furthermore, it is shown that adding time delay to the corresponding autonomous model with standard incidence (considered in [2]) does not alter the qualitative dynamics of the autonomous system (with respect to the elimination or persistence of the disease).

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“…Theorem (case 2) indicates the possibility of backward bifurcation (where the locally asymptotically stable DFE coexists with a locally asymptotically stable endemic equilibrium when ) in the model (see, for instance,()). Furthermore, this is investigated using the center manifold theory below.…”

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“…Therefore, we seek for a unique value of θ = yτ , such that θ ∈ [0, 2π] that will satisfy (18). As observed, the signs of the two equations in (18), are positive, hence θ, must satisfy:…”