Dynamical behavior of an epidemic model for a vector-borne disease with direct transmission

An epidemic model of a vector-borne disease with direct transmission is investigated. The reproduction number (R0) of the model is obtained. Rigorous qualitative analysis of the model reveals the presence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium (DFE) coexists with a stable endemic equilibrium when the reproduction number of the disease is less than unity) in the standard incidence model. The phenomenon shows that the classical epidemiological requirement of having the reproduction number less than unity is no longer sufficient, although necessary, for effectively controlling the spread of some vector-borne diseases in a community. The backward bifurcation phenomenon can be removed by substituting the standard incidence with a bilinear mass action incidence. By using Lyapunov function theory and LaSalle invariance principle, it is shown that the unique endemic equilibrium for the model with a mass action incidence is globally stable if the reproduction number Rmass is greater than one in feasible region. This suggests that the use of standard incidence in modelling some vector-borne diseases with direct transmission results in the presence of backward bifurcation. Numerical simulations analyze the effect of the direct transmission and the disease-induced death rate on dynamics of the disease transmission, and also verify our analyzed results.