Stability And Bifurcation For A Class Of Disease Transmitted By Vector

In this paper, a vector-borne disease transmission model is formulated and analyzed. It is assumed that the recovered individuals can be infected again when they contact with infected vector. The explicit expression of the basic reproduction number is obtained. It is shown that the backward bifurcation occurs when R0< 1. It is also proved that there exists an unique endemic equilibrium if R0> 1. The stability of disease free equilibrium, bifurcation equilibrium and endemic equilibrium is discussed by the theory of Lyapunov function, Routh-Hurwitz criterion, center manifold theorem and second additive Matrix. It is shown that the disease free equilibrium is globally asymptotically stable under certain conditions, the lower bifurcation equilibrium is always unstable whenever it exists, the larger bifurcation equilibrium is stable, the endemic equilibrium is globally asymptotically stable under certain conditions. The results obtained in this paper show that, in order to control the disease it is not enough to decrease the basic reproduction number R0 below one.