Backward bifurcation in SIRS malaria model

Abstract

We present a deterministic mathematical model for malaria transmission with waning immunity. The model consists of five non-linear system of differential equations. We used next generation matrix to derive the basic reproduction number $R_0$. The disease free equilibrium was computed and its local stability has been shown by the virtue of the Jacobean matrix. Moreover, using Lyapunov function theory and LaSalle Invariance Principle we have proved that the disease free equilibrium is globally asymptotically stable. Conditions for existence of endemic equilibrium point have been established. A qualitative study based on bifurcation theory reveals that backward bifurcation occur in the model. The stable disease free equilibrium of the model coexists with the stable endemic equilibrium when $R_0<1$. Furthermore, we have shown that bringing the number of disease (malaria) induced death rate below some threshold is sufficient enough to eliminate backward bifurcation in the model.