Stability and bifurcation analysis of a vector-bias model of malaria transmission

The vector-bias model of malaria transmission, recently proposed by Chamchod and Britton, is considered. Nonlinear stability analysis is performed by means of the Lyapunov theory and the LaSalle Invariance Principle. The classical threshold for the basic reproductive number, R₀, is obtained: if R₀>1, then the disease will spread and persist within its host population. If R₀<1, then the disease will die out. Then, the model has been extended to incorporate both immigration and disease-induced death of humans. This modification has been shown to strongly affect the system dynamics. In particular, by using the theory of center manifold, the occurrence of a backward bifurcation at R₀=1 is shown possible. This implies that a stable endemic equilibrium may also exists for R₀<1. When R₀>1, the endemic persistence of the disease has been proved to hold also for the extended model. This last result is obtained by means of the geometric approach to global stability.