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

*(English)*Zbl 1316.92081Summary: The vector-bias model of malaria transmission, recently proposed by F. Chamchod and N. F. Britton [Bull. Math. Biol. 73, No. 3, 639–657 (2011; Zbl 1225.92030)], 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, \(R_{0}\), is obtained: if \(R_{0}>1\), then the disease will spread and persist within its host population. If \(R_{0}<1\), 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 \(R_{0}=1\) is shown possible. This implies that a stable endemic equilibrium may also exists for \(R_{0}<1\). When \(R_{0}>1\), 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.

##### MSC:

92D30 | Epidemiology |