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Stability and bifurcation analysis of a vector-bias model of malaria transmission. (English) Zbl 1316.92081
Summary: 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
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