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On the backward bifurcation of a vaccination model with nonlinear incidence. (English) Zbl 1271.34045
Summary: A compartmental epidemic model, introduced by A. B. Gumel and S. M. Moghadas [Appl. Math. Comput. 143, No. 2–3, 409–419 (2003; Zbl 1018.92029)], is considered. The model incorporates a nonlinear incidence rate and an imperfect preventive vaccine given to susceptible individuals. A bifurcation analysis is performed by applying the bifurcation method introduced in [C. Castillo-Chávez and B. Song, Math. Biosci. Eng. 1, No. 2, 361–404 (2004; Zbl 1060.92041)], which is based on the use of the center manifold theory. Conditions ensuring the occurrence of backward bifurcation are derived. The obtained results are numerically validated and then discussed from both the mathematical and the epidemiological perspective.

MSC:
34C23 Bifurcation theory for ordinary differential equations
92D30 Epidemiology
93A30 Mathematical modelling of systems (MSC2010)
93D30 Lyapunov and storage functions
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