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Global stability of an epidemic model with latent stage and vaccination. (English) Zbl 1220.34069

Summary: For an epidemic model with latent stage and vaccination for the newborns and susceptibles, we establish that the global dynamics are completely determined by the basic reproduction number \(R_{0}\). More specifically, we prove that, if \(R_{0}\leq 1\), then the disease-free equilibrium is globally asymptotically stable, that is, the disease dies out eventually; if \(R_{0}>1\), then there exists a unique endemic equilibrium and it is globally asymptotically stable in the interior of the feasible region, that is, the disease persists in the population. We propose an approach for determining the Lyapunov function and proving the negative definiteness or semidefiniteness of its derivative. Our proof shows that, for a given Lyapunov function, its derivative should be arranged in different forms for the different values of parameters to prove the negative definiteness or semidefiniteness of its derivative.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D30 Epidemiology
34D20 Stability of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
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