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Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates. (English) Zbl 1261.34040

Summary: We obtain sufficient criteria for the existence of periodic solutions to deterministic SIR and SEIR epidemic models with modified saturation incidence rates by means of using the continuation theorem based on coincidence degree theory, and we show that the solution is unique and globally stable. Second, we discuss that their corresponding stochastic epidemic models with random perturbation have a unique global positive solution, and we utilize stochastic Lyapunov functions to investigate the asymptotic behavior of the solution.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34C25 Periodic solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
34F05 Ordinary differential equations and systems with randomness
92D30 Epidemiology
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