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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:
34C60Qualitative investigation and simulation of models (ODE)
34C25Periodic solutions of ODE
47N20Applications of operator theory to differential and integral equations
34D05Asymptotic stability of ODE
34D23Global stability of ODE
34F05ODE with randomness
92D30Epidemiology