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Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination. (English) Zbl 1111.92049

Summary: A new SEIRS epidemic disease model with two profitless delays and nonlinear incidence are proposed, and the dynamic behavior of the model under pulse vaccination is analyzed. Using a discrete dynamical system determined by the stroboscopic map, we show that there exist ‘infection-free’ periodic solutions; further we show that the ‘infection-free’ periodic solution is globally attractive when the period of impulsive effect is less than some critical value.

Using a new modeling method, we obtain sufficient conditions for the permanence of the epidemic model with pulse vaccination. We show that time delays, pulse vaccination and nonlinear incidence can bring different effects on the dynamic behavior of the model by numerical analysis. Our results also show the delays are “profitless”. The main feature is to introduce two discrete time delays and impulse into the SEIRS epidemic model and to give pulse vaccination strategies.

MSC:
92D30Epidemiology
34C25Periodic solutions of ODE
34A37Differential equations with impulses
34K60Qualitative investigation and simulation of models
37N25Dynamical systems in biology
65L99Numerical methods for ODE