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The dynamics of a new SIR epidemic model concerning pulse vaccination strategy. (English) Zbl 1131.92056

Summary: A new SIR epidemic model with vertical and horizontal transmission is proposed, and the dynamics of this disease model under constant and pulse vaccination are analyzed. Firstly, global stability of the endemic equilibrium states of the model with constant vaccination is established. Further, we show that there exists a stable ‘infection-free’ periodic solution when the period of the impulsive effect is less than some critical value. A condition for the permanence of the system with pulse vaccination is also given, which implies that periodic bursts of the epidemic occur. Numerical simulations show that a system with pulse vaccination has more complex dynamic behavior for positive periodic oscillations, ‘infection free’ quasi-periodic oscillations, than a system with constant vaccination. Finally, we compare the validity of the strategy of pulse vaccination with that of no vaccination and constant vaccination, and conclude that the pulse vaccination strategy is more effective than the no vaccination and continuous vaccination.

MSC:

92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
37N25 Dynamical systems in biology
34C60 Qualitative investigation and simulation of ordinary differential equation models
65C20 Probabilistic models, generic numerical methods in probability and statistics
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