A delayed SEIRS epidemic model with pulse vaccination and nonlinear incidence rate is proposed. We analyze the dynamical behaviors of this model and point out that there exists an infection-free periodic solution which is globally attractive ifR1<1,R2>1, and the disease is permanent. Our results indicate that a short period of pulse or a large pulse vaccination rate is the sufficient condition for the eradication of the disease. The main feature of this paper is to introduce time delay and impulse into SEIRS model and give pulse vaccination strategies.

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A prey-predator model with Beddington-DeAngelis functional response and impulsive state feedback control is investigated. We obtain the sufficient conditions of the global asymptotical stability of the system without impulsive effects. By using the geometry theory of semicontinuous dynamic system and the method of successor function, we obtain the system with impulsive effects that has an order one periodic solution, and sufficient conditions for existence and stability of order one periodic solution are also obtained. Finally, numerical simulations are performed to illustrate our main results.

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In this paper, we propose and discuss a stochastic SIRS epidemic model with non-monotone incidence rate under regime-switching. First of all, we show that there is a unique positive solution, which is a prerequisite for analyzing the long-term behavior of the stochastic model. Then, a threshold dynamic determined by the basic reproduction number R s 0 is established: the disease can be eradicated almost surely if R s 0 < 1 and under mild extra conditions, whereas if R s 0 > 1 , the densities of the distributions of the solution can converge in L 1 to an invariant density by using the Markov semigroups theory. Finally, based on realistic parameters obtained from previous literatures, numerical simulations have been performed to verify our analytical results.

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