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This paper studies the dynamical behaviors of a diffusion epidemic SIRI system with distinct dispersal rates. The overall solution of the system is derived by using theory and the Young’s inequality. The uniformly boundedness of the solution is obtained for the system. The asymptotic smoothness of the semi-flow and the existence of the global attractor are discussed. Moreover, the basic reproduction number is defined in a spatially uniform environment and the threshold dynamical behaviors are obtained for extinction or continuous persistence of disease. When the spread rate of the susceptible individuals or the infected individuals is close to zero, the asymptotic profiles of the system are studied. This can help us to better understand the dynamic characteristics of the model in a bounded space domain with zero flux boundary conditions.

This paper employs a discontinuous temporary immunity control to obtain the periodic orbit for a class of delayed malicious signal transmission model in wireless sensor networks under the framework of differential inclusion. The positivity and boundedness of the solution for the discontinuous system is proved first. Then, by using the Kakutani's fixed point theorem of set‐valued maps, the existence of a periodic orbit is obtained under some assumptions and constraints. Furthermore, the globally exponentially stable ω$$ \omega $$‐periodic orbit is investigated using the Lyapunov functional method. The obtained results can help us better understand the dynamic characteristics of discontinuous delayed systems and have direct applications to the wireless sensor networks for guaranteeing fast response to malicious signals. Finally, the numerical simulations of three examples are given to validate the correctness of the theoretical results.

This paper investigates the global dynamic behavior and bifurcations of a classical nonlinear transmission SIR epidemic model with discontinuous threshold strategy. Different from previous results, we not only consider the general nonlinear transmission, but also adopt the discontinuity control. First, the positivity and boundedness of the model are given. Second, by employing Lyapunov LaSalle approach and using Green Theorem, we perform the globally stable the three types of equilibria of the system. We analytically show the orbit can tend to the disease-free equilibrium point, the endemic equilibrium point or the sliding equilibrium point in discontinuous surfaces of the system. In addition, we also analyze the sliding bifurcations of the model when consider the special transmission. Finally, some numerical simulations are worked out to confirm the results obtained in this paper.