<abstract><p>This study aims to analyze a class of SIV systems considering the transmission rate influenced by media coverage and protective measures, in which the transmission rate is represented by a piecewise-smooth function. Firstly, for the SIV Filippov system, we take the dynamic behaviors of two subsystems into consideration, and obtain the basic reproduction number and the equilibria of the subsystems respectively. Secondly, based on the Filippov convex method, we calculate the sliding domain and the sliding mode equation, and further analyze the global dynamic behaviors of the system, through which we verify that there is no closed orbit in the system. Furthermore, we prove the global asymptotical stability of the disease-free equilibrium, two real equilibria, and the pseudo-equilibrium under certain conditions. The results demonstrate that the threshold value, the protective measures, and the media coverage could affect the number of infected individuals and the final scale of the disease. To prevent the spread of the disease, it is necessary to select an appropriate threshold and take applicable protective measures combined with media coverage. Lastly, we verify the validity of the results by numerical simulations.</p></abstract>
In this article, we consider a SIV infectious disease control system with two-threshold guidance, in which infection rate and vaccination rate are represented by a piecewise threshold function. We analyze the global dynamics of the discontinuous system using the theory of differential equations with discontinuous right-hand sides. We find some dynamical behaviors, including the disease-free equilibrium and endemic equilibria of three subsystems, a globally asymptotically stable pseudo-equilibrium and two endemic equilibria bistable, one of the two pseudo-equilibria or pseudo-attractor is stable. Conclusions can be used to guide the selection of the most appropriate threshold and parameters to achieve the best control effect under different conditions. We hope to minimize the scale of the infection so that the system can eventually stabilize at the disease-free equilibrium, pseudo-equilibrium or pseudo-attractor, corresponding to the disease disappearing or becoming endemic to a minimum extent, respectively.
We analyze a class of SIV system considering effective vaccine protection and protective measures, in which the effective vaccine protection rate is represented by a piecewise-smooth function. This piecewise function is composed of three segments. When the number of infected people exceeds a set threshold, the effective vaccine protection rate is increased based on the first paragraph. To further increase the effective vaccine protection rate to 1, we should take into account not only the influence of the number of infected people, but also the diagonal dividing line of the interaction between the number of infected people and that of vaccinated people. We analyze the global dynamical behavior of the SIV system using the discontinuous differential equation theory, Filippov convex method and other methods, and the theoretical results could provide recommendations for vaccination. Then, in order to achieve the best vaccine protection effect, we analyze the selection of the threshold and the two values of the vaccination effective protection rate using numerical simulation.
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