Abstract Given imperfect vaccination we extend the existing non-smooth models by considering susceptible and vaccinated individuals enhance the protection and control strategies once the number of infected individuals exceeds a certain level. On the basis of global dynamics of two subsystems, for the formulated Filippov system, we examine the sliding mode dynamics, the boundary equilibrium bifurcations, and the global dynamics. Our main results show that it is possible that the pseudo-equilibrium exists and is globally stable, or the pseudo-equilibrium, the disease-free equilibrium and the real equilibrium are tri-stable, or the pseudo-equilibrium and the real equilibrium are bi-stable, or the pseudo-equilibrium and disease-free equilibrium are bi-stable, which depend on the threshold value and other parameter values. The global stability of the disease-free equilibrium or pseudo-equilibrium reveals that we may eradicate the disease or maintain the number of infected individuals at a previously given value. Further, the bi-stability and tri-stability imply that whether the number of infected individuals tends to zero or a previously given value or other positive values depends on the parameter values and the initial states of the system. This emphasizes the importance of threshold policy and challenges in the control of infectious diseases if without perfect vaccines.

中文翻译：

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具有不完善疫苗接种的菲利波夫流行系统的全球动态

摘要 考虑到不完善的疫苗接种，我们通过考虑易感和接种个体来扩展现有的非平滑模型，一旦感染个体的数量超过一定水平，就会增强保护和控制策略。在两个子系统的全局动力学的基础上，对于公式化的 Filippov 系统，我们检查了滑模动力学、边界平衡分岔和全局动力学。我们的主要结果表明，可能存在伪平衡且全局稳定，或伪平衡、无病平衡和实平衡是三稳的，或伪平衡和实平衡是双稳态的。 -stable，或者伪平衡和无病平衡是双稳态的，这取决于阈值和其他参数值。无病平衡或伪平衡的全局稳定性表明我们可以根除疾病或将受感染的个体数量保持在先前给定的值。此外，双稳态和三稳态意味着感染个体的数量是趋于零还是先前给定的值或其他正值取决于参数值和系统的初始状态。如果没有完美的疫苗，这强调了门槛政策和控制传染病的挑战的重要性。双稳态和三稳态意味着感染个体的数量是趋于零还是先前给定的值或其他正值取决于参数值和系统的初始状态。如果没有完美的疫苗，这强调了门槛政策和控制传染病的挑战的重要性。双稳态和三稳态意味着感染个体的数量是趋于零还是先前给定的值或其他正值取决于参数值和系统的初始状态。如果没有完美的疫苗，这强调了门槛政策和控制传染病的挑战的重要性。