In this paper, we develop a new disease-resistant mathematical model with a fraction of the susceptible class under imperfect vaccine and treatment of both the symptomatic and quarantine classes. With standard incidence when the associated reproduction threshold is less than unity, the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium. It is then proved that this phenomenon vanishes either when the vaccine is assumed to be 100% potent and perfect or the Standard Incidence is replaced with a Mass Action Incidence in the model development. Furthermore, the model has a unique endemic and disease-free equilibria. Using a suitable Lyapunov function, the endemic equilibrium and disease free equilibrium are proved to be globally-asymptotically stable depending on whether the control reproduction number is less or greater than unity. Some numerical simulations are presented to validate the analytic results.

在本文中，我们开发了一种新的抗病数学模型，该模型具有不完全疫苗接种以及症状和检疫分类治疗的部分易感性分类。对于标准的发病率，当相关的繁殖阈值小于1时，该模型表现出向后分叉的现象，其中稳定的无病平衡与稳定的地方性平衡并存。然后证明，在模型开发过程中，假设疫苗是100％有效且完美的，或者将标准事件替换为质量行为事件，则该现象就消失了。此外，该模型具有独特的地方性和无病平衡。使用适当的Lyapunov函数，根据控制繁殖数是小于还是大于1，地方病平衡和无病平衡被证明是全局渐近稳定的。提出了一些数值模拟来验证分析结果。