A novel nonlinear delayed susceptible–vaccinated–infected–recovered–susceptible (SVIRS) epidemic model with a Holling type II incidence rate for fully susceptible and vaccinated classes, a saturated treatment rate, and an imperfect vaccine given to susceptibles is proposed herein. Analysis of the model shows that it exhibits two equilibria, namely disease-free and endemic. The basic reproduction number $$R_0$$ R 0 is derived, and it is demonstrated that the disease-free equilibrium is locally asymptotically stable when $$R_0<1$$ R 0 < 1 and linearly neutrally stable when $$R_0=1$$ R 0 = 1 . Furthermore, bifurcation analysis is performed for the undelayed model, revealing that it exhibits backward and forward bifurcation when the basic reproduction number varies from unity. The stability behavior of the endemic equilibrium is also discussed, revealing that oscillatory and periodic solutions may appear via Hopf bifurcation when regarding delay as the bifurcation parameter. Moreover, numerical simulations are carried out to illustrate the theoretical findings.

中文翻译：

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具有发生率和饱和处理的确定性时滞 SVIRS 流行模型

本文提出了一种新的非线性延迟易感 - 接种 - 感染 - 恢复 - 易感（SVIRS）流行模型，该模型具有完全易感和已接种疫苗类别的 Holling II 型发病率、饱和治疗率和对易感者的不完善疫苗。对模型的分析表明，它表现出两种平衡，即无病和地方病。推导出基本再生数$$R_0$$R 0 ，证明$$R_0<1$$ R 0 < 1时无病平衡局部渐近稳定，$$R_0=1时线性中性稳定$$ R 0 = 1 。此外，对未延迟模型进行分叉分析，表明当基本再生数从 1 变化时，它表现出前后分叉。还讨论了地方性平衡的稳定性行为，揭示当将延迟视为分岔参数时，振荡和周期解可能会通过 Hopf 分岔出现。此外，还进行了数值模拟以说明理论发现。