We consider the SIRWJS epidemiological model that includes the waning and boosting of immunity via secondary infections. We carry out combined analytical and numerical investigations of the dynamics. The formulae describing the existence and stability of equilibria are derived. Combining this analysis with numerical continuation techniques, we construct global bifurcation diagrams with respect to several epidemiological parameters. The bifurcation analysis reveals a very rich structure of possible global dynamics. We show that backward bifurcation is possible at the critical value of the basic reproduction number, $\mathcal{R}_0 = 1$. Furthermore, we find stability switches and Hopf bifurcations from steady states forming multiple endemic bubbles, and saddle-node bifurcations of periodic orbits. Regions of bistability are also found, where either two stable steady states, or a stable steady state and a stable periodic orbit coexist. This work provides an insight to the rich and complicated infectious disease dynamics that can emerge from the waning and boosting of immunity.

翻译：我们提出了包含通过二次感染增强和减弱免疫力的SIRWJS流行病模型，并对其动力学进行了分析与数值联合研究。推导了平衡态存在性与稳定性的解析表达式。结合该分析与数值延拓技术，我们构建了关于多个流行病学参数的全局分岔图。分岔分析揭示了可能全局动力学的丰富结构。结果表明，在基本再生数的临界值$\mathcal{R}_0 = 1$处可能发生反向分岔。此外，我们发现了稳态的稳定性切换和产生多个地方病泡的Hopf分岔，以及周期轨的鞍结分岔。还发现了双稳态区域，其中两个稳定稳态或一个稳定稳态与一个稳定周期轨道共存。本研究为免疫力减弱与增强所引发的复杂传染病动力学提供了深入见解。