Some infectious diseases produce lifelong immunity while others only produce temporary immunity. In the case of short-lived immunity, the level of protection wanes over time and may be boosted upon re-exposure, via infection or vaccination. Previous work developed a simple model capturing waning and boosting immunity, known as the Susceptible-Infectious-Recovered-Waned-Susceptible (SIRWS) model, which exhibits rich dynamical behavior including supercritical and subcritical Hopf bifurcations among other structures. Here, we extend the bifurcation analyses of the SIRWS model to examine the influence of all parameters on these bifurcation structures. We show that the bistable region, involving both a stable fixed point and a stable limit cycle, exists only for a small region of biologically realistic parameter space. Furthermore, we contrast the SIRWS model with a modified version, where immune boosting may involve the occurrence of a secondary infection. Analysis of this extended model shows that oscillations and bistability, as found in the SIRWS model, depend on strong assumptions about infectivity and recovery rate from secondary infection. Understanding the dynamics of models of waning and boosting immunity is important for accurately assessing epidemiological data.

中文翻译：

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重复感染在免疫力减弱和增强的简单模型动力学中的作用

有些传染病会产生终身免疫，而另一些只会产生暂时性免疫。在短期免疫的情况下，保护水平会随着时间的推移而减弱，并且可能会在再次暴露后通过感染或疫苗接种得到加强。以前的工作开发了一个简单的模型来捕捉减弱和增强免疫力，称为易感-感染-恢复-减弱-易感 (SIRWS) 模型，该模型表现出丰富的动力学行为，包括超临界和亚临界 Hopf 分岔等结构。在这里，我们扩展了 SIRWS 模型的分叉分析，以检查所有参数对这些分叉结构的影响。我们表明，涉及稳定不动点和稳定极限环的双稳态区域仅存在于生物学现实参数空间的一小部分区域。此外，我们将 SIRWS 模型与修改版本进行对比，其中免疫增强可能涉及继发感染的发生。对该扩展模型的分析表明，在 SIRWS 模型中发现的振荡和双稳态取决于对传染性和继发感染恢复率的强假设。了解减弱和增强免疫力模型的动态对于准确评估流行病学数据非常重要。