Department of Mathematics, University of Western Ontario, London, ON, N6A 5B7, Canada.
J Math Biol. 2024 May 6;89(1):1. doi: 10.1007/s00285-024-02100-0.
In this paper, we introduce the notion of practically susceptible population, which is a fraction of the biologically susceptible population. Assuming that the fraction depends on the severity of the epidemic and the public's level of precaution (as a response of the public to the epidemic), we propose a general framework model with the response level evolving with the epidemic. We firstly verify the well-posedness and confirm the disease's eventual vanishing for the framework model under the assumption that the basic reproduction number . For , we study how the behavioural response evolves with epidemics and how such an evolution impacts the disease dynamics. More specifically, when the precaution level is taken to be the instantaneous best response function in literature, we show that the endemic dynamic is convergence to the endemic equilibrium; while when the precaution level is the delayed best response, the endemic dynamic can be either convergence to the endemic equilibrium, or convergence to a positive periodic solution. Our derivation offers a justification/explanation for the best response used in some literature. By replacing "adopting the best response" with "adapting toward the best response", we also explore the adaptive long-term dynamics.
在本文中，我们引入了实际易感人群的概念，它是生物学易感人群的一部分。假设该比例取决于疫情的严重程度和公众的防范水平（作为公众对疫情的反应），我们提出了一个具有随疫情演变的响应水平的一般框架模型。我们首先验证了框架模型在基本繁殖数 的假设下的适定性，并证实了疾病的最终消失。对于 ，我们研究了行为反应如何随疫情演变，以及这种演变如何影响疾病动态。具体来说，当预防水平被视为文献中的瞬时最佳反应函数时，我们表明地方病动态收敛到地方病平衡点；而当预防水平是延迟的最佳反应时，地方病动态可以收敛到地方病平衡点，也可以收敛到正的周期解。我们的推导为文献中使用的最佳反应提供了一个解释。通过将“采用最佳反应”替换为“适应最佳反应”，我们还探索了自适应长期动态。
