基于信息指数的行为流行病模型的积分更新方程方法
An integral renewal equation approach to behavioural epidemic models with information index
AI总结：
提出包含信息指数的积分流行病模型，发现弱 Erlang 核保证地方病平衡点稳定，强 Erlang 核引发自持振荡，并验证了流感和 SARS 等场景。
摘要：
我们提出了一个描述传染病流行的积分模型。该模型是行为性的，即感染力的本构律包含一个分布式时滞，称为“信息指数”，用于描述观点驱动的人类行为变化。信息指数包含一个记忆核，以模拟个体如何保持对过去感染值的记忆。我们获得了地方病平衡点局部稳定的充分条件。特别地，当传染力函数由指数分布表示时，弱 Erlang 记忆核可保证稳定性。然而，通过数值模拟，我们表明当记忆更集中于疾病的过去历史时（如强 Erlang 核所示），可能会出现自持振荡。我们还展示了在描述流感和 SARS 等传染病的不同传染力函数下的模型解。
英文摘要：
We propose an integral model describing an epidemic of an infectious disease. The model is behavioural in the sense that the constitutive law for the force of infection includes a distributed delay, called "information index", that describes the opinion-driven human behavioural changes. The information index, in turn, contains a memory kernel to mimic how the individuals maintain memory of the past values of the infection. We obtain sufficient conditions for the endemic equilibrium to be locally stable. In particular, we show that when the infectivity function is represented by an exponential distribution, stability is guaranteed by the weak Erlang memory kernel. However, through numerical simulations, we show that self-sustained oscillations may arise when the memory is more focused in the disease's past history, as exemplified by the strong Erlang kernel. We also show the model solutions in cases of different infectivity functions describing infectious diseases like influenza and SARS.
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