We propose a tractable epidemic model that includes containment measures. In the absence of containment measures, the epidemics spread exponentially fast whenever the infectivity rate is positive $\lambda >0$. The containment measures are modeled by considering a time-dependent modulation of the bare infectivity $\lambda$ leading to effective infectivity that decays in time for each infected individual, mimicking, for instance, the combined effect of the asymptomatic onset of the disease, testing policies, and quarantine. We consider a wide range of temporal kernels for effective infectivity, and we investigate the effect of the considered containment measures. We find that not all kernels are able to push the epidemic dynamics below the epidemic threshold with some containment measures only able to reduce the rate of the exponential growth of newly infected individuals. We also propose a pandemic model caused by a growing number of separated foci.

中文翻译：

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流行病与遏制措施

我们提出了一种易于控制的流行病模型，其中包括遏制措施。在没有遏制措施的情况下，只要传染率是正数，流行病就会迅速呈指数传播$\lambda >0$。通过考虑裸露传染性的时间依赖性调制来对遏制措施进行建模$\lambda$导致有效感染力随时间的推移而下降，例如模仿疾病无症状发作，检测策略和隔离的综合效果。我们考虑了有效的传染性的各种暂时性核仁，并研究了所考虑的围堵措施的效果。我们发现，并非所有内核都能通过某些遏制措施将流行病动态推到流行阈值以下，这些措施只能降低新感染个体的指数增长速度。我们还提出了由越来越多的分离疫源引起的大流行模型。