A non-smooth epidemic model with piecewise incidence rate dependent on the derivative of the case number is proposed for the transmission dynamics of an infectious disease with media coverage, enhanced vaccination and treatment policy. This is an implicitly defined system, which is converted into an explicit system with three thresholds by employing the properties of the Lambert W function. We first analyze the dynamics of the proposed model for the limiting case, which induces two non-smooth but continuous models. The dynamic analysis of the model demonstrates that either one of the two generalized equilibria or the pseudo-equilibrium is globally asymptotically stable if the disease does not die out. This suggests that the case number can be contained either at an a priori level or at a high/low level, depending on the threshold, which governs whether the enhanced vaccination and treatment policies are implemented. Media coverage cannot help eradicate the disease, but it significantly delays the epidemic peak and lowers the peak case number. Hence, a good threshold policy and continuously updating the awareness of case numbers are required to combat the disease successfully.

中文翻译：

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具有三个阈值的非光滑流行病模型的动力学。

针对具有媒体覆盖率，加强疫苗接种和治疗策略的传染病的传播动力学，提出了一种分段发病率取决于病例数导数的非光滑流行模型。这是一个隐式定义的系统，通过使用Lambert W的属性将其转换为具有三个阈值的显式系统功能。我们首先分析极限情况下所提出模型的动力学，从而得出两个非平滑但连续的模型。该模型的动态分析表明，如果疾病没有消亡，那么两个广义平衡或伪平衡之一在全局渐近稳定。这表明取决于阈值，病例数可以包含在先验水平或高/低水平，该阈值决定是否实施增强的疫苗接种和治疗政策。媒体报道无法帮助根除该疾病，但会大大延迟该流行病高峰并降低高峰病例数。因此，需要一个好的门槛政策并不断更新对病例数的认识，才能成功地抗击该疾病。