为了更准确地预测传染病的传播趋势,寻求控制疾病发展的最优策略,建立一类具有部分免疫和饱和发病率的反应扩散SVEIQR传染病模型。确定了传染病模型的基本再生数R0,建立了模型全局动力学阈值准则。通过构造Lyapunov函数,证明当R0<1时无病平衡点的全局稳定性和当R0>1时地方病平衡点的全局稳定性。最后通过数值模拟检验理论推导的正确性并且发现扩散效应不会从根本上改变传染病模型的全局阈值稳定性,但较大扩散系数会加速疫情传播从而加快疫情达到稳态的速度。
In order to more accurately predict the spread trend of infectious diseases and seek the best strategy to control the development of diseases. In this paper, a spatial diffusion SVEIQR epidemic model with a saturated incidence rate and temporary immunity is presented. We define the basic reproduction number R0 and establish the threshold criteria for the global dynamics. By constructing a Lyapunov function, we ascertain that the disease-free equilibrium point is globally asymptotically stable when R0<1, and when R0>1, the endemic equilibrium point exhibits the same property. Finally, the correctness of the theoretical derivation was verified by numerical simulation and it was found that the diffusion effect will not fundamentally change the global threshold stability of the epidemic model. However, a large diffusion coefficient will accelerate the spread of the epidemic and thus accelerate the speed of the epidemic to reach a stable state.
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