用“动力学型反应”方法模拟封锁和隔离措施下SARS-CoV-2的传播
Modelling the Spreading of the SARS-CoV-2 in Presence of the Lockdown and Quarantine Measures by a "Kinetic-Type Reactions" Approach
摘要：
我们提出了一个受封锁和隔离措施影响的COVID-19疫情演变的现实模型，该模型考虑了康复或死亡过程的时间延迟。采用动力学型反应方法，推导了整个过程的动力学方程。更具体地说，封锁和隔离措施是以某种抑制剂反应为模型的，其中易感和受感染的个体可以被困在非活动状态。康复者的动态是通过计算那些只追溯到住院感染者的人来获得的。为了得到演化方程，我们从米氏酶-底物反应模型（所谓的MM反应）中获得灵感，其中酶分别与可用的医院病床、感染者的底物和康复者的产物相关联。换句话说，一切的发生就好像医院的病床在医院的恢复过程中起着催化剂的作用。当然，在我们的情况下，逆MM反应在我们的情况下没有意义，因此，动力学常数等于零。最后，COVID-19检测呈阳性的人的O.D.E.s简单地由以下动力学方案S+I=> 2 I，I=>R或I=>D建模，其中S、I、R和D分别表示易感人群、感染人群、死亡人群和死亡人群。由此产生的动力学型方程提供了基本反应步骤的O.D.E.s，描述了受感染者的数量、先前住院的康复者的总数、受封锁和隔离措施的影响以及死亡的总数。该模型还预测了第二波冠状病毒感染。对比利时、法国和德国的真实数据进行的测试证实了我们模型的正确性。
英文摘要：
We propose a realistic model for the evolution of the COVID-19 pandemic subject to the lockdown and quarantine measures, which takes into account the time-delay for recovery or death processes. The dynamic equations for the entire process are derived by adopting a kinetic-type reaction approach. More specifically, the lockdown and the quarantine measures are modelled by some kind of inhibitor reactions where susceptible and infected individuals can be trapped into inactive states. The dynamics for the recovered people is obtained by accounting people who are only traced back to hospitalised infected people. To get the evolution equation we take inspiration from the Michaelis- Menten enzyme-substrate reaction model (the so-called MM reaction) where the enzyme is associated to the available hospital beds, the substrate to the infected people, and the product to the recovered people, respectively. In other words, everything happens as if the hospitals beds act as a catalyser in the hospital recovery process. Of course, in our case the reverse MM reactions has no sense in our case and, consequently, the kinetic constant is equal to zero. Finally, the O.D.E.s for people tested positive to COVID-19 is simply modelled by the following kinetic scheme S+I=>2I with I=>R or I=>D, with S, I, R, and D denoting the compartments Susceptible, Infected, Recovered, and Deceased people, respectively. The resulting kinetic-type equation} provide the O.D.E.s, for elementary reaction steps, describing the number of the infected people, the total number of the recovered people previously hospitalised, subject to the lockdown and the quarantine measure, and the total number of deaths. The model foresees also the second wave of Infection by Coronavirus. The tests carried out on real data for Belgium, France and Germany confirmed the correctness of our model.
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