Bounkaicha Chouaib, Allali Karam
Laboratory of Mathematics and Applications, Faculty of Sciences and Technologies, Hassan II University of Casablanca, PO Box 146, 20650 Mohammedia, Morocco.
Model Earth Syst Environ. 2023 Apr 8:1-13. doi: 10.1007/s40808-023-01773-8.
The global analysis of a spatio-temporal fractional order SIR infection model with saturated incidence function is suggested and studied in this paper. The dynamics of the infection is described by three partial differential equations including a time-fractional derivative order for each one of them. The equations of our model describe the evolution of the susceptible, the infected and the recovered individuals with taking into account the spatial diffusion for each compartment. We will choose a saturated incidence rate in order to describe the nonlinear force of the infection. First, we will prove the well-posedness of our suggested model in terms of existence and uniqueness of the solution. Also in this context, the boundedness and the positivity of solutions are established. Afterward, we will give the forms of the disease-free equilibrium and the endemic one. It was demonstrated that the global stability of the each equilibrium depends mainly on the basic reproduction number. Finally, numerical simulations are performed to validate the theoretical results and to show the effect of vaccination in reducing the infection severity. It was shown that the fractional derivative order has no effect on the equilibria stability but only on the convergence speed towards the steady states. It was also observed that vaccination is amongst the good strategies in controlling the disease spread.
本文提出并研究了一个具有饱和发生率函数的时空分数阶SIR感染模型的全局分析。感染的动态由三个偏微分方程描述，每个方程都包含一个时间分数阶导数。我们模型的方程描述了易感者、感染者和康复者的演化，并考虑了每个 compartment 的空间扩散。我们将选择一个饱和发生率来描述感染的非线性力。首先，我们将从解的存在性和唯一性方面证明我们提出的模型的适定性。在此背景下，还建立了解的有界性和正性。之后，我们将给出无病平衡点和地方病平衡点的形式。结果表明，每个平衡点的全局稳定性主要取决于基本再生数。最后，进行了数值模拟以验证理论结果，并展示疫苗接种在降低感染严重程度方面的效果。结果表明，分数阶导数阶数对平衡点稳定性没有影响，只对向稳态的收敛速度有影响。还观察到疫苗接种是控制疾病传播的良好策略之一。
