论文标题
反应扩散的一类离散分数流行模型的全局动力学
Global dynamics for a class of discrete fractional epidemic model with reaction-diffusion
论文作者
论文摘要
近年来，具有反应扩散的离散分数流行模型在文献中变得越来越流行，不仅是由于其数值模拟的必要性，而且还因为其确定的物理过程。在本文中，通过二阶中央差异方案和L1非标准有限差方案，考虑了与广义发病率的时间分解反应 - 扩散流行模型的离散对应。更重要的是，选择非标准有限差异方案的主要思想是在拟议的系统中获得无条件积极的效率，这导致随时间延迟的提议。此外，还研究了提出的离散系统的全局属性，包括阳性解决方案的全局界限，平衡点的存在和全局稳定性，这与相应的连续系统一致。同时，它表明L1非标准有限差方案和二阶中央差异方案可以使相应连续系统的属性保持良好。值得注意的是，与整数阶的离散流行模型不同，本文构建了内存Lyapunov函数，这取决于所提出的系统的先前历史信息。这与Caputo分数衍生物的非本地性质一致。最后，给出数值结果以验证理论结果。
In recent years, discrete fractional epidemic models with reaction-diffusion have become increasingly popular in the literature, not only for its necessity of numerical simulation, but also for its defined physical processes. In this paper, by second order central difference scheme and L1 nonstandard finite difference scheme, a discrete counterpart of time-fractional reaction-diffusion epidemic model with generalized incidence rate is considered. More importantly, the main idea in choosing an nonstandard finite difference scheme is to obtain unconditionally positivity in the proposed system, which leads to the proposal of the discrete epidemic model with time delay. Furthermore, the global properties of the proposed discrete system are studied, including the global boundedness of positive solutions, the existence and the global stability of equilibrium points, which are consistent with the corresponding continuous systems. Meanwhile, it shows that L1 nonstandard finite difference scheme and second order central difference scheme can keep the properties of the corresponding continuous system well. It is worth noting that, different from discrete epidemic model with integer-order, the memory Lyapunov function is constructed in this paper, which depends on the previous historical information of the proposed system. This is consistent with the non-local property of Caputo fractional derivatives. Finally, numerical results are given to verify the theoretical results.
