SIS流行病模型的数学研究：全局渐近稳定性分析与非标准数值格式的构造
A Mathematical Study of an SIS Epidemic Model: Global Asymptotic Stability Analysis and Construction of Nonstandard Numerical Schemes
- FPT University（FPT大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究对具有饱和接触率的SIS模型进行严格数学分析，通过Lyapunov函数和Poincaré–Bendixson定理证明全局渐近稳定性，并构造保持正性和稳定性的非标准有限差分格式，数值实验验证了理论结果。
AI中文摘要：
本研究旨在对具有饱和接触率的经典SIS流行病模型进行严格的数学分析。首先，我们通过构造合适的Lyapunov函数，并结合Poincaré–Bendixson定理和Bendixson–Dulac准则，建立了模型平衡点的全局渐近稳定性（GAS）。所得的GAS结果改进了先前关于该SIS模型的研究结论，并可能适用于具有更一般饱和接触率的扩展模型。其次，我们构造了一阶和二阶非标准有限差分（NSFD）格式族，这些格式在任意步长下均保持SIS模型的正性和渐近稳定性。所有格式均在Mickens框架内构建；然而，与一阶格式相比，二阶格式采用了更精细的构造，将右端项的加权非局部近似与适当重整化的分母函数相结合。加权非局部近似保证了动态一致性，而分母函数则确保了二阶收敛性。最后，我们进行了数值实验以验证理论结果并展示所提出的NSFD格式的优势。数值结果与理论结果吻合良好。本研究开发的方法不仅适用于其他流行病学系统，而且更广泛地适用于各种实际应用中出现的数学模型。
英文摘要：
The aim of this work is to provide a rigorous mathematical analysis of a well-known SIS epidemic model with a saturating contact rate. First, we establish the global asymptotic stability (GAS) of the model's equilibria by employing a suitable Lyapunov function in combination with the Poincaré--Bendixson theorem and the Bendixson--Dulac criterion. The resulting GAS results improve upon previous findings for this SIS model and may also be applicable to its extensions with more general saturating contact rates. Second, we construct families of first- and second-order nonstandard finite difference (NSFD) schemes that preserve the positivity and asymptotic stability properties of the SIS model for arbitrary step sizes. All these schemes are formulated within Mickens' framework; however, compared with their first-order counterparts, the second-order schemes employ a more elaborate construction that combines a weighted nonlocal approximation of the right-hand side with suitably renormalized denominator functions. The weighted nonlocal approximation guarantees dynamic consistency, whereas the denominator functions ensure second-order convergence. Finally, we conduct numerical experiments to validate the theoretical results and demonstrate the advantages of the proposed NSFD schemes. The numerical results are in good agreement with the theoretical results. The approach developed in this work is applicable not only to other epidemiological systems but also, more generally, to mathematical models arising in a wide range of real-world applications.
