The aim of this paper is to propose and analyze a fractional-order SIS epidemic model with saturating contact rate that is a generalization of a recognized deterministic SIS epidemic model. First, we investigate positivity, boundedness, and asymptotic stability of the proposed fractional-order model. Secondly, we construct positivity-preserving nonstandard finite difference (NSFD) schemes for the model using the Mickens’ methodology. We prove theoretically and confirm by numerical simulations that the proposed NSFD schemes are unconditionally positive. Consequently, we obtain NSFD schemes preserving not only the positivity but also essential dynamical properties of the fractional-order model for all finite step sizes. Meanwhile, standard schemes fail to correctly reflect the essential properties of the continuous model for a given finite step size, and therefore, they can generate numerical approximations which are completely different from the solutions of the continuous model. Finally, a set of numerical simulations are performed to support and confirm the validity of theoretical results as well as advantages and superiority of the constructed NSFD schemes. The results indicate that there is a good agreement between the numerical simulations and the theoretical results and the NSFD schemes are appropriate and effective to solve the fractional-order model.

中文翻译：

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具有饱和接触率的分数阶SIS流行病模型的动力学和数值近似

本文的目的是提出和分析具有饱和接触率的分数阶SIS流行病模型，该模型是公认的确定性SIS流行病模型的概括。首先，我们研究提出的分数阶模型的正性，有界性和渐近稳定性。其次，我们使用Mickens的方法为模型构造了保持阳性的非标准有限差分（NSFD）方案。我们从理论上证明并通过数值模拟证实所提出的NSFD方案是无条件正的。因此，对于所有有限步长，我们获得的NSFD方案不仅保留了分数阶模型的正性，而且还保留了其动力学性质。同时，对于给定的有限步长，标准方案无法正确反映连续模型的基本属性，因此，它们可以生成与连续模型的解完全不同的数值近似值。最后，进行了一组数值模拟，以支持和确认理论结果的有效性以及所构造的NSFD方案的优缺点。结果表明，数值模拟与理论结果之间有很好的一致性，并且NSFD方案对于求解分数阶模型是合适且有效的。