Dynamics and numerical approximations for a fractional-order SIS epidemic model with saturating contact rate
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Dynamics and numerical approximations for a fractional-order SIS epidemic model with saturating contact rate Manh Tuan Hoang1 · Zain Ul Abadin Zafar2 · Thi Kim Quy Ngo3 Received: 24 May 2020 / Revised: 15 August 2020 / Accepted: 5 September 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract 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.
Communicated by José Tenreiro Machado.
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Manh Tuan Hoang [email protected]; [email protected] Zain Ul Abadin Zafar [email protected] Thi Kim Quy Ngo [email protected]
1
Institute of Information Technology, Vietnam Academy of Science and Technology (VAST), 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
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Faculty of Information Technology, University of Central Punjab, Lahore, Pakistan
3
Posts and Telecommunications Institute of Technology (PTIT), Hanoi, Vietnam 0123456789().: V,-vol
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Keywords Stability analysis · Fractional-order SIS epidemic model · Nonstandard finite difference schemes · Unconditionally positive numerical schemes · Lyapunov stability theory Mathematics Subject Classification 34C60; 37M05; 37M15; 65Z05; 92-10
1 Introduction The SIS epidemic models in particular and the epidemic models in general have played an essential role both in theory and practice (see Allen 2007; Brauer and Castillo-Chavez 2001; Martcheva 2015 and references therein). These models have attracted the attention of many mathematicians and biologists for many years. The study of the models can help us to understand transmission mechanism of viruses in infectious diseases. Based on this basis, effective strategies to prevent and control the spread of viruses and to protect the public h
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