Nonstandard finite difference schemes for solving an SIS epidemic model with standard incidence
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Nonstandard finite difference schemes for solving an SIS epidemic model with standard incidence Manh Tuan Hoang1,2 · Oluwaseun Francis Egbelowo3 Received: 4 July 2019 / Accepted: 22 July 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract We extend the nonstandard finite difference (NSFD) method of solutions to the study of an SIS epidemic model with standard incidence. We show that the proposed NSFD schemes preserve two essential properties of the continuous model: positivity and global asymptotic stability properties. The reproduction number of the model is calculated by the next generation matrix approach. It is worth noting that the global asymptotic stability of the disease-free equilibrium point of the proposed numerical schemes is proved theoretically by the use of an extension of the Lyapunov stability theorem. Besides, the global asymptotic stability of the endemic equilibrium point is investigated by the use of the Lyapunov indirect method and numerical simulations. Consequently, NSFD schemes which are dynamically consistent with the continuous model are obtained. Some numerical simulations are presented to validate the theoretical results and to show that the NSFD schemes are effective and appropriate for solving the continuous model. We employ the standard finite difference (SFD) method as a means of comparison to NSFD schemes. The numerical simulations indicate that the use of SFD method is not suitable as it produced solutions that do not correspond exactly to solutions of the continuous model. Keyword SIS epidemic model · Global asymptotic stability · Nonstandard finite difference schemes · Dynamics consistency · Lyapunov stability theorem Mathematics Subject Classification 65L05 · 65L12 · 65L20 · 37M05
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Manh Tuan Hoang [email protected]; [email protected] Oluwaseun Francis Egbelowo [email protected]
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Institute of Information Technology, Vietnam Academy of Science and Technology (VAST), 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
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Department of Mathematics, FPT University, Hoa Lac Hi-Tech Park, Km29 Thang Long Blvd, Hanoi, Vietnam
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Division of Clinical Pharmacology, Department of Medicine, University of Cape Town, Cape Town, South Africa
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M. T. Hoang, O. F. Egbelowo
1 Introduction Many essential processes and phenomena in various fields of applied sciences and the real world are described by systems of differential equations [2,22]. These systems are very complicated and in most cases, it is impossible to find their exact solutions. Therefore, the construction of numerical methods for solving the systems has an essential role in both theory and practice [5,23–26]. It is well known that numerical instabilities may occur when using the standard finite difference (SFD) methods to solve nonlinear differential systems [23– 26]. According to Mickens, numerical instabilities are an indication that the discrete models are not able to provide the correct mathematical properties of the solutions to the differential equations of interest [23–26]. However,
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