TAILIEUCHUNG - Nonstandard finite difference schemes for solving a modified epidemiological model for computer viruses
In this paper we construct two families of nonstandard finite difference (NSFD) schemes preserving the essential properties of a computer virus propagation model, such as positivity, boundedness and stability. The first family of NSFD schemes is constructed based on the nonlocal discretization and has first order of accuracy, while the second one is based on the combination of a classical Runge-Kutta method and selection of a nonstandard denominator function and it is of fourth order of accuracy. | Journal of Computer Science and Cybernetics, , (2018), 171–185 DOI NONSTANDARD FINITE DIFFERENCE SCHEMES FOR SOLVING A MODIFIED EPIDEMIOLOGICAL MODEL FOR COMPUTER VIRUSES* DANG QUANG A1 , HOANG MANH TUAN2,a , DANG QUANG LONG2 1 Centre for Informatics and Computing, VAST of Information Technology, VAST a hmtuan01121990@ 2 Institute Abstract. In this paper we construct two families of nonstandard finite difference (NSFD) schemes preserving the essential properties of a computer virus propagation model, such as positivity, boundedness and stability. The first family of NSFD schemes is constructed based on the nonlocal discretization and has first order of accuracy, while the second one is based on the combination of a classical Runge-Kutta method and selection of a nonstandard denominator function and it is of fourth order of accuracy. The theoretical study of these families of NSFD schemes is performed with support of numerical simulations. The numerical simulations confirm the accuracy and the efficiency of the fourth order NSFD schemes. They hint that the disease-free equilibrium point is not only locally stable but also globally stable, and then this fact is proved theoretically. The experimental results also show that the global stability of the continuous model is preserved. Keywords. Computer viruses, high order NSFD schemes, Lyapunov stability theorem, NSFD schemes, numerical simulations. 1. INTRODUCTION The mathematical models describing the computer virus propagation play especially important role in both theory and practice. The study of properties of these models helps us to understand the law governing the spread of computer viruses. Based on this we can make appropriate policies for controlling and preventing the spread of them. In the last two decades some authors proposed different mathematical models for computer viruses through differential equations systems, ., [23, 25, 26, 27, 28, 29, .
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