An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

Abstract

We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the other conventional approaches that are routinely used for such problems.

A class of nonstandard numerical methods for autonomous dynamical systems

“…[2,3,4,5,6,7,8,9]). Here, a finite difference method is constructed in Section 2 which preserves the aforementioned qualitative properties for the system (1).…”

An unconditionally stable nonstandard finite difference method to solve a mathematical model describing Visceral Leishmaniasis

“…The NSFD schemes are used to solve many biological problems. Standard numerical methods such as Euler and Runge-Kutta methods are usually applied for the comparison with many of NSFD schemes models [2,3,12,16,20,23]. In models [2,20] the matlab solvers are also applied for comparison purposes.…”

“…Standard numerical methods such as Euler and Runge-Kutta methods are usually applied for the comparison with many of NSFD schemes models [2,3,12,16,20,23]. In models [2,20] the matlab solvers are also applied for comparison purposes. It was noticed by these researchers that standard methods like those mentioned above often fail to reflect some essential qualitative features, such as, positivity and invariance of a solution, backward bifurcation, convergence to the correct equilibrium for relatively large step-sizes, etc., as stated in [12].…”

Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay

“…Numerous mathematical models represented by means of a system of differential equations, with or without delay, have been discretized by means of the non-standard finite difference method proposed by Ronald Mickens, see [24][25][26][27][28][29][30][31][32][33][34]. Their use is mainly because they are very effective in preserving certain qualitative properties of the original differential equations and the convergence, consistency and stability of their solutions have been demonstrated, see [35][36][37][38][39][40][41][42][43][44][45][46].…”