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

Abstract

In this paper, a mathematical model of Visceral Leishmaniasis is considered. The model incorporates three populations, the human, the reservoir and the vector host populations. A detailed analysis of the model is presented. This analysis reveals that the model undergoes a backward bifurcation when the associated reproduction threshold is less than unity. For the case where the death rate due to VL is negligible, the disease-free equilibrium of the model is shown to be globally-asymptotically stable if the reproduction number is less than unity. Noticing that the governing model is a system of highly nonlinear differential equations, its analytical solution is hard to obtain. To this end, a special class of numerical methods, known as the nonstandard finite difference (NSFD) method is introduced. Then a rigorous theoretical analysis of the proposed numerical method is carried out. We showed that this method is unconditionally stable. The results obtained by NSFD are compared with other well-known standard numerical methods such as forward Euler method and the fourth-order Runge–Kutta method. Furthermore, the NSFD preserves the positivity of the solutions and is more efficient than the standard numerical methods.