A Galerkin-Type Method for Solving a Delayed Model on HIV Infection of CD $$\mathbf{4^+}$$ 4 + T-cells

Abstract

The topic of this study is one of the most important models of mathematical biology; namely, a nonlinear first order delayed model about the infection of CD $$4^+$$ T-cells by HIV. In this model, the effect of virus on human immune system is described through depletion of healthy CD $$4^+$$ T-cells, where the delayed term representing the eclipse phase of virus makes the model more realistic. To obtain approximate solutions of this model, we develop a method reminiscent of the discrete Galerkin method. Considering the approximate solutions in the form of polynomials, we first substitute these approximate solutions in the original model. Some relations are thus obtained, which we express in terms of matrices. Following Galerkin’s path, we take inner product of a set of monomials with these matrix expressions, thus obtaining a nonlinear system of algebraic equations. The solution of this system gives the approximate solutions of the model. Additionally, the technique of residual correction, which aims to reduce the error of the approximate solution by estimating this error, is discussed in some detail. The method and the residual correction technique are illustrated with an example problem.