A finite-difference discretization preserving the structure of solutions of a diffusive model of type-1 human immunodeficiency virus

• Published in: Advances in difference equations 2021-03, Vol.2021 (1), p.1-19, Article 158
• Main Authors: Alba-Pérez, Joel, Macías-Díaz, Jorge E.
• Format: Article
• Language: English
• Subjects: Analysis; Convergence; Density; Difference and Functional Equations; Diffusive mathematical model; Finite difference method; Functional Analysis; HIV; Human immunodeficiency virus; Human tissues; Lymphocytes; Mathematical models; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Structure-preserving finite-difference scheme
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-021-03322-y

We investigate a model of spatio-temporal spreading of human immunodeficiency virus HIV-1. The mathematical model considers the presence of various components in a human tissue, including the uninfected CD4 + T cells density, the density of infected CD4 + T cells, and the density of free HIV infection particles in the blood. These three components are nonnegative and bounded variables. By expressing the original model in an equivalent exponential form, we propose a positive and bounded discrete model to estimate the solutions of the continuous system. We establish conditions under which the nonnegative and bounded features of the initial-boundary data are preserved under the scheme. Moreover, we show rigorously that the method is a consistent scheme for the differential model under study, with first and second orders of consistency in time and space, respectively. The scheme is an unconditionally stable and convergent technique which has first and second orders of convergence in time and space, respectively. An application to the spatio-temporal dynamics of HIV-1 is presented in this manuscript. For the sake of reproducibility, we provide a computer implementation of our method at the end of this work.