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.

In this work, we investigate numerically a system of partial differential equations that describes the interactions between populations of predators and preys. The system considers the effects of anomalous diffusion and generalized Michaelis–Menten-type reactions. For the sake of generality, we consider an extended form of that system in various spatial dimensions and propose two finite-difference methods to approximate its solutions. Both methodologies are presented in alternative forms to facilitate their analyses and computer implementations. We show that both schemes are structure-preserving techniques, in the sense that they can keep the positive and bounded character of the computational approximations. This is in agreement with the relevant solutions of the original population model. Moreover, we prove rigorously that the schemes are consistent discretizations of the generalized continuous model and that they are stable and convergent. The methodologies were implemented efficiently using MATLAB. Some computer simulations are provided for illustration purposes. In particular, we use our schemes in the investigation of complex patterns in some two- and three-dimensional predator–prey systems with anomalous diffusion.

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