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

**Abraham J. Arenas 1, Gilberto González-Parra 2,\*, Jhon J. Naranjo 1, Myladis Cogollo <sup>1</sup> and Nicolás De La Espriella <sup>3</sup>**


**Abstract:** We propose a mathematical model based on a set of delay differential equations that describe intracellular HIV infection. The model includes three different subpopulations of cells and the HIV virus. The mathematical model is formulated in such a way that takes into account the time between viral entry into a target cell and the production of new virions. We study the local stability of the infection-free and endemic equilibrium states. Moreover, by using a suitable Lyapunov functional and the LaSalle invariant principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable. In addition, we designed a non-standard difference scheme that preserves some relevant properties of the continuous mathematical model.

**Keywords:** HIV infection; mathematical delay model; eclipse phase; NSFD; numerical simulation
