Mathematical modeling of HIV transmission in a heterosexual population: incorporating memory conservation

HIV disease is a major global public health concern since its appearance in the early 1980s. Many mathematical models have been conducted to understand, control, and predict its spread. In this paper we propose a mathematical model with memory effect modelling HIV transmission in a heterosexual population divided into two age classes; young-class (15-24 years old) and grown-class (25 years old and over). The goal of dividing the population according to age is to identify the most vulnerable class to the virus based on their sexual activity and make accurate predictions about HIV transmission. First, we determine the biologically significant space for the study, and we prove the existence of a unique solution. Then we divide the principal model into four sub-models: young-people, grown-people, young-men linked to grown-women, grown-men linked to young-women. The basic reproduction number associated to each sub-model is derived. According to the four sub-models, we have found that, if the basic reproduction number is below unity, then the free disease equilibrium state is locally asymptotically stable. Numerical simulations are provided to validate the theoretical results and discuss the local stability of the endemic equilibrium states of each sub-model. We conclude that incorporating memory conservation gives more realistic results, where reaching a stable state takes higher time. As well, memory effect can play the role of prior knowledge about the disease and experience accumulated over years.