Modeling adult heterosexual HIV transmission and its impact on mother-to-child infection

The Human Immunodeficiency Virus (HIV) has been a major focus of research for several decades due to its multiple transmission pathways, its high virulence and the absence, to date, of both curative and preventive treatments. Although sexual transmission remains by far the most dominant route, mother-to-child (vertical) transmission-which may occur during pregnancy, childbirth or breastfeeding-is also significant and accounts for approximately 15%-30% of cases in the absence of antiretroviral therapy (ART). In this work, we develop a heterosexual HIV transmission model that incorporates vertical transmission, arising from sexual contact. The model is analysed both qualitatively and quantitatively. We establish the global asymptotic stability of the disease-free equilibrium when the basic reproduction number, $\mathscr{R}_{0}$, is less than one. When $\mathscr{R}_{0}$ is greater than one, we show the existence, uniqueness and local asymptotic stability of a unique endemic equilibrium. To perform numerical simulations, we construct a nonstandard finite difference scheme, which preserves the positivity, boundedness and stability properties of the continuous model. Numerical results are presented to support the theoretical findings and to assess the impact of abstinence, safe sexual practices, ART intervention and some joint public health strategies in reducing HIV infections. In particular, we found that if the men's and women's yearly contact rates are less than 120 and 100, respectively, HIV will be eliminated from the community.