Stability and bifurcation analysis of an HIV model with pre-exposure prophylaxis and treatment interventions

Oral pre-exposure prophylaxis (PrEP) is a highly effective method of HIV prevention. However, despite its effectiveness, the use of PrEP is not without challenges. These challenges include issues such as adherence, societal stigma, as well as accessibility and cost. PrEP effectiveness measures the drug's performance in the real world while efficacy measures the degree to which PrEP inhibits HIV infection and its transmission. In this paper, we address the problem of reduced effectiveness of PrEP, primarily due to non-adherence, and the impact on the dynamics of HIV infection. The study aims to determine the critical threshold of PrEP effectiveness required to avoid the occurrence of a backward bifurcation. The conditions for a stable infection -free equilibrium are stated. Through a rigorous analysis of the proposed mathematical model using Castillo-Chavez and Song's bifurcation theorem, it was found that backward bifurcation occurs when PrEP effectiveness falls below 100%. In contrast, if the effectiveness is 100%, which implies full adherence to PrEP, the model undergoes forward bifurcation. The existence of a backward bifurcation bears significant consequences, most notably the co-existence of the infection-free equilibrium and the endemic equilibrium. Under these circumstances, the eradication of HIV within a particular community becomes very difficult. The results of the numerical analysis demonstrate the important role of proper adherence in augmenting the effectiveness of PrEP and, consequently, curbing HIV transmission within communities. Nevertheless, intensive efforts are required to boost adherence to PrEP. Therefore, other HIV control measures need to be promoted to further reduce transmission.