The effect of PrEP uptake and adherence on the spread of HIV in the presence of casual and long-term partnerships.

A model with both casual and long-term partnerships is considered with respect to the impact of a pre-exposure prophylaxis (PrEP) on the spread of HIV. We consider the effect of the effectiveness of PrEP, the rate that susceptible individuals choose to take PrEP, and compliance with the daily dose of the pre-exposure prophylaxis. The rate of infection in long-term partnerships is computed using a linearized expected value as a means for including the nonlocal effects of long-term partnerships while maintaining computational feasibility. The reproduction numbers for models with casual partnerships, long-term partnerships, and a combination of both are analytically computed and global stability of both disease-free and endemic equilibria is shown. Sensitivity and PRCC analysis results suggest that increasing the compliance among the current PrEP users is a more effective strategy in the fight against the HIV epidemic than increased coverage with poor compliance. Furthermore, an analysis of the reproduction number shows that models with either casual or monogamous long-term partnerships can reach the desired $ R_0 < 1 $ threshold for high enough levels of compliance and uptake, however, a model with both casual and monogamous long-term partnerships will require additional interventions. Methods highlighted in this manuscript are applicable to other incurable diseases or diseases with imperfect vaccines effected by long-term partnerships.