Dynamics and optimal control for an HIV/AIDS model with VCT and PrEP treatments among MSM

Considering the important role of Voluntary counseling and testing (VCT) and Pre-exposure prophylaxis (PrEP) in controlling the spread of Acquired Immunodeficiency Syndrome (AIDS), in this paper, we formulate a Human Immunodeficiency Virus (HIV)/AIDS model among men who have sex with men (MSM) to quantitatively describe the impact of VCT and PrEP treatments on the transmission dynamics. First, we analyze the dynamic properties of the model, including proving the global asymptotic stability of the disease-free equilibrium and the uniform persistence of the disease. Then, we explore the optimal control problem of the model using the Pontryagin maximum principle. Finally, numerical simulations and a sensitivity analysis are conducted. The results indicate that enhancing VCT awareness among susceptible and infected individuals and promoting PrEP treatment among susceptible individuals can significantly reduce the number of HIV/AIDS cases. These efforts could include strengthening knowledge dissemination, organizing knowledge contests, and establishing platforms for AIDS education. Therefore, actively promoting VCT awareness and PrEP treatment education is essential to control the spread of HIV/AIDS.