Transmission dynamics of mpox virus and HIV co-infection and its optimal control in MSM population

Sexual contact is one of the main ways for Mpox virus to spread among humans. Men who have sex with men (MSM) individuals carrying the Mpox virus have a higher probability of being co-infected with HIV, seriously threatening human health. Therefore, it is essential to examine the impact of HIV on Mpox virus transmission dynamics among MSM. To address this, we developed a mathematical model to analyze the co-infection dynamics of Mpox virus and HIV. The effective reproduction numbers for the Mpox virus and HIV sub-models were calculated separately using the next-generation matrix method, and the dynamical behaviors of the two sub-models were analyzed in turn. The disease-free equilibrium of each sub-model is globally stable if the corresponding effective reproduction number is less than 1 and unstable if it is greater than 1, with an endemic equilibrium emerging and becoming globally asymptotically stable. Furthermore, the effective reproduction number and the invasive reproduction number of the Mpox virus-HIV co-infection model were defined. Sensitivity analysis reveals that both a decrease in sexual contact rate or transmission probability and an increase in condom usage or pre-exposure prophylaxis (PrEP) measures can significantly reduce the effective and invasive reproduction numbers, thereby lowering the risk of co-infection with the Mpox virus and HIV. Subsequently, we proposed a time-dependent optimal control model based on Pontryagin’s Maximum Principle and conducted a cost-benefit analysis. The results show that a hybrid strategy integrating Mpox virus prevention, HIV prevention, and Mpox virus antiviral therapy is more effective in reducing Mpox virus infection risk among people living with HIV compared to single-intervention approaches.