Mathematical analysis of HPV and gonorrhea co-infection model with optimal control strategies for effective intervention

Human papillomavirus (HPV) and gonorrhea are sexually transmitted infections that pose significant challenges to global public health. This study introduces a novel mathematical model with eleven compartments to capture the dynamics between HPV and gonorrhea co-infection. We explore the basic analytical properties such as nonnegativity and boundedness of the solutions, stability analysis of disease-equilibrium point, existence of endemic equilibrium points for the co-infection model. The basic reproduction number is derived using next generation matrix method. Also, the model is calibrated and validated using real-world gonorrhea infection data from 2012 to 2022, collected in the United States to ensure its accuracy and relevance to current epidemiological trends. Sensitivity analysis is carried out to identify key parameters that influence the basic reproduction number. The findings reveal that reducing contact rates, increasing gonorrhea treatment rates, enhancing HPV vaccine efficacy, and accelerating HPV vaccination efforts substantially lower the disease transmission. Additionally, the effective contact rates of HPV and gonorrhea have been projected until 2062, providing valuable insights into the significance of reducing these infections. To further curb the transmission of these infections, three time-dependent control variables such as promoting safe sex practices, implementing a time-varying HPV vaccination rate and enhancing gonorrhea treatment are incorporated into the model. We then define an optimal control problem to minimize the total number of infected individuals and examine the existence of optimal control along with the necessary conditions for the optimality system. Numerical simulations are carried out to illustrate the practical implications of the model and the effectiveness of the proposed control strategies. The obtained results suggest that the simultaneous implementation of all control measures can significantly lower both single and co-infection rates, leading to improved public health outcomes.