Optimal Control Analysis of Vertically Transmitted Infectious Diseases With Saturated Incidence and Nonlinear Treatment

ABSTRACT This study proposes a novel mathematical model that integrates key epidemiological features, including vertical transmission, saturated incidence, and nonlinear treatment into the dynamics of infectious diseases in a human population. The mathematical and epidemiological relevance of the model is established through the theory of positivity and boundedness of solutions. The basic reproduction number, a key metric for assessing disease spread, is calculated using the next‐generation matrix approach, and the existence of model equilibria is established under specific conditions. Bifurcation analysis and long‐term behavior are investigated using centre manifold theory and Lyapunov functions, respectively. Two time‐dependent control strategies are considered: prevention (blocking mother‐to‐child transmission) and treatment (using effective drugs), which are analyzed using Pontryagin's maximum principle. The study highlights the impact of implementing each control individually and in combination on reducing disease spread within a community. Key epidemiological properties and optimal control scenarios are quantitatively examined using MATLAB simulations. The results provide valuable insights into curbing vertically transmitted diseases and promoting recovery, emphasizing the importance of combining prevention and treatment strategies to effectively control infectious disease spread.