Bifurcation analysis and optimal control in a predator-prey-infection model with additional food

We propose and analyze a three-dimensional eco-epidemiological model involving susceptible and infected prey and predators, in which the predators are supplemented with a constant externally supplied food supply. The model incorporates nonlinear disease transmission and predator feeding saturation through a generalized Holling type II functional response. We investigate the system's dynamics analytically and numerically by examining the existence and stability of equilibria, as well as Hopf, transcritical, and saddle-node bifurcations. One- and two-parameter bifurcation analyses reveal rich dynamics, including limit cycles, period doubling, and chaotic oscillations. Our findings indicate that disease transmission can destabilize the system, while the inclusion of additional food enhances stability and can suppress chaos. Furthermore, we extend the model by introducing a time-dependent optimal control variable representing additional food supply, and derive an optimal strategy using Pontryagin's Maximum Principle. Numerical simulations show that optimal control effectively reduces disease prevalence and stabilizes population dynamics. This study highlights the potential of ecological interventions, such as strategic food supplementation, in regulating complex eco-epidemiological systems.