Optimizing a delayed predator-prey model to account for infection in both populations

In this study, we propose and analyze a delayed eco-epidemiological prey–predator model that simultaneously incorporates infection in both prey and predator populations together with a discrete time delay in predator response. The model assumes a Holling type-II functional response and considers irreversible infection dynamics without recovery. Biologically, infected prey are assumed to be more vulnerable and form the primary source of predation, while delay represents handling or incubation effects influencing predator growth. We first establish the positivity and boundedness of solutions and determine the existence of biologically feasible equilibrium points. Stability analysis of the interior equilibrium is conducted in both delayed and non-delayed cases. By treating the delay as a bifurcation parameter, we derive explicit threshold conditions under which a pair of complex conjugate eigenvalues crosses the imaginary axis, leading to Hopf bifurcation. The direction and stability of the bifurcating periodic solutions are further determined using normal form and center manifold theory. Unlike many existing eco-epidemiological models that consider infection in only one trophic level or neglect time delay effects, the proposed framework integrates dual infection dynamics, delay-induced oscillations, and optimal intervention strategies within a unified mathematical setting. An optimal control problem is formulated using Pontryagin’s Maximum Principle to determine effective treatment and isolation strategies that minimize infection prevalence and control costs. Numerical simulations are provided to validate the theoretical findings and to illustrate the role of delay in generating sustained oscillatory behavior. The results highlight how delay and targeted control measures jointly influence the long-term dynamics and stability of infected prey–predator ecosystems.