Stability Analysis of a Time-Delayed Disease Transmission Model in Prey–Predator Populations Incorporating a Holling Type II Functional Response

Abstrak. This article presents a comprehensive study of a mathematical model describing the spread of infectious disease within a prey–predator population, incorporating the Holling type II functional response and a delay parameter, denoted as τ, representing the incubation or infection period. The model captures the interactions among four population groups: susceptible prey, infected prey, susceptible predators, and infected predators. Through analytical investigation, six fixed points (equilibrium points) of the system were identified. The stability of these fixed points was examined using the eigenvalues of the Jacobian matrix, and one locally stable fixed point was found, while the others were identified as saddle points or unstable. To gain deeper insights into the model’s behavior over time, numerical simulations were conducted for different values of the delay parameter . The results indicate that the presence of a time delay significantly affects the dynamics of all four population groups. Specifically, the infection delay can suppress or slow the spread of the disease by delaying the transition from susceptible to infected classes. Oscillatory behavior emerged in certain population groups when the delay was introduced, especially among infected prey and predators, before gradually stabilizing toward the disease-endemic equilibrium. These findings highlight the critical role of time delay in disease transmission dynamics in ecological systems and provide a framework for further research on delay-induced phenomena in epidemiological models. Keywords: Prey–Predator, Disease Spread, Delay Time, Functional Response, Stability Analysis.