This article investigates a tri-trophic food chain model with the epidemic at the bottom level (prey) and gestation delays for the next two levels( the intermediate predator and top predator). The steady states’ positivity, boundedness, existence, and stability have been analyzed. The study of Hopf bifurcation has been carried out for different steady states. It is shown that when the gestation delay for intermediate predator \(\tau _1\) is absent, the interior equilibrium is locally stable for gestation delays of top predators \(\tau _2\in (0,\tau _{20}^{+})\) and performs Hopf bifurcation for all \(\tau _2 >\tau _{20}^{+}\) . Further, in the presence both the delay \(\tau _1\) and \(\tau _2\) , the system becomes unstable when \(\tau _1\) , \(\tau _2\) crosses their critical values \(\tau _{10}^{++}\) , \(\tau _{20}^{++}\) respectively and executes Hopf bifurcation \(\tau _1>\tau _{10}^{++}\) , \(\tau _2>\tau _{20}^{++}\) . Moreover, the proposed system is stable for \(\tau _1>\tau _{10}^{++}\) , \(\tau _2<\tau _{20}^{++}\) and unstable for \(\tau _1<\tau _{10}^{++}\) , \(\tau _2>\tau _{20}^{++}\) . The sensitivity analysis of the system at coexistence steady state are presented to identify the effect of system parameters on the variables. Finally, we have presented a numerical simulation to verify our analytical findings.
