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In this paper, a delayed eco-epidemiological model with disease in prey, anti-predator factor for the prey and stage structure for the predator is proposed. The conditions for the stability and coexistence of the populations have been derived. Positivity and boundedness of all solutions with positive initial conditions has been proved. Sufficient conditions for the occurrence of several Hopf and Bogdanov–Takens bifurcations about the endemic steady state are derived. By taking the delay as the bifurcation parameter, it is shown that Hopf and Bogdanov–Takens bifurcations occur at positive steady state for some critical values of the delay. The normal form of the Bogdanov–Takens singularity is obtained by performing a center manifold reduction and by using the normal form theory of FDEs. Some numerical simulations of solutions in different parametric regions illustrating Hopf and homoclinic bifutcations are given to support the analytical results.