Effect of fear and refuge in eco-epidemiological model with predator supported by additional food

Abstract We propose a three-compartment eco-epidemiological model in which prey experience fear from susceptible predators and may seek refuge, predator follows a modified Leslie–Gower growth. The predator population is divided into susceptible and infected class, with disease transmission occurring horizontally together with saturated treatment of infected class. We analyze positivity and boundedness, compute biologically feasible equilibria, and derive the basic reproduction number ( $$\mathcal {R}_0$$ R 0 ) along with conditions for local and global stability. Sensitivity is explored by partial rank correlation coefficient (PRCC) method; bifurcation behaviours are illustrated analytically and numerically. Furthermore, we formulate and numerically solve an optimal control problem aimed at minimizing infected predator biomass and treatment cost using the forward–backward sweep method. The results reveal that larger refuge size and stronger treatment efforts effectively reduce infection prevalence, while fear on prey species control by the refuge. Additionally, the system exhibits parameter-dependent bi-stability and multi-stability, as demonstrated through bifurcation diagrams.