Impact of fear on an eco-epidemiological model

In the present paper, we investigate the impact of fear in a predator-prey model with disease in the prey species. The logistically growing prey population is divided into two groups: susceptible and infected. We take the fear of predators among prey population into consideration, which costs lowering of prey’s growth rate and slims down the interactions among prey individuals. In our model, it is reflected by two decreasing factors of the fear parameter and the predator population: one in the growth term of the susceptibles and the other in the disease transmission term. We choose a general disease transmission function for which mass action, standard incidence, and saturation laws are particular cases. The predator-prey interactions are described by generalized Holling type-II functional response. We explore the effect of fear for three subcases of our model, all of which happen to be identical to three published works (without fear effect). Apart from the preliminary mathematical analysis of our model (e.g. positivity, boundedness, etc.), we find the conditions for existence and local stability of the equilibrium points and study Hopf-bifurcation around the endemic equilibrium point w.r.t. the fear parameters. We observe that fear can eliminate the chaotic oscillations of the system, produced in the absence of fear, by either making the endemic equilibrium regular (stable limit cycle or stable equilibrium point) or moving it towards the disease-free state. We also observe the presence of multiple attractors in the phase-space and different types of bistabilities. We perform extensive numerical simulations to explore the rich dynamics of our model.