RESEARCH PAPER Dynamical Behaviour of an Infected Predator-Prey Model with Fear Effect Dipesh Barman 1 • Jyotirmoy Roy 1 • Shariful Alam 1 Received: 15 March 2020 / Accepted: 22 October 2020 Ó Shiraz University 2020 Abstract In this article, we have presented an infected predator-prey model with Holling type II functional response where the predator population is divided into two sub-classes, namely susceptible and infected due to disease. Here, we have assumed that the fear induced by susceptible and infected predators are of different levels. Well-posedness of the model system along with persistence criterion and the conditions of local stability of each equilibrium point have been established. Direction of Hopf bifurcation near the interior equilibrium point has been investigated. From model analysis, it is observed that fear induced by both susceptible and infected predators jointly determine dynamical complexity of the system. Fear induced by susceptible predators enhances the stable coexistence of the system whereas high amount of fear induced by infected predators destabilizes the system. It is also observed that the ratio of the birth rate of prey and the level of fear induced by both the susceptible and infected predators actively determine the topological behaviour of the system. We have performed comprehensive and meticulous numerical simulations to verify and validate the analytical findings of our model system, and finally, the article is ended up with a conclusion. Keywords Eco-epidemiological model  Fear factor  Periodic solution  Super-critical Hopf bifurcation 1 Introduction In the literature of mathematical ecology, predator-prey interactions have been received noticeably attention by many researchers. In this context, Lotka (1925) and Vol- terra (1926) were first to represent the prey-predator interactions in mathematical ecology. After that, several researchers have given intensive attention to functionally signify the predator-prey interactions considering the direct effect of predator species on prey species (Khan et al. 2004; Bera et al. 2015; Liu and Beretta 2006; Wo ¨rz- Busekros 1978; Chen et al. 2009; Chakraborty et al. 2012). In literature, functional response (Holling 1965; Kot 2001; Murray 1993; Pielou 1977) of predator species, i.e. how predator consumes the prey species becomes the centre of attraction. Several studies (Khan et al. 2004; Bera et al. 2015; Liu and Beretta 2006; Wo ¨rz-Busekros 1978; Chen et al. 2009; Chakraborty et al. 2012) have been done by considering different kind of functional responses to investigate the effect of direct predation. But, the outcomes of recent field experiments reveal the fact that the indirect impact of predator species on prey species sometimes become more crucial than the direct effect of predators in determining the dynamics of the predator-prey interactions (Cresswell 2011; Creel and Christianson 2008; Lima 1998, 2009). One can easily observe the effect of direct predation on the dynamics of an ecological system, but the indirect impact of predator species on the dynamical complexity of a predator-prey system has not been inves- tigated with prior attention. It is to be noted that the physiological behaviour of prey species may be changed due to the fear of being consumed by the predator species. As a result, they respond to discerned predation risk which compelled them to exhibit various anti-predator behaviours like new habitat selection, search for a secure area for food & Dipesh Barman dipesh.rs2018@math.iiests.ac.in Jyotirmoy Roy jroy.rs2015@math.iiests.ac.in Shariful Alam salam@math.iiests.ac.in 1 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India 123 Iran J Sci Technol Trans Sci https://doi.org/10.1007/s40995-020-01014-y