Investigating the dynamics of Lotka$-$Volterra model with disease in the prey and predator species

‎In this paper, a  predator$-$prey model  with logistic growth rate in the prey population was proposed.  It included an SIS infection in the prey and predator population.  The stability of the positive equilibrium point, the existence of Hopf and transcortical  bifurcation with parameter $a$ were investigated, where $a$ was regarded as  predation rate. It was found that when the parameter $a$ passed through a critical value,  stability changed and Hopf bifurcation occurred.  Biologically, the population  is positive and bounded. In the present article,  it was also shown that the model was bounded and that it had the positive solution.  Moreover, the current researchers came to the conclusion that although  the disease was present in the system, none of the species would be extinct. In other words, the system was persistent. Important thresholds, $R_{0}, R_{1}$ and $R_{2}$, were identified in the study. This theoretical study indicated that under certain conditions of $R_{0}, R_{1}$ and $R_{2}$,  the disease remained in the system or disappeared.