In this paper we study a predator-prey model with logistic growth in the prey population, where a disease spreads among the prey according to an susceptible-infected-susceptible (SIS) epidemic model. The predators do not consume infected prey. After a review of the literature we formulate the basic mathematical model. For simplicity, we work initially with a model involving the fractions of prey susceptible and infected and then translate the results back to the model with absolute numbers. Both local and global stability results are examined. For the model working with absolute numbers, we find six possible equilibria and three important threshold values determining the behaviour of the system. There is always a unique locally stable equilibrium. We make conjectures concerning the global behaviour of the system. Next the effect of predator removal on the ecoepidemiological system is examined. The penultimate section describes numerical simulations using realistic parameter values for a real-life situation. This is humans predating on fish (Atlantic cod) infected by bacterial fin rot. The simulations confirm our conjectures. A discussion concludes the paper.