To better describe the spread of a disease, we extend a discrete time stochastic SIR-type epidemic model of Tuckwell and Williams. We assume the dependence on time of the number of daily encounters and include a parameter to represent a possible quarantine of the infectious individuals. We provide an analytic description of this Markovian model and investigate its dynamics. Both a diffusion approximation and the basic reproduction number are derived. Through several simulations, we show how the evolution of a disease is affected by the distribution of the number of daily encounters and its dependence on time. Finally, we show how the appropriate choice of this parameter allows a suitable application of our model to two real diseases.