In this paper, we construct and analyze a stochastic model for infectious disease spread in service systems. Our model considers a $M/M/1$ queue where a proportion, $p$, of the arriving population is infectious and the remaining population is susceptible. When an infectious person overlaps in the queue with a susceptible customer for more than $\ell$ units of time, then the susceptible person becomes infected. We provide both a personalized perspective of how an individual susceptible customer is infected and a community perspective of how one infectious customer can infect the others around them. From the personalized perspective, we derive the probability that a susceptible person becomes infected and the probability they are infected by at least $k$ different people. From the community perspective we compute the mean and higher moments of the number of susceptible customers that are infected by one infectious individual. Finally, we use stochastic simulation to verify our analytical results and show that our new formulas accurately describe the behavior our stochastic model of infection and to expand our model into the consideration of different arrival and service regimes.

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