Recurrences in an infection model: A medical application of GI/M/s loss systems

We consider a simple model for the evolution of a bacterial infection and its treatment for a population of s susceptible individuals. When an individual catches the disease, he or she is treated (e.g., by antibiotics) and is cured after a random amount of time. While being treated, an individual cannot be infected and will not transmit the disease to others. Every healthy individual is equally likely to be the next to catch the infection. The interarrival times of the occurrences of new bacteria are assumed to be i.i.d. random variables. We derive the distribution of the time S until some individual is infected for the second time and thus has to be treated again. The Laplace transform of S is given explicitly and in a more convenient recursive form. The model can be formulated as a GI/M/s loss system, and S can be viewed as the first time some service station starts working for its second customer. In the M/M/s case, the results simplify.