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

Article Excerpt
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.

Key words: infection model; loss system; recurrence; Laplace transform

1. Introduction. We consider a simple model for the evolution of a bacterial infection and its treatment. Let the population consist of s susceptible individuals. When one of them catches the disease, he or she is immediately administered antibiotics and/or is isolated and is cured after an (exponentially distributed) random amount of time. During the treatment time an individual cannot be infected again, and transmission to others is prevented by isolation or the effect of the antibiotics, so that the disease can be considered to be noncontagious. All healthy individuals are equally likely to be the next to catch the infection. We assume that the interarrival times of the occurrences of new bacteria are i.i.d, random variables. In our simple model, every bacterium infects one and only one individual. In this paper, we derive the distribution of the time until some individual is infected for the second time and thus has to be treated again.

The reinfection time is particularly important in situations where patients acquire immunity to certain antibiotics. The well-known phenomenon that bacteria can become resistant in the course of antibiotic treatment is often aggravated when patients do not take the prescribed amount or stop taking their medication too early. In some infections, e.g., those with one of the various types of campylobacter, such an immunity can already be the result of a single insufficient treatment. In these cases, reinfected patients need special attention.

The model can be described in...

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