Random walks on a complete graph: A model for infection

We introduce a new model for the infection of one or more subjects by a single agent, and calculate the probability of infection after a fixed length of time. We model the agent and subjects as random walkers on a complete graph of N sites, jumping with equal rates from site to site. When one of the walkers is at the same site as the agent for a length of time tau, we assume that the infection probability is given by an exponential law with parameter gamma, i.e. q(tau) = 1 - e(-gammatau). We introduce the boundary condition that all walkers return to their initial site ('home') at the end of a fixed period T. We also assume that the incubation period is longer than T, so that there is no immediate propagation of the infection. In this model, we find that for short periods T, i.e. such that gammaT << 1 and T << 1, the infection probability is remarkably small and behaves like T-3. On the other hand, for large T, the probability tends to 1 (as might be expected) exponentially. However, the dominant exponential rate is given approximately by 2gamma/[(2 + gamma)N] and is therefore small for large N.