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 τ , we assume that the infection probability is given by an exponential law with parameter γ , i.e. q ( τ ) = 1 - e - γ τ . 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 γ T ≪ 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 2 γ /[(2+ γ ) N ] and is therefore small for large N .