Possible velocities for a simple epidemic

Abstract

It is here shown that for a deterministic simple epidemic in which the spatial distribution of the contacts made by an infectious individual is negative exponential, propagation of the epidemic as a travelling wave is only possible for velocities at or above a certain minimal velocity; and that there exists exactly one waveform for each such velocity. The method employed is to reduce the equation of propagation to a three-dimensional autonomous differential equation, and attack this mainly using topological considerations. This result is similar to those obtained by Kolmogoroff et al. (1937) for the advance of an advantageous gene, and by Kendall (1965) for the deterministic “general” epidemic, but these were achieved using the considerable simplification of approximating the spatial dependence by a diffusion term in the equation of propagation. Taken together with Mollison (1972), in which it was shown that when the spatial distribution of contacts (V) is more spread than negative exponential (∫e ks dV(s) divergent for all k > 0), no finite bound can be set on the velocity of propagation: the present work suggests that the diffusion-term approximations may be reasonable if, as well as only if, V is less spread than negative exponential.