Asymptotic results for a multi-type contact birth–death process and related SIS epidemic

Exact results concerning the asymptotic speed of propagation of infection have recently been obtained for the multi-type SIS epidemic in continuous space when the contact distributions are assumed to be symmetric with the Laplace transforms finite for all entries. There is a link between the equations for this epidemic and the equations for a multi-type contact birth–death process. This enables methods developed for the epidemic to be used to obtain the asymptotic speed of translation for the contact birth–death process. Symmetry of the contact distributions is required but no existence constraint is placed on their Laplace transforms. The method for removing this constraint may also be used for the SIS epidemic. Results are given for both processes when the basic reproduction ratio is at most one.