The derivation of certain pandemic bounds

Exact results have previously been obtained concerning the spread of infection in continuous space contact models describing a class of multitype epidemics. The Pandemic Theorem gave a lower bound for the spatial final size. A discrete space model is considered. A simpler, more direct proof based on an infinite matrix formulation of the final size equations is used to obtain the pandemic result for this model. An upper bound is obtained, which is valid for both continuous and discrete space models. This enables a limiting result to be obtained for the spatial final size when the amount of initial infection tends to zero.