Horizontal versus vertical transmission of parasites in a stochastic spatial model

A number of pathogens may be transmitted from parent to child at or before birth (vertically) or from one individual to another by contact (horizontally). A natural deterministic and non-spatial model, introduced by Lipsitch et al. [Proc. Roy. Soc. London Ser. B 260 (1995) 3211 shows that an epidemic is possible if the vertical transmission or the horizontal transmission is high enough. In contrast, we introduce a stochastic spatial model that shows that, on a particular graph, if the vertical transmission is not high enough, then the infected individuals disappear even for very high horizontal transmission. This illustrates the fact that introducing space may greatly change the qualitative behavior of a model.