Persistence of vertically transmitted parasite strains which protect against more virulent horizontally transmitted strains

The question whether a vertically and a horizontally transmitted parasite strain can coexist under complete cross-protection is investigated in a host-parasite model with susceptibles and infectives only. It is shown that coexistence is possible even if the vertically transmitted strain would go extinct on its own provided that it is considerably less virulent than the horizontally transmitted strain. While the vertical transmitted strain is without benefit to the host as such, it protects the host against the more harmful horizontally transmitted strain. The coexistence is shown in the form of uniform strong persistence of the host and both parasite strains.