Analysis of a vaccine model with cross-immunity: when can two competing infectious strains coexist?

We analyse here the vaccine model with cross-immunity proposed by Porco and Blower [1]. Porco and Blower [1] show that vaccination can shift the competitive balance in favour of a strain that, without vaccination, would be out-competed and that vaccination can also promote coexistence of different strains, something that normally is not expected [2]. Their results have been mainly obtained through numerical simulations, so that the conditions under which a shift in competitive balance or coexistence occurs have not been fully established. We give a rather complete description of its behaviour, at least in terms of equilibria. We find the exact conditions under which vaccination may lead to a shift in competitive balance and show that, under these conditions, there always exist a range of vaccination rates under which a coexistence equilibrium exists. We also find that a coexistence equilibrium exists (and is unstable) in a 'bi-stability' region, where both monomorphic equilibria are stable. This fact has been rarely observed in models of competition between pathogen strains.