For an SI type endemic model with one host and two parasite strains, we study the
stability of the endemic coexistence equilibrium, where the host and both parasite strains
are present. Our model, which is a system of three ordinary differential equations,
assumes complete cross-protection between the parasite strains and reduced fertility and
increased mortality of infected hosts. It also assumes that one parasite strain is
exclusively vertically transmitted and cannot persists just by itself. We give several
sufficient conditions for the equilibrium to be locally asymptotically stable. One of them
is that the horizontal transmission is of density-dependent (mass-action) type. If the
horizontal transmission is of frequency-dependent (standard) type, we show that, under
certain conditions, the equilibrium can be unstable and undamped oscillations can occur.
We support and extend our analytical results by numerical simulations and by
two-dimensional plots of stability regions for various pairs of parameters.