Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects

•We study SI models with Allee effects and different modes of disease transmissions.•We identify sufficient conditions of disease-free and endemic disease patterns.•The SI density-dependent system can support rich dynamics such as limit cycles.•The SI frequency-dependent model can only support equilibrium dynamics.•High levels of asymmetry and Allee effects can generate diffusive instability.

A general SI (Susceptible-Infected) epidemic system of host–parasite interactions operating under Allee effects, horizontal and/or vertical transmission, and where infected individuals experience pathogen-induced reductions in reproductive ability, is introduced. The initial focus of this study is on the analyses of the dynamics of density-dependent and frequency-dependent effects on SI models (SI-DD and SI-FD). The analyses identify conditions involving horizontal and vertical transmitted reproductive numbers, namely those used to characterize and contrast SI-FD and SI-DD dynamics. Conditions that lead to disease-driven extinction, or disease-free dynamics, or susceptible-free dynamics, or endemic disease patterns are identified. The SI-DD system supports richer dynamics including limit cycles while the SI-FD model only supports equilibrium dynamics. SI models under “small” horizontal transmission rates may result in disease-free dynamics. SI models under with and inefficient reproductive infectious class may lead to disease-driven extinction scenarios. The SI-DD model supports stable periodic solutions that emerge from an unstable equilibrium provided that either the Allee threshold and/or the disease transmission rate is large; or when the disease has limited influence on the infectives growth rate; and/or when disease-induced mortality is low. Host-parasite systems where diffusion or migration of local populations manage to destabilize them are examples of what is known as diffusive instability. The exploration of SI-dynamics in the presence of dispersal brings up the question of whether or not diffusive instability is a possible outcome. Here, we briefly look at such possibility within two-patch coupled SI-DD and SI-FD systems. It is shown that relative high levels of asymmetry, two modes of transmission, frequency dependence, and Allee effects are capable of supporting diffusive instability.