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.