Phase portraits of SIR epidemic models with vertical transmissions and linear treatment

Susceptible-infective-removed epidemic models with horizontal and vertical transmissions and linear treatment rates are investigated. All the ranges of the parameters involved in the models for the infection-free equilibrium and the epidemic equilibrium to be positive are found. Like the previous results on the models without vertical transmissions or the linear treatments, we study stability of these equilibria. The novelty is that we justify that these positive equilibria are stable focuses or stable nodes under suitable conditions on the parameters. These results provide more detailed descriptions of behaviours of the epidemic diseases near the equilibria. Our results will exhibit the effect of the vertical transmissions and the linear treatment rates on the epidemic models. Some simulations results are provided to understand the phase portraits near the equilibria.