A vertically transmitted epidemic model with two state-dependent pulse controls

Vertical transmission refers to the process in which a mother transmits bacteria or viruses to her offspring through childbirth, and this phenomenon takes place commonly in nature. This paper formulates an SIR epidemic model where the impact of vertical transmission and two state-dependent pulse controls are both taken into consideration. Using the Poincare´map, an analogue of Poincare´ criterion and the method of related qualitative analysis, the existence and the stability of a positive order-1 or order-2 periodic solution for the epidemic model are proved. Furthermore, phase diagrams are demonstrated by means of numerical simulations, illustrating the feasibility and correctness of our main results. It can be further implied that the epidemic can be controlled to a certain extent, with vertical transmission reduced and timely state-dependent pulse controls carried out.