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.

A stochastic mathematical model of two different infectious epidemic under vertical transmission.

In this study, considering the effect of environment perturbation which is usually embodied by the alteration of contact infection rate, we formulate a stochastic epidemic mathematical model in which two different kinds of infectious diseases that spread simultaneously through both horizontal and vertical transmission are described. To indicate our model is well-posed and of biological significance, we prove the existence and uniqueness of positive solution at the beginning. By constructing suitable Lyapunov functions (which can be used to prove the stability of a certain fixed point in a dynamical system or autonomous differential equation) and applying Itô's formula as well as Chebyshev's inequality, we also establish the sufficient conditions for stochastic ultimate boundedness.