On stability and bifurcation of solutions of an SEIR epidemic model with vertical transmission

A four-dimensional SEIR epidemic model is considered. The stability of the equilibria is established. Hopf bifurcation and center manifold theories are applied for a reduced three-dimensional epidemic model. The boundedness, dissipativity, persistence, global stability , and Hopf-Andronov-Poincaré bifurcation for the four-dimensional epidemic model are studied. 1. Introduction. Many infectious diseases in nature transmit through both horizontal and vertical models. These include such human diseases as Rubella, Herpes Simplex, Hepatitis B, Chagas, and the most notorious AIDS (see [8, 9]). For human and animal diseases, horizontal transmission typically occurs through direct or indirect physical contact with hosts, or through a disease vector such as mosquitos, ticks, or other biting insects. Vertical transmission can be accomplished through transplacental transfer of disease agents. Li et al. [10] discussed vertical and horizontal models. In standard SIR compartmental models the vertical transmission can be incorporated by assuming that the fraction q of the offspring from the infectious I class is infectious at birth, and hence birth flux, qbI, enters the I class and the remaining birth b − qbI enters the susceptible S class. In this paper we study an SEIR model in which vertical transmission is incorporated based on the above assumption. The total host population is partitioned into susceptible , exposed, infectious, and recovered with densities denoted, respectively, by S(t), E(t), I(t), and R(t). The natural birth, and death rates are assumed to be identical and denoted by b. The horizontal transmission is assumed to take the form of direct contact between infectious and susceptible hosts. The incidence rate term H(I, S) is assumed to be differentiable, ∂H/∂I and ∂H/∂S are nonnegative and finite for all I b is the natural birth rate of the host population which is assumed to have a constant density 1. For the vertical transmission, we assume that a fraction p and a fraction q of the offspring from the exposed and infectious classes, respectively, are born into the exposed class E. Consequently, the birth flux into the exposed class is given by pbE + qbI and the birth flux into the susceptible class is given by b − pbE − qbI, naturally 0 ≤ p ≤ 1 and 0 ≤ q ≤ 1. The above assumptions lead to the following system of