Bifurcation of a Delayed SEIS Epidemic Model with a Changing Delitescence and Nonlinear Incidence Rate

Abstract

This paper is concerned with a delayed SEIS (Susceptible-Exposed-Infectious-Susceptible) epidemic model with a changing delitescence and nonlinear incidence rate. First of all, local stability of the endemic equilibrium and the existence of a Hopf bifurcation are studied by choosing the time delay as the bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are determined based on the normal form theory and the center manifold theorem. At last, numerical simulations are carried out to illustrate the obtained theoretical results.Hindawi

Hopf Bifurcation of a Delayed Ecoepidemic Model with Ratio-Dependent Transmission Rate

“…In the present paper, we propose a delayed ecoepidemic model by incorporating the time delay due to the gestation of the predator in the model studied in [31]. Compared with the work in [31], we mainly consider the effect of the time delay on the stability of system (2). The model investigated in our paper is more general since the time required for the gestation of the predator and the results we obtained are suitable complements to the literature [31].…”

“…Theorem 1. Suppose that the conditions ( 1 )-( 3 ) hold for system (2). The positive equilibrium * ( * , * , * ) is locally asymptotically stable when ∈ [0, 0 ) and a Hopf bifurcation occurs at the positive equilibrium * ( * , * , * ) when = 0 .…”

“…Let = 0 + , ∈ ; then = 0 is the Hopf bifurcation value of system (2). Rescaling the time delay → ( / ), then system (2) can be transformed into a functional differential equation in = ([−1, 0], 3 ) aṡ…”

“…Theorem 2. For system (2), If 2 > 0 ( 2 < 0), then the Hopf bifurcation is supercritical (subcritical). If 2 < 0 ( 2 > 0), then the bifurcating periodic solutions are stable (unstable).…”

“…In recent years, many dynamical models characterizing the propagation of infectious disease [1][2][3], spread of computer viruses [4][5][6], and dynamics of some other systems [7][8][9][10] are studied by scholars. Ecoepidemiological research deals with the study of the spread of diseases among interacting populations, where the epidemic and demographic aspects are merged within one model.…”