Dynamics of SEIS epidemic models with varying population size

In this paper, SEIS epidemic models with varying population size are considered. Firstly, we consider the case when births of population are throughout the year. A threshold sigma is identified, which determines the outcome of disease, that is, when sigma < 1, the disease dies out; whereas when s > 1, the disease persists and the unique endemic equilibrium is globally asymptotically stable; when sigma = 1, bifurcation occurs and leads to '' the change of stability ''. Two other thresholds sigma' and (sigma) over bar are also identified, which determine the dynamics of epidemic model with varying population size, when the disease dies out or it is endemic. Secondly, we consider the other case, birth pulse. The population density is increased by an amount B(N) N at the discrete time n tau, where n is any non-negative integer and tau is a positive constant, B(N) is density-dependent birth rate. By applying the corresponding stroboscopic map, we obtain the existence of infection-free periodic solution with period tau. Lastly, through numerical simulations, we show the dynamic complexities of SEIS epidemic models with varying population size, there is a sequence of bifurcations, leading to chaotic strange attractors. Non-unique attractors also appear, which implies that the dynamics of SEIS epidemic models with varying population size can be very complex.