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Dynamics of SEIS epidemic models with varying population size. (English) Zbl 1145.34028
SEIS models with varying population size are considered. After rescaling, the main system has the following form \left\{ \begin{aligned} \dot{s} = & \gamma i-(\lambda-\alpha)si,\\ \dot{e} = & bi+\alpha ei+\lambda si-\varepsilon e,\\ \dot{i} = & \varepsilon e-(\alpha+\gamma+b)i+\alpha i^2, \end{aligned} \right. subject to $$s+e+i=1$$. Here $$s,e,i$$ denote the proportions of susceptible, exposed, and infected individuals in a population respectively, $$b$$ is the natural birth rate of the population, $$\alpha$$ is the disease-related death rate, $$\gamma$$ is the recovery rate of the infected, $$\lambda$$ is the contact rate between the infected and other individuals, and $$\varepsilon$$ is the rate at which the exposed become infected. Letting $$\sigma=\lambda/(\alpha+\gamma)$$, the authors show that if $$\sigma<1$$, the disease-free equilibrium $$(1,0,0)$$ of the system is globally asymptotically stable, while if $$\sigma>1$$, $$(1,0,0)$$ is unstable and the system is uniformly persistent, meaning that the disease persists. If $$\sigma=1$$, bifurcation occurs and leads to “the change of stability”. The authors also consider the system equipped with birth pulse and explore the dynamic complexity of SEIS epidemic models with varying population size through numerical simulation.

##### MSC:
 34C60 Qualitative investigation and simulation of ordinary differential equation models 34D23 Global stability of solutions to ordinary differential equations 37N25 Dynamical systems in biology 92D30 Epidemiology
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