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Global dynamics of delay epidemic models with nonlinear incidence rate and relapse. (English) Zbl 1208.34125
Summary: A mathematical model for a disease with a general exposed distribution, the possibility of relapse and nonlinear incidence rate is proposed. By the method of Lyapunov functionals, it is shown that the disease dies out if $$\operatorname{Re}_{0}\leq 1$$ and that the disease becomes endemic if $$\operatorname{Re}_{0}>1$$. Applications are also made to the special case with a discrete delay and the result confirms that the endemic equilibrium is globally asymptotically stable.

##### MSC:
 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K20 Stability theory of functional-differential equations 92D30 Epidemiology
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