## Dynamical behavior of a new epidemiological model.(English)Zbl 1406.92634

Summary: A new epidemiological model is introduced with nonlinear incidence, in which the infected disease may lose infectiousness and then evolves to a chronic noninfectious disease when the infected disease has not been cured for a certain time $$\tau$$. The existence, uniqueness, and stability of the disease-free equilibrium and endemic equilibrium are discussed. The basic reproductive number $$R_0$$ is given. The model is studied in two cases: with and without time delay. For the model without time delay, the disease-free equilibrium is globally asymptotically stable provided that $$R_0\leq 1$$; if $$R_0> 1$$, then there exists a unique endemic equilibrium, and it is globally asymptotically stable. For the model with time delay, a sufficient condition is given to ensure that the disease-free equilibrium is locally asymptotically stable. Hopf bifurcation in endemic equilibrium with respect to the time $$\tau$$ is also addressed.

### MSC:

 92D30 Epidemiology 92C60 Medical epidemiology
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