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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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