Abstract
We study the dynamics of a SEIR epidemic model with nonlinear treatment function, that takes into account the limited availability of resources in community. Under some conditions we prove the existence of two possible equilibria: the disease-free equilibrium and the endemic equilibrium. Using Lyapunov's method and Li's geometrical approach, We also show that the reproduction number R0 is a threshold parameter: the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than unity and the unique endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than this critical value. In the end, we give some concluding remarks concerning the role of treatment on the epidemic propagation.
Highlights
Infectious diseases remain to be one of the main sources of deaths for the human beings
The modeling of this effect may be taken into account by introducing a treatment function in an epidemiological model
We extend the model from [3, 4] to include general nonlinear incidence function f(S, I), and we establish the complete global dynamics
Summary
Infectious diseases remain to be one of the main sources of deaths for the human beings. The authors investigated the existence of equilibrium and proved the global asymptotical stability of the endemic equilibrium by using Dulac’s criteria and Poincare-Bendixson Theorem They obtained that the model (1.9) undergoes a Hopf bifurcation. In (2015, [3]), Dubey et al considered the incidence function as Beddington-DeAngelis type and the treatment rate as Holling type II (saturated) They showed that the disease-free equilibrium is locally asymptotically stable when reproduction number is less than one and obtained the global stability of the endemic equilibrium using Lyapunov function. They investigated the existence of Hopf bifurcation by using Andronov-Hopf bifurcation theorem.
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