Abstract
An epidemic model with saturated incidence rate and vaccination is investigated. The model exhibits two equilibria namely disease-free and endemic equilibria. It is shown that if the basic reproduction number (R0) is less than unity, the disease-free equilibrium is locally asymptotically stable and in such case, the endemic equilibrium does not exist. Also, it is shown that if R0 > 1, the disease is persistent and the unique endemic equilibrium of the system with saturation incidence is locally asymptotically stable. Lyapunov function and Dulac’s criterion plus Poincare-Bendixson theorem are applied to prove the global stability of the disease-free and endemic equilibria respectively. The effect of vaccine in the model is critically looked into.
Highlights
Vaccinating susceptible against disease infections is an effective measure to control and prevent the spread of the infection
Lyapunov function and Dulac’s criterion plus Poincare-Bendixson theorem are applied to prove the global stability of the disease-free and endemic equilibria respectively
Some of the main findings of this study are: i) The model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity; ii) The model has a unique endemic equilibrium under certain conditions
Summary
Vaccinating susceptible against disease infections is an effective measure to control and prevent the spread of the infection. Adebimpe [7] investigated a SEIV epidemic model with saturated incidence rate that incorporates polynomial information on current and past states of the disease. He showed that if the basic reproduction number R0 1 , the Disease-Free Equilibrium (DFE) is locally asymptotically stable and by the use of Lyapunov function, DFE is globally asymptotically stable and in such a case, the Endemic Equilibrium (EE) is unstable. Adebimpe et al [8] investigated the global stability of a SEIR epidemic model with saturating incidence rate They identified a threshold R0 which determines the outcome of the disease.
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