On the Lyapunov stability for SIRS epidemic models with general nonlinear incidence rate

2010 ◽  
Vol 217 (8) ◽  
pp. 4010-4016 ◽  
Author(s):  
Bruno Buonomo ◽  
Salvatore Rionero
Keyword(s):  
Incidence Rate ◽  
Lyapunov Stability ◽  
Epidemic Models ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence

scholarly journals Global Behaviors of a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Lei Wang ◽  
Zhidong Teng ◽  
Long Zhang
Keyword(s):  
Incidence Rate ◽  
Endemic Equilibrium ◽  
Epidemic Models ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
General Incidence Rate ◽  
Special Cases ◽  
Disease Free

We study a class of discrete SIRS epidemic models with nonlinear incidence rateF(S)G(I)and disease-induced mortality. By using analytic techniques and constructing discrete Lyapunov functions, the global stability of disease-free equilibrium and endemic equilibrium is obtained. That is, if basic reproduction numberℛ0<1, then the disease-free equilibrium is globally asymptotically stable, and ifℛ0>1, then the model has a unique endemic equilibrium and when some additional conditions hold the endemic equilibrium also is globally asymptotically stable. By using the theory of persistence in dynamical systems, we further obtain that only whenℛ0>1, the disease in the model is permanent. Some special cases ofF(S)G(I)are discussed. Particularly, whenF(S)G(I)=βSI/(1+λI), it is obtained that the endemic equilibrium is globally asymptotically stable if and only ifℛ0>1. Furthermore, the numerical simulations show that for general incidence rateF(S)G(I)the endemic equilibrium may be globally asymptotically stable only asℛ0>1.

scholarly journals Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate

2010 ◽  
Vol 72 (5) ◽  
pp. 1192-1207 ◽  
Author(s):  
Gang Huang ◽  
Yasuhiro Takeuchi ◽  
Wanbiao Ma ◽  
Daijun Wei
Keyword(s):  
Global Stability ◽  
Incidence Rate ◽  
Epidemic Models ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence

scholarly journals Lyapunov Functions for a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate and Varying Population Sizes

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Ying Wang ◽  
Zhidong Teng ◽  
Mehbuba Rehim
Keyword(s):  
Incidence Rate ◽  
Lyapunov Functions ◽  
Epidemic Models ◽  
Inductive Method ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
All Solutions ◽  
Population Sizes ◽  
Sufficient And Necessary Conditions ◽  
Varying Population

We investigate the dynamical behaviors of a class of discrete SIRS epidemic models with nonlinear incidence rate and varying population sizes. The model is required to possess different death rates for the susceptible, infectious, recovered, and constant recruitment into the susceptible class, infectious class, and recovered class, respectively. By using the inductive method, the positivity and boundedness of all solutions are obtained. Furthermore, by constructing new discrete type Lyapunov functions, the sufficient and necessary conditions on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium are established.

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