A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse

Georgescu, Paul; Zhang, Hong

doi:10.1016/j.amc.2013.02.044

Cited by 44 publications

(40 citation statements)

References 20 publications

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“…Substituting (9) and R * = 2 +k 1 +k 2 I * into (8b), we see that, for an endemic equilibrium E * , I * is a zero of F(I) on J ≜ ( 0,…”

Section: A Positively Invariant Set and Equilibriamentioning

confidence: 99%

“…(ii) If ℛ 0 > 1, then, besides E 0 , model (5) also has a unique endemic equilibrium E * = (S * , I * , R * ), where I * is the unique positive zero of F defined by (10), R * = 2 +k 1 +k 2 I * , and S * is given by (9).…”

Section: Theoremmentioning

confidence: 99%

“…Since then, a lot of SIRI epidemic models described by ordinary differential equations and functional differential equations have been proposed and analyzed. To name a few, see Feng et al, Georgescu and Zhang, Liu et al, van den Driessche et al, van den Driessche and Zou, Wang and Shu, Xu, and Yang et al Although nonlinear incidences were used in Liu et al (depending on S ), Xu (depending on I ), Feng et al, Georgescu and Zhang, and Wang and Shu (depending on both S and I but separably), the recovered individuals are always assumed to stay at the immune state unless relapse; that is, the recovered are immune for infection.…”

Section: Introductionmentioning

confidence: 99%

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Global dynamics of an epidemic model with relapse and nonlinear incidence

Chen

Zou

2018

Math Methods in App Sciences

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On the basis of a basic SIR epidemic model, we propose and study an epidemic model with nonlinear incidence. The model also incorporates many features of the recovered such as relapse and with/without immunity. A threshold dynamics is established, which is completely determined by the basic reproduction number. The global stability of the disease‐free equilibrium is proved by means of the fluctuation lemma. To prove the global stability of the endemic equilibrium, we need some novel techniques including the transformation of variables, the construction of a new type of Lyapunov functions, and the seeking of an appropriate positively invariant set of the model.

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“…Substituting (9) and R * = 2 +k 1 +k 2 I * into (8b), we see that, for an endemic equilibrium E * , I * is a zero of F(I) on J ≜ ( 0,…”

Section: A Positively Invariant Set and Equilibriamentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Global dynamics of an epidemic model with relapse and nonlinear incidence

Chen

Zou

2018

Math Methods in App Sciences

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“…The system was extended to include nonlinear incidence rates by Moreira and Wang ; they used the technique of Liénard's equation to establish conditions for the global stability of the unique endemic steady state (when it exists). Georgescu and Zhang developed a generalized model with a nonlinear incidence of infection and a relapse/recurrence period from a disease. They investigated its global stability.…”

Section: Introductionmentioning

confidence: 99%

Global stability properties of age‐dependent epidemic models with varying rates of recurrence

Vargas‐De‐León

2015

Math Methods in App Sciences

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Communicated by M. A. LachowiczThe purpose of this paper is to study the global stability properties of equilibria for age-dependent epidemiological models in presence of recurrence phenomenon. In these systems, the recurrence rate depends on asymptomatic-infectionage. The models are appropriate for human herpes virus (HSV-1 and HSV-2) and varicella-zoster virus. We derived explicit formulas for the basic reproductive number, which completely characterizes the global behaviour of solutions to the models: if the basic reproductive number is less than or equal to unity, the disease will die out; if the basic reproductive number is greater than unity, the disease will be persistent. Volterra-type Lyapunov functions are constructed to establish the global asymptotic stability of the infection-free and endemic steady states.

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“…Trajectories of system (3) with time delay h = 2, initial conditions (4) and(5), and parameters Λ = 20, µ = 0.4, β = 0.02, c = 0.1, δ = 0.006, and γ = 0.7, for which R 0 = 0.8406 < 1, converging to the disease free equilibrium E 0 . Trajectories of system (3) with time delay h = 2, initial conditions (4) and(5), and parameters Λ = 18, µ = 0.65, β = 0.2, c = 0.77, δ = 0.02, and γ = 0.75, for which R 0 = 2.2923 > 1, converging to the endemic equilibrium E * .…”

mentioning

confidence: 99%

Analysis of a SIRI Epidemic Model with Distributed Delay and Relapse

Elazzouzi

Alaoui

Tilioua³

et al. 2019

Stat., optim. inf. comput.

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We investigate the global behaviour of a SIRI epidemic model with distributed delay and relapse. From the theory of functional differential equations with delay, we prove that the solution of the system is unique, bounded, and positive, for all time. The basic reproduction number R 0 for the model is computed. By means of the direct Lyapunov method and LaSalle invariance principle, we prove that the disease free equilibrium is globally asymptotically stable when R 0 < 1. Moreover, we show that there is a unique endemic equilibrium, which is globally asymptotically stable, when R 0 > 1.

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A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse

Cited by 44 publications

References 20 publications

Global dynamics of an epidemic model with relapse and nonlinear incidence

Global dynamics of an epidemic model with relapse and nonlinear incidence

Global stability properties of age‐dependent epidemic models with varying rates of recurrence

Analysis of a SIRI Epidemic Model with Distributed Delay and Relapse

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