Lyapunov functions for SIR and SIRS epidemic models

O’Regan, Suzanne M.; Kelly, Thomas C.; Korobeinikov, Andrei; O’Callaghan, Michael; Pokrovskiĭ, A. V.

doi:10.1016/j.aml.2009.11.014

Cited by 91 publications

(68 citation statements)

References 14 publications

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“…Lyapunov functions for classical SIR and SIS epidemiological models were introduced in [24], and the global stability of the endemic equilibrium states of the models were thereby established. A new Lyapunov function for a variety of SIR and SIRS models in epidemiology was introduced in [25]. Traveling wave trains in generalized two-species predator-prey models and two-component reaction-diffusion equations were considered in [26], and the stability of the fixed points of the traveling wave ordinary differential equations in the usual "spatial" variable was analyzed.…”

Section: Introductionmentioning

confidence: 99%

Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates

Harkó

Lobo

Mak

2014

Applied Mathematics and Computation

430

376

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In this paper, the exact analytical solution of the Susceptible-Infected-Recovered (SIR) epidemic model is obtained in a parametric form. By using the exact solution we investigate some explicit models corresponding to fixed values of the parameters, and show that the numerical solution reproduces exactly the analytical solution. We also show that the generalization of the SIR model, including births and deaths, described by a nonlinear system of differential equations, can be reduced to an Abel type equation. The reduction of the complex SIR model with vital dynamics to an Abel type equation can greatly simplify the analysis of its properties. The general solution of the Abel equation is obtained by using a perturbative approach, in a power series form, and it is shown that the general solution of the SIR model with vital dynamics can be represented in an exact parametric form.

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Section: Introductionmentioning

confidence: 99%

Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates

Harkó

Lobo

Mak

2014

Applied Mathematics and Computation

430

376

View full text Add to dashboard Cite

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“…Then there is an endemic equilibrium * for system (2) but not the endemic equilibrium * for system (3), as there is no endemic equilibrium in system (3). Similarly, we also expect to find out whether or not the solution goes around * .…”

Section: Asymptotic Behavior Around the Endemic Equilibrium Of The Dementioning

confidence: 99%

“…Our main purpose is to investigate how the solution of system (3) spirals around the disease-free equilibrium and endemic equilibrium of the corresponding deterministic system under some conditions. We first show the existence and uniqueness of a global positive solution of model (3). Then the main results will be seen in Sections 3 and 4.…”

Section: Introductionmentioning

confidence: 99%

“…Since system (3) is constructed by adding stochastic perturbation in a deterministic system (2), it seems reasonable to investigate whether there are similar properties as in system (2). But there is neither a disease-free equilibrium 0 nor an endemic equilibrium * for system (3). Hence, in order to show the stability to some extent, we discuss the behavior around 0 and * , respectively.…”

Section: Introductionmentioning

confidence: 99%

“…There have been a large number of works on the dynamics of epidemic models described by ordinary differential equations [1][2][3][4][5][6][7][8][9] and the references cited therein. In most of the literatures, the incubation period is often to be ignored usually.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a Stochastic SEIS Model with Treatment Rate of Latent Population

Gao

Dai

Zhang

et al. 2014

Abstract and Applied Analysis

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The asymptotic dynamics of a stochastic SEIS epidemic model with treatment rate of latent population is investigated. First, we show that the system provides a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: if 0 , which is called the basic reproduction number of the corresponding deterministic model, is not more than unity, the solution of the model is oscillating around the disease-free equilibrium of the corresponding deterministic system, whereas if 0 is larger than unity, we show how the solution spirals around the endemic equilibrium of deterministic system under certain parametric restrictions. Finally, numerical simulations are carried out to support our theoretical findings.

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First integrals and exact solutions of the SIRI and tuberculosis models

Naz

Mahomed

Naeem

2016

Math Methods in App Sciences

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Communicated by J. BanasiakThe first integrals and exact solutions of mathematical models of epidemiology: a susceptible-infected-recovered-infected (SIRI) model and a tuberculosis model with demographic growth are analyzed. These models are represented by systems of first-order nonlinear ordinary differential equations, and this system is replaced by one which contains a secondorder ordinary differential equation. The partial Lagrangian approach is then utilized to derive the first integrals of these models. Several cases arise. Then, we utilize the derived first integrals to construct exact solutions for the models under investigation and determine new solutions. The dynamic properties of these models are studied too.

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Lyapunov functions for SIR and SIRS epidemic models

Cited by 91 publications

References 14 publications

Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates

Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates

On a Stochastic SEIS Model with Treatment Rate of Latent Population

First integrals and exact solutions of the SIRI and tuberculosis models

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