On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules

Sen, Manuel De la; Alonso‐Quesada, S.; Ibeas, Asier

doi:10.1016/j.amc.2015.08.099

Cited by 49 publications

(29 citation statements)

References 31 publications

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“…A number of epidemic models with time delay have been developed in the literature to gain insights into the effect of time delay on the dynamic behavior of the model (see, e.g., [15][16][17][18][19][20][21][22][23][24][25][26][27]). Li et al [15] investigated the existence of a positive solution and local stability for the steady state of an age-structured SEIR epidemic model.…”

Section: Introductionmentioning

confidence: 99%

Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period

Sirijampa

Chinviriyasit

2018

Adv Differ Equ

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In this paper, we analyze a delayed SEIR epidemic model in which the latent and infected states are infective. The model has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number R 0 , is less than or equal to unity. We investigate the effect of the time delay on the stability of endemic equilibrium when R 0 > 1. We give criteria that ensure that endemic equilibrium is asymptotically stable for all time delays and a Hopf bifurcation occurs as time delay exceeds the critical value. We give formulae for the direction of Hopf bifurcations and the stability of bifurcated periodic solutions by applying the normal form theory and the center manifold reduction for functional differential equations. Numerical simulations are presented to illustrate the analytical results.

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Section: Introductionmentioning

confidence: 99%

Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period

Sirijampa

Chinviriyasit

2018

Adv Differ Equ

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“…The total population is divided into four classes: susceptible class S, latent class E, infectious class I, and removed class R. Here, an individual may undergo a latent phase and then becomes infectious; meanwhile, a removed individual can relapse into the infectious class due to the reactivation of the infectious diseases. There are many studies on multigroup epidemic models (see other works [1][2][3][4][5][6] and the references therein) and SEIR epidemic models (see other works [7][8][9][10][11] and the references therein). Wang and Shu 12 established a class of multigroup SEIR models with latency and relapse, where there are two delays in latency and relapse, respectively, and they proved that the global stability of equilibria is completely determined by the basic reproduction number by using Lyapunov functionals.…”

Section: Introductionmentioning

confidence: 99%

A multigroup SEIR epidemic model with age‐dependent latency and relapse

Liu

Feng

2018

Math Methods in App Sciences

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Different multigroup epidemic models have been proposed, but few models include the latent class that becomes infectious at different rates and the fact that the removed class may relapse into an infectious class at different rates.In this paper, a multigroup SEIR epidemic model is constructed to study the transmission dynamics of infectious diseases with age-dependent latency and relapse. The model is realistic for some infectious diseases, such as TB and herpes virus. The sharp threshold condition, which corresponds to the well-known basic reproduction number ℜ 0 , is derived, and it determines the global stability of each equilibrium. In particular, if ℜ 0 < 1, then the disease-free equilibrium is globally asymptotically stable, whereas if ℜ 0 > 1, the endemic equilibrium exists uniquely and is globally asymptotically stable. We utilize appropriate Lyapunov functionals, graph-theoretical results, and the LaSalle's invariance principle to prove these results. Two specific examples and their corresponding numerical simulations are provided to explain the obtained results. KEYWORDS age-dependent latency, age-dependent relapse, global stability, multigroup epidemic model, threshold condition 6814

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“…General description of epidemic models and some mathematical analysis on them is given in some classical books. See, for instance, [1][2][3] and for more recent models, see, for instance, [4][5][6][7][8][9][10][11] and references therein. The positivity of the solution is investigated in a number of works.…”

Section: Introductionmentioning

confidence: 99%

“…The positivity of the solution is investigated in a number of works. See, for instance, [6][7][8][9]12] and some references therein. The use of nonlinear incidence rates in the models is also investigated in a number of papers.…”

Section: Introductionmentioning

confidence: 99%

“…See, for instance, [9,[15][16][17] 2 Discrete Dynamics in Nature and Society to give some of them. Also, certain robustness studies of stability and positivity under deviations of the equilibrium points due to Wiener noise are performed in [9]. The stability properties and the convergence of the solutions to equilibrium states are a major analysis tool in most of the works.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On a New Epidemic Model with Asymptomatic and Dead-Infective Subpopulations with Feedback Controls Useful for Ebola Disease

Sen

Ibeas

Alonso‐Quesada

et al. 2017

Discrete Dynamics in Nature and Society

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This paper studies the nonnegativity and local and global stability properties of the solutions of a newly proposed SEIADR model which incorporates asymptomatic and dead-infective subpopulations into the standard SEIR model and, in parallel, it incorporates feedback vaccination plus a constant term on the susceptible and feedback antiviral treatment controls on the symptomatic infectious subpopulation. A third control action of impulsive type (or "culling") consists of the periodic retirement of all or a fraction of the lying corpses which can become infective in certain diseases, for instance, the Ebola infection. The three controls are allowed to be eventually time varying and contain a total of four design control gains. The local stability analysis around both the disease-free and endemic equilibrium points is performed by the investigation of the eigenvalues of the corresponding Jacobian matrices. The global stability is formally discussed by using tools of qualitative theory of differential equations by using Gauss-Stokes and Bendixson theorems so that neither Lyapunov equation candidates nor the explicit solutions are used. It is proved that stability holds as a parallel property to positivity and that disease-free and the endemic equilibrium states cannot be simultaneously either stable or unstable. The periodic limit solution trajectories and equilibrium points are analyzed in a combined fashion in the sense that the endemic periodic solutions become, in particular, equilibrium points if the control gains converge to constant values and the control gain for culling the infective corpses is asymptotically zeroed.Hindawi

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On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules

Cited by 49 publications

References 31 publications

Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period

Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period

A multigroup SEIR epidemic model with age‐dependent latency and relapse

On a New Epidemic Model with Asymptomatic and Dead-Infective Subpopulations with Feedback Controls Useful for Ebola Disease

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