Global dynamics of an SEIRI epidemiological model with time delay

Xu, Rui

doi:10.1016/j.amc.2014.01.100

Cited by 19 publications

(17 citation statements)

References 11 publications

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“…[6], we proved that the endemic equilibrium is globally asymptotically stable. The obtained result extended the Theorem (3.1) in [38]. But the Lyapunov function proposed in [38] is not valid for our model (1).…”

Section: Proof Consider the Following Lyapunov Functionalsupporting

confidence: 49%

“…The obtained result extended the Theorem (3.1) in [38]. But the Lyapunov function proposed in [38] is not valid for our model (1).…”

Section: Proof Consider the Following Lyapunov Functionalsupporting

confidence: 49%

“…In this paper we extend the global stability results presented in [6,38] to an SIRI epidemic model with a distributed time delay and a general incidence function. It is established that the basic reproduction number R 0 depends on the distribution of the latent period, the incidence function and the relapse rate and it is shown that the global stability can be attained under suitable sufficient conditions on the incidence function and the threshold parameter R 0 .…”

Section: Introductionmentioning

confidence: 74%

“…The basic reproduction number for this model is identified and some global results are obtained and the disease is shown to be uniformly persistent with the infective population size either approaching or oscillating about the endemic level. The effect of the discrete time delay in the epidemic models with a relapse rate has been taken into account in some recent studies ( see, for example [6,38,39]).…”

Section: Introductionmentioning

confidence: 99%

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Stability Analysis of an SIRI Epidemic Model with Distributed Latent Period

Bernoussi¹,

Kaddar²,

Asserda³

2016

JAAM

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In this paper, we investigate the global asymptotic stability of the possible equilibria for an SIRI epidemic model with a distributed latent period and a general incidence function. By using the method of Lyapunov functions and the LaSalle invariance principle, we show that the global asymptotic stability is completely determined by the basic reproduction number R0. The originality of this work is to have a basic reproduction number R0 which depends on the distribution of the latent period. Finally, we use a nonstandard discretization method of Mickens to illustrate numerically our theoretical results.

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Section: Proof Consider the Following Lyapunov Functionalsupporting

confidence: 49%

“…The obtained result extended the Theorem (3.1) in [38]. But the Lyapunov function proposed in [38] is not valid for our model (1).…”

Section: Proof Consider the Following Lyapunov Functionalsupporting

confidence: 49%

Section: Introductionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability Analysis of an SIRI Epidemic Model with Distributed Latent Period

Bernoussi¹,

Kaddar²,

Asserda³

2016

JAAM

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show abstract

“…In this paper we extend the global stability results presented in [31] to a delayed SIRI epidemic model (system (1)) with a general nonlinear incidence function. It is shown that global stability can be attained under suitable monotonicity conditions and it is established that the basic reproduction number 0 is a threshold parameter for the stability of a delayed SIRI model.…”

Section: Introductionmentioning

confidence: 75%

Global Stability of a Delayed SIRI Epidemic Model with Nonlinear Incidence

Bernoussi

Kaddar²,

Asserda

2014

International Journal of Engineering Mathematics

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In this paper we propose the global dynamics of an SIRI epidemic model with latency and a general nonlinear incidence function. The model is based on the susceptible-infective-recovered (SIR) compartmental structure with relapse (SIRI). Sufficient conditions for the global stability of equilibria (the disease-free equilibrium and the endemic equilibrium) are obtained by means of Lyapunov-LaSalle theorem. Also some numerical simulations are given to illustrate this result.

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Multigroup deterministic and stochastic SEIRI epidemic models with nonlinear incidence rates and distributed delays: A stability analysis

Zhang

Xia

Georgescu

2017

Math Methods in App Sciences

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In this paper, we investigate the dynamics of a multigroup disease propagation model with distributed delays and nonlinear incidence rates, which accounts for the relapse of recovered individuals. The main concern is the stability of the equilibria, sufficient conditions for global stability being obtained by applying Lyapunov-LaSalle invariance principle and using Lyapunov functionals, which are constructed using their single-group counterparts. The situation in which the deterministic model is subject to perturbations of white noise type is also investigated from a stability viewpoint. KEYWORDSdelay differential equations, disease propagation, disease relapse, Lyapunov stability, multigroup model, nonlinear incidence 6254 ;40:6254-6275. ZHANG ET AL. 6255onset of symptoms, and an individual may be infective prior to the onset of symptoms. Prolonged latency between exposure and infectiousness is a characteristic of tuberculosis, for instance, latent compartments being incorporated even in the earliest models of tuberculosis transmission. 1 Also, several models have included both fast and slow pathways from susceptible to actively infected, with a proportion of exposed susceptibles progressing immediately to active infection or several sequential latent compartments to simulate the increased risk of progression to active disease in the years immediately following initial infection. 2,3 As noted in Lloyd, 4 an accurate description of the distribution of the latent and infectious periods is an important step towards the construction of an appropriate model. Assuming that the latency is exponentially distributed, the corresponding disease propagation model is represented by a system of nonlinear ODEs. This assumption, however, is equivalent to the fact that the chance of recovery within a given time interval is constant, regardless of the time since infection. This is sometimes unrealistic, as observed through the statistical studies of the transmission dynamics of measles in small communities (see, for instance, Lloyd 4 or Krylova and Earn 5 ), the chance of recovery being initially small, but increasing over time and corresponding to a distribution of the infectious period, which is less dispersed than an exponential and more centered around the mean. If, however, a general distribution is assumed for the latent period, then one obtains a delayed integro-differential system. 6,7 The most commonly used distributions of the latent period are the Gamma distribution, used, for instance, to model the spread of avian influenza (H7N7) in chicken 8 and the log-normal distribution, used, for instance, to model the spread of Ebola. 9 Since the host population can be divided geographically into communities, cities, and countries, or epidemiologically, to incorporate factors such as modes of transmission, contact patterns, and genetic susceptibility, a heterogeneous host population can subsequently be divided into several homogeneous groups, the within-group dynamics and the mixing patterns being then modeled separately. One o...

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Global dynamics of an SEIRI epidemiological model with time delay

Cited by 19 publications

References 11 publications

Stability Analysis of an SIRI Epidemic Model with Distributed Latent Period

Stability Analysis of an SIRI Epidemic Model with Distributed Latent Period

Global Stability of a Delayed SIRI Epidemic Model with Nonlinear Incidence

Multigroup deterministic and stochastic SEIRI epidemic models with nonlinear incidence rates and distributed delays: A stability analysis

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