Dynamics of an ultra-discrete SIR epidemic model with time delay

Sekiguchi, Masaki; Ishiwata, Emiko; Nakata, Yukihiko

doi:10.3934/mbe.2018029

Cited by 5 publications

(3 citation statements)

References 29 publications

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“…Today, the serious epidemics, such as SARS and H1N1, are still threatening the life of people continually. Plenty of mathematical models have been proposed to analyze the spread and the control of these diseases [1][2][3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Stability and Hopf Bifurcation Analysis of an Epidemic Model with Time Delay

Zhang

et al. 2021

Computational and Mathematical Methods in Medicine

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Epidemic models are normally used to describe the spread of infectious diseases. In this paper, we will discuss an epidemic model with time delay. Firstly, the existence of the positive fixed point is proven; and then, the stability and Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equations. Thirdly, the theory of normal form and manifold is used to drive an explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions. Finally, some simulation results are carried out to validate our theoretic analysis.

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Section: Introductionmentioning

confidence: 99%

Stability and Hopf Bifurcation Analysis of an Epidemic Model with Time Delay

Zhang

et al. 2021

Computational and Mathematical Methods in Medicine

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“…Generally speaking, the solutions of problems ( 62) and (70) are different. In contrast to solving problem (62), for solving (70), one can apply the well-known methods for estimating the parameters of nonlinear regression and analyzing their accuracy [8,11,19,21], etc.…”

Section: Solutions For Model With Latent Period With Different Proces...mentioning

confidence: 99%

Dynamic models of epidemiology in discrete time taking into account processes with lag

Knopov

Korkhin

2023

Int. J. Dynam. Control

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Models of epidemic dynamics in the form of systems of differential equations of the type SIR and its generalizations, for example SEIR and SIRS, have become widespread in epidemiology. Their coefficients are averages of some epidemic indicators, for example the time when a person is contagious. Statistical data about spreading of the epidemic are known in discrete periods of time, for example twenty-four hours. Therefore, adjustment of the differential equations system under such data comes across cleanly calculable difficulties. They can be avoided, initially to build a model in discrete time as a system of difference equations. Such initial consideration allows, as it shown in the article, to get a general model. On its basis, the models of development of epidemics can be built taking into account their specific. There is another way to obtain a model in discrete time. It consists in discretizing the original model in continuous time. The model obtained in this way is inaccurate, and it is only an approximation to the original one, which makes it possible to simplify calculations and increase the stability of the calculation process. This model is inappropriate, for example, for fitting the model to statistical data. Another argument against the use of systems of differential equations is that the coefficients of such a model may not be the same during a day. For example, the number of contacts of an infected person with susceptible people during a day differs from that at night. However, there is no such difference for daily data. It is possible depending on the day of the week. KeywordsDifference equation • Lag • Random value • Stability • Identification 1 Model of Kermack and McKendrick and their system of differential equations SIRWe will use the set denotations according to that all population of some territory (region, country) in quantity N is divided by categories: S (susceptible, receptive are people that were not infected or lost immunity after infecting), I (infected, infected and being contagious) and R (recovered). Let S(t) is a number of susceptible people in the moment of time t, where t is a continuous value. The values I (t) and B A. S. Korkhin

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“…In recent years, great attention has been paid to the study of SIR type models, which have been formulated to describe the propagation and evolution of some human or animal diseases. In such models, the population is subdivided into compartments or classes, in particular the compartment of susceptible (S), the compartment of infective (I), and the compartment of recovered individuals (R) [18,19]. When recovered individuals may experience a relapse of the disease, due to an incomplete treatment or due to the reactivation of a latent infection, and then re-enter the class of infective, a SIRI model is more convenient to model the dynamic of the diseases.…”

Section: Introductionmentioning

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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Dynamics of an ultra-discrete SIR epidemic model with time delay

Cited by 5 publications

References 29 publications

Stability and Hopf Bifurcation Analysis of an Epidemic Model with Time Delay

Stability and Hopf Bifurcation Analysis of an Epidemic Model with Time Delay

Dynamic models of epidemiology in discrete time taking into account processes with lag

Analysis of a SIRI Epidemic Model with Distributed Delay and Relapse

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