Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate

Kyrychko, Yuliya N.; Blyuss, Konstantin B.

doi:10.1016/j.nonrwa.2004.10.001

Cited by 146 publications

(98 citation statements)

References 9 publications

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“…It has been suggested by several authors that the disease transmission process may have a nonlinear incidence rate (see Derrick and van den Driessche, 2003;Korobeinikov and Maini, 2005;Korobeinikov, 2006Korobeinikov, , 2007Kyrychko andBlyuss, 2005, andLiu et al, 1987 for survey of the possible reasons for nonlinearity of the incidence rate). To study the impact of the non-linearity, Korobeinikov and Maini (2005) considered a variety of models with the incidence rate of the form g(I )h(S) and constructed Lyapunov functions that enable them to establish global properties for some of SIR and SEIR models.…”

Section: Introductionmentioning

confidence: 99%

Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate

et al. 2010

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In this paper, based on SIR and SEIR epidemic models with a general nonlinear incidence rate, we incorporate time delays into the ordinary differential equation models. In particular, we consider two delay differential equation models in which delays are caused (i) by the latency of the infection in a vector, and (ii) by the latent period in an infected host. By constructing suitable Lyapunov functionals and using the Lyapunov-LaSalle invariance principle, we prove the global stability of the endemic equilibrium and the disease-free equilibrium for time delays of any length in each model. Our results show that the global properties of equilibria also only depend on the basic reproductive number and that the latent period in a vector does not affect the stability, but the latent period in an infected host plays a positive role to control disease development.

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Section: Introductionmentioning

confidence: 99%

Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate

et al. 2010

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“…So many epidemics were controlled. Generally speaking, when epidemics spread, there are many kinds of delays, which include immunity period delay [7][8][9], infectious period delay, incubation period delay. In [10], Enatsu et al studied stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates, at the same time, they proved disease-free equilibrium was globally asymptotically stable and endemic equilibrium was permanent under certain conditions.…”

Section: Introductionmentioning

confidence: 99%

Global Stability Analysis of a Delayed SEIQR Epidemic Model with Quarantine and Latent

Li¹,

Xue²

2013

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In this paper, we study a kind of the delayed SEIQR infectious disease model with the quarantine and latent, and get the threshold value which determines the global dynamics and the outcome of the disease. The model has a disease-free equilibrium which is unstable when the basic reproduction number is greater than unity. At the same time, it has a unique endemic equilibrium when the basic reproduction number is greater than unity. According to the mathematical dynamics analysis, we show that disease-free equilibrium and endemic equilibrium are locally asymptotically stable by using Hurwitz criterion and they are globally asymptotically stable by using suitable Lyapunov functions for any . Besides, the SEIQR model with nonlinear incidence rate is studied, and the 0  that the basic reproduction number is a unity can be found out. Finally, numerical simulations are performed to illustrate and verify the conclusions that will be useful for us to control the spread of infectious diseases. Meanwhile, the 1 , k 3 k will effect changing trends of , S , E , I , Q R in system (1), which is obvious in simulations. Here, we take 3 k as an example to explain that.

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“…To solve the system is the autile decomposition method and for this we use the references of [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. In order to describe the effects of disease immunity temporal delays are often incorporated in such models [15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Stability of a Delayed SIQRS Model with Temporary Immunity

Chahrazed¹,

Lazhar²

2013

APM

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This paper addresses a time-delayed SIQRS model with a linear incidence rate. Immunity gained by experiencing the disease is temporary; whenever infected, the disease individuals will return to the susceptible class after a fixed period of time. First, the local and global stabilities of the infection-free equilibrium are analyzed, respectively. Second, the endemic equilibrium is formulated in terms of the incidence rate, and locally asymptotic stability. Finally we use the adomian decomposition method is applied to the system epidemiologic. This method yields an analytical solution in terms of convergent infinite power series.

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Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate

Cited by 146 publications

References 9 publications

Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate

Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate

Global Stability Analysis of a Delayed SEIQR Epidemic Model with Quarantine and Latent

Stability of a Delayed SIQRS Model with Temporary Immunity

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