scholarly journals Global Stability of Two-Group Epidemic Models with Distributed Delays and Random Perturbation

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaoming Fan ◽  
Zhigang Wang ◽  
Xuelian Xu
Keyword(s):  
Lyapunov Function ◽  
Stability Condition ◽  
Epidemic Model ◽  
Random Perturbation ◽  
Endemic Equilibrium ◽  
Distributed Delays ◽  
Random Fluctuation ◽  
Asymptotically Stable ◽  
Sufficient Stability Condition ◽  
Sufficient Stability

We discuss a two-group SEIR epidemic model with distributed delays, incorporating random fluctuation around the endemic equilibrium. Our research shows that the endemic equilibrium of the model with distributed delays and random perturbation is stochastically asymptotically stable in the large. In addition, a sufficient stability condition is obtained by constructing suitable Lyapunov function.

scholarly journals Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 328 ◽  
Author(s):  
Yanli Ma ◽  
Jia-Bao Liu ◽  
Haixia Li
Keyword(s):  
Lyapunov Function ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

scholarly journals Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yakui Xue ◽  
Tiantian Li
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

Stability condition for a single-server retrial queue

1993 ◽  
Vol 25 (03) ◽  
pp. 690-701 ◽  
Author(s):  
Huei-Mei Liang ◽  
V. G. Kulkarni
Keyword(s):  
Stability Condition ◽  
Service Time ◽  
Retrial Queue ◽  
Arrival Rate ◽  
Retrial Queues ◽  
Single Server ◽  
Sufficient Stability Condition ◽  
Sufficient Stability ◽  
The Mean ◽  
The Stability

A single-server retrial queue consists of a primary queue, an orbit and a single server. Assume the primary queue capacity is 1 and the orbit capacity is infinite. Customers can arrive at the primary queue either from outside the system or from the orbit. If the server is busy, the arriving customer joins the orbit and conducts a retrial later. Otherwise, he receives service and leaves the system. We investigate the stability condition for a single-server retrial queue. Let λ be the arrival rate and 1/μ be the mean service time. It has been proved that λ / μ &lt; 1 is a sufficient stability condition for the M/G /1/1 retrial queue with exponential retrial times. We give a counterexample to show that this stability condition is not valid for general single-server retrial queues. Next we show that λ /μ &lt; 1 is a sufficient stability condition for the stability of a single-server retrial queue when the interarrival times and retrial times are finite mixtures of Erlangs.

A VECTOR-HOST EPIDEMIC MODEL WITH HOST MIGRATION

2011 ◽  
Vol 04 (01) ◽  
pp. 93-108
Author(s):  
QINGKAI KONG ◽  
ZHIPENG QIU ◽  
YUN ZOU
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Uniform Persistence ◽  
Asymptotically Stable ◽  
Host Disease ◽  
Disease Free

The host migration is one of the important elements that cause the worldwide diffusion and outbreak of many vector-host diseases. In this paper, we formulate a patchy model to investigate the effect of host migration between two patches on the spread of a vector-host disease. The results of the paper show that the reproduction number R0 is a threshold value that determines the uniform persistence and extinction of the disease. If the reproduction number R0 < 1 the disease free equilibrium (DFE) is locally asymptotically stable. If the reproduction number R0 > 1 then the DFE is unstable and the system is uniformly persistent. It is also shown that a unique endemic equilibrium, which exists when R0 > 1, is locally stable if both regions are identical.

The Control of Leontief Model on Industry Manufacturing Process

2011 ◽  
Vol 187 ◽  
pp. 287-290
Author(s):  
Yong Liang Cui
Keyword(s):  
Linear System ◽  
Discrete Time ◽  
Stability Condition ◽  
Manufacturing Process ◽  
Input Output ◽  
Sufficient Stability Condition ◽  
Sufficient Stability ◽  
Output System ◽  
Leontief Model ◽  
The Stability

The classic Leontief model on industry manufacturing process is investigated. A kind of discrete-time singular dynamic input-output model of industry manufacturing process based on the classic Leontief Model is provided and the stability of this kind of model is researched. By the new mathematic method, the singular dynamic input-output system will not be converted into the general linear system. Finally, a sufficient stability condition under which the discrete-time singular Extended Leontief Model is admissible is proved.

STABILITY OF SVIR SYSTEM WITH RANDOM PERTURBATIONS

2012 ◽  
Vol 05 (04) ◽  
pp. 1250025 ◽  
Author(s):  
CHAO DENG ◽  
HONGJUN GAO
Keyword(s):  
Stability Analysis ◽  
Equilibrium State ◽  
Transmission Rate ◽  
Random Perturbation ◽  
Epidemic Models ◽  
Random Fluctuation ◽  
Random Perturbations ◽  
Asymptotically Stable ◽  
Vaccination Strategies ◽  
Lyapunov Stability Analysis

In this paper, the SVIR epidemic models with continuous vaccination strategies investigated by Liu, Takeuchi and Iwamo, [SVIR epidemic models with vaccination strategies, J. Theor. Biol.253 (2008) 1–11], allowing random fluctuation around the endemic equilibrium and the transmission rate β are analyzed. The equilibrium state of the model with random perturbation is locally asymptotically stable as shown by a Lyapunov stability analysis.

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