Stability on coupling SIR epidemic model with vaccination

Liu, Helong; Xu, Houbao; Yu, Jingyuan; Zhu, Guangtian

doi:10.1155/jam.2005.301

Cited by 12 publications

(9 citation statements)

References 14 publications

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“…A strategy to control infectious diseases is vaccination [14,15]. One can investigate under what conditions a given agent can invade a (partially) vaccinated population, i.e., how large a fraction of the population has to be kept vaccinated in order to prevent the agent from establishing.…”

Section: Introductionmentioning

confidence: 99%

Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate

Zhou

Cui

2010

Nonlinear Dyn

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In this paper, an SEIV epidemic model with vaccination and nonlinear incidence rate is formulated. The analysis of the model is presented in terms of the basic reproduction number R 0 . It is shown that the model has multiple equilibria and using the center manifold theory, the model exhibits the phenomenon of backward bifurcation where a stable diseasefree equilibrium coexists with a stable endemic equilibrium for a certain defined range of R 0 . We also discuss the global stability of the endemic equilibrium by using a generalization of the Poincaré-Bendixson criterion. Numerical simulations are presented to illustrate the results.

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Section: Introductionmentioning

confidence: 99%

Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate

Zhou

Cui

2010

Nonlinear Dyn

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“…Apply the same way to (17), (18), and (19) in [T − , T], therefore, we see that (21) holds for all t ∈ [T − 2 , T] by a finite number of iterations, Equation 16 can be rewritten as follows:…”

Section: Some Priori Estimates Of the Susceptible Infected And Recomentioning

confidence: 99%

“…To better study these control measures, there are a few research results of control strategy. [17][18][19][20][21][22][23] For example, Kar and Batabyal 20 considered an SIR epidemic model using vaccination as control and analyzed stability. Lashari 22 used optimal control to study an SIR epidemic model with a saturated treatment.…”

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confidence: 99%

Optimal strategy of vaccination and treatment in an SIRS model with Markovian switching

Zhang

2018

Math Methods in App Sciences

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Medical treatment and vaccination decisions are often sequential and uncertain. Markov decision process is an appropriate means to model and handle such stochastic dynamic decisions. This paper studies the near-optimality of a stochastic SIRS epidemic model that incorporates vaccination and saturated treatment with regime switching. The stochastic model takes white noises and color noise into account. We first prove some priori estimates of the susceptible, infected, and recovered populations. Moreover, we establish some sufficient and necessary conditions of the near-optimality by Pontryagin stochastic maximum principle. Our results show that the two kinds of environmental noises have great impacts on the infectious diseases. Finally, we illustrate our conclusions through numerical simulations. 767 768 MU AND ZHANG Strategies for prevention and control of infectious diseases include vaccination, treatment, quarantine, isolation, and prophylaxis. Vaccination and treatment control are two effective ways to control diseases. For most infectious diseases, treatments exist that can cure or lessen the effects of the diseases and improve the life of the patients. To better study these control measures, there are a few research results of control strategy. [17][18][19][20][21][22][23] For example, Kar and Batabyal 20 considered an SIR epidemic model using vaccination as control and analyzed stability. Lashari 22 used optimal control to study an SIR epidemic model with a saturated treatment. However, these systems are the deterministic epidemic model. Moreover, the exact solution of the state equation and the adjoint equation are difficult to be found and the precise control is difficult to be given. Hence, it is more practical to study the near-optimal control problem of the epidemic model. This paper is aimed at considering a stochastic SIRS epidemic model 6 with Markov chains, which includes vaccination and saturated treatment control. The objective function is formulated as an optimal control problem with two control variables to minimize the susceptible and infected individuals as well as the cost of implementing the two controls. Moreover, we will establish the sufficient and necessary conditions for the near-optimal control problems. The novelty of this study are as follows:(i) The Markov chains, vaccination control, and saturated treatment are taken into account the SIRS epidemic model simultaneously, which will obtain a novel model. (ii) The near-optimal control problem of a stochastic SIRS model is discussed.The layout of this paper is as follows: In Section 2, some basic concepts are presented and the SIRS model is formulated. The sufficient and necessary conditions of the near-optimal control for the stochastic SIRS model are obtained in Sections 3 and 4, respectively. In Section 6, we make simulations to confirm our results. Section 7 is given some conclusions to finish the paper. BASIC CONCEPTS AND THE MODEL FORMULATIONWe first introduce the following definitions and notations in Section 2.1 before we formu...

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“…Therefore (16) implies that dU 2 dt (15) ≤ 0, equality is valid if and only if S = S * , y ¼ u ¼ z. Lasalle invariance principle indicates that the system (14), the solution of the limit set is contained in {(S, E 1 , E 2 , I) [ R + 4 |S = S * } the largest invariant set change focus, it is clear by the first type of (14) and S = S * , we can obtain I = I * , using the second type of (14) and S = S * and I = I * , it is seen when time t 1, E 1 E * 1 , use the third type of (14), and E 1 E * 1 , we can obtain when the t 1,…”

Section: Constructing a Dynamic Model For Tbmentioning

confidence: 99%

“…The number of asymptomatic, called latent; J indicates the number of confirmed cases, namely, the number of being diagnosed; S 1 and S 2 , are not vulnerable to infection and vulnerable to infection are not infected two people, respectively, S 2 is less susceptible to infected people, whose value is p; q is no symptoms yet having the potential proportion of contagious; k is the proportion of patients becoming latent, l for the isolation rate; a is the proportion for patients to be diagnosed and being diagnosed; the m is divided into two parts m 1 and m 2 ; m 1 expressed by the patient recovered as immunisation, the proportion, and m 2 is the proportion of immune persons recovered by the diagnosis; other parameters are the same as the previous model introduced and will not be detailed [15,16]. A new theoretical model is constructed.…”

Section: New Theoretical Model Is Constructed and Computer Simulationmentioning

confidence: 99%

Analysis of a bio-dynamic model via Lyapunov principle and small-world network for tuberculosis

Chung

2012

IET Syst. Biol.

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The study will apply Lyapunov principle to construct a dynamic model for tuberculosis (TB). The Lyapunov principle is commonly used to examine and determine the stability of a dynamic system. To simulate the transmissions of vector-borne diseases and discuss the related health policies effects on vector-borne diseases, the authors combine the multi-agent-based system, social network and compartmental model to develop an epidemic simulation model. In the identity level, the authors use the multi-agent-based system and the mirror identity concept to describe identities with social network features such as daily visits, long-distance movement, high degree of clustering, low degree of separation and local clustering. The research will analyse the complex dynamic mathematic model of TB epidemic and determine its stability property by using the popular Matlab/Simulink software and relative software packages. Facing the current TB epidemic situation, the development of TB and its developing trend through constructing a dynamic bio-mathematical system model of TB is investigated. After simulating the development of epidemic situation with the solution of the SMIR epidemic model, the authors will come up with a good scheme to control epidemic situation to analyse the parameter values of a model that influence epidemic situation evolved. The authors will try to find the quarantining parameters that are the most important factors to control epidemic situation. The SMIR epidemic model and the results via numerical analysis may offer effective prevention with reference to controlling epidemic situation of TB.

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Stability on coupling SIR epidemic model with vaccination

Cited by 12 publications

References 14 publications

Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate

Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate

Optimal strategy of vaccination and treatment in an SIRS model with Markovian switching

Analysis of a bio-dynamic model via Lyapunov principle and small-world network for tuberculosis

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