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December  2016, 21(10): 3429-3440. doi: 10.3934/dcdsb.2016105

Long-time behavior of an SIR model with perturbed disease transmission coefficient

Nguyen Huu Du 1, and  Nguyen Thanh Dieu 2,

1. 

Faculty of Mathematics, Mechanics, and Informatics, University of Science-VNU, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
2. 

Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam

Received  December 2015 Revised  June 2016 Published  November 2016

In this paper, we consider a stochastic SIR model with the perturbed disease transmission coefficient. We determine the threshold $\lambda$ that is used to classify the extinction and permanence of the disease. Precisely, $\lambda<0$ implies that the disease-free $(\frac{\alpha}{\mu}, 0, 0)$ is globally asymptotic stable, i.e., the disease will disappear and the entire population will become susceptible individuals. If $\lambda>0$ the epidemic takes place. In this case, we derive that the Markov process $(S(t), I(t))$ has a unique invariant probability measure. We also characterize the support of a unique invariant probability measure and prove that the transition probability converges to this invariant measures in total variation norm. Our result is considered as an significant improvement over the results in [6,7,11,18].
Keywords: SIR model, stationary distribution, ergodicity., permanence, extinction.
Mathematics Subject Classification: 34C12, 60H10, 92D2.
Citation: Nguyen Huu Du, Nguyen Thanh Dieu. Long-time behavior of an SIR model with perturbed disease transmission coefficient. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3429-3440. doi: 10.3934/dcdsb.2016105
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show all references

References:
[1]

SIAM J. Appl. Dyn. Syst., 15 (2016), 1062-1084. doi: 10.1137/15M1043315.  Google Scholar

[2]

J. Appl. Probab., 53 (2016), 187-202, Available from: http://projecteuclid.org/euclid.jap/1457470568. doi: 10.1017/jpr.2015.18.  Google Scholar

[3]

SIAM J. Appl. Math., 71 (2011), 876-902. doi: 10.1137/10081856X.  Google Scholar

[4]

Z. Wahrsch. Verw. Gebiete, 30 (1974), 235-254; Corrections in 39 (1977), 81-84. doi: 10.1007/BF00533476.  Google Scholar

[5]

second edition, North-Holland Publishing Co., Amsterdam, 1989.  Google Scholar

[6]

Appl. Math. Model., 38 (2014), 5067-5079. doi: 10.1016/j.apm.2014.03.037.  Google Scholar

[7]

Stochastic Anal. Appl., 30 (2012), 755-773. doi: 10.1080/07362994.2012.684319.  Google Scholar

[8]

J. IFAC, 48 (2012), 121-131. doi: 10.1016/j.automatica.2011.09.044.  Google Scholar

[9]

Springer-Verlag Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar

[10]

Ann. Probab., 15 (1987), 690-707. doi: 10.1214/aop/1176992166.  Google Scholar

[11]

Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1873-1887. doi: 10.3934/dcdsb.2013.18.1873.  Google Scholar

[12]

Horwood Publishing Chichester, 1997.  Google Scholar

[13]

Adv. Appl. Prob., 25 (1993), 518-548. doi: 10.2307/1427522.  Google Scholar

[14]

In: Séminaire de probabilitiés XX, Lecture Notes in Mathematics, Springer, New York, 1204 (1986), 101-130. doi: 10.1007/BFb0075716.  Google Scholar

[15]

Springer-Verlag, Berlin Heidelberg, 2006.  Google Scholar

[16]

J. Dynam. Differential Equations, 27 (2015), 69-82. doi: 10.1007/s10884-014-9415-9.  Google Scholar

[17]

LCDS Report, No. 86-16, April 1986, Brown University, Providence. Available from: https://www.amazon.co.uk/existence-uniqueness-invariant-continuous-processes/dp/B000722C66 Google Scholar

[18]

Phys. A, 354 (2005), 111-126. doi: 10.1016/j.physa.2005.02.057.  Google Scholar

[19]

Math. Biosci. Eng., 11 (2014), 1003-1025. doi: 10.3934/mbe.2014.11.1003.  Google Scholar

[20]

Journal of Control Science and Engineering, 1 (2013), 13-22. Google Scholar

[21]

Appl. Math. Comput., 244 (2014), 118-131. doi: 10.1016/j.amc.2014.06.100.  Google Scholar

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