THE DYNAMIC BEHAVIOR OF DETERMINISTIC AND STOCHASTIC DELAYED SIQS MODEL

Xiaobing Zhang; Haifeng Huo; Hong Xiang; Dungang Li

doi:10.11948/2018.1061

2018 Volume 8 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(4): 1061-1084

Next Article Previous Article

Xiaobing Zhang, Haifeng Huo, Hong Xiang, Dungang Li. THE DYNAMIC BEHAVIOR OF DETERMINISTIC AND STOCHASTIC DELAYED SIQS MODEL[J]. Journal of Applied Analysis & Computation, 2018, 8(4): 1061-1084. doi: 10.11948/2018.1061

Citation:

Xiaobing Zhang, Haifeng Huo, Hong Xiang, Dungang Li. THE DYNAMIC BEHAVIOR OF DETERMINISTIC AND STOCHASTIC DELAYED SIQS MODEL[J]. Journal of Applied Analysis & Computation, 2018, 8(4): 1061-1084. doi: 10.11948/2018.1061

THE DYNAMIC BEHAVIOR OF DETERMINISTIC AND STOCHASTIC DELAYED SIQS MODEL

1 College of Electrical and Information engineering, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China;
2 Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China

Fund Project:

Abstract

Abstract

In this paper, we present the deterministic and stochastic delayed SIQS epidemic models. For the deterministic model, the basic reproductive number R₀ is given. Moreover, when R₀<1, the disease-free equilibrium is globally asymptotical stable. When R₀>1 and additional conditions hold, the endemic equilibrium is globally asymptotical stable. For the stochastic model, a sharp threshold R₀ which determines the extinction or persistence in the mean of the disease is presented. Sufficient conditions for extinction and persistence in the mean of the epidemic are established. Numerical simulations are also conducted in the analytic results.
- Random perturbations /
- Itô's formula /
- the threshold /
- time delay
MSC: 34K45;37G15;92C45

FullText(HTML)

References

[1]	E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comput. Simulation., 1998, 45, 269-277. Google Scholar
[2]	E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 2002, 33, 1144-1165. Google Scholar
[3]	Y. Cai, Y. Kang, M. Banerjee and W. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differential Equations., 2015, 259, 7463-7502. Google Scholar
[4]	X. Chen, J. Cao and H. Jug, Stability analysis and estimation of domain of attraction for the endemic equilibrium of an SEIQ epidemic model, Nonlinear Dynamics., 2016, 1-11. Google Scholar
[5]	G. Chowell, C. Castillo-Chavez, P. Fenimore et al., Model parameters and outbreak control for SARS, EID., 2004, 10, 1258-1263. Google Scholar
[6]	T. Day, A. Park, N. Madras et al., When is quarantine a useful control strategy for emerging infectious diseases?, Amer. J. Epidemiol., 2006, 163, 479-485. Google Scholar
[7]	A. Dobay, G. Gall and D. Rankin, Renaissance model of an epidemic with quarantine, Journal of Theoretical Biology, 2013, 317(1), 348-358. Google Scholar
[8]	G. Gensini, M. Yacoub and A. Conti, The concept of quarantine in history:from plague to SARS, J. Infect., 2004, 49(4), 257-261. Google Scholar
[9]	A. Gray, D. Greenhalgh, L. Hu et al., A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 2011, 71, 876-902. Google Scholar
[10]	A. Gray, D. Greenhalgh and X. Mao, The SIS epidemic model with markovian switching, J. Math. Anal. Appl., 2012, 394, 496-516. Google Scholar
[11]	Z. Han and J. Zhao, Stochastic SIRS model under regime switching, Nonlinear Anal. Real World Appl., 2013, 14, 352-364. Google Scholar
[12]	H. Herbert, Z. Ma and S. Liao, Effects of quarantine in six endemic models for infectious diseases, Mathematical Biosciences, 2002, 180, 141-160. Google Scholar
[13]	Y. Hsieh and C. King, Impact of quarantine on the 2003 SARS outbreak:A retrospective modeling study, Journal of Theoretical Biology, 2007, 244(4), 729-736. Google Scholar
[14]	C. Ji and D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model., 2014, 38, 5067-5079. Google Scholar
[15]	C. Ji, D. Jiang and N. Shi, Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comput. Simulation., 2011, 45, 1747-1762. Google Scholar
[16]	D. Jiang, J. Yu, C. Ji and N. Shi, Asymptotic behavior of global positive solution to a stochastic SIR model, Mathematical and Computer Modelling, 2011, 54, 221-232. Google Scholar
[17]	R. Kao and M. Roberts, Quarantine-based disease control in domesticated animal herds, Appl. Math. Lett., 1998, 4, 115-120. Google Scholar
[18]	A. Lahrouz, L. Omari and D. Kiouach, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statistics and Probability Letters, 2013, 83, 960-968. Google Scholar
[19]	A. Lahrouz and A. Settati, Asymptotic properties of switching diffusion epidemic model with varying population size, Applied Mathematics Computation, 2013, 219(24), 11134-11148. Google Scholar
[20]	Y. Lin, D. Jiang and S. Wang, Stationary distribution of a stochastic SIS epidemic model with vaccination, Phys. A, 2014, 394, 187-197. Google Scholar
[21]	Q. Liu, The threshold of a stochastic Susceptible-Infective epidemic model under regime switching, Nonlinear Analysis Hybrid Systems, 2016, 21, 49-58. Google Scholar
[22]	Q. Liu and Q. Chen, Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Phys. A, 2015, 428, 140-153. Google Scholar
[23]	Q. Liu and Q. Chen, The threshold of a stochastic delayed SIR epidemic model with temporary immunity, Phys. A, 2016, 450, 115-125. Google Scholar
[24]	X. Liu, X. Chen and Y. Takeuchi, Dynamics of an SIQS epidemic model with transport-related infection and exit-entry screenings, Journal of Theoretical Biology, 2011, 285(1), 25-35. Google Scholar
[25]	Q. Lu, Stability of SIRS system with random perturbations, Phys. A, 2009, 388, 3677-3686. Google Scholar
[26]	X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997. Google Scholar
[27]	X. Mao, Stationary distribution of stochastic population systems, Systems and Control Letters, 2011, 60, 398-405. Google Scholar
[28]	M. Safi and A. Gumel, Global asymptotic dynamics of a model for quarantine and isolation, Discrete Contin. Dyn. Syst. Ser. B., 2010, 14, 209-231. Google Scholar
[29]	M. Safi and A. Gumel, Qualitative study of a quarantine/isolation model with multiple disease stages, Appl. Math. Comput., 2011, 218(5), 1941-1961. Google Scholar
[30]	M. Safi and A. Gumel, Dynamics of a model with quarantine-adjusted incidence and quarantine of susceptible individuals, J. Math. Anal. Appl., 2013, 399, 565-575. Google Scholar
[31]	G. Sahu and J. Dhar, Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with preexisting immunity, J. Math. Anal. Appl., 2015, 421, 1651-1672. Google Scholar
[32]	H. Sato, R. Y. H. Nakada, S. M. S. Imoto and M. Kami, When should we intervene to control the 2009 influenza A(H1N1) pandemic?, Rapid Communications, Euro. Surveill., 2010, 15(1), 9-12. Google Scholar
[33]	Z. Teng and L. Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Phys. A, 2016, 451, 507-518. Google Scholar
[34]	X. Wang, T. Zhao and X. Qin, Model of epidemic control based on quarantine and message delivery, Phys. A, 2016, 458, 168-178. Google Scholar
[35]	F. Wei and F. Chen, Stochastic permanence of an SIQS epidemic model with saturated incidence and independent random perturbations, Commun Nonlinear Sci Numer Simulat, 2016, 453, 99-107. Google Scholar
[36]	H. Xiang, Y. L. Tang and H. F. Huo, A viral model with intracellular delay and humoral immunity, Bulletin of the Malaysian Mathematical Sciences Society, 2016, doi:10.1007/s40840-016-0326-2. Google Scholar
[37]	X. Yan and Y. Zou, Optimal and sub-optimal quarantine and isolation control in sars epidemics, Math. Comput. Modelling., 2008, 47, 235-245. Google Scholar
[38]	Q. Yang, D. Jiang, N. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, Journal of Mathematical Analysis and Applications, 2012, 388, 248-271. Google Scholar
[39]	J. Yu, D. Jiang and N. Shi, Global stability of two-group SIR model with random perturbation, Journal of Mathematical Analysis and Applications, 2009, 360, 235-244. Google Scholar
[40]	C. Yuan, D. Jiang, D. O'Regan and R. Agarwal, Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation, Commun Nonlinear Sci Numer Simulat, 2012, 17, 2501-2516. Google Scholar
[41]	X. B. Zhang, H. F. Huo, H. Xiang and X. Y. Meng, Dynamics of the deterministic and stochastic SIQS epidemic model with non-linear incidence, Applied Mathematics Computation, 2014, 243, 546-558. Google Scholar
[42]	Y. Zhao and D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence, Appl. Math. Lett., 2014, 34, 90-93. Google Scholar
[43]	Y. Zhao and D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 2014, 243, 18-27. Google Scholar
[44]	Y. Zhao, D. Jiang and X. Mao, The threshold of a stochastic SIRS epidemic model in a population with varying size, Discrete Contin. Dyn. Syst. Ser. B., 2015, 20(2), 1289-1307. Google Scholar