##
**Dynamic behavior of a stochastic SIQS epidemic model with Lévy jumps.**
*(English)*
Zbl 1398.37096

Summary: In this paper, we propose a stochastic SIQS epidemic model with Lévy jumps and investigate sufficient conditions of the extinction and persistence of the disease. Then, we analyze the asymptotic behavior of the solution of the model around the endemic equilibrium of the corresponding deterministic model. We find that Lévy jumps can suppress the disease outbreak. Numerical simulations are carried out and approve our results.

### MSC:

37N25 | Dynamical systems in biology |

37H10 | Generation, random and stochastic difference and differential equations |

92D25 | Population dynamics (general) |

34F05 | Ordinary differential equations and systems with randomness |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

PDF
BibTeX
XML
Cite

\textit{X.-B. Zhang} et al., Nonlinear Dyn. 93, No. 3, 1481--1493 (2018; Zbl 1398.37096)

Full Text:
DOI

### References:

[1] | Yang, Q; Jiang, D; Shi, N; Ji, C, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388, 248-271, (2012) · Zbl 1231.92058 |

[2] | Lin, Y; Jiang, D; Wang, S, Stationary distribution of a stochastic SIS epidemic model with vaccination, Phys. A, 394, 187-197, (2014) · Zbl 1395.34064 |

[3] | Jiang, D; Yu, J; Ji, C; Shi, N, Asymptotic behavior of global positive solution to a stochastic SIR model, Math. Comput. Model., 54, 221-232, (2011) · Zbl 1225.60114 |

[4] | Zhang, XB; Huo, HF; Xiang, H; Meng, XY, Dynamics of the deterministic and stochastic SIQS epidemic model with non-linear incidence, Appl. Math. Comput., 243, 546-558, (2014) · Zbl 1335.92107 |

[5] | Meng, X; Zhao, S; Feng, T; Zhang, T, Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 433, 227-242, (2016) · Zbl 1354.92089 |

[6] | Huo, HF; Cui, FF; Xiang, H, Dynamics of an saits alcoholism model on unweighted and weighted networks, Phys. A Stat. Mech. Appl., 496, 249-262, (2018) |

[7] | Meng, XY; Qin, NN; Huo, HF, Dynamics analysis of a predator-prey system with harvesting prey and disease in prey species, J. Biol. Dyn., 12, 342-374, (2018) |

[8] | Zhao, W; Li, J; Zhang, T; Meng, X, Persistence and ergodicity of plant disease model with Markov conversion and impulsive toxicant input, Commun. Nonlinear Sci. Numer. Simul., 48, 70-84, (2017) |

[9] | Rifhat, R; Wang, L; Teng, Z, Dynamics for a class of stochastic SIS epidemic models with nonlinear incidence and periodic coefficients, Phys. A, 481, 176-190, (2017) |

[10] | Gray, A; Greenhalgh, D; Hu, L; Mao, X; Pan, J, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71, 876-902, (2011) · Zbl 1263.34068 |

[11] | Teng, Z; Wang, L, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Phys. A, 451, 507-518, (2016) · Zbl 1400.92542 |

[12] | Liu, Q; Chen, Q, Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Phys. A, 428, 140-153, (2015) · Zbl 1400.92515 |

[13] | Wei, F; Chen, F, Stochastic permanence of an SIQS epidemic model with saturated incidence and independent random perturbations, Commun. Nonlinear Sci. Numer. Simul., 453, 99-107, (2016) · Zbl 1400.92555 |

[14] | Zhang, XB; Huo, HF; Xiang, H; Shi, Q; Li, D, The threshold of a stochastic SIQS epidemic model, Phys. A, 482, 362-374, (2017) |

[15] | Herbert, H; Ma, Z; Liao, S, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180, 141-160, (2002) · Zbl 1019.92030 |

[16] | Cai, Y; Kang, Y; Banerjee, M; Wang, W, A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differ. Equ., 259, 7463-7502, (2015) · Zbl 1330.35464 |

[17] | Arqub, OAbu; El-Ajou, Ahmad, Solution of the fractional epidemic model by homotopy analysis method, J. King Saud Univ. Sci., 25, 73-81, (2013) |

[18] | Ji, C; Jiang, D, Threshold behaviour of a stochastic SIR model, Appl. Math. Model., 38, 5067-5079, (2014) · Zbl 1428.92109 |

[19] | Zhao, Y; Jiang, D, The threshold of a stochastic SIRS epidemic model with saturated incidence, Appl. Math. Lett., 34, 90-93, (2014) · Zbl 1314.92174 |

[20] | Zhao, Y; Jiang, D, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 243, 18-27, (2014) · Zbl 1335.92108 |

[21] | Zhao, Y; Jiang, D; Mao, X, The threshold of a stochastic SIRS epidemic model in a population with varying size, Discret. Contin. Dyn. Syst. Ser. B, 20, 1289-1307, (2015) · Zbl 1333.60148 |

[22] | Zhao, D; Zhang, T; Yuan, S, The threshold of a stochastic SIVS epidemic model with nonlinear saturated incidence, Phys. A, 443, 372-379, (2016) · Zbl 1400.92565 |

[23] | Liu, Q; Chen, Q; Jiang, D, The threshold of a stochastic delayed SIR epidemic model with temporary immunity, Phys. A, 450, 115-125, (2016) · Zbl 1400.92516 |

[24] | Liu, M; Bai, C, Optimal harvesting of a stochastic mutualism model with levy jumps, Appl. Math. Comput., 276, 301-309, (2016) · Zbl 1410.92158 |

[25] | Liu, M; Bai, C, Dynamics of a stochastic one-prey two-predator model with levy jumps, Appl. Math. Comput., 248, 308-321, (2016) · Zbl 1410.92102 |

[26] | liu, Q; Jiang, D; Shi, N; Hayat, T; Alsaedi, A, Stochastic mutualism model with levy jumps, Commun. Nonlinear Sci. Numer. Simul., 43, 78-90, (2017) |

[27] | Zhao, Y; Yuan, S; Zhang, Q, The effect of levy noise on the survival of a stochastic competitive model in an impulsive polluted environment, Appl. Math. Model., 40, 7583-7600, (2017) |

[28] | Bao, J; Mao, X; Yin, G; Yuan, C, Competitive lotkacvolterra population dynamics with jumps, Nonlinear Anal., 74, 6601-6616, (2011) · Zbl 1228.93112 |

[29] | Liu, M; Bai, C; Deng, M; Du, B, Analysis of stochastic two-prey one-predator model with levy jumps, Phys. A, 445, 176-188, (2016) · Zbl 1400.92437 |

[30] | Zhang, X; Wang, K, Stochastic SIR model with jumps, Appl. Math. Lett., 26, 867-874, (2013) · Zbl 1308.92107 |

[31] | Zhou, Y; Zhang, W, Threshold of a stochastic SIR epidemic model with levy jumps, Phys. A, 446, 204-216, (2016) · Zbl 1400.92566 |

[32] | Guo, Y, Stochastic regime switching sis epidemic model with vaccination driven by levy noise, Adv. Differ. Equ., 2017, 375, (2017) · Zbl 1444.92113 |

[33] | Arqub, O.Abu: Numerical solutions for the robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm. Int. J. Numer. Methods Heat Fluid Flow (2016) |

[34] | Arqub, OA, Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions, Comput. Math. Appl., 73, 1243-1261, (2016) · Zbl 1412.65174 |

[35] | Ge, Q; Ji, G; Xu, J; Fan, X, Extinction and persistence of a stochastic nonlinear SIS epidemic model with jumps, Phys. A, 462, 1120-1127, (2016) · Zbl 1400.92482 |

[36] | Zhang, X; Jiang, D; Hayat, T; Ahmad, B, Dynamics of a stochatic SIS model with double epidemic disease driven by levy jumps, Phys. A, 471, 767-777, (2017) · Zbl 1400.92564 |

[37] | Liu, Q; Jiang, D; Hayat, T; Ahmad, B, Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and lvy jumps, Nonlinear Anal. Hybrid Syst., 27, 29-43, (2018) · Zbl 1382.92240 |

[38] | Liu, Q; Jiang, D; Shi, N; Hayat, T, Dynamics of a stochastic delayed SIR epidemic model with vaccination and double diseases driven by lvy jumps, Phys. A, 492, 2010-2018, (2018) |

[39] | Leng, X; Tao, F; Meng, X, Stochastic inequalities and applications to dynamics analysis of a novel SIVS epidemic model with jumps, J. Inequal. Appl., 2017, 138, (2017) · Zbl 1379.92062 |

[40] | Hattaf, K; Mahrouf, M; Adnani, J; Yousfi, N, Qualitative analysis of a stochastic epidemic model with specific functional response and temporary immunity, Phys. A Stat. Mech. Appl., 490, 591-600, (2018) |

[41] | Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997) · Zbl 0892.60057 |

[42] | Lipster, R, A strong law of large numbers for local martingales, Stochastics, 3, 217-228, (1980) · Zbl 0435.60037 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.