DYNAMICS OF A STOCHASTIC SIR MODEL WITH BOTH HORIZONTAL AND VERTICAL TRANSMISSION

Anqi Miao; Tongqian Zhang; Jian Zhang; Chaoyang Wang

doi:10.11948/2018.1108

2018 Volume 8 Issue 4

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Anqi Miao, Tongqian Zhang, Jian Zhang, Chaoyang Wang. DYNAMICS OF A STOCHASTIC SIR MODEL WITH BOTH HORIZONTAL AND VERTICAL TRANSMISSION[J]. Journal of Applied Analysis & Computation, 2018, 8(4): 1108-1121. doi: 10.11948/2018.1108

Citation:

Anqi Miao, Tongqian Zhang, Jian Zhang, Chaoyang Wang. DYNAMICS OF A STOCHASTIC SIR MODEL WITH BOTH HORIZONTAL AND VERTICAL TRANSMISSION[J]. Journal of Applied Analysis & Computation, 2018, 8(4): 1108-1121. doi: 10.11948/2018.1108

DYNAMICS OF A STOCHASTIC SIR MODEL WITH BOTH HORIZONTAL AND VERTICAL TRANSMISSION

1 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China;
2 State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Fund Project:

Abstract

Abstract

A stochastic mathematical model with both horizontal and vertical transmission is proposed to investigate the dynamical behavior of SIR disease. By employing theories of stochastic differential equation and inequality techniques, the threshold associating on extinction and persistence of infectious diseases is deduced for the case of the small noise. Our results show that the threshold completely depends on the stochastic perturbation and the basic reproductive number of the corresponding deterministic model. Moreover, we find that large noise is conducive to control the spread of diseases and the persistent disease in deterministic model may eliminate ultimately due to the effect of large noise. Finally, numerical simulations are performed to illustrate the theoretical results.
- Stochastic SIR epidemic model /
- vertical transmission /
- extinction /
- persistence /
- threshold
MSC: 60H10;92F99

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References

[1]	L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Second Edition, CRC Press, 2015. Google Scholar
[2]	R. Anderson and R. May, Infectious Diseases in Humans:Dynamics and Control, Oxford University Press, Oxford, 1992. Google Scholar
[3]	N. Bacaër, On the stochastic SIS epidemic model in a periodic environment, Journal of Mathematical Biology, 2014, 71(2), 491-511. Google Scholar
[4]	E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Mathematics and Computers in Simulation, 1998, 45(3-4), 269-277. Google Scholar
[5]	F. Bian, W. Zhao, Y. Song and R. Yue, Dynamical analysis of a class of prey-predator model with Beddington-Deangelis functional response, stochastic perturbation, and impulsive toxicant input, Complexity, 2017, 2017, Article ID 3742197. Google Scholar
[6]	Y. Cai, Y. Kang, M. Banerjee and W. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, Journal of Differential Equations, 2015, 259(12), 7463-7502. Google Scholar
[7]	N. T. Dieu, D. H. Nguyen, N. H. Du and G. Yin, Classification of asymptotic behavior in a stochastic SIR model, SIAM Journal on Applied Dynamical Systems, 2016, 15(2), 1062-1084. Google Scholar
[8]	N. H. Du and N. N. Nhu, Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises, Applied Mathematics Letters, 2017, 64, 223-230. Google Scholar
[9]	T. Feng, X. Meng, L. Liu and S. Gao, Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model, Journal of Inequalities and Applications, 2016, 2016(1), 327. Google Scholar
[10]	A. Gray, D. Greenhalgh, X. Mao and J. Pan, The SIS epidemic model with markovian switching, Journal of Mathematical Analysis and Applications, 2012, 394(2), 496-516. Google Scholar
[11]	C. Ji, D. Jiang and N. Shi, The behavior of an SIR epidemic model with stochastic perturbation, Stochastic Analysis and Applications, 2012, 30(5), 755-773. Google Scholar
[12]	W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London A:Mathematical, Physical and Engineering Sciences, 1927, 115(772), 700-721. Google Scholar
[13]	P. E. Kloeden and E. Platen, Higher-order implicit strong numerical schemes for stochastic differential equations, Journal of Statistical Physics, 1992, 66(1), 283-314. Google Scholar
[14]	A. Lahrouz, A. Settati and A. Akharif, Effects of stochastic perturbation on the SIS epidemic system, Journal of Mathematical Biology, 2017, 74(1), 469-498. Google Scholar
[15]	X. Leng, T. Feng and X. Meng, Stochastic inequalities and applications to dynamics analysis of a novel SIVS epidemic model with jumps, Journal of Inequalities and Applications, 2017, 2017(1), 138. Google Scholar
[16]	Y. Lin and D. Jiang, Long-time behaviour of a perturbed SIR model by white noise, Discrete and Continuous Dynamical Systems-Series B, 2013, 18(7), 1873-1887. Google Scholar
[17]	G. Liu, X. Wang, X. Meng and S. Gao, Extinction and persistence in mean of a novel delay impulsive stochastic infected predator-prey system with jumps, Complexity, 2017, 2017, Article ID 1950970. Google Scholar
[18]	L. Liu and X. Meng, Optimal harvesting control and dynamics of two-species stochastic model with delays, Advances in Difference Equations, 2017, 2017(1), 18. Google Scholar
[19]	M. Liu, C. Bai and K. Wang, Asymptotic stability of a two-group stochastic SEIR model with infinite delays, Communications in Nonlinear Science and Numerical Simulation, 2014, 19(10), 3444-3453. Google Scholar
[20]	Q. Liu and Q. Chen, Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Physica A:Statistical Mechanics and its Applications, 2015, 428, 140-153. Google Scholar
[21]	Q. Lu, Stability of SIRS system with random perturbations, Physica A:Statistical Mechanics and its Applications, 2009, 388(18), 3677-3686. Google Scholar
[22]	Z. Lu, X. Chi and L. Chen, The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission, Mathematical and Computer Modelling, 2002, 36(9), 1039-1057. Google Scholar
[23]	X. Lv, L. Wang and X. Meng, Global analysis of a new nonlinear stochastic differential competition system with impulsive effect, Advances in Difference Equations, 2017, 2017(1), 296. Google Scholar
[24]	X. Mao, Stochastic Differential Equations and Applications. 2nd Edition, Horwood Publishing, Chichester, UK, 2007. Google Scholar
[25]	A. Miao, X. Wang, T. Zhang et al., Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis, Advances in Difference Equations, 2017, 2017(1), 226. Google Scholar
[26]	A. Miao, J. Zhang, T. Zhang and B. G. S. Pradeep, Threshold dynamics of a stochastic SIR model with vertical transmission and vaccination, Computational and Mathematical Methods in Medicine, 2017, 2017, Article ID 4820183. Google Scholar
[27]	H. Qi, L. Liu and X. Meng, Dynamics of a non-autonomous stochastic SIS epidemic model with double epidemic hypothesis, Complexity, 2017, 2017, Article ID 4861391. Google Scholar
[28]	W. Wang and W. Ma, A diffusive HIV infection model with nonlocal delayed transmission, Applied Mathematics Letters, 2018, 75, 96-101. Google Scholar
[29]	W. Wang and W. Ma, Hepatitis C virus infection is blocked by HMGB1:A new nonlocal and time-delayed reaction-diffusion model, Applied Mathematics and Computation, 2018, 320, 633-653. Google Scholar
[30]	W. Wang and W. Ma, Travelling wave solutions for a nonlocal dispersal HIV infection dynamical model, Journal of Mathematical Analysis and Applications, 2018, 457(1), 868-889. Google Scholar
[31]	W. Wang and T. Zhang, Caspase-1-mediated pyroptosis of the predominance for driving CD4⁺ T cells death:a nonlocal spatial mathematical model, Bulletin of Mathematical Biology, 2018, 80(3), 540-582. Google Scholar
[32]	C. Xu, Global threshold dynamics of a stochastic differential equation SIS model, Journal of Mathematical Analysis and Applications, 2017, 447(2), 736-757. Google Scholar
[33]	S. Zhang, X. Meng, T. Feng and T. Zhang, Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects, Nonlinear Analysis:Hybrid Systems, 2017, 26, 19-37. Google Scholar
[34]	T. Zhang, X. Meng and T. Zhang, Global dynamics of a virus dynamical model with cell-to-cell transmission and cure rate, Computational and Mathematical Methods in Medicine, 2015, 2015, Article ID 758362. Google Scholar
[35]	T. Zhang, X. Meng and T. Zhang, Global analysis for a delayed SIV model with direct and environmental transmissions, Journal of Applied Analysis & Computation, 2016, 6(2), 479-491. Google Scholar
[36]	T. Zhang, X. Meng, T. Zhang and Y. Song, Global dynamics for a new highdimensional SIR model with distributed delay, Applied Mathematics and Computation, 2012, 218(24), 11806-11819. Google Scholar
[37]	X. Zhang, D. Jiang, T. Hayat and B. Ahmad, Dynamics of a stochastic SIS model with double epidemic diseases driven by lévy jumps, Physica A:Statistical Mechanics and its Applications, 2017, 471, 767-777. Google Scholar
[38]	D. Zhao and S. Yuan, Break-even concentration and periodic behavior of a stochastic chemostat model with seasonal fluctuation, Communications in Nonlinear Science and Numerical Simulation, 2017, 46, 62-73. Google Scholar
[39]	J. Zhao, L. Wang and Z. Han, Stability analysis of two new SIRS models with two viruses, International Journal of Computer Mathematics, 2017, DOI:10.1080/00207160.2017.1364369. Google Scholar
[40]	W. Zhao, J. Li and X. Meng, Dynamical analysis of SIR epidemic model with nonlinear pulse vaccination and lifelong immunity, Discrete Dynamics in Nature and Society, 2015, 2015, Article ID 848623. Google Scholar
[41]	W. Zhao, T. Zhang, Z. Chang et al., Dynamical analysis of SIR epidemic models with distributed delay, Journal of Applied Mathematics, 2013, 2013, Article ID 154387. Google Scholar
[42]	Y. Zhao, Y. Lin, D. Jiang et al., Stationary distribution of stochastic SIRS epidemic model with standard incidence, Discrete and Continuous Dynamical Systems-Series B, 2016, 21(7), 2363-2378. Google Scholar
[43]	Y. Zhao, S. Yuan and T. Zhang, Stochastic periodic solution of a nonautonomous toxic-producing phytoplankton allelopathy model with environmental fluctuation, Communications in Nonlinear Science and Numerical Simulation, 2017, 44, 266-276. Google Scholar
[44]	J. Zhou, Y. Yang and T. Zhang, Global stability of a discrete multigroup SIR model with nonlinear incidence rate, Mathematical Methods in the Applied Sciences, 2017, 40(14), 5370-5379. Google Scholar
[45]	Y. Zhou, S. Yuan and D. Zhao, Threshold behavior of a stochastic SIS model with jumps, Applied Mathematics and Computation, 2016, 275, 255-267. Google Scholar