### Dynamics of a stochastic SIS epidemic model with different incidences and double epidemic hypothesis

ZHANG Dao-xiang1,2, HU Wei1, TAO Long1, ZHOU Wen1

1. 1. School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, Anhui, China;
2. Department of Mathematics and Statistics, University of Helsinki, Helsinki 00014, Finland
• Received:2016-11-18 Online:2017-05-20 Published:2017-05-15

Abstract: We propose a new mathematical model with two different incidence rates and double epidemic hypothesis. By the Lyapunov function and Itôs formula, we explore and obtain the threshold of a stochastic SIS system for the extinction and thepermanence in mean of two epidemic diseases. The results show that not only a large stochastic disturbance but also a small stochastic disturbance can cause infectious diseases to go to extinction.

CLC Number:

• Q332
  马知恩, 周义仓, 王稳地, 等. 传染病动力学的数学建模与研究[M]. 北京:科学出版社, 2004. MA Zhien, ZHOU Yicang, WANG Wendi, et al. Mathematical modeling and research of infectious diease dynamics[M]. Beijing: Science Press, 2004. 张道祥, 丁伟伟. 具非连续收获策略的Gilpin-Ayala竞争系统周期解的存在性[J]. 安徽师范大学学报(自然科学版), 2014, 37(6):515-519. ZHANG Daoxiang, DING Weiwei. Existence of periodic solutions of gilpin-ayala competitive system with discontinuous harvesting[J]. Journal of Anhui Normal University(Natural Science), 2014, 37(6):515-519. GREENHALGH D, AKHAN Q J, Lewis F I. Recurrent epidemic cycles in an infectious disease model with a time delay in loss of vaccine immunity[J]. Nonlinear Anal, 2005, 63:779-788. KERMAC W O, MCKENDRICK A G. Contributions to the mathematical theory of epidemic[J]. Proceedings of the Royal Society, 1932, A138:55-83. CHEN Q L, TENG Z D, WANG L, et al. The existence of codimension-two bifurcation in a discrete SIS epidemic model with standard incidence[J]. Nonlinear Dynam, 2013, 719(1/2):55-73. KUNIYA T, INABA H. Endemic threshold results for an age-structured SIS epidemic model with periodic parameters[J]. J Math Anal Appl, 2013, 402(2):477-492. LI X Z, LI W S, GOHOS M. Stability and bifurcation of an SIS epidemic model with treatment[J]. Chaos Solitons Fractals, 2009, 42(5):2822-2832. ZHANG X, LIU X. Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment[J]. Nonlinear Anal, 2009, 10(2):565-575. GRAY A, GREENHALGH D, HU L, et al. A stochastic differential equation SIS epidemic model[J]. SIAM J Appl Math, 2001, 71(3):876-902. ZHAO Y N, JIANG D Q, REGAN D O. The extinction and persistence of stochastic SIS epidemic model with Vaccination[J]. Phy A, 2013, 392(20):4916-4927. LIN Y G, JIANG D Q, WANG S. Stationary distribution of a stochastic SIS epidemic model with Vaccination[J]. Phy A: Statistical Mechanics and its Applications, 2014, 394(2):187-197. 周艳丽, 张卫国. 非线性传染率的随机SIS传染病模型的持久性和灭绝性[J]. 山东大学学报(理学版), 2013, 48(10): 68-77. ZHOU Yanli, ZHANG Weiguo. Persistence and extinction in stochastic SIS epidemic model with nonlinear incidence rate[J]. Journal of Shandong University(Natural Science), 2013, 48(10):68-77. MENG X Z, ZHAO S N, FENG T, et al. Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis[J]. J Math Anal Appl, 2016, 433(1):227-242. XIAO D, RUAN S. Global analysis of an epidemic model with nonmonotone incidence rate[J]. Math Biosci, 2007, 208(2):419-429.
  GAO Rui-mei, CHU Ying. Freeness of arrangements between the Weyl arrangements of types An-1 and Bn [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 70-75.  ZHANG Qian, LI Hai-yang. The iterative fraction thresholding algorithm in sparse information processing [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(9): 76-82.  SHI Zhang-lei, LI Wei-guo. A† graded hard thresholding pursuit algorithm [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(8): 58-64.  WANG Ya-qi, WANG Jing. Rumor spreading on dynamic complex networks with curious psychological mechanism [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(6): 99-104.  LIANG Lei, LIU Gui-rong. Analysis of an epidemic model with information on overlay network [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(11): 107-114.  ZHU Xi, DONG Xi-shuang, GUAN Yi, LIU Zhi-guang. Sentiment analysis of Chinese Micro-blog based on semi-supervised learning [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(11): 37-42.  CHEN Gui-ying. Stability analysis of generalized fuzzy bidirectional associative #br# memory networks with thresholds [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(1): 80-85.  ZHAO Xiu-feng, WANG Ai-lan, WANG Xiang. Threshold scheme for LWE inversion [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(07): 34-37.  WU Fang-lan1, ZUO Lian-cui2*. Equitable colorings of a special class of Cartesian products of graphs [J]. J4, 2013, 48(4): 20-24.  ZHOU Yan-li1,2, ZHANG Wei-guo1. Persistence and extinction in stochastic SIS epidemic model with nonlinear incidence rate [J]. J4, 2013, 48(10): 68-77.  MA Yan, LIU Jian-wei, ZHANG Yu-fei. Threshold signature-based lightweight clustering handover scheme for Ad hoc networks [J]. J4, 2012, 47(11): 78-82.  SUN Jing-Yun. Absolute ruin for the compound Piosson risk model with  a threshold dividend strategy [J]. J4, 2010, 45(3): 105-110.  . Global qualitative analysis of a new SIR epidemic disease model with vertical transmission and pulse vaccination [J]. J4, 2009, 44(5): 67-73.  ZHU Yi,DAI Tao,ZHANG Xian-feng . A state-tree based (t, n) secret sharing scheme [J]. J4, 2008, 43(9): 68-72 .  HUA Zhao-xiu,NIU Ming-fei . A threshold dividend strategy in a risk model with inter-claim-dependent claim sizes [J]. J4, 2008, 43(10): 91-96 .
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