Dynamics of a stochastic heroin epidemic model

Abstract In this paper, we investigate a stochastic heroin epidemic model. By constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the extinction of the drug users. Moreover, we also establish sufficient conditions for the existence of ergodic stationary distribution to the model, which reveals that the drug users will persist. The sensitivity analysis indicates that prevention would be more important than cure.

[1]  Chuanzhi Bai,et al.  Analysis of a stochastic tri-trophic food-chain model with harvesting , 2016, Journal of Mathematical Biology.

[2]  B. Stanton,et al.  Illicit drug initiation among institutionalized drug users in China. , 2002, Addiction.

[3]  Marija Krstić,et al.  The influence of time-dependent delay on behavior of stochastic population model with the Allee effect , 2015 .

[4]  Gang George Yin,et al.  Asymptotic Properties of Hybrid Diffusion Systems , 2007, SIAM J. Control. Optim..

[5]  L. Yanhong,et al.  Drug Abuse in China , 2004 .

[6]  Hai-Feng Huo,et al.  The threshold of a stochastic SIQS epidemic model , 2017 .

[7]  Bin Fang,et al.  Global stability for a heroin model with age-dependent susceptibility , 2015, J. Syst. Sci. Complex..

[8]  M. Carletti,et al.  On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment. , 2002, Mathematical biosciences.

[9]  S. Faraone,et al.  Genome‐wide linkage analysis of heroin dependence in Han Chinese: Results from wave one of a multi‐stage study , 2006, American journal of medical genetics. Part B, Neuropsychiatric genetics : the official publication of the International Society of Psychiatric Genetics.

[10]  K. Sato,et al.  Evolution of predator–prey systems described by a Lotka–Volterra equation under random environment , 2006 .

[11]  Dingshi Li,et al.  The stationary distribution and ergodicity of a stochastic generalized logistic system , 2013 .

[12]  Zhixia Han,et al.  Stochastic SIRS model under regime switching , 2013 .

[13]  D. Vlahov,et al.  Rapid transmission of hepatitis C virus among young injecting heroin users in Southern China. , 2004, International journal of epidemiology.

[14]  Y. Takeuchi,et al.  Dynamical behavior of Lotka-Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise , 2004 .

[15]  Baocang Ding,et al.  The existence and asymptotic behaviour of energy solutions to stochastic age-dependent population equations driven by Levy processes , 2015, Appl. Math. Comput..

[16]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[17]  D. Jiang,et al.  Threshold Behavior in a Stochastic SIS Epidemic Model with Standard Incidence , 2014 .

[18]  X. Mao,et al.  A stochastic model of AIDS and condom use , 2007 .

[19]  Meng Liu,et al.  Permanence of Stochastic Lotka–Volterra Systems , 2017, J. Nonlinear Sci..

[20]  Tasawar Hayat,et al.  Stationary distribution and extinction of a stochastic SEIR epidemic model with standard incidence , 2017 .

[21]  Xuerong Mao,et al.  A stochastic model for internal HIV dynamics , 2008 .

[22]  Qimin Zhang,et al.  The effect of Lévy noise on the survival of a stochastic competitive model in an impulsive polluted environment , 2016 .

[23]  K. Sporer.  Acute Heroin Overdose , 1999, Annals of Internal Medicine.

[24]  Daqing Jiang,et al.  Stationary distribution of a stochastic SIS epidemic model with vaccination , 2014 .

[25]  X. Mao,et al.  Environmental Brownian noise suppresses explosions in population dynamics , 2002 .

[26]  Hai-Feng Huo,et al.  Dynamic behavior of a stochastic SIQS epidemic model with Lévy jumps , 2018 .

[27]  Hong Xiang,et al.  Dynamics of the deterministic and stochastic SIQS epidemic model with non-linear incidence , 2014, Appl. Math. Comput..

[28]  Shengqiang Liu,et al.  Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching , 2017, 1707.06380.

[29]  Ahmed Alsaedi,et al.  Periodic solution for a stochastic non-autonomous competitive Lotka–Volterra model in a polluted environment ☆ , 2017 .

[30]  E. White,et al.  Heroin epidemics, treatment and ODE modelling. , 2007, Mathematical biosciences.

[31]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[32]  Hai-Feng Huo,et al.  The dynamic behavior of deterministic and stochastic delayed SIQS model , 2018 .

[33]  Ravi P. Agarwal,et al.  Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation , 2012 .

[34]  Daqing Jiang,et al.  Multigroup SIR epidemic model with stochastic perturbation , 2011 .

[35]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[36]  Xuerong Mao,et al.  The SIS epidemic model with Markovian switching , 2012 .

[37]  Yu Zhao,et al.  Stability in distribution of a stochastic hybrid competitive Lotka–Volterra model with Lévy jumps , 2016 .

[38]  Z. Teng,et al.  Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate , 2016 .

[39]  V. B. Kolmanovskii,et al.  Stability of epidemic model with time delays influenced by stochastic perturbations 1 This paper was , 1998 .