Numerical treatment of stochastic heroin epidemic model

Abstract

We have presented the numerical analysis of a stochastic heroin epidemic model in this paper. The mean of stochastic heroin model is itself a deterministic solution. The effect of reproduction number has also been observed in the stochastic heroin epidemic model. We have developed some stochastic explicit and implicitly driven explicit methods for this model. But stochastic explicit methods have flopped for certain values of parameters. In support, some theorems and graphical illustrations are presented.difficult to quit. If one decides to stop using heroin, the after effects include: extreme craving for the drug, restlessness, muscle, bone pain, diarrhea, and vomiting. These can persist between 48 to 72 hours. It also may take several attempts to get rid of it. Approximately, seventeen million peoples are directly affected by these types of drugs in the form of health problems all over the world. Literature surveyIn the last decade of the twentieth century various mathematical models were proposed to discuss the epidemic dynamics of heroin model. In 2009, Mulone and Straughan suggested in [4] that the model proposed by White and Comiskey in [5] has stable steady states. Wang, Yang, and Li in 2011 preferred bilinear incidence law over standard incidence in the heroin epidemic model. They proposed that the population is not constant with time in [6]. Samanta extended the model in [5] to non-autonomous epidemic form, which was an improved version of the periodic epidemic model. Here, the population has been treated periodically in [7]. Liu and Zhang worked on time delayed heroin epidemic model, which led to the formulation of a delay differential equation system in [8]. In 2013, Haung and Liu in [9] found that, under specific condition, the delay differential equation model can be converted into an ordinary differential equation model which resembles the renowned SIR epidemic model. Abdurahman, Zhang, and Teng in [10] changed a non-autonomous time delayed heroin epidemic model to an autonomous model. Here the non-standard finite difference pattern was applied to get the discretized heroin epidemic model. In 2015 the authors Fang et al. in [2,11] formulated age-dependent susceptible and treat-age heroine epidemic models respectively. In 2016 Yang, Li, and Zhang in [12] inquired an age structured heroin model. Non-linear incidence rate was discussed instead of ordinary incidence rate. The time delayed heroin epidemic model was also proposed by the authors in [13]. Ma, Liu, and Li in [14] discussed various types of bifurcation of heroin epidemic model with non-linear contact rate. Besides, the age structured heroin epidemic model was also proposed in [15]. Wangari and Stone discussed the backward bifurcations and stability in the heroin model, and they considered that heroin users can be rehabilitated quickly in [16]. In 2018, a stochastic heroin epidemic model with bilinear incidence and varying population size was obtained from a deterministic version by Liu, Zhang, and Li in [17]. Liu, Zhang, and Xing inspected...

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“…Opium has been a popular narcotic for millennia. During the 1800 s, opium dens were prevalent, and the first drug, morphine, was derived from the opium poppy [2]. Later, heroin became an opiate substance derived from morphine [1][2][3].…”

“…During the 1800 s, opium dens were prevalent, and the first drug, morphine, was derived from the opium poppy [2]. Later, heroin became an opiate substance derived from morphine [1][2][3]. Illicit drug exploitation is destructive to society.…”

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