Analysis of a Heroin Epidemic Model with Saturated Treatment Function

Wangari, Isaac Mwangi; Stone, Lewi

doi:10.1155/2017/1953036

Cited by 25 publications

(37 citation statements)

References 43 publications

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“…Since the heroin addiction was first defined as epidemic in [1981][1982][1983] in Ireland, the heroin-using can be modeled in a similar way to the epidemic models [6] and mathematical modeling has been used extensively to address issues of public health importance. For example, Ma et al developed different forms of heroin epidemic models [3,[6][7][8][9][10] to study the transmission of heroin epidemics. Sharomi et al formulated different smoking models [11][12][13][14][15][16][17] for giving up smoking.…”

Section: Introductionmentioning

confidence: 99%

Influence of time delay on bifurcation of a synthetic drug transmission model with psychological addicts

2020

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A synthetic drug transmission model with psychological addicts and time delay is proposed in this paper. By analyzing the corresponding characteristic equation and choosing the time delay as the bifurcation parameter, a set of sufficient criteria guaranteeing local stability of the synthetic drug addiction equilibrium and the appearance of a Hopf bifurcation of the model is established. Further, the direction and stability of the Hopf bifurcation are investigated with the aid of normal form theory and center manifold theory. Finally, numerical simulations are performed to support the analytical results.

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Section: Introductionmentioning

confidence: 99%

Influence of time delay on bifurcation of a synthetic drug transmission model with psychological addicts

2020

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“…Thus, it is more realistic to investigate the heroin model with nonlinear treatment function. To this end, Wangari and Stone [21] formulated the following heroin model with a saturated treatment function:…”

Section: Introductionmentioning

confidence: 99%

“…A is the constant input rate of the susceptibles through immigration and birth; β is the contact rate of the susceptibles with heroin users; p is the rate at which the heroin users undergoing treatment relapse; δ 1 and δ 2 are the heroin-induced death rates of heroin users not in treatment and heroin users undergoing treatment, respectively; ε is the self-cure rate of heroin users not in treatment; σ is the successful treatment rate of heroin users undergoing treatment; α is the rate at which heroin users are treated; η accounts for the extent of saturation of heroin users; μ is the natural death rate of all the subpopulations. Wangari and Stone [21] studied the stability and backward bifurcations of system (1). It should be pointed out that Wangari and Stone [21] assume that the heroin users are cured instantaneously, which is not consistent with reality.…”

Section: Introductionmentioning

confidence: 99%

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Hopf bifurcation of a heroin model with time delay and saturated treatment function

Zhang

Wang

2019

Adv Differ Equ

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In this paper, local stability and Hopf bifurcation of a delayed heroin model with saturated treatment function are discussed. First of all, sufficient conditions for local stability and existence of Hopf bifurcation are obtained by regarding the time delay as a bifurcation parameter and analyzing the distribution of the roots of the associated characteristic equation. Directly afterward, properties of the Hopf bifurcation, such as the direction and stability, are investigated with the aid of the normal form theory and the manifold center theorem. Finally, numerical simulations are presented to justify the obtained theoretical results, and some suggestions are offered for controlling heroin abuse in populations.

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“…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].…”

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confidence: 99%

Numerical treatment of stochastic heroin epidemic model

et al. 2019

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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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Analysis of a Heroin Epidemic Model with Saturated Treatment Function

Cited by 25 publications

References 43 publications

Influence of time delay on bifurcation of a synthetic drug transmission model with psychological addicts

Influence of time delay on bifurcation of a synthetic drug transmission model with psychological addicts

Hopf bifurcation of a heroin model with time delay and saturated treatment function

Numerical treatment of stochastic heroin epidemic model

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