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

Zhang, Zizhen; Wang, Yougang

doi:10.1186/s13662-019-2009-4

Cited by 11 publications

(6 citation statements)

References 22 publications

(38 reference statements)

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“…( 2015 ), Huang and Liu ( 2013 ), Liu and Zhang ( 2011 ), Wang et al. ( 2011 ), Wangari and Stone ( 2017 ), Zhang and Wang ( 2019 ), Maji ( 2021 ), White and Comiskey ( 2007 ), Yang et al. ( 2016a ), Elragig ( 2013 ), Friedman ( 2008 ), Guo et al.…”

Section: Discussionmentioning

confidence: 99%

“… 2015 ; Huang and Liu 2013 ; Liu and Zhang 2011 ; Wang et al. 2011 ; Wangari and Stone 2017 ; Zhang and Wang 2019 ; Maji 2021 ). In 1979, an exponential model has been simplified from Kermack and McKendrick’s disease model by MacKintosh and Stewart ( 1979 ).…”

Section: Introductionmentioning

confidence: 99%

“…Considering that it usually takes some time for a heroin user to be susceptible. Zhang and Wang ( 2019 ) developed a heroin model with time delay and saturated treatment function based on the model proposed by Wangari and Stone ( 2017 ). They have derived the sufficient conditions for local stability and existence of Hopf bifurcation of the model by taking the time delay due to the period used to cure heroin users as the bifurcation parameter and investigated the direction and stability of the Hopf bifurcation by using the normal form theory and the manifold center theorem.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability and bifurcation analysis of a heroin model with diffusion, delay and nonlinear incidence rate

Kundu

Kumari

Kouachi

et al. 2021

Model. Earth Syst. Environ.

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As we all know, the use of heroin and other drugs in Europe and more specifically in Ireland and the resulting prevalence are well documented. A huge population is still dying using heroin every day. This may happen due to, several reasons like, excessive use of painkiller, lack of awareness etc. It has also inspired mathematical modelers to develop dynamical systems predicting the use of heroin in long run. In this work, the effect of heroin in Europe has been discussed by constructing a suitable mathematical model. Our model describes the process of treatment for heroin users by consolidating a sensible utilitarian structure that speaking to the restricted accessibility of treatment. In the treatment time frame, because of the discretion of the medication clients, some kind of time delay called immunity delay might be found. The effect of immunity delay on the system’s stability has been examined. The existence of positive solution and its boundedness has been established. Also, the local stability of the interior equilibrium point has been studied. Taking the immunity delay as the key parameter, the condition for Hopf-bifurcation has been studied. Using normal form theory and center manifold theorem, we have likewise talked about the direction and stability of delay induced Hopf-bifurcation. The corresponding reaction diffusion system with Dirichlet boundary condition has been considered and the Turing instability has been studied. Obtained solutions have also been plotted by choosing a suitable value of the parameters as the support of our obtained analytical results.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stability and bifurcation analysis of a heroin model with diffusion, delay and nonlinear incidence rate

Kundu

Kumari

Kouachi

et al. 2021

Model. Earth Syst. Environ.

View full text Add to dashboard Cite

show abstract

“…On the other hand, there should be a period for a drug user in treatment before he relapses into the physiological addicts due to the effect of treatment and his self-control. With an eye to such considerations and motivated by the work of some other dynamical systems with time delay [26][27][28][29][30], we investigate a more realistic synthetic drugs transmission model as follows:…”

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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“…Inspired by the previous work contributions, Zhang and Wang acquired a delayed heroin model in 2019. The authors studied the impact of the time delay on heroin model and concluded that the addicts are not cured quickly, rather they need a time period [19]. Random perturbations were introduced in the deterministic heroin epidemic model by Wei, Yang, and Li in [20].Although huge improvements have been made towards the utilization of mathematical methods to model infectious disease, very little efforts have been made to impose this work on heroin epidemic models.…”

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

Cited by 11 publications

References 22 publications

Stability and bifurcation analysis of a heroin model with diffusion, delay and nonlinear incidence rate

Stability and bifurcation analysis of a heroin model with diffusion, delay and nonlinear incidence rate

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

Numerical treatment of stochastic heroin epidemic model

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