# American Institute of Mathematical Sciences

July  2020, 25(7): 2407-2432. doi: 10.3934/dcdsb.2020016

## Stability of stochastic heroin model with two distributed delays

 Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia

* Corresponding author: Miljana Jovanović

Received  May 2019 Revised  July 2019 Published  July 2020 Early access  April 2020

Fund Project: The first author is supported by Grant No 174007 of MNTRS

In this paper a stability of stochastic heroin model with two distributed delays is studied. Precisely, the deterministic model for dynamics of heroin users is extended by random perturbation that briefly describe how a environmental fluctuations lead an individual to become a heroin user. By using a suitable Lyapunov function stability conditions for heroin use free equilibrium are obtained. Furthermore, asymptotic behavior around the heroin spread equilibrium of the deterministic model is investigated by using appropriate Lyapunov functional. Theoretical studies, based on real data, are applied on modeling of number of heroin users in the USA from $01.01.2014.$

Citation: Miljana Jovanović, Vuk Vujović. Stability of stochastic heroin model with two distributed delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2407-2432. doi: 10.3934/dcdsb.2020016
##### References:

show all references

##### References:
Initial condition $U_1(\theta) = \varphi(\theta)$, $\theta \in [-10,0]$
CDF of truncated Weibull distribution over interval $[0,10]$ with parameters $(0.25,1.3)$ (left) and CDF of double truncated Cauchy distribution over interval $[0,0.25]$ with parameters $(-0.011,0.005)$ (right)
The graph of the deterministic model (1) and the stochastic trajectory of the number of susceptible individuals in USA from 1.1.2014
The graph of the deterministic model (1) and the stochastic trajectory of the number of heroin users not in treatment in USA from 1.1.2014
Stochastic trajectories of the number of susceptible individuals and heroin users not in treatment in USA from 1.1.2014. (left); stochastic trajectory of the number of heroin users not in treatment and real data (right)
 [1] Bin Fang, Xue-Zhi Li, Maia Martcheva, Li-Ming Cai. Global stability for a heroin model with two distributed delays. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 715-733. doi: 10.3934/dcdsb.2014.19.715 [2] Pham Huu Anh Ngoc. New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021040 [3] Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521 [4] Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial and Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002 [5] Weiyi Zhang, Ling Zhou. Global asymptotic stability of constant equilibrium in a nonlocal diffusion competition model with free boundaries. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022062 [6] Xiong Li. The stability of the equilibrium for a perturbed asymmetric oscillator. Communications on Pure and Applied Analysis, 2006, 5 (3) : 515-528. doi: 10.3934/cpaa.2006.5.515 [7] Xiong Li. The stability of the equilibrium for a perturbed asymmetric oscillator. Communications on Pure and Applied Analysis, 2007, 6 (1) : 69-82. doi: 10.3934/cpaa.2007.6.69 [8] Abdennasser Chekroun, Mohammed Nor Frioui, Toshikazu Kuniya, Tarik Mohammed Touaoula. Mathematical analysis of an age structured heroin-cocaine epidemic model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4449-4477. doi: 10.3934/dcdsb.2020107 [9] Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048 [10] Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529 [11] Franco Maceri, Michele Marino, Giuseppe Vairo. Equilibrium and stability of tensegrity structures: A convex analysis approach. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 461-478. doi: 10.3934/dcdss.2013.6.461 [12] L. Bakker. The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1183-1200. doi: 10.3934/cpaa.2013.12.1183 [13] Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361 [14] Edoardo Beretta, Dimitri Breda. Discrete or distributed delay? Effects on stability of population growth. Mathematical Biosciences & Engineering, 2016, 13 (1) : 19-41. doi: 10.3934/mbe.2016.13.19 [15] Lili Liu, Xianning Liu, Jinliang Wang. Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2615-2630. doi: 10.3934/dcdsb.2016064 [16] Wei Wang, Kai Liu, Xiulian Wang. Sensitivity to small delays of mean square stability for stochastic neutral evolution equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2403-2418. doi: 10.3934/cpaa.2020105 [17] Elena Braverman, Sergey Zhukovskiy. Absolute and delay-dependent stability of equations with a distributed delay. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2041-2061. doi: 10.3934/dcds.2012.32.2041 [18] Sibel Senan, Eylem Yucel, Zeynep Orman, Ruya Samli, Sabri Arik. A Novel Lyapunov functional with application to stability analysis of neutral systems with nonlinear disturbances. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1415-1428. doi: 10.3934/dcdss.2020358 [19] Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715 [20] Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1447-1462. doi: 10.3934/cpaa.2011.10.1447

2020 Impact Factor: 1.327