Stability of stochastic heroin model with two distributed delays

Miljana Jovanović; Vuk Vujović

doi:10.3934/dcdsb.2020016

Article Contents

2020, Volume 25, Issue 7: 2407-2432. Doi: 10.3934/dcdsb.2020016

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Stability of stochastic heroin model with two distributed delays

Miljana Jovanović^, and
Vuk Vujović

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

^* Corresponding author: Miljana Jovanović
^* Corresponding author: Miljana Jovanović

Received: April 30, 2019

Revised: June 30, 2019

Early access: April 2020

Published: July 2020

The first author is supported by Grant No 174007 of MNTRS

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Abstract

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. $

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Mathematics Subject Classification: Primary: 60H10; Secondary: 92D30, 93E15.

Citation:

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Figure 1. Initial condition $ U_1(\theta) = \varphi(\theta) $, $ \theta \in [-10,0] $

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Figure 2. 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)

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Figure 3. The graph of the deterministic model (1) and the stochastic trajectory of the number of susceptible individuals in USA from 1.1.2014

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Figure 4. 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

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Figure 5. 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)

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