• Previous Article
    Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart
  • DCDS-B Home
  • This Issue
  • Next Article
    Statistical solution and Kolmogorov entropy for the impulsive discrete Klein-Gordon-Schrödinger-type equations
doi: 10.3934/dcdsb.2022026
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

The effect of irreversible drug abuse in a dynamic model

Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran

*Corresponding author: Malek Pourhosseini

Received  May 2021 Revised  December 2021 Early access February 2022

In this paper, we analyze a mathematical model of the SIER type which includes susceptible and infected and removed people. In this model, we compute $ {\mathcal{R}_1} $ in strain one and $ {\mathcal{R}_2} $ in strain two. Then we compute the equilibrium points and then determine the global stability.

Citation: Malek Pourhosseini, Reza Memarbashi. The effect of irreversible drug abuse in a dynamic model. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022026
References:
[1]

J. ArinoC. C. McCluskey and P. Van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260-276.  doi: 10.1137/S0036139902413829.

[2]

D. BentalebS. HarroudiS. Amine and K. Allali, Analysis and optimal control of a multistrain SEIR epidemic model with saturated incidence rate and treatment, Differential Equations and Dynamical Systems, 13 (2020), 1-7.  doi: 10.1007/s12591-020-00544-6.

[3]

C. P. Bhunu and W. Garira, A two strain tuberculosis transmission model with therapy and quarantine, Math. Mod. Anal., 14 (2009), 291-312.  doi: 10.3846/1392-6292.2009.14.291-312.

[4]

S. Bonhoeffer and M. Nowak, Mutation and the evolution of parasite virulence, Proc. R. Soc. London. B., 258 (1994), 133-140. 

[5]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Bio. Eng., 1 (2004), 361-404.  doi: 10.3934/mbe.2004.1.361.

[6]

A. DenesY. Muroya and G. Rost, Global stability of a multistrain SIS model with superinfection, Math. Bio. Eng., 14 (2017), 421-435.  doi: 10.3934/mbe.2017026.

[7]

B. FangX. Z. LiM. Martcheva and L. M. Cai, Global asymptotic properties of a heroin epidemic model with treat-age, Appl. Math. Comput., 263 (2015), 315-331.  doi: 10.1016/j.amc.2015.04.055.

[8]

A. S. Kalula and F. Nyabadza, A theoretical model for substance abuse in the presence of treatment, South African Journal of Science, 108 (2012), 1-12. 

[9]

O. Khyar and K. Allali, Global dynamics of a multi-strain SEIR epidemic model with general incidence rate: Application to COVID-19 pandemic, Nonlinear Dynamics, 102 (2020), 489-509.  doi: 10.1007/s11071-020-05929-4.

[10]

C. Li, Y. Zhang and Y. Zhou, Competitive coexistence in a two-strain epidemic model with a periodic infection rate, Discrete Dyn. Nat. Soc., 2020 (2020), Art. ID 7541861, 10 pp. doi: 10.1155/2020/7541861.

[11]

M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083.  doi: 10.1137/S0036141094266449.

[12]

M. Y. Li and J. S. Muldowney, On R.A. Smith's autonomous convergence theorem, Rocky Mountain J. Math., 25 (1995), 365-378.  doi: 10.1216/rmjm/1181072289.

[13]

S. LiuL. Zhang and Y. Xing, Dynamics of a stochastic heroin epidemic model, J. Comput. Appl. Math., 351 (2019), 260-269.  doi: 10.1016/j.cam.2018.11.005.

[14]

W. M. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.

[15]

M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epi- demiology, J. Math. Biol., 35 (1996), 21-36.  doi: 10.1007/s002850050040.

[16]

M. MaS. Liu and J. Li, Bifurcation of a heroin model with nonlinear incidence rate, Nonlinear Dynamics, 88 (2017), 555-565.  doi: 10.1007/s11071-016-3260-9.

[17]

D. R. Mackintosh and G. T. Stewart, A mathematical model of a heroin epidemic: Implications for control policies, J. Epidemiology and Community Health, 33 (1979), 299-304.  doi: 10.1136/jech.33.4.299.

[18]

M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.

[19]

M. Martcheva and X. Z. Li, Linking immunological and epidemiological dynamics of HIV: The case of superinfection, J. Biol. Dyn., 7 (2013), 161-182.  doi: 10.1080/17513758.2013.820358.

[20]

R. M. May and M. Nowak, Coinfection and the evolution of parasite virulence, Proc. R. Soc. London. B., 261 (1995), 209-215. 

[21]

J. MushanyuF. NyabadzaG. Muchatibaya and A. G. Stewart, Modeling multiple relapses in drug epidemics, Ric. Mat., 65 (2016), 37-63.  doi: 10.1007/s11587-015-0241-0.

[22]

H. J. B. Njagarah and F. Nyabadza, Modeling the impact of rehabilitation, amelioration, and relapse on the prevalence of drug epidemics, J. Biol. Systems, 21 (2013), 1350001, 23 pp. doi: 10.1142/S0218339013500010.

[23]

M. Nowak and R. M. May, Superinfection and the evolution of parasite virulence, Proc. R. Soc. London. B., 255 (1994), 81-89. 

[24]

M. NunoZ. FengM. Martcheva and C. Castillo-Chavez, Dynamics of two-strain influenza with isolation and partial cross-immunity, SIAM J. Appl. Math., 65 (2005), 964-982.  doi: 10.1137/S003613990343882X.

[25]

F. NyabadzaJ. B. Njagarah and R. J. Smith, Modeling the dynamics of crystal meth ('tik') abuse in the presence of drug-supply chains in South Africa, Bull. Math. Biol., 75 (2013), 24-48.  doi: 10.1007/s11538-012-9790-5.

[26]

M. Pourhosseini and R. Memarbashi, Relationship between addicts with incurable diseases in an epidemic model, Math. Meth. Appl. Sci., 44 (2021), 7820-7833.  doi: 10.1002/mma.7011.

[27]

P. Van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[28]

E. White and C. Comiskey, Heroin epidemics, treatment and ODE modeling, Math. Biosci., 208 (2007), 312-324.  doi: 10.1016/j.mbs.2006.10.008.

show all references

References:
[1]

J. ArinoC. C. McCluskey and P. Van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260-276.  doi: 10.1137/S0036139902413829.

[2]

D. BentalebS. HarroudiS. Amine and K. Allali, Analysis and optimal control of a multistrain SEIR epidemic model with saturated incidence rate and treatment, Differential Equations and Dynamical Systems, 13 (2020), 1-7.  doi: 10.1007/s12591-020-00544-6.

[3]

C. P. Bhunu and W. Garira, A two strain tuberculosis transmission model with therapy and quarantine, Math. Mod. Anal., 14 (2009), 291-312.  doi: 10.3846/1392-6292.2009.14.291-312.

[4]

S. Bonhoeffer and M. Nowak, Mutation and the evolution of parasite virulence, Proc. R. Soc. London. B., 258 (1994), 133-140. 

[5]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Bio. Eng., 1 (2004), 361-404.  doi: 10.3934/mbe.2004.1.361.

[6]

A. DenesY. Muroya and G. Rost, Global stability of a multistrain SIS model with superinfection, Math. Bio. Eng., 14 (2017), 421-435.  doi: 10.3934/mbe.2017026.

[7]

B. FangX. Z. LiM. Martcheva and L. M. Cai, Global asymptotic properties of a heroin epidemic model with treat-age, Appl. Math. Comput., 263 (2015), 315-331.  doi: 10.1016/j.amc.2015.04.055.

[8]

A. S. Kalula and F. Nyabadza, A theoretical model for substance abuse in the presence of treatment, South African Journal of Science, 108 (2012), 1-12. 

[9]

O. Khyar and K. Allali, Global dynamics of a multi-strain SEIR epidemic model with general incidence rate: Application to COVID-19 pandemic, Nonlinear Dynamics, 102 (2020), 489-509.  doi: 10.1007/s11071-020-05929-4.

[10]

C. Li, Y. Zhang and Y. Zhou, Competitive coexistence in a two-strain epidemic model with a periodic infection rate, Discrete Dyn. Nat. Soc., 2020 (2020), Art. ID 7541861, 10 pp. doi: 10.1155/2020/7541861.

[11]

M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083.  doi: 10.1137/S0036141094266449.

[12]

M. Y. Li and J. S. Muldowney, On R.A. Smith's autonomous convergence theorem, Rocky Mountain J. Math., 25 (1995), 365-378.  doi: 10.1216/rmjm/1181072289.

[13]

S. LiuL. Zhang and Y. Xing, Dynamics of a stochastic heroin epidemic model, J. Comput. Appl. Math., 351 (2019), 260-269.  doi: 10.1016/j.cam.2018.11.005.

[14]

W. M. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.

[15]

M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epi- demiology, J. Math. Biol., 35 (1996), 21-36.  doi: 10.1007/s002850050040.

[16]

M. MaS. Liu and J. Li, Bifurcation of a heroin model with nonlinear incidence rate, Nonlinear Dynamics, 88 (2017), 555-565.  doi: 10.1007/s11071-016-3260-9.

[17]

D. R. Mackintosh and G. T. Stewart, A mathematical model of a heroin epidemic: Implications for control policies, J. Epidemiology and Community Health, 33 (1979), 299-304.  doi: 10.1136/jech.33.4.299.

[18]

M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.

[19]

M. Martcheva and X. Z. Li, Linking immunological and epidemiological dynamics of HIV: The case of superinfection, J. Biol. Dyn., 7 (2013), 161-182.  doi: 10.1080/17513758.2013.820358.

[20]

R. M. May and M. Nowak, Coinfection and the evolution of parasite virulence, Proc. R. Soc. London. B., 261 (1995), 209-215. 

[21]

J. MushanyuF. NyabadzaG. Muchatibaya and A. G. Stewart, Modeling multiple relapses in drug epidemics, Ric. Mat., 65 (2016), 37-63.  doi: 10.1007/s11587-015-0241-0.

[22]

H. J. B. Njagarah and F. Nyabadza, Modeling the impact of rehabilitation, amelioration, and relapse on the prevalence of drug epidemics, J. Biol. Systems, 21 (2013), 1350001, 23 pp. doi: 10.1142/S0218339013500010.

[23]

M. Nowak and R. M. May, Superinfection and the evolution of parasite virulence, Proc. R. Soc. London. B., 255 (1994), 81-89. 

[24]

M. NunoZ. FengM. Martcheva and C. Castillo-Chavez, Dynamics of two-strain influenza with isolation and partial cross-immunity, SIAM J. Appl. Math., 65 (2005), 964-982.  doi: 10.1137/S003613990343882X.

[25]

F. NyabadzaJ. B. Njagarah and R. J. Smith, Modeling the dynamics of crystal meth ('tik') abuse in the presence of drug-supply chains in South Africa, Bull. Math. Biol., 75 (2013), 24-48.  doi: 10.1007/s11538-012-9790-5.

[26]

M. Pourhosseini and R. Memarbashi, Relationship between addicts with incurable diseases in an epidemic model, Math. Meth. Appl. Sci., 44 (2021), 7820-7833.  doi: 10.1002/mma.7011.

[27]

P. Van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[28]

E. White and C. Comiskey, Heroin epidemics, treatment and ODE modeling, Math. Biosci., 208 (2007), 312-324.  doi: 10.1016/j.mbs.2006.10.008.

Figure 1.  The flowchart of the model
Figure 2.  The occurence of $ R_0<1 $ when, $ \beta_1 = 0.2 $, $ \mu = 0.3 $, $ \nu_1 = 0.4 $ and $ \beta_2 = 0.2 $
Figure 3.  The occurence of $ \mathcal{\widehat{R}}^1_2<1 $ when, $ \beta_1 = 0.5 $, $ \mu = 0.2 $, $ \nu_1 = 0.25 $ and $ \beta_2 = 0.15 $
Figure 4.  The occurence of $ \mathcal{\widehat{R}}^2_1<1 $ when, $ \beta_1 = 0.4 $, $ \mu = 0.1 $, $ \nu_1 = 0.3 $ and $ \beta_2 = 0.35 $
Figure 5.  The occurence of backward bifurcation when, $ \beta_1 = 0.2 $, $ \mu = 0.1 $, $ \nu_1 = 0.1 $ and $ \beta_2 = 0.01 $, $ \beta_3 = 0.85 $, $ \nu_2 = 0.05 $ and $ \nu_3 = 0.025 $
[1]

M. W. Hirsch, Hal L. Smith. Asymptotically stable equilibria for monotone semiflows. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 385-398. doi: 10.3934/dcds.2006.14.385

[2]

Scipio Cuccagna. Orbitally but not asymptotically stable ground states for the discrete NLS. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 105-134. doi: 10.3934/dcds.2010.26.105

[3]

François Genoud. Orbitally stable standing waves for the asymptotically linear one-dimensional NLS. Evolution Equations and Control Theory, 2013, 2 (1) : 81-100. doi: 10.3934/eect.2013.2.81

[4]

Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3629-3650. doi: 10.3934/dcds.2021010

[5]

G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Globally stable quasistatic evolution in plasticity with softening. Networks and Heterogeneous Media, 2008, 3 (3) : 567-614. doi: 10.3934/nhm.2008.3.567

[6]

Wenjie Li, Lihong Huang, Jinchen Ji. Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2639-2664. doi: 10.3934/dcdsb.2020026

[7]

Gui-Dong Li, Yong-Yong Li, Xiao-Qi Liu, Chun-Lei Tang. A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1351-1365. doi: 10.3934/cpaa.2020066

[8]

Serafin Bautista, Yeison Sánchez. Sectional-hyperbolic Lyapunov stable sets. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2011-2016. doi: 10.3934/dcds.2020103

[9]

Ranjit Bhattacharjee, Robert L. Devaney, R.E. Lee Deville, Krešimir Josić, Monica Moreno-Rocha. Accessible points in the Julia sets of stable exponentials. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 299-318. doi: 10.3934/dcdsb.2001.1.299

[10]

Wen-Chiao Cheng, Yun Zhao, Yongluo Cao. Pressures for asymptotically sub-additive potentials under a mistake function. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 487-497. doi: 10.3934/dcds.2012.32.487

[11]

Carlos H. Vásquez. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. Journal of Modern Dynamics, 2009, 3 (2) : 233-251. doi: 10.3934/jmd.2009.3.233

[12]

Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393

[13]

Bruno Buonomo, Alberto d’Onofrio, Deborah Lacitignola. Rational exemption to vaccination for non-fatal SIS diseases: Globally stable and oscillatory endemicity. Mathematical Biosciences & Engineering, 2010, 7 (3) : 561-578. doi: 10.3934/mbe.2010.7.561

[14]

Andrew M. Zimmer. Compact asymptotically harmonic manifolds. Journal of Modern Dynamics, 2012, 6 (3) : 377-403. doi: 10.3934/jmd.2012.6.377

[15]

Juraj Földes, Peter Poláčik. On asymptotically symmetric parabolic equations. Networks and Heterogeneous Media, 2012, 7 (4) : 673-689. doi: 10.3934/nhm.2012.7.673

[16]

Marcela Mejía, J. Urías. An asymptotically perfect pseudorandom generator. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 115-126. doi: 10.3934/dcds.2001.7.115

[17]

Gang Li, Xiaoqi Yang, Yuying Zhou. Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces. Journal of Industrial and Management Optimization, 2013, 9 (3) : 671-687. doi: 10.3934/jimo.2013.9.671

[18]

Byung-Soo Lee. A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 557-565. doi: 10.3934/naco.2013.3.557

[19]

B. San Martín, Kendry J. Vivas. Asymptotically sectional-hyperbolic attractors. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4057-4071. doi: 10.3934/dcds.2019163

[20]

José F. Alves. Stochastic behavior of asymptotically expanding maps. Conference Publications, 2001, 2001 (Special) : 14-21. doi: 10.3934/proc.2001.2001.14

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (261)
  • HTML views (157)
  • Cited by (0)

Other articles
by authors

[Back to Top]