The effect of irreversible drug abuse in a dynamic model

Malek Pourhosseini; Reza Memarbashi

doi:10.3934/dcdsb.2022026

Article Contents

2022, Volume 27, Issue 11: 6907-6926. Doi: 10.3934/dcdsb.2022026

This issue Previous Article Stability and Hopf bifurcation in a prey-predator model with memory-based diffusion Next Article Existence of complete Lyapunov functions with prescribed orbital derivative

The effect of irreversible drug abuse in a dynamic model

Malek Pourhosseini^, and
Reza Memarbashi

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

^*Corresponding author: Malek Pourhosseini
^*Corresponding author: Malek Pourhosseini

Received: April 30, 2021

Revised: November 30, 2021

Published: October 2022

Abstract Full Text(HTML) Figure(5) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 37N99, 34D23; Secondary: 34C23.

Citation:

Full Text(HTML)

Figure 1. The flowchart of the model

Download: Full-size image PowerPoint slide

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 $

Download: Full-size image PowerPoint slide

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 $

Download: Full-size image PowerPoint slide

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 $

Download: Full-size image PowerPoint slide

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 $

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	J. Arino, C. 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. Bentaleb, S. Harroudi, S. 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. Denes, Y. 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. Fang, X. Z. Li, M. 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. Liu, L. 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. Liu, S. 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. Ma, S. 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. Mushanyu, F. Nyabadza, G. 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. Nuno, Z. Feng, M. 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. Nyabadza, J. 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.