November  2020, 25(11): 4449-4477. doi: 10.3934/dcdsb.2020107

Mathematical analysis of an age structured heroin-cocaine epidemic model

1. 

Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées, University of Tlemcen, Tlemcen 13000, Algeria

2. 

Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan

* Corresponding author

Received  December 2018 Revised  December 2019 Published  November 2020 Early access  March 2020

This paper is devoted to studying the dynamics of a certain age structured heroin-cocaine epidemic model. More precisely, this model takes into account the following unknown variables: susceptible individuals, heroin users, cocaine users and recovered individuals. Each one of these classes can change or interact with others. In this paper, firstly, we give some results on the existence, uniqueness and positivity of solutions. Next, we obtain a threshold value $ r(\Psi'[0]) $ such that an endemic equilibrium exists if $ r(\Psi'[0]) > 1 $. We then show that if $ r(\Psi'[0]) < 1 $, then the disease-free equilibrium is globally asymptotically stable, whereas if $ r(\Psi'[0]) > 1 $, then the system is uniformly persistent. Moreover, for $ r(\Psi'[0]) > 1 $, we show that the endemic equilibrium is globally asymptotically stable under an additional assumption that epidemic parameters for heroin users and cocaine users are same. Finally, some numerical simulations are presented to illustrate our theoretical results.

Citation: 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
References:
[1]

R. M. Anderson, Discussion: The Kermack-McKendrick epidemic threshold theorem, Bull. Math. Biol., 53 (1991), 3-32.  doi: 10.1016/s0092-8240(05)80039-4.

[2]

Z. Bai and S. Zhang, Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1370-1381.  doi: 10.1016/j.cnsns.2014.07.005.

[3]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, Philadelphia, 1994. doi: 10.1137/1.9781611971262.

[4]

M. N. BurattiniE. MassadF. A. B. CoutinhoR. S. Azevedo-NetoR. X. Menezes and L. F. Lopes, A mathematical model of the impact of crack-cocaine use on the prevalence of HIV/AIDS among drug users, Math. Comput. Model., 28 (1998), 21-29.  doi: 10.1016/S0895-7177(98)00095-8.

[5]

A. ChekrounM. N. FriouiT. Kuniya and T. M. Touaoula, Global stability of an age-structured epidemic model with general Lyapunov functional, Math. Biosci. Eng., 16 (2019), 1525-1553.  doi: 10.3934/mbe.2019073.

[6]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000.

[7]

S. DjilaliT. M. Touaoula and S. E. H. Miri, A heroin epidemic model: Very general non linear incidence, treat-age, and global stability, Acta Appl. Math., 152 (2017), 171-194.  doi: 10.1007/s10440-017-0117-2.

[8]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.  doi: 10.1017/S0308210507000455.

[9]

B. FangX. LiM. Martcheva and L. M. Cai, Global stability for a heroin model with two distributed delays, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 715-733.  doi: 10.3934/dcdsb.2014.19.715.

[10]

B. FangX. LiM. Martcheva and L. M. Cai, Global stability for a heroin model with age-dependent susceptibility, J. Syst. Sci. Complex., 28 (2015), 1243-1257.  doi: 10.1007/s11424-015-3243-9.

[11]

B. FangX. 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.

[12]

Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504.

[13]

G. Huang and A. Liu, A note on global stability for a heroin epidemic model with distributed delay, Appl. Math. Lett., 26 (2013), 687-691.  doi: 10.1016/j.aml.2013.01.010.

[14]

H. Inaba, Kermack and McKendrick revisited: The variable susceptibility model for infectious diseases, Japan J. Indust. Appl. Math., 18 (2001), 273-292.  doi: 10.1007/BF03168575.

[15]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721.  doi: 10.1098/rspa.1927.0118.

[16]

M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, Noordhoff Ltd., Groningen, 1964.

[17]

W. T. LiG. LinC. Ma and F. Y. Yang, Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 467-484.  doi: 10.3934/dcdsb.2014.19.467.

[18]

L. Liu and X. Liu, Mathematical analysis for an age-structured heroin epidemic model, Acta Appl. Math., 164 (2019), 193-217.  doi: 10.1007/s10440-018-00234-0.

[19]

J. Liu and T. Zhang, Global behaviour of a heroin epidemic model with distributed delays, Appl. Math. Lett., 24 (2011), 1685-1692.  doi: 10.1016/j.aml.2011.04.019.

[20]

L. LiuX. Liu and J. Wang, Threshold dynamics of a delayed multigroup heroin epidemic model in heterogeneous populations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2615-2630.  doi: 10.3934/dcdsb.2016064.

[21]

X. Liu and J. Wang, Epidemic dynamics on a delayed multi-group heroin epidemic model with nonlinear incidence rate, J. Nonlinear Sci. Appl., 9 (2016), 2149-2160.  doi: 10.22436/jnsa.009.05.20.

[22]

P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 2001 (2001), 35pp.

[23]

C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841.  doi: 10.3934/mbe.2012.9.819.

[24]

G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 218 (2009), 138-141.  doi: 10.1016/j.mbs.2009.01.006.

[25]

G. P. Samanta, Dynamic behaviour for a nonautonomous heroin epidemic model with time delay, J. Appl. Math. Comput., 35 (2011), 161-178.  doi: 10.1007/s12190-009-0349-z.

[26]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118, American Mathematical Society, Providence, RI, 2011.

[27]

H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53 (1993), 1447-1479.  doi: 10.1137/0153068.

[28]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066. 

[29]

W. Wang and X. Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168.  doi: 10.1137/090775890.

[30]

Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.

[31]

P. Weng and X. Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.

[32]

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

[33]

Z. Xu, Traveling waves in a Kermack-Mckendrick epidemic model with diffusion and latent period, Nonlinear Anal., 111 (2014), 66-81.  doi: 10.1016/j.na.2014.08.012.

[34]

J. Yang, X. Li and F. Zhang, Global dynamics of a heroin epidemic model with age structure and nonlinear incidence, Int. J. Biomath., 9 (2016), 20pp. doi: 10.1142/S1793524516500339.

[35]

L. ZhangB. Li and J. Shang, Stability and travelling waves for a time-delayed population system with stage structure, Nonlinear Anal. Real World Appl., 13 (2012), 1429-1440.  doi: 10.1016/j.nonrwa.2011.11.007.

show all references

References:
[1]

R. M. Anderson, Discussion: The Kermack-McKendrick epidemic threshold theorem, Bull. Math. Biol., 53 (1991), 3-32.  doi: 10.1016/s0092-8240(05)80039-4.

[2]

Z. Bai and S. Zhang, Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1370-1381.  doi: 10.1016/j.cnsns.2014.07.005.

[3]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, Philadelphia, 1994. doi: 10.1137/1.9781611971262.

[4]

M. N. BurattiniE. MassadF. A. B. CoutinhoR. S. Azevedo-NetoR. X. Menezes and L. F. Lopes, A mathematical model of the impact of crack-cocaine use on the prevalence of HIV/AIDS among drug users, Math. Comput. Model., 28 (1998), 21-29.  doi: 10.1016/S0895-7177(98)00095-8.

[5]

A. ChekrounM. N. FriouiT. Kuniya and T. M. Touaoula, Global stability of an age-structured epidemic model with general Lyapunov functional, Math. Biosci. Eng., 16 (2019), 1525-1553.  doi: 10.3934/mbe.2019073.

[6]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000.

[7]

S. DjilaliT. M. Touaoula and S. E. H. Miri, A heroin epidemic model: Very general non linear incidence, treat-age, and global stability, Acta Appl. Math., 152 (2017), 171-194.  doi: 10.1007/s10440-017-0117-2.

[8]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.  doi: 10.1017/S0308210507000455.

[9]

B. FangX. LiM. Martcheva and L. M. Cai, Global stability for a heroin model with two distributed delays, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 715-733.  doi: 10.3934/dcdsb.2014.19.715.

[10]

B. FangX. LiM. Martcheva and L. M. Cai, Global stability for a heroin model with age-dependent susceptibility, J. Syst. Sci. Complex., 28 (2015), 1243-1257.  doi: 10.1007/s11424-015-3243-9.

[11]

B. FangX. 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.

[12]

Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504.

[13]

G. Huang and A. Liu, A note on global stability for a heroin epidemic model with distributed delay, Appl. Math. Lett., 26 (2013), 687-691.  doi: 10.1016/j.aml.2013.01.010.

[14]

H. Inaba, Kermack and McKendrick revisited: The variable susceptibility model for infectious diseases, Japan J. Indust. Appl. Math., 18 (2001), 273-292.  doi: 10.1007/BF03168575.

[15]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721.  doi: 10.1098/rspa.1927.0118.

[16]

M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, Noordhoff Ltd., Groningen, 1964.

[17]

W. T. LiG. LinC. Ma and F. Y. Yang, Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 467-484.  doi: 10.3934/dcdsb.2014.19.467.

[18]

L. Liu and X. Liu, Mathematical analysis for an age-structured heroin epidemic model, Acta Appl. Math., 164 (2019), 193-217.  doi: 10.1007/s10440-018-00234-0.

[19]

J. Liu and T. Zhang, Global behaviour of a heroin epidemic model with distributed delays, Appl. Math. Lett., 24 (2011), 1685-1692.  doi: 10.1016/j.aml.2011.04.019.

[20]

L. LiuX. Liu and J. Wang, Threshold dynamics of a delayed multigroup heroin epidemic model in heterogeneous populations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2615-2630.  doi: 10.3934/dcdsb.2016064.

[21]

X. Liu and J. Wang, Epidemic dynamics on a delayed multi-group heroin epidemic model with nonlinear incidence rate, J. Nonlinear Sci. Appl., 9 (2016), 2149-2160.  doi: 10.22436/jnsa.009.05.20.

[22]

P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 2001 (2001), 35pp.

[23]

C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841.  doi: 10.3934/mbe.2012.9.819.

[24]

G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 218 (2009), 138-141.  doi: 10.1016/j.mbs.2009.01.006.

[25]

G. P. Samanta, Dynamic behaviour for a nonautonomous heroin epidemic model with time delay, J. Appl. Math. Comput., 35 (2011), 161-178.  doi: 10.1007/s12190-009-0349-z.

[26]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118, American Mathematical Society, Providence, RI, 2011.

[27]

H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53 (1993), 1447-1479.  doi: 10.1137/0153068.

[28]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066. 

[29]

W. Wang and X. Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168.  doi: 10.1137/090775890.

[30]

Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.

[31]

P. Weng and X. Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.

[32]

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

[33]

Z. Xu, Traveling waves in a Kermack-Mckendrick epidemic model with diffusion and latent period, Nonlinear Anal., 111 (2014), 66-81.  doi: 10.1016/j.na.2014.08.012.

[34]

J. Yang, X. Li and F. Zhang, Global dynamics of a heroin epidemic model with age structure and nonlinear incidence, Int. J. Biomath., 9 (2016), 20pp. doi: 10.1142/S1793524516500339.

[35]

L. ZhangB. Li and J. Shang, Stability and travelling waves for a time-delayed population system with stage structure, Nonlinear Anal. Real World Appl., 13 (2012), 1429-1440.  doi: 10.1016/j.nonrwa.2011.11.007.

Figure 1.  Transfer diagram for model (1)
Figure 2.  The functions $ \beta $, $ \theta_1 $ and $ \theta_2 $ with respect to age $ a $. (left) $ \theta_1 \equiv \theta_2 $ as in (40), (right) $ \theta_1 $ greater than $ \theta_2 $
Figure 3.  The evolution of solution $ S $ and $ R $ with respect to time $ t $ are drawn. The case of disease-free equilibrium with $ r(\Psi'[0]) < 1 $
Figure 4.  The evolution of solutions $ i_1 $ and $ i_2 $ with respect to time $ t $ and age $ a $. The case of disease-free equilibrium with $ r(\Psi'[0]) < 1 $
Figure 5.  The evolution of solution $ S $ and $ R $ with respect to time $ t $ are drawn. The case of endemic equilibrium with $ r(\Psi'[0]) > 1 $
Figure 6.  The evolution of solutions $ i_1 $ and $ i_2 $ with respect to time $ t $ and age $ a $. The case of endemic equilibrium with $ r(\Psi'[0]) > 1 $
Table 1.  Description of each symbol in model (1)
Symbol Description
$ S(t) $ Density of susceptible individuals at time $ t $
$ i_1(t, \xi_1) $ Density of heroin users at time $ t $ and age $ \xi_1 $
$ i_2(t, \xi_2) $ Density of cocaine users at time $ t $ and age $ \xi_2 $
$ R(t) $ Density of recovered individuals at time $ t $
$ A $ Number of all newborns per unit time
$ \mu $ Natural death rate per capita and unit time
$ \beta(\xi_i) $ $ (i=1, 2) $ Transmission rate for drug users with age $ \xi_i $ $ (i=1, 2) $
$ \theta_1(\xi_1) $ Recovery rate from the consumption of heroin at age $ \xi_1 $
$ \theta_2(\xi_2) $ Recovery rate from the consumption of cocaine at age $ \xi_2 $
$ k_1 $ Rate at which an individual recovered from the consumption
of heroin becomes a cocaine user
$ k_2 $ Rate at which an individual recovered from the consumption
of cocaine becomes a heroin user
$ \delta_1 $ Rate at which a recovered individual becomes a heroin user
$ \delta_2 $ Rate at which a recovered individual becomes a cocaine user
Symbol Description
$ S(t) $ Density of susceptible individuals at time $ t $
$ i_1(t, \xi_1) $ Density of heroin users at time $ t $ and age $ \xi_1 $
$ i_2(t, \xi_2) $ Density of cocaine users at time $ t $ and age $ \xi_2 $
$ R(t) $ Density of recovered individuals at time $ t $
$ A $ Number of all newborns per unit time
$ \mu $ Natural death rate per capita and unit time
$ \beta(\xi_i) $ $ (i=1, 2) $ Transmission rate for drug users with age $ \xi_i $ $ (i=1, 2) $
$ \theta_1(\xi_1) $ Recovery rate from the consumption of heroin at age $ \xi_1 $
$ \theta_2(\xi_2) $ Recovery rate from the consumption of cocaine at age $ \xi_2 $
$ k_1 $ Rate at which an individual recovered from the consumption
of heroin becomes a cocaine user
$ k_2 $ Rate at which an individual recovered from the consumption
of cocaine becomes a heroin user
$ \delta_1 $ Rate at which a recovered individual becomes a heroin user
$ \delta_2 $ Rate at which a recovered individual becomes a cocaine user
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