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Research Article

Dynamical analysis of a heroin–cocaine epidemic model with nonlinear incidence and spatial heterogeneity

Article: 2189026 | Received 02 Sep 2022, Accepted 04 Mar 2023, Published online: 15 Mar 2023

References

  • L.J.S. Allen, B.M. Bolker, Y. Lou, and A.L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A. 21 (1) (2008), pp. 1–20.
  • S. Bentout, S. Djilali, and B. Ghanbari, Backward, Hopf bifurcation in a heroin epidemic model with treat age, Int. J. Model. Simul. Sci. Comput. 12(02) (2021), pp. 1–22.
  • A. Chekroun, M.N. Frioui, T. Kuniya, and T.M. Touaoula, Mathematical analysis of an age structured heroin-cocaine epidemic model, Discrete Contin. Dyn. Syst. B. 25(11) (2020), pp. 4449–4477.
  • R. Cui, K. Lam, and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differ. Equ. 263 (4) (2017), pp. 2343–2373.
  • A. Din and Y. Li, Controlling heroin addiction via age-structured modeling, Adv. Differ. Equ. 2020 (1) (2020), pp. 1–17.
  • S. Djilali, A. Zeb, and T. Saeed, Effect of occasional heroin consumers on the spread of heroin addiction, Fractals 30 (05) (2022), pp. 1–12.
  • Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Vol. 1, in: Maximum Principles and Applications, World Scientific, New Jersey, 2006.
  • X. Duan, X. Li, and M. Martcheva, Qualitative analysis on a diffusive age-structured heroin transmission model, Nonlinear Anal. RWA. 54 (2020), pp. 1–26.
  • B. Fang, X. Li, M. Martcheva, and L.M. Cai, Global stability for a heroin model with two distributed delays, Discrete Contin. Dyn. Syst. Ser. B. 19 (2014), pp. 715–733.
  • B. Fang, X. Li, M. Martcheva, and L.M. Cai, Global asymptotic properties of a heroin epidemic model with treat-age, Appl. Math. Comput. 263 (2015), pp. 315–331.
  • B. Fang, X. Li, M. Martcheva, and L.M. Cai, Global stability for a heroin model with age-dependent susceptibility, J. Syst. Sci. Complex. 28 (6) (2015), pp. 1243–1257.
  • R.B. Guenther and J.W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Dover Publica-tions, 1996.
  • Z. Guo, F.B. Wang, and X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and nonlocal infections, J. Math. Biol. 65 (6-7) (2012), pp. 1387–1410.
  • J.K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988.
  • P.T. Harrell, B.E. Manchaa, H. Petras, R.C. Trenz, and W.W. Latimer, Latent classes of heroin and cocaine users predict unique HIV/HCV risk factors, Drug. Alcohol. Depend. 122 (3) (2012), pp. 220–227.
  • P Hess, Pitman Research Notes in Mathematics Series; Vol. 247, Periodic-Parabolic Boundary Value Problems and Positivity. Longman Scientific and Technical; New York: 1991.
  • G. Huang and A. Liu, A note on global stability for a heroin epidemic model with distributed delay, Appl. Math. Lett. 26 (7) (2013), pp. 687–691.
  • S. Kundu, N. Kumari, and S. Kouachi, et al. Stability and bifurcation analysis of a heroin model with diffusion, delay and nonlinear incidence rate, Model. Earth Syst. Environ. 8 (1) (2022), pp. 1351–1362.
  • L. Liu and X. Liu, Mathematical analysis for an age-structured heroin epidemic model, Acta. Appl. Math. 164 (1) (2019), pp. 193–217.
  • L. Liu, X. Liu, and J. Wang, Threshold dynamics of a delayed multigroup heroin epidemic model in heterogeneous populations, Discrete Contin. Dyn. Syst. Ser. B. 21 (8) (2016), pp. 2615–2630.
  • M. Ma, S. Liu, and J. Li, Bifurcation of a heroin model with nonlinear incidence rate, Nonlinear Dyn.88(1) (2017), pp. 555–565.
  • M. Ma, S. Liu, and J. Li, Does media coverage influence the spread of drug addiction?, Commun. Nonlinear Sci. Numer. Simul. 50 (2017), pp. 169–179.
  • M. Ma, S. Liu, and H. Xiang, et al. Dynamics of synthetic drugs transmission model with psychological addicts and general incidence rate, Physica A. 491 (2018), pp. 641–649.
  • D.R. Mackintosh and G.T. Stewart, A mathematical model of a heroin epidemic: implications for control policies, J. Epidemiol. Community Health. 33(4) (1979), pp. 299–304.
  • P. Magal and X.Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal. 37 (1) (2005), pp. 251–275.
  • P. Magal, G. Webb, and Y. Wu, On a vector-host epidemic model with spatial structure, Nonlinearity31 (12) (2018), pp. 5589–5614.
  • R.H. Martin and H.L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc. 321 (1990), pp. 1–44.
  • M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
  • X. Ren, Y. Tian, L. Liu, and X. Liu, A reaction-diffusion within host HIV model with cell-to-cell transmission, J. Math. Biol. 76 (7) (2018), pp. 1831–1872.
  • G.P. Samanta, Dynamic behaviour for a nonautonomous heroin epidemic model with time delay, J. Appl. Math. Comput. 35 (1-2) (2011), pp. 161–178.
  • H.L. Smith, Monotone dynamic systems: an introduction to the theory of competitive and cooperative systems, in Math Surveys Monogr, Am Math Soc Providence RI, 1995.
  • H.L. Smith and X.Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal. 47 (9) (2001), pp. 6169–6179.
  • P. Song, Y. Lou, and Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differ. Equ. 267 (9) (2019), pp. 5084–5114.
  • H.R. Thieme, Convergence results and a Poincar–Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol. 30 (7) (1992), pp. 755–763.
  • M. Wang, Nonlinear Elliptic Equations, Science Press, Beijing, 2010.
  • J. Wang and R. Cui, Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates, Adv. Nonlinear Anal. 10(1) (2021), pp. 922–951.
  • J. Wang and H. Sun, Analysis of a diffusive heroin epidemic model in a heterogeneous environment, Complexity 2020 (2020), pp. 1–12.
  • W. Wang and X.Q. Zhao, Basic reproduction numbers for reaction–diffusion epidemic models, SIAM J. Appl. Dyn. Syst. 11 (4) (2012), pp. 1652–1673.
  • J. Wang, J. Wang, and T. Kuniya, Analysis of an age-structured multi-group heroin epidemic model, Appl. Math. Comput. 347 (2019), pp. 78–100.
  • G Webb, Monographs and Textbooks in Pure and Applied Math Series; Vol. 89, Theory of Nonlinear Age-Dependent Population Dynamics. Dekker; New York: 1985.
  • E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci. 208 (1) (2007), pp. 312–324.
  • J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.
  • Y. Wu and X. Zou, Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates, J. Differ. Equ. 264 (8) (2018), pp. 4989–5024.
  • J. Xu and Y. Geng, Global stability for a heroin epidemic model in a critical case, Appl. Math. Lett.121 (2021), pp. 1–6.
  • Y. Yang, J. Zhou, and C.H. Hsu, Threshold dynamics of a diffusive SIRI model with nonlinear incidence rate, J. Math. Anal. Appl. 478(2) (2019), pp. 874–896.
  • Z. Zhang and Y. Wang, Hopf bifurcation of a heroin model with time delay and saturated treatment function, Adv. Differ. Equ. 2019 (2019), pp. 1–16.
  • Z. Zhang, F. Yang, and W. Xia, Hopf bifurcation analysis of a synthetic drug transmission model with time delays, Complexity 2019 (2019), pp. 1–17.