# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] M. N. Burattini, E. Massad, F. A. B. Coutinho, R. S. Azevedo-Neto, R. 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.  Google Scholar [5] A. Chekroun, M. N. Frioui, T. 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.  Google Scholar [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.  Google Scholar [7] S. Djilali, T. 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.  Google Scholar [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.  Google Scholar [9] 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), 715-733.  doi: 10.3934/dcdsb.2014.19.715.  Google Scholar [10] 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 (2015), 1243-1257.  doi: 10.1007/s11424-015-3243-9.  Google Scholar [11] 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), 315-331.  doi: 10.1016/j.amc.2015.04.055.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, Noordhoff Ltd., Groningen, 1964.  Google Scholar [17] W. T. Li, G. Lin, C. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] 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 (2016), 2615-2630.  doi: 10.3934/dcdsb.2016064.  Google Scholar [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.  Google Scholar [22] P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 2001 (2001), 35pp.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118, American Mathematical Society, Providence, RI, 2011.  Google Scholar [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.  Google Scholar [28] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] L. Zhang, B. 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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] M. N. Burattini, E. Massad, F. A. B. Coutinho, R. S. Azevedo-Neto, R. 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.  Google Scholar [5] A. Chekroun, M. N. Frioui, T. 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.  Google Scholar [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.  Google Scholar [7] S. Djilali, T. 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.  Google Scholar [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.  Google Scholar [9] 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), 715-733.  doi: 10.3934/dcdsb.2014.19.715.  Google Scholar [10] 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 (2015), 1243-1257.  doi: 10.1007/s11424-015-3243-9.  Google Scholar [11] 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), 315-331.  doi: 10.1016/j.amc.2015.04.055.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, Noordhoff Ltd., Groningen, 1964.  Google Scholar [17] W. T. Li, G. Lin, C. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] 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 (2016), 2615-2630.  doi: 10.3934/dcdsb.2016064.  Google Scholar [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.  Google Scholar [22] P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 2001 (2001), 35pp.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118, American Mathematical Society, Providence, RI, 2011.  Google Scholar [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.  Google Scholar [28] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] L. Zhang, B. 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.  Google Scholar
Transfer diagram for model (1)
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 $\theta_2$
The evolution of solution $S$ and $R$ with respect to time $t$ are drawn. The case of disease-free equilibrium with $r(\Psi'[0]) < 1$
The evolution of solutions $i_1$ and $i_2$ with respect to time $t$ and age $a$. The case of disease-free equilibrium with $r(\Psi'[0]) < 1$
The evolution of solution $S$ and $R$ with respect to time $t$ are drawn. The case of endemic equilibrium with $r(\Psi'[0]) > 1$
The evolution of solutions $i_1$ and $i_2$ with respect to time $t$ and age $a$. The case of endemic equilibrium with $r(\Psi'[0]) > 1$
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
 [1] Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859 [2] Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age. Mathematical Biosciences & Engineering, 2014, 11 (3) : 449-469. doi: 10.3934/mbe.2014.11.449 [3] Bin Fang, Xue-Zhi Li, Maia Martcheva, Li-Ming Cai. Global stability for a heroin model with two distributed delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 715-733. doi: 10.3934/dcdsb.2014.19.715 [4] Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva. Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1159-1186. doi: 10.3934/mbe.2017060 [5] Miljana Jovanović, Vuk Vujović. Stability of stochastic heroin model with two distributed delays. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2407-2432. doi: 10.3934/dcdsb.2020016 [6] Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641 [7] Jinliang Wang, Xiu Dong. Analysis of an HIV infection model incorporating latency age and infection age. Mathematical Biosciences & Engineering, 2018, 15 (3) : 569-594. doi: 10.3934/mbe.2018026 [8] Andrey V. Melnik, Andrei Korobeinikov. Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility. Mathematical Biosciences & Engineering, 2013, 10 (2) : 369-378. doi: 10.3934/mbe.2013.10.369 [9] Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643 [10] C. Connell McCluskey. Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 381-400. doi: 10.3934/mbe.2015008 [11] Jinliang Wang, Jiying Lang, Yuming Chen. Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3721-3747. doi: 10.3934/dcdsb.2017186 [12] Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971 [13] Fred Brauer, Zhisheng Shuai, P. van den Driessche. Dynamics of an age-of-infection cholera model. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1335-1349. doi: 10.3934/mbe.2013.10.1335 [14] Jinliang Wang, Ran Zhang, Toshikazu Kuniya. A note on dynamics of an age-of-infection cholera model. Mathematical Biosciences & Engineering, 2016, 13 (1) : 227-247. doi: 10.3934/mbe.2016.13.227 [15] Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525 [16] Yu Yang, Yueping Dong, Yasuhiro Takeuchi. Global dynamics of a latent HIV infection model with general incidence function and multiple delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 783-800. doi: 10.3934/dcdsb.2018207 [17] Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with generalized nonlinear incidence and vaccination age. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 977-996. doi: 10.3934/dcdsb.2016.21.977 [18] Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083 [19] Shanjing Ren. Global stability in a tuberculosis model of imperfect treatment with age-dependent latency and relapse. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1337-1360. doi: 10.3934/mbe.2017069 [20] C. Connell McCluskey. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Mathematical Biosciences & Engineering, 2012, 9 (4) : 819-841. doi: 10.3934/mbe.2012.9.819

2020 Impact Factor: 1.327

## Metrics

• PDF downloads (250)
• HTML views (275)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]