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Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations
1. | Key Laboratory of Eco-environments in Three Gorges Reservoir Region, (Ministry of Education), School of Mathematics and Statistics, Southwest University, Chongqing 400715, China, China |
2. | School of Mathematical Science, Heilongjiang University, Harbin 150080, China |
References:
[1] |
N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Springer, Berlin, 1967. |
[2] |
O. Diekmann, J. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
[3] |
M. C. Eisenberg, Z. Shuai, J. H. Tien and P. van den Driessche, A cholera model in a patchy environment with water and human movement, Math. Biosci., 246 (2013), 105-112.
doi: 10.1016/j.mbs.2013.08.003. |
[4] |
B. Fang, X. Li, M. Martcheva and L. Cai, Global stability for a heroin model with two distributed delays, Discrete. Cont. Dyn. Sys. B., 19 (2014), 715-733.
doi: 10.3934/dcdsb.2014.19.715. |
[5] |
X. Feng, Z. Teng and F. Zhang, Global dynamics of a general class of multi-group epidemic models with latency and relapse, Math. Biosci. Eng., 12 (2015), 99-115. |
[6] |
H. Guo, M. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, P. Am. Math. Soc., 136 (2008), 2793-2802.
doi: 10.1090/S0002-9939-08-09341-6. |
[7] |
Y. Hino, T. Naito and S. Murakami, Functional Differential Equations with Infinite Delay, Springer, 1991.
doi: 10.1007/BFb0084432. |
[8] |
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. |
[9] |
G. Huang, J. Wang and J. Zu, Global dynamics of multi-group dengue disease model with latency distributions, Math. Meth. Appl. Sci., 38 (2015), 2703-2718.
doi: 10.1002/mma.3252. |
[10] |
M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equations, 248 (2010), 1-20.
doi: 10.1016/j.jde.2009.09.003. |
[11] |
M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47.
doi: 10.1016/j.jmaa.2009.09.017. |
[12] |
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. |
[13] |
D. R. MacKintosh and G. T. Stewart, A mathematical model of a heroin epidemic: Implications for control policies, J. Epidemiol. Commun. H., 33 (1979), 299-304.
doi: 10.1136/jech.33.4.299. |
[14] |
C. C. McCluskey, Global stability for an {SEIR} epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.
doi: 10.3934/mbe.2009.6.603. |
[15] |
G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 218 (2009), 138-141.
doi: 10.1016/j.mbs.2009.01.006. |
[16] |
Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes, Nonlinear Anal. RWA., 14 (2013), 1693-1704.
doi: 10.1016/j.nonrwa.2012.11.005. |
[17] |
Y. Muroya and T. Kuniya, Further stability analysis for a multi-group {SIRS} epidemic model with varying total population size, Appl. Math. Lett., 38 (2014), 73-78.
doi: 10.1016/j.aml.2014.07.005. |
[18] |
G. 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. |
[19] |
H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. RWA., 13 (2012), 1581-1592.
doi: 10.1016/j.nonrwa.2011.11.016. |
[20] |
Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513-1532.
doi: 10.1137/120876642. |
[21] |
H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.
doi: 10.1137/0524026. |
[22] |
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[23] |
J. Wang and X. Liu, Modeling diseases with latency and nonlinear incidence rates: Global dynamics of a multi-group model, Math. Meth. Appl. Sci., 39 (2016), 1964-1976.
doi: 10.1002/mma.3613. |
[24] |
J. Wang, X. Liu, J. Pang and D. Hou, Global dynamics of a multi-group epidemic model with general exposed distribution and relapse, Osaka J. Math., 52 (2015), 117-138. |
[25] |
J. Wang, J. Pang and X. Liu, Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model, J. Biol. Dynam., 8 (2014), 99-116.
doi: 10.1080/17513758.2014.912682. |
[26] |
J. Wang and H. Shu, Global dynamics of a multi-group epidemic model with latency, relapse and nonlinear incidence rate, Math. Biosci. Eng., 13 (2016), 209-225.
doi: 10.3934/mbe.2016.13.209. |
[27] |
W. Wang and X.-Q. Zhao, An epidemic model with population dispersal and infection period, SIAM J. Appl. Math., 66 (2006), 1454-1472.
doi: 10.1137/050622948. |
[28] |
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. |
show all references
References:
[1] |
N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Springer, Berlin, 1967. |
[2] |
O. Diekmann, J. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
[3] |
M. C. Eisenberg, Z. Shuai, J. H. Tien and P. van den Driessche, A cholera model in a patchy environment with water and human movement, Math. Biosci., 246 (2013), 105-112.
doi: 10.1016/j.mbs.2013.08.003. |
[4] |
B. Fang, X. Li, M. Martcheva and L. Cai, Global stability for a heroin model with two distributed delays, Discrete. Cont. Dyn. Sys. B., 19 (2014), 715-733.
doi: 10.3934/dcdsb.2014.19.715. |
[5] |
X. Feng, Z. Teng and F. Zhang, Global dynamics of a general class of multi-group epidemic models with latency and relapse, Math. Biosci. Eng., 12 (2015), 99-115. |
[6] |
H. Guo, M. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, P. Am. Math. Soc., 136 (2008), 2793-2802.
doi: 10.1090/S0002-9939-08-09341-6. |
[7] |
Y. Hino, T. Naito and S. Murakami, Functional Differential Equations with Infinite Delay, Springer, 1991.
doi: 10.1007/BFb0084432. |
[8] |
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. |
[9] |
G. Huang, J. Wang and J. Zu, Global dynamics of multi-group dengue disease model with latency distributions, Math. Meth. Appl. Sci., 38 (2015), 2703-2718.
doi: 10.1002/mma.3252. |
[10] |
M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equations, 248 (2010), 1-20.
doi: 10.1016/j.jde.2009.09.003. |
[11] |
M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47.
doi: 10.1016/j.jmaa.2009.09.017. |
[12] |
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. |
[13] |
D. R. MacKintosh and G. T. Stewart, A mathematical model of a heroin epidemic: Implications for control policies, J. Epidemiol. Commun. H., 33 (1979), 299-304.
doi: 10.1136/jech.33.4.299. |
[14] |
C. C. McCluskey, Global stability for an {SEIR} epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.
doi: 10.3934/mbe.2009.6.603. |
[15] |
G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 218 (2009), 138-141.
doi: 10.1016/j.mbs.2009.01.006. |
[16] |
Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes, Nonlinear Anal. RWA., 14 (2013), 1693-1704.
doi: 10.1016/j.nonrwa.2012.11.005. |
[17] |
Y. Muroya and T. Kuniya, Further stability analysis for a multi-group {SIRS} epidemic model with varying total population size, Appl. Math. Lett., 38 (2014), 73-78.
doi: 10.1016/j.aml.2014.07.005. |
[18] |
G. 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. |
[19] |
H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. RWA., 13 (2012), 1581-1592.
doi: 10.1016/j.nonrwa.2011.11.016. |
[20] |
Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513-1532.
doi: 10.1137/120876642. |
[21] |
H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.
doi: 10.1137/0524026. |
[22] |
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[23] |
J. Wang and X. Liu, Modeling diseases with latency and nonlinear incidence rates: Global dynamics of a multi-group model, Math. Meth. Appl. Sci., 39 (2016), 1964-1976.
doi: 10.1002/mma.3613. |
[24] |
J. Wang, X. Liu, J. Pang and D. Hou, Global dynamics of a multi-group epidemic model with general exposed distribution and relapse, Osaka J. Math., 52 (2015), 117-138. |
[25] |
J. Wang, J. Pang and X. Liu, Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model, J. Biol. Dynam., 8 (2014), 99-116.
doi: 10.1080/17513758.2014.912682. |
[26] |
J. Wang and H. Shu, Global dynamics of a multi-group epidemic model with latency, relapse and nonlinear incidence rate, Math. Biosci. Eng., 13 (2016), 209-225.
doi: 10.3934/mbe.2016.13.209. |
[27] |
W. Wang and X.-Q. Zhao, An epidemic model with population dispersal and infection period, SIAM J. Appl. Math., 66 (2006), 1454-1472.
doi: 10.1137/050622948. |
[28] |
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. |
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