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September  2017, 22(7): 2795-2812. doi: 10.3934/dcdsb.2017151

Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

2. 

China, School of Mathematical Science, Heilongjiang University, Harbin 150080, China

3. 

Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe-shi, Hyogo 657-0067, Japan

corresponding author: liuxn@swu.edu.cn (X. Liu)

Received  September 2013 Revised  March 2017 Published  May 2017

In this paper, we investigate the global asymptotic stability of multi-group SIR and SEIR age-structured models. These models allow the infectiousness and the death rate of susceptible individuals to vary and depend on the susceptibility, with which we can consider the heterogeneity of population. We establish global dynamics and demonstrate that the heterogeneity does not alter the dynamical structure of the basic SIR and SEIR with age-dependent susceptibility. Our results also demonstrate that, for age structured multi-group models considered, the graph-theoretic approach can be successfully applied by choosing an appropriate weighted matrix as well.

Citation: Jinliang Wang, Xianning Liu, Toshikazu Kuniya, Jingmei Pang. Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2795-2812. doi: 10.3934/dcdsb.2017151
References:
[1]

V. Andreasen, Instability in an SIR-model with age-dependent susceptibility, in: O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, in: Theory of Epidemics, Wuerz Publ., Winnipeg, (1995), 3-14.   Google Scholar

[2]

H. Andersson and T. Britton, Heterogeneity in epidemic models and its effect on the spread of infection, J. App. Prob., 35 (1998), 651-661.  doi: 10.1017/S0021900200016302.  Google Scholar

[3]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control Oxford University Press, Oxford, 1991. Google Scholar

[4]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences Academic Press, New York, 1979.  Google Scholar

[5]

S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics Springer, Berlin, 1993. doi: 10.1007/978-3-642-75301-5.  Google Scholar

[6]

S. N. BusenbergM. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080.  doi: 10.1137/0522069.  Google Scholar

[7]

Y. ChaM. Iannelli and F. A. Milner, Stability change of an epidemic model, Dynam. Systems Appl., 9 (2000), 361-376.   Google Scholar

[8]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation John Wiley and Sons, Chichester, 2000.  Google Scholar

[9]

H. GuoM. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canad. Appl. Math. Quart., 14 (2006), 259-284.   Google Scholar

[10]

H. GuoM. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.  doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar

[11]

T. J. HagenaarsC. A. Donnelly and N. M. Ferguson, Spatial heterogeneity and the persistence of infectious diseases, J. Theor. Biol., 229 (2004), 349-359.  doi: 10.1016/j.jtbi.2004.04.002.  Google Scholar

[12]

J. K. Hale, Asymptotic Behavior of Dissipative Systems in: Math. Surv. Monogr. , vol. 25, Am. Math. Soc. , Providence, RI, 1988.  Google Scholar

[13]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[14]

J. M. HymanJ. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155 (1999), 77-109.  doi: 10.1016/S0025-5564(98)10057-3.  Google Scholar

[15]

H. Inaba, Endemic threshold results in an age-duration-structured population model for HIV infection, Math. Biosci., 201 (2006), 15-47.  doi: 10.1016/j.mbs.2005.12.017.  Google Scholar

[16]

H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434.  doi: 10.1007/BF00178326.  Google Scholar

[17]

A. KorobeinikovP. K. Maini and W. J. Walker, Estimation of effective vaccination rate: Pertussis in New Zealand as a case study, J. Theor. Biol., 224 (2003), 269-275.  doi: 10.1016/S0022-5193(03)00163-2.  Google Scholar

[18]

T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Anal. RWA, 12 (2011), 2640-2655.  doi: 10.1016/j.nonrwa.2011.03.011.  Google Scholar

[19]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236.  doi: 10.1016/0025-5564(76)90125-5.  Google Scholar

[20]

M. Y. LiZ. 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.  Google Scholar

[21]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[22]

A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11.  doi: 10.1006/jtbi.1996.0042.  Google Scholar

[23]

P. MagalC. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.  doi: 10.1080/00036810903208122.  Google Scholar

[24]

P. Magal, Compact attractors for time periodic age-structured population models, Electron. J. Differential Equations, 65 (2001), 1-35.   Google Scholar

[25]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models Memoirs of the American Mathematical Society, 202 (2009), vi+71 pp. doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[26]

P. Magal and H. R. Thieme, Eventual compactness for semiflows generated by nonlinear age-structured models, Commun. Pure Appl. Anal., 3 (2004), 695-727.  doi: 10.3934/cpaa.2004.3.695.  Google Scholar

[27]

A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44 (1925), 98-130.  doi: 10.1017/S0013091500034428.  Google Scholar

[28]

C. C. McCluskey, Delay versus age-of-infection-Global stability, Applied Math. Comput., 217 (2010), 3046-3049.  doi: 10.1016/j.amc.2010.08.037.  Google Scholar

[29]

A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, Math. Biosci. Eng., 10 (2013), 369-378.  doi: 10.3934/mbe.2013.10.369.  Google Scholar

[30]

D. Schenzle, An age-structured model of pre-and post-vaccination measles transmission, IMA J. Math. Appl. Med. Biol., 1 (1984), 169-191.  doi: 10.1093/imammb/1.2.169.  Google Scholar

[31]

H. ShuD. 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.  Google Scholar

[32]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence Amer. Math. Soc. , Providence, 2011.  Google Scholar

[33]

R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010), 2286-2291.  doi: 10.1016/j.camwa.2010.08.020.  Google Scholar

[34]

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

[35]

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.  Google Scholar

[36]

J. WangJ. ZuX. LiuG. Huang and J. Zhang, Global dynamics of a multi-group epidemic model with general relapse distribution and nonlinear incidence rate, J. Biol. Syst., 20 (2012), 235-258.  doi: 10.1142/S021833901250009X.  Google Scholar

[37]

J. WangX. LiuJ. 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.   Google Scholar

[38]

Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. RWA, 11 (2010), 995-1004.  doi: 10.1016/j.nonrwa.2009.01.040.  Google Scholar

[39]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics Marcel Dekker, New York, 1985.  Google Scholar

show all references

References:
[1]

V. Andreasen, Instability in an SIR-model with age-dependent susceptibility, in: O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, in: Theory of Epidemics, Wuerz Publ., Winnipeg, (1995), 3-14.   Google Scholar

[2]

H. Andersson and T. Britton, Heterogeneity in epidemic models and its effect on the spread of infection, J. App. Prob., 35 (1998), 651-661.  doi: 10.1017/S0021900200016302.  Google Scholar

[3]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control Oxford University Press, Oxford, 1991. Google Scholar

[4]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences Academic Press, New York, 1979.  Google Scholar

[5]

S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics Springer, Berlin, 1993. doi: 10.1007/978-3-642-75301-5.  Google Scholar

[6]

S. N. BusenbergM. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080.  doi: 10.1137/0522069.  Google Scholar

[7]

Y. ChaM. Iannelli and F. A. Milner, Stability change of an epidemic model, Dynam. Systems Appl., 9 (2000), 361-376.   Google Scholar

[8]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation John Wiley and Sons, Chichester, 2000.  Google Scholar

[9]

H. GuoM. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canad. Appl. Math. Quart., 14 (2006), 259-284.   Google Scholar

[10]

H. GuoM. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.  doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar

[11]

T. J. HagenaarsC. A. Donnelly and N. M. Ferguson, Spatial heterogeneity and the persistence of infectious diseases, J. Theor. Biol., 229 (2004), 349-359.  doi: 10.1016/j.jtbi.2004.04.002.  Google Scholar

[12]

J. K. Hale, Asymptotic Behavior of Dissipative Systems in: Math. Surv. Monogr. , vol. 25, Am. Math. Soc. , Providence, RI, 1988.  Google Scholar

[13]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[14]

J. M. HymanJ. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155 (1999), 77-109.  doi: 10.1016/S0025-5564(98)10057-3.  Google Scholar

[15]

H. Inaba, Endemic threshold results in an age-duration-structured population model for HIV infection, Math. Biosci., 201 (2006), 15-47.  doi: 10.1016/j.mbs.2005.12.017.  Google Scholar

[16]

H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434.  doi: 10.1007/BF00178326.  Google Scholar

[17]

A. KorobeinikovP. K. Maini and W. J. Walker, Estimation of effective vaccination rate: Pertussis in New Zealand as a case study, J. Theor. Biol., 224 (2003), 269-275.  doi: 10.1016/S0022-5193(03)00163-2.  Google Scholar

[18]

T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Anal. RWA, 12 (2011), 2640-2655.  doi: 10.1016/j.nonrwa.2011.03.011.  Google Scholar

[19]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236.  doi: 10.1016/0025-5564(76)90125-5.  Google Scholar

[20]

M. Y. LiZ. 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.  Google Scholar

[21]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[22]

A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11.  doi: 10.1006/jtbi.1996.0042.  Google Scholar

[23]

P. MagalC. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.  doi: 10.1080/00036810903208122.  Google Scholar

[24]

P. Magal, Compact attractors for time periodic age-structured population models, Electron. J. Differential Equations, 65 (2001), 1-35.   Google Scholar

[25]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models Memoirs of the American Mathematical Society, 202 (2009), vi+71 pp. doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[26]

P. Magal and H. R. Thieme, Eventual compactness for semiflows generated by nonlinear age-structured models, Commun. Pure Appl. Anal., 3 (2004), 695-727.  doi: 10.3934/cpaa.2004.3.695.  Google Scholar

[27]

A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44 (1925), 98-130.  doi: 10.1017/S0013091500034428.  Google Scholar

[28]

C. C. McCluskey, Delay versus age-of-infection-Global stability, Applied Math. Comput., 217 (2010), 3046-3049.  doi: 10.1016/j.amc.2010.08.037.  Google Scholar

[29]

A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, Math. Biosci. Eng., 10 (2013), 369-378.  doi: 10.3934/mbe.2013.10.369.  Google Scholar

[30]

D. Schenzle, An age-structured model of pre-and post-vaccination measles transmission, IMA J. Math. Appl. Med. Biol., 1 (1984), 169-191.  doi: 10.1093/imammb/1.2.169.  Google Scholar

[31]

H. ShuD. 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.  Google Scholar

[32]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence Amer. Math. Soc. , Providence, 2011.  Google Scholar

[33]

R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010), 2286-2291.  doi: 10.1016/j.camwa.2010.08.020.  Google Scholar

[34]

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

[35]

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.  Google Scholar

[36]

J. WangJ. ZuX. LiuG. Huang and J. Zhang, Global dynamics of a multi-group epidemic model with general relapse distribution and nonlinear incidence rate, J. Biol. Syst., 20 (2012), 235-258.  doi: 10.1142/S021833901250009X.  Google Scholar

[37]

J. WangX. LiuJ. 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.   Google Scholar

[38]

Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. RWA, 11 (2010), 995-1004.  doi: 10.1016/j.nonrwa.2009.01.040.  Google Scholar

[39]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics Marcel Dekker, New York, 1985.  Google Scholar

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