• Previous Article
    The vanishing surface tension limit for the Hele-Shaw problem
  • DCDS-B Home
  • This Issue
  • Next Article
    Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes
December  2016, 21(10): 3515-3550. doi: 10.3934/dcdsb.2016109

A multi-group SIR epidemic model with age structure

1. 

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

2. 

School of Mathematical Science, Heilongjiang University, Harbin 150080

3. 

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914

Received  September 2013 Revised  August 2016 Published  November 2016

This paper provides the first detailed analysis of a multi-group SIR epidemic model with age structure, which is given by a nonlinear system of $3n$ partial differential equations. The basic reproduction number $\mathcal{R}_0$ is obtained as the spectral radius of the next generation operator, and it is shown that if $\mathcal{R}_0 < 1$, then the disease-free equilibrium is globally asymptotically stable, while if $\mathcal{R}_0 >1$, then an endemic equilibrium exists. The global asymptotic stability of the endemic equilibrium is also shown under additional assumptions such that the transmission coefficient is independent from the age of infective individuals and the mortality and removal rates are constant. To our knowledge, this is the first paper which applies the method of Lyapunov functional and graph theory to a multi-dimensional PDE system.
Citation: Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109
References:
[1]

V. Andreasen, Instability in an SIR-model with age-dependent susceptibility,, in Mathematical Population Dynamics: Analysis of Heterogeneity (eds. O. Arino, (1995), 3.

[2]

E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model,, in Mathematical Ecology (eds. T.G. Hallam, (1988), 317.

[3]

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

[4]

Y. Cha, M. Iannelli and F. A. Milner, Stability change of an epidemic model,, Dynam. Syst. Appl., 9 (2000), 361.

[5]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. 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. doi: 10.1007/BF00178324.

[6]

O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics,, Princeton University Press, (2013).

[7]

K. Dietz, Transmission and control of arbovirus disease,, in Proc. SIMS Conf. on Epidemiology (eds. D. Ludwig and K.L. Cooke), (1975), 104.

[8]

Z. Feng, W. Huang and C. Castillo-Chavez, Global behavior of a multi-group SIS epidemic model with age structure,, J. Diff. Equat., 218 (2005), 292. doi: 10.1016/j.jde.2004.10.009.

[9]

A. Franceschetti, A. Pugliese and D. Breda, Multiple endemic states in age-structured SIR epidemic models,, Math. Biosci. Eng., 9 (2012), 577. doi: 10.3934/mbe.2012.9.577.

[10]

D. Greenhalgh, Analytical results on the stability of age-structured recurrent epidemic models,, IMA J. Math. Appl. Med. Biol., 4 (1987), 109. doi: 10.1093/imammb/4.2.109.

[11]

G. Gripenberg, On a nonlinear integral equation modelling an epidemic in an age-structured population,, J. reine angew. Math., 341 (1983), 54. doi: 10.1515/crll.1983.341.54.

[12]

H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models,, Canadian Appl. Math. Quart., 14 (2006), 259.

[13]

H. J. A. M. Heijmans, The dynamical behaviour of the age-size-distribution of a cell population,, in The Dynamics of Physiologically Structured Populations (eds. J.A.J. Metz and O. Diekmann), 68 (1986), 185. doi: 10.1007/978-3-662-13159-6_5.

[14]

H. W. Hethcote, An immunization model for a heterogeneous population,, Theor. Popul. Biol., 14 (1978), 338. doi: 10.1016/0040-5809(78)90011-4.

[15]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907.

[16]

F. Hoppensteadt, An age dependent epidemic model,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4.

[17]

H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process,, Math. Popul. Studies, 1 (1988), 49. doi: 10.1080/08898488809525260.

[18]

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

[19]

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

[20]

H. Inaba and H. Nishiura, The basic reproduction number of an infectious disease in a stable population: the impact of population growth rate on the eradication threshold,, Math. Model. Nat. Phenom., 3 (2008), 194. doi: 10.1051/mmnp:2008050.

[21]

H. Inaba, The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments,, Math. Biosci. Eng., 9 (2012), 313. doi: 10.3934/mbe.2012.9.313.

[22]

H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments,, J. Math. Biol., 65 (2012), 309. doi: 10.1007/s00285-011-0463-z.

[23]

T. Kato, Perturbation Theory for Linear Operators,, 2nd edition, (1995).

[24]

K. Kawachi, Deterministic models for rumor transmission,, Nonlinear Analysis RWA., 9 (2008), 1989. doi: 10.1016/j.nonrwa.2007.06.004.

[25]

A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages,, Bull. Math. Biol., 71 (2009), 75. doi: 10.1007/s11538-008-9352-z.

[26]

M. A. Krasnoselskii, Positive Solutions of Operator Equations,, 1st edition, (1964).

[27]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space,, Am. Math. Soc. Transl., 1950 (1950).

[28]

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

[29]

J. P. Lasalle, The Stability of Dynamical Systems,, 2nd edition, (1976).

[30]

X. Z. Li, J. X. Liu and M. Martcheva, An age-structured two-strain epidemic model with super-infection,, Math. Biosci. Eng., 7 (2010), 123. doi: 10.3934/mbe.2010.7.123.

[31]

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

[32]

I. Marek, Frobenius theory of positive operators: Comparison theorems and applications,, SIAM J. Appl. Math., 19 (1970), 607. doi: 10.1137/0119060.

[33]

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. doi: 10.3934/mbe.2012.9.819.

[34]

A. G. McKendrick, Application of mathematics to medical problems,, Proc. Edinburgh Math. Soc., (1925), 98. doi: 10.1017/S0013091500034428.

[35]

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. doi: 10.3934/mbe.2013.10.369.

[36]

R. Nagel, One-Parameter Semigroups of Positive Operators,, 1st edition, (1986). doi: 10.1007/BFb0074922.

[37]

I. Sawashima, On spectral properties of some positive operators,, Nat. Sci. Rep. Ochanomizu Univ., 15 (1964), 53.

[38]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, 1st edition, (2011).

[39]

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

[40]

H. R. Thieme, Stability change of the endemic equilibrium in age-structured models for the spread of S-I-R type infectious diseases,, in Differential Equations Models in Biology, 92 (1991), 139. doi: 10.1007/978-3-642-45692-3_10.

[41]

D. W. Tudor, An age-dependent epidemic model with application to measles,, Math. Biosci., 73 (1985), 131. doi: 10.1016/0025-5564(85)90081-1.

[42]

S. Tuljapurkar and A. M. John, Disease in changing populations: Growth and disequilibrium,, Theor. Popul. Biol., 40 (1991), 322. doi: 10.1016/0040-5809(91)90059-O.

[43]

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. doi: 10.1016/S0025-5564(02)00108-6.

[44]

J. A. Walker, Dynamical Systems and Evolution Equations,, 1st edition, (1980).

[45]

J. Wang, J. Zu, X. Liu, G. 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. doi: 10.1142/S021833901250009X.

[46]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, 1st edition, (1985).

[47]

K. Yosida, Functional Analysis,, 6th edition, (1980).

show all references

References:
[1]

V. Andreasen, Instability in an SIR-model with age-dependent susceptibility,, in Mathematical Population Dynamics: Analysis of Heterogeneity (eds. O. Arino, (1995), 3.

[2]

E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model,, in Mathematical Ecology (eds. T.G. Hallam, (1988), 317.

[3]

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

[4]

Y. Cha, M. Iannelli and F. A. Milner, Stability change of an epidemic model,, Dynam. Syst. Appl., 9 (2000), 361.

[5]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. 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. doi: 10.1007/BF00178324.

[6]

O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics,, Princeton University Press, (2013).

[7]

K. Dietz, Transmission and control of arbovirus disease,, in Proc. SIMS Conf. on Epidemiology (eds. D. Ludwig and K.L. Cooke), (1975), 104.

[8]

Z. Feng, W. Huang and C. Castillo-Chavez, Global behavior of a multi-group SIS epidemic model with age structure,, J. Diff. Equat., 218 (2005), 292. doi: 10.1016/j.jde.2004.10.009.

[9]

A. Franceschetti, A. Pugliese and D. Breda, Multiple endemic states in age-structured SIR epidemic models,, Math. Biosci. Eng., 9 (2012), 577. doi: 10.3934/mbe.2012.9.577.

[10]

D. Greenhalgh, Analytical results on the stability of age-structured recurrent epidemic models,, IMA J. Math. Appl. Med. Biol., 4 (1987), 109. doi: 10.1093/imammb/4.2.109.

[11]

G. Gripenberg, On a nonlinear integral equation modelling an epidemic in an age-structured population,, J. reine angew. Math., 341 (1983), 54. doi: 10.1515/crll.1983.341.54.

[12]

H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models,, Canadian Appl. Math. Quart., 14 (2006), 259.

[13]

H. J. A. M. Heijmans, The dynamical behaviour of the age-size-distribution of a cell population,, in The Dynamics of Physiologically Structured Populations (eds. J.A.J. Metz and O. Diekmann), 68 (1986), 185. doi: 10.1007/978-3-662-13159-6_5.

[14]

H. W. Hethcote, An immunization model for a heterogeneous population,, Theor. Popul. Biol., 14 (1978), 338. doi: 10.1016/0040-5809(78)90011-4.

[15]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907.

[16]

F. Hoppensteadt, An age dependent epidemic model,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4.

[17]

H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process,, Math. Popul. Studies, 1 (1988), 49. doi: 10.1080/08898488809525260.

[18]

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

[19]

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

[20]

H. Inaba and H. Nishiura, The basic reproduction number of an infectious disease in a stable population: the impact of population growth rate on the eradication threshold,, Math. Model. Nat. Phenom., 3 (2008), 194. doi: 10.1051/mmnp:2008050.

[21]

H. Inaba, The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments,, Math. Biosci. Eng., 9 (2012), 313. doi: 10.3934/mbe.2012.9.313.

[22]

H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments,, J. Math. Biol., 65 (2012), 309. doi: 10.1007/s00285-011-0463-z.

[23]

T. Kato, Perturbation Theory for Linear Operators,, 2nd edition, (1995).

[24]

K. Kawachi, Deterministic models for rumor transmission,, Nonlinear Analysis RWA., 9 (2008), 1989. doi: 10.1016/j.nonrwa.2007.06.004.

[25]

A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages,, Bull. Math. Biol., 71 (2009), 75. doi: 10.1007/s11538-008-9352-z.

[26]

M. A. Krasnoselskii, Positive Solutions of Operator Equations,, 1st edition, (1964).

[27]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space,, Am. Math. Soc. Transl., 1950 (1950).

[28]

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

[29]

J. P. Lasalle, The Stability of Dynamical Systems,, 2nd edition, (1976).

[30]

X. Z. Li, J. X. Liu and M. Martcheva, An age-structured two-strain epidemic model with super-infection,, Math. Biosci. Eng., 7 (2010), 123. doi: 10.3934/mbe.2010.7.123.

[31]

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

[32]

I. Marek, Frobenius theory of positive operators: Comparison theorems and applications,, SIAM J. Appl. Math., 19 (1970), 607. doi: 10.1137/0119060.

[33]

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. doi: 10.3934/mbe.2012.9.819.

[34]

A. G. McKendrick, Application of mathematics to medical problems,, Proc. Edinburgh Math. Soc., (1925), 98. doi: 10.1017/S0013091500034428.

[35]

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. doi: 10.3934/mbe.2013.10.369.

[36]

R. Nagel, One-Parameter Semigroups of Positive Operators,, 1st edition, (1986). doi: 10.1007/BFb0074922.

[37]

I. Sawashima, On spectral properties of some positive operators,, Nat. Sci. Rep. Ochanomizu Univ., 15 (1964), 53.

[38]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, 1st edition, (2011).

[39]

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

[40]

H. R. Thieme, Stability change of the endemic equilibrium in age-structured models for the spread of S-I-R type infectious diseases,, in Differential Equations Models in Biology, 92 (1991), 139. doi: 10.1007/978-3-642-45692-3_10.

[41]

D. W. Tudor, An age-dependent epidemic model with application to measles,, Math. Biosci., 73 (1985), 131. doi: 10.1016/0025-5564(85)90081-1.

[42]

S. Tuljapurkar and A. M. John, Disease in changing populations: Growth and disequilibrium,, Theor. Popul. Biol., 40 (1991), 322. doi: 10.1016/0040-5809(91)90059-O.

[43]

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. doi: 10.1016/S0025-5564(02)00108-6.

[44]

J. A. Walker, Dynamical Systems and Evolution Equations,, 1st edition, (1980).

[45]

J. Wang, J. Zu, X. Liu, G. 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. doi: 10.1142/S021833901250009X.

[46]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, 1st edition, (1985).

[47]

K. Yosida, Functional Analysis,, 6th edition, (1980).

[1]

Yoshiaki Muroya. A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model). Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 999-1008. doi: 10.3934/dcdss.2015.8.999

[2]

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

[3]

Toshikazu Kuniya, Yoshiaki Muroya. Global stability of a multi-group SIS epidemic model for population migration. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1105-1118. doi: 10.3934/dcdsb.2014.19.1105

[4]

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

[5]

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

[6]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37

[7]

Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057

[8]

Lili Liu, Xianning Liu, Jinliang Wang. Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2615-2630. doi: 10.3934/dcdsb.2016064

[9]

Jinliang Wang, Hongying Shu. Global analysis on a class of multi-group SEIR model with latency and relapse. Mathematical Biosciences & Engineering, 2016, 13 (1) : 209-225. doi: 10.3934/mbe.2016.13.209

[10]

Hisashi Inaba. Mathematical analysis of an age-structured SIR epidemic model with vertical transmission. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 69-96. doi: 10.3934/dcdsb.2006.6.69

[11]

Bin-Guo Wang, Wan-Tong Li, Liang Zhang. An almost periodic epidemic model with age structure in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291

[12]

Xiaomei Feng, Zhidong Teng, Fengqin Zhang. Global dynamics of a general class of multi-group epidemic models with latency and relapse. Mathematical Biosciences & Engineering, 2015, 12 (1) : 99-115. doi: 10.3934/mbe.2015.12.99

[13]

Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1141-1157. doi: 10.3934/mbe.2017059

[14]

Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101

[15]

Yan Li, Wan-Tong Li, Guo Lin. Traveling waves of a delayed diffusive SIR epidemic model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1001-1022. doi: 10.3934/cpaa.2015.14.1001

[16]

Gunduz Caginalp, Mark DeSantis. Multi-group asset flow equations and stability. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 109-150. doi: 10.3934/dcdsb.2011.16.109

[17]

John Cleveland. Basic stage structure measure valued evolutionary game model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 291-310. doi: 10.3934/mbe.2015.12.291

[18]

Jacek Banasiak, Eddy Kimba Phongi, MirosŁaw Lachowicz. A singularly perturbed SIS model with age structure. Mathematical Biosciences & Engineering, 2013, 10 (3) : 499-521. doi: 10.3934/mbe.2013.10.499

[19]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595

[20]

C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]