A multi-group SIR epidemic model with age structure

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December 2016, 21(10): 3515-3550. doi: 10.3934/dcdsb.2016109

A multi-group SIR epidemic model with age structure

Toshikazu Kuniya ^1, , Jinliang Wang ^2, and Hisashi Inaba ^3,

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

Abstract
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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.

Keywords: SIR epidemic model, age structure, multi-group model, the basic reproduction number..

Mathematics Subject Classification: Primary: 35Q92, 92D30; Secondary: 37N2.

Citation: Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete and 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, D. Axelrod, M. Kimmel and M. Langlais), Wuerz Publ., (1995), 3-14.
[2]	E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, in Mathematical Ecology (eds. T.G. Hallam, L.J. Gross and S.A. Levin), World Scientific, (1988), 317-342.
[3]	S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080. doi: 10.1137/0522069.
[4]	Y. Cha, M. Iannelli and F. A. Milner, Stability change of an epidemic model, Dynam. Syst. Appl., 9 (2000), 361-376.
[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), 365-382. doi: 10.1007/BF00178324.
[6]	O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton University Press, Princeton and Oxford, 2013.
[7]	K. Dietz, Transmission and control of arbovirus disease, in Proc. SIMS Conf. on Epidemiology (eds. D. Ludwig and K.L. Cooke), SIAM, (1975), 104-121.
[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), 292-324. 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), 577-599. 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), 109-144. 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), 54-67. 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), 259-284.
[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), Springer, 68 (1986), 185-202. 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), 338-349. doi: 10.1016/0040-5809(78)90011-4.
[15]	H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907.
[16]	F. Hoppensteadt, An age dependent epidemic model, J. Franklin Inst., 297 (1974), 325-333. 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), 49-77. doi: 10.1080/08898488809525260.
[18]	H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434. doi: 10.1007/BF00178326.
[19]	H. Inaba, Endemic threshold results for age-duration-structured population model for HIV infection, Math. Biosci., 201 (2006), 15-47. 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), 194-228. 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), 313-346. 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), 309-348. doi: 10.1007/s00285-011-0463-z.
[23]	T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer, Berlin, 1995.
[24]	K. Kawachi, Deterministic models for rumor transmission, Nonlinear Analysis RWA., 9 (2008), 1989-2028. 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), 75-83. doi: 10.1007/s11538-008-9352-z.
[26]	M. A. Krasnoselskii, Positive Solutions of Operator Equations, 1st edition, Noordhoff, Groningen, 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), 128pp.
[28]	T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Analysis RWA., 12 (2011), 2640-2655. doi: 10.1016/j.nonrwa.2011.03.011.
[29]	J. P. Lasalle, The Stability of Dynamical Systems, 2nd edition, SIAM, Philadelphia, 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), 123-147. 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), 1109-1140. doi: 10.1080/00036810903208122.
[32]	I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math., 19 (1970), 607-628. 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), 819-841. doi: 10.3934/mbe.2012.9.819.
[34]	A. G. McKendrick, Application of mathematics to medical problems, Proc. Edinburgh Math. Soc., (1925), 98-130. 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), 369-378. doi: 10.3934/mbe.2013.10.369.
[36]	R. Nagel, One-Parameter Semigroups of Positive Operators, 1st edition, Springer, Berlin, 1986. doi: 10.1007/BFb0074922.
[37]	I. Sawashima, On spectral properties of some positive operators, Nat. Sci. Rep. Ochanomizu Univ., 15 (1964), 53-64.
[38]	H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, 1st edition, Amer. Math. Soc., Providence, 2011.
[39]	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.
[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, Epidemiology and Ecology (eds. S. Busenberg and M. Martelli), Springer, 92 (1991), 139-158. 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), 131-147. 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), 322-353. 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), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[44]	J. A. Walker, Dynamical Systems and Evolution Equations, 1st edition, Plenum Press, New York and London, 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), 235-258. doi: 10.1142/S021833901250009X.
[46]	G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, 1st edition, Marcel Dekker, New York and Basel, 1985.
[47]	K. Yosida, Functional Analysis, 6th edition, Springer, Berlin, 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, D. Axelrod, M. Kimmel and M. Langlais), Wuerz Publ., (1995), 3-14.
[2]	E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, in Mathematical Ecology (eds. T.G. Hallam, L.J. Gross and S.A. Levin), World Scientific, (1988), 317-342.
[3]	S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080. doi: 10.1137/0522069.
[4]	Y. Cha, M. Iannelli and F. A. Milner, Stability change of an epidemic model, Dynam. Syst. Appl., 9 (2000), 361-376.
[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), 365-382. doi: 10.1007/BF00178324.
[6]	O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton University Press, Princeton and Oxford, 2013.
[7]	K. Dietz, Transmission and control of arbovirus disease, in Proc. SIMS Conf. on Epidemiology (eds. D. Ludwig and K.L. Cooke), SIAM, (1975), 104-121.
[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), 292-324. 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), 577-599. 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), 109-144. 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), 54-67. 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), 259-284.
[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), Springer, 68 (1986), 185-202. 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), 338-349. doi: 10.1016/0040-5809(78)90011-4.
[15]	H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907.
[16]	F. Hoppensteadt, An age dependent epidemic model, J. Franklin Inst., 297 (1974), 325-333. 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), 49-77. doi: 10.1080/08898488809525260.
[18]	H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434. doi: 10.1007/BF00178326.
[19]	H. Inaba, Endemic threshold results for age-duration-structured population model for HIV infection, Math. Biosci., 201 (2006), 15-47. 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), 194-228. 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), 313-346. 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), 309-348. doi: 10.1007/s00285-011-0463-z.
[23]	T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer, Berlin, 1995.
[24]	K. Kawachi, Deterministic models for rumor transmission, Nonlinear Analysis RWA., 9 (2008), 1989-2028. 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), 75-83. doi: 10.1007/s11538-008-9352-z.
[26]	M. A. Krasnoselskii, Positive Solutions of Operator Equations, 1st edition, Noordhoff, Groningen, 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), 128pp.
[28]	T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Analysis RWA., 12 (2011), 2640-2655. doi: 10.1016/j.nonrwa.2011.03.011.
[29]	J. P. Lasalle, The Stability of Dynamical Systems, 2nd edition, SIAM, Philadelphia, 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), 123-147. 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), 1109-1140. doi: 10.1080/00036810903208122.
[32]	I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math., 19 (1970), 607-628. 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), 819-841. doi: 10.3934/mbe.2012.9.819.
[34]	A. G. McKendrick, Application of mathematics to medical problems, Proc. Edinburgh Math. Soc., (1925), 98-130. 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), 369-378. doi: 10.3934/mbe.2013.10.369.
[36]	R. Nagel, One-Parameter Semigroups of Positive Operators, 1st edition, Springer, Berlin, 1986. doi: 10.1007/BFb0074922.
[37]	I. Sawashima, On spectral properties of some positive operators, Nat. Sci. Rep. Ochanomizu Univ., 15 (1964), 53-64.
[38]	H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, 1st edition, Amer. Math. Soc., Providence, 2011.
[39]	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.
[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, Epidemiology and Ecology (eds. S. Busenberg and M. Martelli), Springer, 92 (1991), 139-158. 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), 131-147. 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), 322-353. 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), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[44]	J. A. Walker, Dynamical Systems and Evolution Equations, 1st edition, Plenum Press, New York and London, 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), 235-258. doi: 10.1142/S021833901250009X.
[46]	G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, 1st edition, Marcel Dekker, New York and Basel, 1985.
[47]	K. Yosida, Functional Analysis, 6th edition, Springer, Berlin, 1980.

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