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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. Google Scholar [2] E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model,, in Mathematical Ecology (eds. T.G. Hallam, (1988), 317. Google Scholar [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. Google Scholar [4] Y. Cha, M. Iannelli and F. A. Milner, Stability change of an epidemic model,, Dynam. Syst. Appl., 9 (2000), 361. Google Scholar [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. Google Scholar [6] O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics,, Princeton University Press, (2013). Google Scholar [7] K. Dietz, Transmission and control of arbovirus disease,, in Proc. SIMS Conf. on Epidemiology (eds. D. Ludwig and K.L. Cooke), (1975), 104. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar [16] F. Hoppensteadt, An age dependent epidemic model,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4. Google Scholar [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. Google Scholar [18] H. Inaba, Threshold and stability results for an age-structured epidemic model,, J. Math. Biol., 28 (1990), 411. doi: 10.1007/BF00178326. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [23] T. Kato, Perturbation Theory for Linear Operators,, 2nd edition, (1995). Google Scholar [24] K. Kawachi, Deterministic models for rumor transmission,, Nonlinear Analysis RWA., 9 (2008), 1989. doi: 10.1016/j.nonrwa.2007.06.004. Google Scholar [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. Google Scholar [26] M. A. Krasnoselskii, Positive Solutions of Operator Equations,, 1st edition, (1964). Google Scholar [27] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space,, Am. Math. Soc. Transl., 1950 (1950). Google Scholar [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. Google Scholar [29] J. P. Lasalle, The Stability of Dynamical Systems,, 2nd edition, (1976). Google Scholar [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. Google Scholar [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. Google Scholar [32] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications,, SIAM J. Appl. Math., 19 (1970), 607. doi: 10.1137/0119060. Google Scholar [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. Google Scholar [34] A. G. McKendrick, Application of mathematics to medical problems,, Proc. Edinburgh Math. Soc., (1925), 98. doi: 10.1017/S0013091500034428. Google Scholar [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. Google Scholar [36] R. Nagel, One-Parameter Semigroups of Positive Operators,, 1st edition, (1986). doi: 10.1007/BFb0074922. Google Scholar [37] I. Sawashima, On spectral properties of some positive operators,, Nat. Sci. Rep. Ochanomizu Univ., 15 (1964), 53. Google Scholar [38] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, 1st edition, (2011). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [44] J. A. Walker, Dynamical Systems and Evolution Equations,, 1st edition, (1980). Google Scholar [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. Google Scholar [46] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, 1st edition, (1985). Google Scholar [47] K. Yosida, Functional Analysis,, 6th edition, (1980). Google Scholar

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. Google Scholar [2] E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model,, in Mathematical Ecology (eds. T.G. Hallam, (1988), 317. Google Scholar [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. Google Scholar [4] Y. Cha, M. Iannelli and F. A. Milner, Stability change of an epidemic model,, Dynam. Syst. Appl., 9 (2000), 361. Google Scholar [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. Google Scholar [6] O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics,, Princeton University Press, (2013). Google Scholar [7] K. Dietz, Transmission and control of arbovirus disease,, in Proc. SIMS Conf. on Epidemiology (eds. D. Ludwig and K.L. Cooke), (1975), 104. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar [16] F. Hoppensteadt, An age dependent epidemic model,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4. Google Scholar [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. Google Scholar [18] H. Inaba, Threshold and stability results for an age-structured epidemic model,, J. Math. Biol., 28 (1990), 411. doi: 10.1007/BF00178326. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [23] T. Kato, Perturbation Theory for Linear Operators,, 2nd edition, (1995). Google Scholar [24] K. Kawachi, Deterministic models for rumor transmission,, Nonlinear Analysis RWA., 9 (2008), 1989. doi: 10.1016/j.nonrwa.2007.06.004. Google Scholar [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. Google Scholar [26] M. A. Krasnoselskii, Positive Solutions of Operator Equations,, 1st edition, (1964). Google Scholar [27] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space,, Am. Math. Soc. Transl., 1950 (1950). Google Scholar [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. Google Scholar [29] J. P. Lasalle, The Stability of Dynamical Systems,, 2nd edition, (1976). Google Scholar [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. Google Scholar [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. Google Scholar [32] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications,, SIAM J. Appl. Math., 19 (1970), 607. doi: 10.1137/0119060. Google Scholar [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. Google Scholar [34] A. G. McKendrick, Application of mathematics to medical problems,, Proc. Edinburgh Math. Soc., (1925), 98. doi: 10.1017/S0013091500034428. Google Scholar [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. Google Scholar [36] R. Nagel, One-Parameter Semigroups of Positive Operators,, 1st edition, (1986). doi: 10.1007/BFb0074922. Google Scholar [37] I. Sawashima, On spectral properties of some positive operators,, Nat. Sci. Rep. Ochanomizu Univ., 15 (1964), 53. Google Scholar [38] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, 1st edition, (2011). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [44] J. A. Walker, Dynamical Systems and Evolution Equations,, 1st edition, (1980). Google Scholar [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. Google Scholar [46] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, 1st edition, (1985). Google Scholar [47] K. Yosida, Functional Analysis,, 6th edition, (1980). Google Scholar
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