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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 |
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.
|
[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. 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. |
[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. 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.
|
[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. 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. |
[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).
|
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