Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate
An infection-age structured epidemic model with a nonlinear incidence rate is investigated.We formulate the model as an abstract non-densely defined Cauchy problem and derive the condition which guarantees the existence and uniqueness for positive age-dependent equilibrium of the model.By analyzing...
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| Published in | Science China. Mathematics Vol. 60; no. 8; pp. 1371 - 1398 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Beijing
Science China Press
01.08.2017
Springer Nature B.V |
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| Abstract | An infection-age structured epidemic model with a nonlinear incidence rate is investigated.We formulate the model as an abstract non-densely defined Cauchy problem and derive the condition which guarantees the existence and uniqueness for positive age-dependent equilibrium of the model.By analyzing the associated characteristic transcendental equation and applying the normal form theory presented recently for non-densely defined semilinear equations,we show that the SIR（susceptible-infected-recovered）epidemic model undergoes Zero-Hopf bifurcation at the positive equilibrium which is the main result of this paper. |
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| AbstractList | An infection-age structured epidemic model with a nonlinear incidence rate is investigated.We formulate the model as an abstract non-densely defined Cauchy problem and derive the condition which guarantees the existence and uniqueness for positive age-dependent equilibrium of the model.By analyzing the associated characteristic transcendental equation and applying the normal form theory presented recently for non-densely defined semilinear equations,we show that the SIR（susceptible-infected-recovered）epidemic model undergoes Zero-Hopf bifurcation at the positive equilibrium which is the main result of this paper. An infection-age structured epidemic model with a nonlinear incidence rate is investigated. We formulate the model as an abstract non-densely defined Cauchy problem and derive the condition which guarantees the existence and uniqueness for positive age-dependent equilibrium of the model. By analyzing the associated characteristic transcendental equation and applying the normal form theory presented recently for non-densely defined semilinear equations, we show that the SIR (susceptible-infected-recovered) epidemic model undergoes Zero-Hopf bifurcation at the positive equilibrium which is the main result of this paper. |
| Author | LIU ZhiHua YUAN Rong |
| AuthorAffiliation | School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China |
| Author_xml | – sequence: 1 givenname: ZhiHua surname: Liu fullname: Liu, ZhiHua email: zhihualiu@bnu.edu.cn organization: School of Mathematical Sciences, Beijing Normal University – sequence: 2 givenname: Rong surname: Yuan fullname: Yuan, Rong organization: School of Mathematical Sciences, Beijing Normal University |
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| Cites_doi | 10.1016/0022-1236(89)90116-X 10.1007/s00033-010-0088-x 10.1016/S0022-0396(02)00089-X 10.1016/j.jmaa.2006.09.061 10.1007/BF00277162 10.1007/s00028-007-0355-2 10.1016/j.jmaa.2007.09.074 10.1016/j.jde.2014.04.018 10.1016/0022-247X(90)90074-P 10.1016/0025-5564(78)90006-8 10.1016/j.mbs.2004.02.004 10.1007/BF00276956 10.1016/0895-7177(95)00029-2 10.57262/ade/1355854784 |
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| Keywords | non-densely defined 92D30 zero-Hopf bifurcation epidemic 34K18 35K90 37G10 infection-age structured normal form stability 35K55 37L10 |
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| Notes | infection-age structured, epidemic, non-densely defined, stability, normal form, zero-Hopf bifurcation 11-5837/O1 An infection-age structured epidemic model with a nonlinear incidence rate is investigated.We formulate the model as an abstract non-densely defined Cauchy problem and derive the condition which guarantees the existence and uniqueness for positive age-dependent equilibrium of the model.By analyzing the associated characteristic transcendental equation and applying the normal form theory presented recently for non-densely defined semilinear equations,we show that the SIR（susceptible-infected-recovered）epidemic model undergoes Zero-Hopf bifurcation at the positive equilibrium which is the main result of this paper. ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
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| References | Ducrot, Liu, Magal (CR2) 2008; 341 Thieme (CR16) 1990; 3 Thieme (CR17) 1990; 152 Thieme (CR18) 1997 Zhang, Peng (CR20) 2007; 331 Kim (CR5) 1996; 74 Liu, Magal, Ruan (CR9) 2011; 62 Ruan, Wang (CR15) 2003; 188 Liu, Hethcote, Levin (CR7) 1987; 25 Liu, Levin, Iwasa (CR8) 1986; 23 Capasso, Serio (CR1) 1978; 42 Magal, Ruan (CR14) 2009; 202 Magal (CR11) 2001; 2001 Magal, Ruan (CR12) 2007; 20 Thieme (CR19) 2008; 8 Kellermann, Hieber (CR4) 1989; 84 Magal, Ruan (CR13) 2009; 14 Liu, Magal, Ruan (CR10) 2014; 257 Inaba, Sekine (CR3) 2004; 190 Kim, Milner (CR6) 1995; 21 P Magal (371_CR12) 2007; 20 H R Thieme (371_CR16) 1990; 3 Z Zhang (371_CR20) 2007; 331 M Y Kim (371_CR6) 1995; 21 W M Liu (371_CR7) 1987; 25 W M Liu (371_CR8) 1986; 23 P Magal (371_CR11) 2001; 2001 P Magal (371_CR13) 2009; 14 P Magal (371_CR14) 2009; 202 H Kellermann (371_CR4) 1989; 84 A Ducrot (371_CR2) 2008; 341 Z Liu (371_CR9) 2011; 62 M Y Kim (371_CR5) 1996; 74 V Capasso (371_CR1) 1978; 42 Z Liu (371_CR10) 2014; 257 S Ruan (371_CR15) 2003; 188 H R Thieme (371_CR18) 1997 H Inaba (371_CR3) 2004; 190 H R Thieme (371_CR17) 1990; 152 H R Thieme (371_CR19) 2008; 8 |
| References_xml | – start-page: 691 year: 1997 end-page: 713 ident: CR18 article-title: Quasi-compact semigroups via bounded perturbation publication-title: Advances in Mathematical Population Dynamics: Molecules, Cells and Man – volume: 84 start-page: 160 year: 1989 end-page: 180 ident: CR4 article-title: Integrated semigroups publication-title: J Funct Anal doi: 10.1016/0022-1236(89)90116-X – volume: 62 start-page: 191 year: 2011 end-page: 222 ident: CR9 article-title: Hopf bifurcation for non-densely defined Cauchy problems publication-title: Z Angew Math Phys doi: 10.1007/s00033-010-0088-x – volume: 188 start-page: 135 year: 2003 end-page: 163 ident: CR15 article-title: Dynamical behavior of an epidemic model with a nonlinear incidence rate publication-title: J Differential Equations doi: 10.1016/S0022-0396(02)00089-X – volume: 331 start-page: 1396 year: 2007 end-page: 1414 ident: CR20 article-title: A SIRS epidemic model with infection-age dependence publication-title: J Math Anal Appl doi: 10.1016/j.jmaa.2006.09.061 – volume: 3 start-page: 1035 year: 1990 end-page: 1066 ident: CR16 article-title: Semiflows generated by Lipschitz perturbations of non-densely defined operators publication-title: Differ Integral Equ – volume: 25 start-page: 359 year: 1987 end-page: 380 ident: CR7 article-title: Dynamical behavior of epidemiological models with nonlinear incidence rates publication-title: J Math Biol doi: 10.1007/BF00277162 – volume: 8 start-page: 283 year: 2008 end-page: 305 ident: CR19 article-title: Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem publication-title: J Evol Equ doi: 10.1007/s00028-007-0355-2 – volume: 341 start-page: 501 year: 2008 end-page: 518 ident: CR2 article-title: Essential growth rate for bounded linear perturbation of non densely defined Cauchy problems publication-title: J Math Anal Appl doi: 10.1016/j.jmaa.2007.09.074 – volume: 14 start-page: 1041 year: 2009 end-page: 1084 ident: CR13 article-title: On semilinear Cauchy problems with non-dense domain publication-title: Adv Differential Equations – volume: 257 start-page: 921 year: 2014 end-page: 1011 ident: CR10 article-title: Normal forms for semilinear equations with non-dense domain publication-title: J Differential Equations doi: 10.1016/j.jde.2014.04.018 – volume: 20 start-page: 197 year: 2007 end-page: 239 ident: CR12 article-title: On integrated semigroups and age structured models in spaces publication-title: Differ Integral Equ – volume: 152 start-page: 416 year: 1990 end-page: 447 ident: CR17 article-title: Integrated semigroups and integrated solutions to abstract Cauchy problems publication-title: J Math Anal Appl doi: 10.1016/0022-247X(90)90074-P – volume: 42 start-page: 43 year: 1978 end-page: 61 ident: CR1 article-title: A generalization of the Kermack-Mckendrick deterministic epidemic model publication-title: Math Biosci doi: 10.1016/0025-5564(78)90006-8 – volume: 2001 start-page: 1 year: 2001 end-page: 35 ident: CR11 article-title: Compact attractors for time-periodic age structured population models publication-title: Electron J Differ Equ Conf – volume: 74 start-page: 37 year: 1996 end-page: 58 ident: CR5 article-title: Existence of steady state solutions to an epidemic model with screening and their asymptotic stability publication-title: Appl Math Comput – volume: 190 start-page: 39 year: 2004 end-page: 69 ident: CR3 article-title: A mathematical model for Chagas disease with infection-age-dependent infectivity publication-title: Math Biosci doi: 10.1016/j.mbs.2004.02.004 – volume: 23 start-page: 187 year: 1986 end-page: 204 ident: CR8 article-title: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models publication-title: J Math Biol doi: 10.1007/BF00276956 – volume: 202 start-page: 2559965 year: 2009 ident: CR14 article-title: Center manifolds for semilinear equations with non-dense domain and applications on Hopf bifurcation in age structured models publication-title: Mem Amer Math Soc – volume: 21 start-page: 29 year: 1995 end-page: 42 ident: CR6 article-title: A mathematical model of epidemics with screening and variable infectivity publication-title: Math Comput Modelling doi: 10.1016/0895-7177(95)00029-2 – volume: 23 start-page: 187 year: 1986 ident: 371_CR8 publication-title: J Math Biol doi: 10.1007/BF00276956 – start-page: 691 volume-title: Advances in Mathematical Population Dynamics: Molecules, Cells and Man year: 1997 ident: 371_CR18 – volume: 152 start-page: 416 year: 1990 ident: 371_CR17 publication-title: J Math Anal Appl doi: 10.1016/0022-247X(90)90074-P – volume: 331 start-page: 1396 year: 2007 ident: 371_CR20 publication-title: J Math Anal Appl doi: 10.1016/j.jmaa.2006.09.061 – volume: 190 start-page: 39 year: 2004 ident: 371_CR3 publication-title: Math Biosci doi: 10.1016/j.mbs.2004.02.004 – volume: 188 start-page: 135 year: 2003 ident: 371_CR15 publication-title: J Differential Equations doi: 10.1016/S0022-0396(02)00089-X – volume: 2001 start-page: 1 year: 2001 ident: 371_CR11 publication-title: Electron J Differ Equ Conf – volume: 42 start-page: 43 year: 1978 ident: 371_CR1 publication-title: Math Biosci doi: 10.1016/0025-5564(78)90006-8 – volume: 84 start-page: 160 year: 1989 ident: 371_CR4 publication-title: J Funct Anal doi: 10.1016/0022-1236(89)90116-X – volume: 20 start-page: 197 year: 2007 ident: 371_CR12 publication-title: Differ Integral Equ – volume: 62 start-page: 191 year: 2011 ident: 371_CR9 publication-title: Z Angew Math Phys doi: 10.1007/s00033-010-0088-x – volume: 202 start-page: 2559965 year: 2009 ident: 371_CR14 publication-title: Mem Amer Math Soc – volume: 74 start-page: 37 year: 1996 ident: 371_CR5 publication-title: Appl Math Comput – volume: 257 start-page: 921 year: 2014 ident: 371_CR10 publication-title: J Differential Equations doi: 10.1016/j.jde.2014.04.018 – volume: 8 start-page: 283 year: 2008 ident: 371_CR19 publication-title: J Evol Equ doi: 10.1007/s00028-007-0355-2 – volume: 21 start-page: 29 year: 1995 ident: 371_CR6 publication-title: Math Comput Modelling doi: 10.1016/0895-7177(95)00029-2 – volume: 341 start-page: 501 year: 2008 ident: 371_CR2 publication-title: J Math Anal Appl doi: 10.1016/j.jmaa.2007.09.074 – volume: 25 start-page: 359 year: 1987 ident: 371_CR7 publication-title: J Math Biol doi: 10.1007/BF00277162 – volume: 14 start-page: 1041 year: 2009 ident: 371_CR13 publication-title: Adv Differential Equations doi: 10.57262/ade/1355854784 – volume: 3 start-page: 1035 year: 1990 ident: 371_CR16 publication-title: Differ Integral Equ |
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| SubjectTerms | Age Applications of Mathematics Cauchy problems Epidemics Externality Hopf bifurcation Hopf分岔 Hopf分支 Incidence Mathematics Mathematics and Statistics Uniqueness 传染率 传染病模型 半线性方程 年龄结构 流行病模型 非线性 |
| Title | Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate |
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