Lyapunov functional and global asymptotic stability for an infection-age model

References

• Anderson , RM and May , RM . 1991 . Infectious Diseases of Humans: Dynamics and Control , Oxford, , UK : Oxford University Press .
   Google Scholar
• Brauer , F and Castillo-Chavez , C . 2000 . Mathematical Models in Population Biology and Epidemiology , New York : Springer .
   Google Scholar
• Diekmann , O and Heesterbeek , JAP . 2000 . Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation , Chichester, , UK : Wiley .
   Google Scholar
• Thieme , HR . 2003 . Mathematics in Population Biology , Princeton, NJ : Princeton University Press .
   Google Scholar
• Thieme , HR and Castillo-Chavez , C . 1989 . “ On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic ” . In Mathematical and Statistical Approaches to AIDS Epidemiology , Lecture Notes in Biomathematics Edited by: Castillo-Chavez , C . Vol. 83 , 157 – 176 . Berlin, New York : Springer-Verlag .
   Google Scholar
• Thieme , HR and Castillo-Chavez , C . 1993 . How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS? . SIAM. J. Appl. Math. , 53 : 1447 – 1479 .
   Web of Science ®Google Scholar
• Adda , P , Dimi , JL , Iggidr , A , Kamgang , JC , Sallet , G and Tewa , JJ . 2007 . General models of host-parasite systems. Global analysis . Discrete Contin. Dyn. Syst. Ser. B , 8 : 1 – 17 .
   Web of Science ®Google Scholar
• Beretta , A and Capasso , V . 1986 . On the general structure of epidemic systems. Global asymptotic stability . Comput. Math. Appl. Ser. A , 12 : 677 – 694 .
   Google Scholar
• Capasso , V . 1993 . Mathematical Structures of Epidemic Systems , Berlin, Heidelberg : Springer .
   Google Scholar
• Georgescu , P and Hsieh , YH . 2006 . Global stability for a virus dynamics model with nonlinear incidence of infection and removal . SIAM J. Appl. Math. , 67 : 337 – 353 .
   Web of Science ®Google Scholar
• Guo , H and Li , MY . 2006 . Global dynamics of a staged progression model for infectious diseases . Math. Biosci. Eng. , 3 : 513 – 525 .
   PubMed Web of Science ®Google Scholar
• Guo , H , Li , MY and Shuai , Z . 2008 . A graph-theoretic approach to the method of global Lyapunov functions . Proc. Amer. Math. Soc. , 136 : 2793 – 2802 .
   Web of Science ®Google Scholar
• Hethcote , HW . 1976 . Qualitive analyses of communicable disease models . Math. Biosci. , 28 : 335 – 356 .
   Google Scholar
• Hethcote , HW . 2000 . The mathematics of infectious diseases . SIAM Rev. , 42 : 599 – 653 .
   Web of Science ®Google Scholar
• Hethcote , HW and Thieme , HR . 1985 . Stability of the endemic equilibrium in epidemic models with subpopulations . Math. Biosci. , 75 : 205 – 227 .
   Web of Science ®Google Scholar
• Korobeinikov , A . 2006 . Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission . Bull. Math. Biol. , 68 : 615 – 626 .
   PubMed Web of Science ®Google Scholar
• Korobeinikov , A . 2007 . Global properties of infectious disease models with nonlinear incidence . Bull. Math. Biol. , 69 : 1871 – 1886 .
   PubMed Web of Science ®Google Scholar
• Korobeinikov , A and Maini , PK . 2004 . A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence . Math. Biosci. Eng. , 1 : 57 – 60 .
   PubMed Web of Science ®Google Scholar
• Korobeinikov , A and Maini , PK . 2005 . Non-linear incidence and stability of infectious disease models . Math. Med. Biol. , 22 : 113 – 128 .
   PubMed Web of Science ®Google Scholar
• Korobeinikov , A and Wake , GC . 2002 . Lyapunov functions and global stability for SIR and SIRS and SIS epidemiolgical models . Appl. Math. Lett. , 15 : 955 – 961 .
   Web of Science ®Google Scholar
• McCluskey , CC . 2006 . Lyapunov functions for tuberculosis models with fast and slow progression . Math. Biosci. Eng. , 3 : 603 – 614 .
   PubMed Web of Science ®Google Scholar
• McCluskey , CC . 2008 . Global stability for a class of mass action systems allowing for latency in tuberculosis . J. Math. Anal. Appl. , 338 : 518 – 535 .
   Web of Science ®Google Scholar
• Hethcote , HW . 1989 . “ Three basic epidemiological models ” . In Applied Mathematical Ecology , Edited by: Gross , L , Hallam , TG and Levin , SA . 119 – 144 . Berlin : Springer-Verlag .
   Google Scholar
• Iggidr , A , Kamgang , JC , Sallet , G and Tewa , JJ . 2006 . Global analysis of new malaria intrahost model with a competitive exclusion principle . SIAM J. Appl. Math. , 67 : 260 – 278 .
   Web of Science ®Google Scholar
• Ma , W , Takeuchi , Y , Hara , T and Beretta , E . 2002 . Permanence of an SIR epidemic model with distributed time delays . Tohoku Math. J. , 54 : 581 – 591 .
   Web of Science ®Google Scholar
• Thieme , HR . 1983 . “ Global asymptotic stability in epidemic models ” . In Equadiff 82 , Lecture Notes in Math 1017 Edited by: Knobloch , W and Schmitt , K . 608 – 615 . Berlin : Springer-Verlag .
   Google Scholar
• Li , MY , Graef , JR , Wang , L and Karsai , J . 1999 . Global dynamics of a SEIR model with varying total population size . Math. Biosci. , 160 : 191 – 213 .
   PubMed Web of Science ®Google Scholar
• Li , MY and Muldowney , JS . 1995 . Global stability for the SEIR model in epidemiology . Math. Biosci. , 125 : 155 – 164 .
   PubMed Web of Science ®Google Scholar
• Li , MY and Muldowney , JS . 1996 . A geometric approach to global-stability problems . SIAM J. Math. Anal. , 27 : 1070 – 1083 .
   Web of Science ®Google Scholar
• Li , MY , Muldowney , JS and van den Driessche , P . 1999 . Global stability of SEIRS models in epidemiology . Can. Appl. Math. Q. , 7 : 409 – 425 .
   Google Scholar
• Li , MY , Smith , HL and Wang , L . 2001 . Global dynamics of an SEIR epidemic model with vertical transmission . SIAM J. Appl. Math. , 62 : 58 – 69 .
   Web of Science ®Google Scholar
• De Leenheer , P and Smith , HL . 2003 . Virus dynamics: A global analysis . SIAM J. Appl. Math. , 63 : 1313 – 1327 .
   Web of Science ®Google Scholar
• Pruss , J , Pujo-Mejouet , L , Webb , GF and Zacher , R . 2006 . Analysis of a model for the dynamics of prions . Discrete Contin. Dyn. Syst. Ser. B , 6 : 225 – 235 .
   Web of Science ®Google Scholar
• Pruss , J , Zacher , R and Schnaubelt , R . 2008 . Global asymptotic stability of equilibria in models for virus dynamics . Math. Model. Nat. Phenom. , 3 : 126 – 142 .
   Web of Science ®Google Scholar
• Bame , N , Bowong , S , Mbang , J and Sallet , G . 2008 . Global stability analysis for SEIS models with n latent classes . Math. Biosci. Eng. , 5 : 20 – 33 .
   PubMed Web of Science ®Google Scholar
• McCluskey , CC . 2009 . Global stability for an SEIR epidemiological model with varying infectivity and infinite delay . Math. Biosci. Eng. , 6 : 603 – 610 .
   PubMed Web of Science ®Google Scholar
• McCluskey , CC . 2010 . Complete global stability for an SIR epidemic model with delay – distributed or discrete, Nonlinear Anal. RWA , 11 ( 1 ) : 55 – 59 .
   Google Scholar
• Röst , G and Wu , J . 2008 . SEIR epidemiological model with varying infectivity and infinite delay . Math. Biosci. Eng. , 5 : 389 – 402 .
   PubMed Web of Science ®Google Scholar
• Takeuchi , Y , Ma , W and Beretta , E . 2000 . Global asymptotic properties of a delay SIR epidemic model with finite incubation times . Nonlinear Anal. , 42 : 931 – 947 .
   Web of Science ®Google Scholar
• Iggidr , A , Mbang , J and Sallet , G . 2007 . Stability analysis of within-host parasite models with delays . Math. Biosci. , 209 : 51 – 75 .
   PubMed Web of Science ®Google Scholar
• D'Agata , E , Magal , P , Ruan , S and Webb , GF . 2006 . Asymptotic behavior in nosocomial epidemic models with antibiotic resistance . Differential Integral Equations , 19 : 573 – 600 .
   Web of Science ®Google Scholar
• Magal , P . 2009 . Perturbation of a globally stable steady state and uniform rersistence . J. Dyn. Diff. Equat. , 21 : 1 – 20 .
   Web of Science ®Google Scholar
• D'Agata , EMC , Magal , P , Olivier , D , Ruan , S and Webb , GF . 2007 . Modeling antibiotic resistance in hospitals: The impact of minimizing treatment duration . J. Theoret. Biol. , 249 : 487 – 499 .
   PubMed Web of Science ®Google Scholar
• Webb , GF , D'Agata , E , Magal , P and Ruan , S . A model of antibiotic resistant bacterial epidemics in hospitals . Proceedings of the National Academics of Sciences of the USA . Vol. 102 , pp. 13343 – 13348 .
   Google Scholar
• Webb , GF . 1985 . Theory of Nonlinear Age-dependent Population Dynamics , New York : Marcel Dekker .
   Google Scholar
• Webb , GF . 1987 . An operator-theoretic exponential growth in differential equations . Trans. Amer. Math. Soc. , 303 : 751 – 763 .
   Web of Science ®Google Scholar
• Engel , K-J and Nagel , R . 2000 . One Parameter Semigroups for Linear Evolution Equations , New York : Springer-Verlag .
   Google Scholar
• Hale , JK and Waltman , P . 1989 . Persistence in infinite dimensional systems . SIAM J. Math. Anal. , 20 : 388 – 395 .
   Web of Science ®Google Scholar
• Magal , P and Zhao , X-Q . 2005 . Global attractors in uniformly persistent dynamical systems . SIAM J. Math. Anal. , 37 : 251 – 275 .
   Web of Science ®Google Scholar
• Iannelli , M . 1994 . “ Mathematical Theory of Age-structured Population Dynamics ” . In Applied Mathematics Monographs CNR , Vol. 7 , Pisa : Giadini Editori e Stampatori .
   Google Scholar
• Webb , GF . 2008 . “ Population models structured by age, size, and spatial position ” . In Structured Population Models in Biology and Epidemiology , Lecture Notes in Mathematics Edited by: Maga , P and Ruan , S . Vol. 1936 , 1 – 49 . Berlin-New York : Springer-Verlag .
   Google Scholar
• Magal , P . 2001 . Compact attractors for time periodic age-structured population models . Electron. J. Differential Equations , 2001 ( 65 ) : 1 – 35 .
   Google Scholar
• Magal , P and Ruan , S . 2007 . On integrated semigroups and age-structured models in L p space . Differential Integral Equations , 20 : 197 – 239 .
   Web of Science ®Google Scholar
• Magal , P and Ruan , S . 2009 . Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models , : 951 Memoirs of the American Mathematical Society 202
   Google Scholar
• Thieme , HR . 1990 . Semiflows generated by Lipschitz perturbations of non-densely defined operators . Differential Integral Equations , 3 : 1035 – 1066 .
   Google Scholar
• Magal , P and Thieme , HR . 2004 . Eventual compactness for a semiflow generated by an age-structured models . Commun. Pure Appl. Anal. , 3 : 695 – 727 .
   Web of Science ®Google Scholar
• Thieme , HR and Vrabie , JI . 2003 . Relatively compact orbits and compact attractors for a class of nonlinear evolution equations . J. Dynam. Differential Equations , 15 : 731 – 750 .
   Google Scholar
• Hale , JK . 1988 . “ Asymptotic Behavior of Dissipative Systems ” . In Mathematical Surveys and Monographs 25 , Providence, RI : American Mathematical Society .
   Google Scholar
• H.R. Thieme, Quasi-compact semigroups via bounded perturbation, in Advances in Mathematical Population Dynamics – Molecules, Cells and Man, Houston, TX, 1995, Series in Mathematical Biology and Medicine, Vol. 6, World Scientific Publishing, River Edge, NJ, 1997, 691–711.
   Google Scholar
• Ducrot , A , Liu , Z and Magal , P . 2008 . Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems . J. Math. Anal. Appl. , 341 : 501 – 518 .
   Web of Science ®Google Scholar