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Original Articles

Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion

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Pages 406-439 | Received 07 Oct 2010, Accepted 10 Aug 2011, Published online: 21 Sep 2011

References

  • Adams , R. 1975 . Sobolev Spaces , New York : Academic Press .
  • Allen , L. J.S. , Bolker , B. , Lou , Y. and Nevai , A. 2008 . Asymptotic profiles of the steady states for an SIS epidemic reaction–diffusion model . Discrete Contin. Dyn. Syst. , 21 : 1 – 20 .
  • Babuska , I. and Osborn , J. E. 1987 . Estimates for the errors in eigenvalue and eigenvector approximation by Galerkin methods, with particular attention to the case of multiple eigenvalues . SIAM J. Numer. Anal. , 24 : 1249 – 1276 .
  • Babuska , I. and Osborn , J. E. 1989 . Finite element Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems . Math. Comput. , 52 : 275 – 297 .
  • Berman , A. and Plemmons , R. J. 1979 . “ Nonnegative Matrices in the Mathematical Sciences ” . In Computer Science and Applied Mathematics , New York : Academic Press [Harcourt Brace Jovanovich Publishers] .
  • Brenner , S. and Scott , L. R. 2007 . The Mathematical Theory of Finite Element Methods , New York : Springer-Verlag .
  • Cantrell , R. and Cosner , C. 2003 . Spatial Ecology via Reaction–Diffusion Equations , Chichester : Wiley .
  • Cantrell , R. S. and Cosner , C. 1998 . On the effects of spatial heterogeneity on the persistence of interacting species . J. Math. Biol. , 37 ( 2 ) : 103 – 145 .
  • Ciarlet , P. 2002 . The Finite Element Method for Elliptic Problems , SIAM, Philadelphia .
  • Ciarlet , P. G. and Raviart , P.-A. 1973 . Maximum principle and uniform convergence for the finite element method . Comput. Methods Appl. Mech. Eng. , 2 : 17 – 31 .
  • Cosner , C. , Beier , J. , Cantrell , R. , Impoinvil , D. , Kapitanski , L. , Potts , M. , Troyo , A. and Ruan , S. 2009 . The effects of human movement on the persistence of vector-borne diseases . J. Theor. Biol. , 258 : 550 – 560 .
  • Diekmann , O. and Heesterbeek , J. A.P. 2000 . Mathematical Epidemiology of Infectious Diseases , Chichester : Wiley .
  • Evans , L. C. 1998 . “ Partial Differential Equations ” . In Graduate Studies in Mathematics , Vol. 19 , Providence , RI : American Mathematical Society .
  • Fiedler , M. 1986 . Special Matrices and Their Applications in Numerical Mathematics , Dordrecht : Martinus Nijhoff Publishers . Translated from the Czech by Petr Přikryl and Karel Segeth
  • Fitzgibbon , W. and Langlias , M. “ Simple models for the transmission of microparasites between host populations living on noncoincident spatial domains in Structured Population Models in Biology and Epidemiology ” . In Lecture Notes in Mathematics , Edited by: Magal , P. and Ruan , S. 115 – 164 . Berlin : Springer .
  • Fitzgibbon , W. , Langlais , M. and Morgan , J. 2001 . A mathematical model of the spread of feline leukemia virus (FELV) through a highly heterogeneous spatial domain . SIAM J. Math. Anal. , 33 : 570 – 588 .
  • Fitzgibbon , W. , Langlais , M. and Morgan , J. 2004 . A reaction--diffusion system modeling direct and indirect transmission of diseases . Discrete Contin. Dyn. Syst. B , 4 : 893 – 910 .
  • Henry , D. 1981 . Geometric Theory of Semilinear Parabolic Equations , New York : Springer-Verlag .
  • Hsieh , Y.-H. , van den Driessche , P. and Wang , L. 2007 . Impact of travel between patches for spatial spread of disease . Bull. Math. Biol. , 69 : 1355 – 1375 .
  • Johnson , C. 2009 . Numerical Solution of Partial Differential Equations by the Finite Element Method , Mineola , NY : Dover Publications .
  • Kim , M. 2006 . Global dynamics of approximate solutions to an age-structured epidemic model with diffusion . Adv. Comput. Math. , 25 : 451 – 474 .
  • Kim , K. , Lin , Z. and Zhang , L. 2010 . Avian--human influenza epidemic model with diffusion . Nonlinear Anal.: Real World Appl. , 11 : 313 – 322 .
  • Martcheva , M. , Bolker , B. and Holt , R. 2008 . Vaccine-induced pathogen strain replacement: What are the mechanisms? . J. Roy. Soc. Interface , 5 : 3 – 13 .
  • Mora , X. 1983 . Semilinear parabolic problems define semiflows on C k spaces . Trans. Amer. Math. Soc. , 278 ( 1 ) : 21 – 55 .
  • Pacala , S. W. and Roughgarden , J. 1982 . Spatial heterogeneity and interspecific competition . Theor. Popul. Biol. , 21 ( 1 ) : 92 – 113 .
  • Pao , C. V. 1992 . Nonlinear Parabolic and Elliptic Equations , New York : Plenum Press .
  • Smoller , J. 1983 . Shock Waves and Reaction--Diffusion Equations , New York : Springer-Verlag .
  • Strang , G. and Fix , G. 1973 . An Analysis of the Finite Elment Method , Englewood Cliffs , NJ : Prentice-Hall .
  • Strauss , W. A. 1992 . An introduction in Partial Differential Equations , New York : John Wiley & Sons .
  • Thomèe , V. 1997 . Galerkin Finite Element Method for Parabolic Problems , Berlin : Springer .
  • Varga , R. S. Matrix Iterative Analysis . expanded ed., Springer Series in Computational Mathematics Vol. 27, Springer-Verlag, Berlin, 2000.
  • Vejchodskyacute; , T. 2004 . “ On the nonnegativity conservation in semidiscrete parabolic problems in Conjugate Gradient Algorithms and Finite Element Methods ” . In Scientific Computing , 197 – 210 . Berlin : Springer .