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Analysis of a Reaction-Diffusion Epidemic Model

  • B.  Cantó
  • C.  Coll
  • S. Romero-VivóEmail author
  • E.  Sánchez
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 471)

Abstract

A model of an epidemic is introduced to describe an indirect transmission of the disease through the density of pathogens in the environment. The scenario of an emerging disease in a contaminated environment is assumed and the possibility that an initial infection can spread in the population living in that environment is analyzed.

Keywords

Epidemic model Stability Equilibrium points Basic reproduction number Discrete-time system 

References

  1. 1.
    Beaumont, C., Burie, J., Ducrot, A., Zongo, P.: Propagation of salmonella within an industrial hen house. SIAM J. Appl. Math. 72(4), 1113–1148 (2012)Google Scholar
  2. 2.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in Mathematical Sciences. Academic Press, New York (1979)zbMATHGoogle Scholar
  3. 3.
    Capasso, V.: Mathematical Structures of Epidemic Systems. Lecture Notes in Biomathematics, vol. 97. Springer, Heidelberg (1993)Google Scholar
  4. 4.
    Capasso, V., Wilson, R.E.: Analysis of a reaction-diffusion system modeling man-environment-man epidemics. SIAM J. Appl. Math. 57(2), 327–346 (1997)Google Scholar
  5. 5.
    Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J.: On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneus populations. J. Math. Biol. 28, 365–382 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Farina, L., Rinaldi, S.: Positive Linear Systems, Theory and Applications. Pure and Applied Mathematics. Wiley (2000)Google Scholar
  7. 7.
    Gantmacher, F.R.: Applications of the Theory of Matrices. Interscience Publishers, New York (1959)zbMATHGoogle Scholar
  8. 8.
    Kaczorek, T.: Linear Control System, vol. 2. Wiley, New York (1991)zbMATHGoogle Scholar
  9. 9.
    Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans and Animals. Priceton University Press, Princeton and Oxford (2008)zbMATHGoogle Scholar
  10. 10.
    Li, C.K., Schneider, H.: Applications of Perron-Frobenius theory to population dynamics. J. Math. Biol. 44, 450–462 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  12. 12.
    Prévost, K., Beaumont, C., Magal, P.: Asymptotic behavior in a Salmonella infection model. Math. Modell. Nat. Phenomena Epidemiology 2(1), 1–22 (2006)Google Scholar
  13. 13.
    Smith, H.L.: Monotone Dynamical system: an introduction to the theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, American Mathematical Society, Providence (1995)Google Scholar
  14. 14.
    Varga, R.S.: Matrix Iterative Analysis. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  15. 15.
    Wang, W., Zhao, X.Q.: Basic reproduction numbers for reaction-diffusion epidemic models. SIAM J. Appl. Dyn. Syst 11, 1652–1673 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • B.  Cantó
    • 1
  • C.  Coll
    • 1
  • S. Romero-Vivó
    • 1
    Email author
  • E.  Sánchez
    • 1
  1. 1.Instituto de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValenciaSpain

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