Remarks About Spatially Structured SI Model Systems with Cross Diffusion

  • Verónica Anaya
  • Mostafa Bendahmane
  • Michel LanglaisEmail author
  • Mauricio Sepúlveda
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 47)


One of the simplest deterministic mathematical model for the spread of an epidemic disease is the so-called SI system made of two Ordinary Differential Equations. It exhibits simple dynamics: a bifurcation parameter \(\mathscr {T}_0\) yielding persistence of the disease when \(\mathscr {T}_0 > 1\), else extinction occurs. A natural question is whether this gentle dynamic can be disturbed by spatial diffusion. It is straightforward to check it is not feasible for linear/nonlinear diffusions. When cross diffusion is introduced for suitable choices of the parameter data set this persistent state of the ODE model system becomes linearly unstable for the resulting initial and no-flux boundary value problem. On the other hand “natural” weak solutions can be defined for this initial and no-flux boundary value problem and proved to exist provided nonlinear and cross diffusivities satisfy some constraints. These constraints are not fully met for the parameter data set yielding instability. A remaining open question is: to which solutions does this apply? Periodic behaviors are observed for a suitable range of cross diffusivities.


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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Verónica Anaya
    • 1
  • Mostafa Bendahmane
    • 2
  • Michel Langlais
    • 3
    Email author
  • Mauricio Sepúlveda
    • 4
  1. 1.Departamento de Matemática and GIMNAPUniversidad del Bío-BíoConcepciónChile
  2. 2.Institut de Mathématiques de Bordeaux, UMR CNRS 5251Université de BordeauxTALENCE Cedex, BordeauxFrance
  3. 3.BordeauxFrance
  4. 4.Universidad de Concepción, DIM and CI2 MA, Esteban Iturra s/n, Barrio UniversitarioConcepciónChile

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