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Remarks about spatially structured SI model systems with cross diffusion. (English) Zbl 1416.35272

Chetverushkin, B. N. (ed.) et al., Contributions to partial differential equations and applications. Invited papers of the conferences ‘Contributions to partial differential equations’, Université Pierre et Marie Curie, Paris, France, August 31 – September 1, 2015 and ‘Applied and computational mathematics’, University of Houston, Texas, USA, February 26–27, 2016. Cham: Springer. Comput. Methods Appl. Sci. 47, 43-64 (2019).
Summary: 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.
For the entire collection see [Zbl 1411.35011].

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
35K51 Initial-boundary value problems for second-order parabolic systems
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