Stefan problem, traveling fronts, and epidemic spread

Karl P. Hadeler

doi:10.3934/dcdsb.2016.21.417

Article Contents

2016, Volume 21, Issue 2: 417-436. Doi: 10.3934/dcdsb.2016.21.417

This issue Previous Article Computation of $\mathcal R $ in age-structured epidemiological models with maternal and temporary immunity Next Article Classification of potential flows under renormalization group transformation

Stefan problem, traveling fronts, and epidemic spread

Karl P. Hadeler^1,

1.
Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen

Received: October 31, 2014

Revised: March 31, 2015

Published: February 2016

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Abstract

The scalar reaction diffusion equation with a nonlinearity of logistic type has a minimal speed $c_0$ for standard traveling fronts. It is shown that also for speeds $0 < c < c_0$ there are traveling fronts but these are solutions to free boundary value (Stefan) problems. Furthermore, these speeds depend in a monotone way on the Stefan coefficient which links the loss of matter at the free boundary to the displacement per time. The results are extended to correlated random walks, Cattaneo systems and, in particular, to models for epidemic spread. In the epidemic problems a dichotomy phenomenon shows up: For small values of the Stefan coefficient there are no fronts indicating that for such values and certain data the free boundary stays bounded.

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Mathematics Subject Classification: Primary: 35C07, 92D30; Secondary: 35K57, 35L60.

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