Traveling fronts in a spatial epidemic model with slow loss of immunity
Abstract
We investigate the emergence of traveling front solutions in a spatial SIRS epidemic model with diffusion acting on the infected population. The model exhibits a natural slow-fast structure due to the presence of a small parameter governing the loss of immunity, which induces a separation of scales in the dynamics. Using a traveling wave reduction, the PDE system is transformed into a singularly perturbed system of ODEs, which we analyze within the framework of Geometric Singular Perturbation Theory. In the singular limits, we study the fast excursions governed by the layer problem, and the slow evolution close to the critical manifold. In particular, we identify an entry-exit mechanism tracking the transitions between slow and fast regimes, and derive a quantitative characterization of the entry-exit dynamics. Numerical simulations of the full system confirm the validity of the proposed geometric picture. The traveling front is shown to consist of a concatenation of local, fast, and slow segments, in agreement with the theoretical analysis.
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BibTeX
@misc{marca2026traveling,
title = {{Traveling fronts in a spatial epidemic model with slow loss of immunity}},
author = {Rossella Della Marca and Gabriele Grifò and Annalisa Iuorio and Mattia Sensi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.04594},
eprint = {2608.04594},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.04594},
}
RIS
TY - UNPB
TI - Traveling fronts in a spatial epidemic model with slow loss of immunity
AU - Rossella Della Marca
AU - Gabriele Grifò
AU - Annalisa Iuorio
AU - Mattia Sensi
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2608.04594
UR - https://arxiv.org/abs/2608.04594
AB - We investigate the emergence of traveling front solutions in a spatial SIRS epidemic model with diffusion acting on the infected population. The model exhibits a natural slow-fast structure due to the presence of a small parameter governing the loss of immunity, which induces a separation of scales in the dynamics. Using a traveling wave reduction, the PDE system is transformed into a singularly perturbed system of ODEs, which we analyze within the framework of Geometric Singular Perturbation Theory. In the singular limits, we study the fast excursions governed by the layer problem, and the slow evolution close to the critical manifold. In particular, we identify an entry-exit mechanism tracking the transitions between slow and fast regimes, and derive a quantitative characterization of the entry-exit dynamics. Numerical simulations of the full system confirm the validity of the proposed geometric picture. The traveling front is shown to consist of a concatenation of local, fast, and slow segments, in agreement with the theoretical analysis.
ER -