Abstract:The basic reproduction ratio is a crucial threshold parameter in infectious disease models. In nonlocal dispersal systems, its variational characterization is challenging due to the possible absence of a principal eigenvalue caused by non-compactness. In this paper, we aim to establish such a characterization even when the principal eigenvalue does not exist. To this end, we first study the spectral bound of a class of nonlocal dispersal operators, establishing a Collatz-Wielandt characterization as well as a Rayleigh-Ritz characterization when the operator is self-adjoint. Using this, we characterize the unique parameter value at which the spectral bound equals zero, covering both non-degenerate and partially degenerate cases, and subsequently obtain an explicit expression for the basic reproduction ratio. To demonstrate the utility of our theoretical framework, we apply it to a nonlocal dispersal SIS epidemic model with saturated incidence rate. The analysis shows that, in the degenerate case of the saturation coefficient, the limiting behavior of the basic reproduction ratio as the total population tends to zero is strikingly different from that in local diffusion case.
| Comments: | 25 pages |
| Subjects: | Analysis of PDEs (math.AP) |
| MSC classes: | 45C05, 35K57, 45P05, 35R20 |
| Cite as: | arXiv:2603.14256 [math.AP] |
| (or arXiv:2603.14256v2 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2603.14256 arXiv-issued DOI via DataCite |
From: Qi-Ru Wang [view email]
[v1]
Sun, 15 Mar 2026 07:26:39 UTC (22 KB)
[v2]
Tue, 26 May 2026 12:22:06 UTC (22 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。