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The infection propagates with independent random time increments (i.e., edge-delays) between neighboring nodes, while only the infection times of a subset of nodes can be observed. We show that a reduced set of observers may be sufficient, in the statistical sense, to localize the source and characterize its identifiability via the joint Laplace transform of the observers\u2019 infection times. Using the explicit form of these transforms in terms of the edge-delay probability distributions, we propose scale-invariant estimators of the source. We evaluate their performance on synthetic trees and on a river network, demonstrating accurate localization under diverse edge-delay models.<\/jats:p>","DOI":"10.1007\/s11538-026-01640-4","type":"journal-article","created":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T01:51:37Z","timestamp":1777254697000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Observer-Based Source Localization in Tree Infection Networks via Laplace Transforms"],"prefix":"10.1007","volume":"88","author":[{"given":"Graham Kesler","family":"O\u2019Connor","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Julia M.","family":"Jess","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Devlin","family":"Costello","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6843-6845","authenticated-orcid":false,"given":"Manuel E.","family":"Lladser","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,27]]},"reference":[{"key":"1640_CR1","volume-title":"Infectious diseases of humans: Dynamics and control","author":"RM Anderson","year":"1992","unstructured":"Anderson RM, May RM (1992) Infectious diseases of humans: Dynamics and control. 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