{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T17:57:28Z","timestamp":1792000648336,"version":"4.4.0"},"reference-count":43,"publisher":"Public Library of Science (PLoS)","issue":"10","license":[{"start":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T00:00:00Z","timestamp":1791936000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"<jats:p>\n                    We extend the delayed reaction\u2013diffusion SI model with nonlinear incidence proposed by Song et al. by replacing the fixed transmission lag with weak and strong gamma-distributed memory kernels. Because the model has no exposed compartment, the memory term is interpreted phenomenologically as a distributed timing of transmission-related effects of past infectious states rather than as an explicit latent-stage mechanism. Using the same parameter set adopted by Song et al., we show that the distributed-memory models have a stability structure that is qualitatively different from that of the discrete-delay model. For the spatially homogeneous mode, both gamma kernels exhibit a small stabilising spectral Hopf crossing at a mean memory time near 0.08. The discrete-delay model has a comparable small stabilising crossing but, in addition, a much larger destabilising crossing near\n                    <jats:inline-formula id=\"pone.0360508.e001\">\n                      <jats:alternatives>\n                        <jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" id=\"pone.0360508.e001g\" mimetype=\"image\" position=\"anchor\" xlink:href=\"info:doi\/10.1371\/journal.pone.0360508.e001\" xlink:type=\"simple\"\/>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" id=\"M1\">\n                          <mml:mrow>\n                            <mml:mi>\u03c4<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1.335<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . Consequently, at the nominal values 1.2 and 1.6 used in the reference simulations, the gamma models converge to the selected endemic equilibrium, whereas the discrete-delay model crosses into an oscillatory regime. The first spatial mode is different: for the gamma formulations its modal constant term is negative for every positive memory rate, so this mode is unstable for every positive mean-memory value. We therefore interpret this behavior as a memory-independent spatial-mode instability, distinct from both a memory-induced Hopf crossing and a classical Turing instability.\n                  <\/jats:p>","DOI":"10.1371\/journal.pone.0360508","type":"journal-article","created":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T17:38:40Z","timestamp":1791999520000},"page":"e0360508","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":0,"title":["Gamma\u2013distributed transmission memory in a reaction\u2013diffusion epidemic model with nonlinear incidence"],"prefix":"10.1371","volume":"21","author":[{"given":"Martin","family":"Anokye","sequence":"first","affiliation":[{"name":"University of Cape Coast","place":["Cape Coast, Ghana"],"department":["Department of Mathematics"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luca","family":"Guerrini","sequence":"additional","affiliation":[{"name":"Polytechnic University of Marche","place":["Ancona, Italy"],"department":["Department of Management"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"John Amoah","family":"Mensah","sequence":"additional","affiliation":[{"name":"Sunyani Technical University","place":["Sunyani, Ghana"],"department":["Department of Computer Science"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4201-1993","authenticated-orcid":true,"given":"Michael Fosu","family":"Ofori","sequence":"additional","affiliation":[{"name":"Kumasi Technical University","place":["Kumasi, Ghana"],"department":["Department of Statistical Sciences"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefania","family":"Ragni","sequence":"additional","affiliation":[{"name":"University of Ferrara","place":["Ferrara, Italy"],"department":["Department of Economics and Management"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2026,10,14]]},"reference":[{"issue":"6","key":"pone.0360508.ref001","article-title":"Hopf Bifurcation of an Epidemic Model with Delay","volume":"11","author":"L-P Song","year":"2016","journal-title":"PLoS One"},{"issue":"772","key":"pone.0360508.ref002","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1098\/rspa.1927.0118","article-title":"A contribution to the mathematical theory of epidemics","volume":"115","author":"WO Kermack","year":"1927","journal-title":"Proc R Soc Lond A"},{"key":"pone.0360508.ref003","doi-asserted-by":"crossref","DOI":"10.1007\/b98869","volume-title":"Mathematical Biology II: Spatial Models and Biomedical Applications","author":"JD Murray","year":"2003"},{"key":"pone.0360508.ref004","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/1-4020-3647-7_11","article-title":"Delay differential equations in single species dynamics.","volume-title":"Delay Differential Equations and Applications","author":"S. 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