{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:40:43Z","timestamp":1760175643758,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,21]],"date-time":"2020-02-21T00:00:00Z","timestamp":1582243200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["PTDC\/ MAT-NAN\/ 6890\/ 2014","PTDC\/MAT-APL\/31753\/2017","UID\/ EEA\/ 50014\/ 2013","NORTE-01-0145-FEDER-000016"],"award-info":[{"award-number":["PTDC\/ MAT-NAN\/ 6890\/ 2014","PTDC\/MAT-APL\/31753\/2017","UID\/ EEA\/ 50014\/ 2013","NORTE-01-0145-FEDER-000016"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011929","name":"Programa Operacional Tem\u00e1tico Factores de Competitividade","doi-asserted-by":"publisher","award":["POCI-01-0145-FEDER-006961"],"award-info":[{"award-number":["POCI-01-0145-FEDER-006961"]}],"id":[{"id":"10.13039\/501100011929","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["Bolsa Pesquisador Visitante Especial \u2013 PVE: 401068\/2014-5 MEC\/MCTI\/CAPES\/CNPQ\/FAPS"],"award-info":[{"award-number":["Bolsa Pesquisador Visitante Especial \u2013 PVE: 401068\/2014-5 MEC\/MCTI\/CAPES\/CNPQ\/FAPS"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>We study the equilibria of an Ordinary Differencial Equation (ODE) system where CD4     +     effector or helper T cells and Regulatory T cells (Tregs) are present. T cells trigger an immune response in the presence of their specific antigen. Regulatory T cells (Tregs) play a role in limiting auto-immune diseases due to their immune-suppressive ability. Here, we present explicit exact formulas that give the relationship between the concentration of T cells, the concentration of Tregs, and the antigenic stimulation of T cells, when the system is at equilibria, stable or unstable. We found a parameter region of bistability, limited by two thresholds of antigenic stimulation of T cells (hysteresis). Moreover, there are values of the slope parameter of the tuning for which an isola-center bifurcation appears, and, for some other values, there is a transcritical bifurcation. We also present time evolutions of the ODE system.<\/jats:p>","DOI":"10.3390\/math8020293","type":"journal-article","created":{"date-parts":[[2020,3,12]],"date-time":"2020-03-12T12:22:51Z","timestamp":1584015771000},"page":"293","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["The Effect of a Linear Tuning between the Antigenic Stimulations of CD4+ T Cells and CD4+ Tregs"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2634-295X","authenticated-orcid":false,"given":"Aliyu","family":"Yusuf","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias, Universidade do Porto, 4169-007 Porto, Portugal"},{"name":"Laborat\u00f3rio de Intelig\u00eancia Artificial e Apoio \u00e0 Decis\u00e3o (LIAAD)\u2013Instituto de Engenharia de Sistemas e Computadores, Tecnologia e Ci\u00eancia (INESC TEC), 4200-465 Porto, Portugal"},{"name":"Department of Mathematics, Kano University of Science and Technology, Wudil, 3244 Kano, Nigeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3653-0842","authenticated-orcid":false,"given":"Isabel","family":"Figueiredo","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Instituto Superior de Engenharia do Porto, Instituto Polit\u00e9cnico do Porto, 4200-072 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8743-1812","authenticated-orcid":false,"given":"Atefeh","family":"Afsar","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias, Universidade do Porto, 4169-007 Porto, Portugal"},{"name":"Laborat\u00f3rio de Intelig\u00eancia Artificial e Apoio \u00e0 Decis\u00e3o (LIAAD)\u2013Instituto de Engenharia de Sistemas e Computadores, Tecnologia e Ci\u00eancia (INESC TEC), 4200-465 Porto, Portugal"}]},{"given":"Nigel","family":"Burroughs","sequence":"additional","affiliation":[{"name":"Systems Biology Centre, University of Warwick, Coventry CV4 7AL, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2953-6688","authenticated-orcid":false,"given":"Alberto","family":"Pinto","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias, Universidade do Porto, 4169-007 Porto, Portugal"},{"name":"Laborat\u00f3rio de Intelig\u00eancia Artificial e Apoio \u00e0 Decis\u00e3o (LIAAD)\u2013Instituto de Engenharia de Sistemas e Computadores, Tecnologia e Ci\u00eancia (INESC TEC), 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7710-4284","authenticated-orcid":false,"given":"Bruno","family":"Oliveira","sequence":"additional","affiliation":[{"name":"Laborat\u00f3rio de Intelig\u00eancia Artificial e Apoio \u00e0 Decis\u00e3o (LIAAD)\u2013Instituto de Engenharia de Sistemas e Computadores, Tecnologia e Ci\u00eancia (INESC TEC), 4200-465 Porto, Portugal"},{"name":"Faculdade de Ci\u00eancias da Nutri\u00e7\u00e3o e Alimenta\u00e7\u00e3o, Universidade do Porto, 4200-465 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/0167-5699(95)80093-X","article-title":"A quantitative model of auto-immune disease and T-cell vaccination: Does more mean less?","volume":"16","author":"Segel","year":"1995","journal-title":"Immunol. 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