{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,5]],"date-time":"2025-07-05T14:02:23Z","timestamp":1751724143861},"reference-count":34,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2020,7,28]],"date-time":"2020-07-28T00:00:00Z","timestamp":1595894400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,3,15]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A predator-prey model is used to investigate the interactions between phages and bacteria by considering the lytic and lysogenic life cycles of phages and the prophage induction. We provide answers to the following conflictual research questions: (1) what are conditions under which the presence of phages can purify a bacterial infected environment? (2) Can the presence of phages triggers virulent bacterial outbreaks? We derive the basic offspring number $\\mathcal N_0$ that serves as a threshold and the bifurcation parameter to study the dynamics and bifurcation of the system. The model exhibits three equilibria: an unstable environment-free equilibrium, a globally asymptotically stable (GAS) phage-free equilibrium (PFE) whenever $\\mathcal N_0&amp;lt;1$, and a locally asymptotically stable environment-persistent equilibrium (EPE) when $\\mathcal N_0&amp;gt;1$. The Lyapunov\u2013LaSalle techniques are used to prove the GAS of the PFE and estimate the EPE basin of attraction. Through the center manifold approximation, topological types of the PFE are precised. Existence of transcritical and Hopf bifurcations are established. Precisely, when $\\mathcal N_0&amp;gt;1$, the EPE loses its stability and periodic solutions arise. Furthermore, increasing $\\mathcal N_0$ can purify an environment where bacteriophages are introduced. Purposely, we prove that for large values of $\\mathcal N_0$, the overall bacterial population asymptotically approaches zero, while the phage population sustains. Ecologically, our results show that for small values of $\\mathcal N_0$, the existence of periodic solutions could explain the occurrence of repetitive bacteria-borne disease outbreaks, while large value of $\\mathcal N_0$ clears bacteria from the environment. Numerical simulations support our theoretical results.<\/jats:p>","DOI":"10.1093\/imammb\/dqaa010","type":"journal-article","created":{"date-parts":[[2020,6,29]],"date-time":"2020-06-29T19:17:18Z","timestamp":1593458238000},"page":"28-58","source":"Crossref","is-referenced-by-count":9,"title":["Bifurcation analysis of a phage-bacteria interaction model with prophage induction"],"prefix":"10.1093","volume":"38","author":[{"given":"H M","family":"Ndongmo Teytsa","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Dschang, PO Box 67, Dschang, Cameroon"},{"name":"IRD UMI 209 UMMISCO, University of Yaounde I, PO Box 337, Yaounde, Cameroon and LIRIMA-EPITAG Team Project, University of Yaounde I, PO Box 812, Yaounde, Cameroon"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"B","family":"Tsanou","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Dschang, PO Box 67, Dschang, Cameroon"},{"name":"Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa"},{"name":"IRD UMI 209 UMMISCO, University of Yaounde I, PO Box 337, Yaounde, Cameroon and LIRIMA-EPITAG Team Project, University of Yaounde I, PO Box 812, Yaounde, Cameroon"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S","family":"Bowong","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Douala, PO Box 24157, Cameroon"},{"name":"IRD UMI 209 UMMISCO, University of Yaounde I, PO Box 337, Yaounde, Cameroon and LIRIMA-EPITAG Team Project, University of Yaounde I, PO Box 812, Yaounde, Cameroon"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J M-S","family":"Lubuma","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2020,7,28]]},"reference":[{"key":"2021022305195367000_ref1","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198545996.001.0001","volume-title":"Infectious Disease of Human: Dynamics and Control","author":"Anderson","year":"1991"},{"key":"2021022305195367000_ref2","article-title":"Biocontrol of V. cholerae using bacteriophages","volume-title":"Ph.D. Thesis","author":"Bhandare","year":"2005"},{"key":"2021022305195367000_ref3","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1080\/17513758.2016.1229817","article-title":"A simple mathematical model of Ebola in Africa","volume":"11","author":"Berge","year":"2016","journal-title":"J. Biol. Dyn."},{"key":"2021022305195367000_ref4","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/S0025-5564(97)10015-3","article-title":"Modelling and analysis of a marine bacteriophages infection","volume":"149","author":"Beretta","year":"1998","journal-title":"Math. Biosci."},{"volume-title":"Ordinary Differential Equations","year":"1989","author":"Birkhoff","key":"2021022305195367000_ref5"},{"key":"2021022305195367000_ref6","first-page":"9271","article-title":"Global dynamics and bifurcation in a stage structured prey\u2013predator fishery model with harvesting","volume":"218","author":"Chakraborty","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"2021022305195367000_ref7","doi-asserted-by":"crossref","first-page":"361","DOI":"10.3934\/mbe.2004.1.361","article-title":"Dynamical models of tuberculosis and their applications","volume":"1","author":"Castillo","year":"2004","journal-title":"Math. Biosci. Eng."},{"key":"2021022305195367000_ref8","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.mbs.2017.12.005","article-title":"On the intrinsic dynamics of bacteria in waterborne infection","volume":"296","author":"Chayu","year":"2018","journal-title":"Math. Biosci."},{"key":"2021022305195367000_ref9","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/S1369-5274(02)00005-X","article-title":"Filamentous phages linked to virulence of vibrio cholerae","volume":"6","author":"Davis","year":"2003","journal-title":"Curr. Opin. Microbiol."},{"key":"2021022305195367000_ref10","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","article-title":"Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission","volume":"180","author":"Driessche","year":"2002","journal-title":"Math. Biosci."},{"key":"2021022305195367000_ref11","first-page":"372","article-title":"Role of media coverage and delay in controlling infectious diseases: a mathematical model","volume":"337","author":"Fahad","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"2021022305195367000_ref12","doi-asserted-by":"crossref","first-page":"556","DOI":"10.4161\/viru.22351","article-title":"Phage\u2013bacterial interactions in the evolution of toxigenic vibrio cholerae","volume":"3","author":"Faruque","year":"2012","journal-title":"Virulence"},{"key":"2021022305195367000_ref13","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1007\/BF02218848","article-title":"Uniform persistence and flows near a closed positively invariant set","volume":"6","author":"Fremann","year":"1994","journal-title":"J. Diff. Equ."},{"key":"2021022305195367000_ref14","first-page":"29","volume-title":"Bacteriophage: Biology and Application","author":"Guttman","year":"2005"},{"key":"2021022305195367000_ref15","doi-asserted-by":"crossref","first-page":"21","DOI":"10.3934\/mbe.2005.2.753","article-title":"The role of vaccination in the control of SARS","volume":"2","author":"Gjorgjieva","year":"2005","journal-title":"Math. Biosci. Eng."},{"key":"2021022305195367000_ref16","doi-asserted-by":"crossref","first-page":"4652","DOI":"10.1073\/pnas.0600166103","article-title":"Modeling the role of bacteriophage in the control of cholera outbreaks","volume":"103","author":"Jensen","year":"2006","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"2021022305195367000_ref17","doi-asserted-by":"crossref","first-page":"2025","DOI":"10.1007\/s11538-014-9996-9","article-title":"Dynamics of a cholera transmission model with immunological threshold and natural phage control in reservoir","volume":"76","author":"Jude","year":"2014","journal-title":"Bull. Math. Biol."},{"volume-title":"Stability Analysis of Non Linear System","year":"1989","author":"Lakshmikantham","key":"2021022305195367000_ref18"},{"key":"2021022305195367000_ref19","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970432","volume-title":"The Stability of Dynamical Systems","author":"LaSalle","year":"1976"},{"key":"2021022305195367000_ref20","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/0022-0396(68)90048-X","article-title":"Stability theory for ordinary differential equations","volume":"4","author":"LaSalle","year":"1968","journal-title":"J. Differ. Equ."},{"key":"2021022305195367000_ref21","first-page":"68","article-title":"Mechanism of multi-strain coexistence in host-phage system with nested infection network","volume":"332","author":"Lui","year":"2013","journal-title":"J. Theor. Biol."},{"key":"2021022305195367000_ref22","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1006\/jmaa.1994.1079","article-title":"Criterion of Hopf bifurcation without using eigenvalues","volume":"182","author":"Liu","year":"1994","journal-title":"J. Math. Anal. Appl."},{"key":"2021022305195367000_ref23","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1017\/CBO9780511541483.008","article-title":"Contribution of lysogeny, pseudo-lysogeny, and starvation to phage ecology","volume-title":"Bacteriophage Ecology","author":"Miller","year":"2008"},{"key":"2021022305195367000_ref24","first-page":"610","article-title":"Cholera dynamics with bacteriophage infection: a mathematical study, Chaos Sol","volume":"91","author":"Misra","year":"2016","journal-title":"Frac."},{"key":"2021022305195367000_ref25","doi-asserted-by":"crossref","first-page":"1379","DOI":"10.1137\/060670377","article-title":"Global dynamics of a predator\u2013prey model with stage structure for predator","volume":"67","author":"Pau","year":"2007","journal-title":"SIAM J. Appl. Math."},{"key":"2021022305195367000_ref26","first-page":"373","article-title":"Stability analysis of coexistence of three species prey\u2013predator model","volume":"81","author":"Prabir","year":"2005","journal-title":"J. Non. Dyn."},{"key":"2021022305195367000_ref27","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1016\/j.chaos.2017.08.036","article-title":"Effect of toxicants on pytoplankton-zooplankton-fish dynamics and harvesting","volume":"104","author":"Prabir","year":"2017","journal-title":"Chaos. Sol. Frac."},{"key":"2021022305195367000_ref28","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.jtbi.2008.04.011","article-title":"A methodology for performing global uncertainty and sensitivity analysis in system biology","volume":"254","author":"Ray","year":"2008","journal-title":"J. Theor. Biol."},{"key":"2021022305195367000_ref29","first-page":"1513","article-title":"Global stability of infectious disease models using Lyapunov functions","volume":"73","author":"Shuia","year":"2013","journal-title":"J. Appl. Math."},{"key":"2021022305195367000_ref30","first-page":"215","article-title":"Analysis of deterministic and stochastic mathematical models with resistant bacteria and bacteria debris for bacteriophage dynamics","volume":"316","author":"Vidurupola","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"2021022305195367000_ref31","first-page":"241","article-title":"Global dynamics of a predator\u2013prey system with Holling type II functional response","volume":"16","author":"Xiabong","year":"2011","journal-title":"Nonlinear Anal. Model."},{"key":"2021022305195367000_ref32","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1080\/17513758.2016.1269957","article-title":"Modelling the within-host dynamics of cholera: bacteria-viral interaction","volume":"11","author":"Xueying","year":"2017","journal-title":"J. Biol. Dyn."},{"key":"2021022305195367000_ref33","first-page":"9","article-title":"Global stability and sliding bifurcation of a non-smooth Gause predator\u2013prey system","volume":"224","author":"Yang","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"2021022305195367000_ref34","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1007\/s00332-016-9351-4","article-title":"The impact of prophage on the equilibria and stability of phages and host","volume":"27","author":"Yu","year":"2017","journal-title":"J. Nonlinear. Sci."}],"container-title":["Mathematical Medicine and Biology: A Journal of the IMA"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/imammb\/article-pdf\/38\/1\/28\/36337363\/dqaa010.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/imammb\/article-pdf\/38\/1\/28\/36337363\/dqaa010.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T19:09:27Z","timestamp":1723144167000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/imammb\/article\/38\/1\/28\/5877074"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,28]]},"references-count":34,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,7,28]]},"published-print":{"date-parts":[[2021,3,15]]}},"URL":"https:\/\/doi.org\/10.1093\/imammb\/dqaa010","relation":{},"ISSN":["1477-8599","1477-8602"],"issn-type":[{"type":"print","value":"1477-8599"},{"type":"electronic","value":"1477-8602"}],"subject":[],"published-other":{"date-parts":[[2021,3]]},"published":{"date-parts":[[2020,7,28]]}}}