Gompertz model with delays and treatment: Mathematical analysis
In this paper we study the delayed Gompertz model, as a typical model of tumor growth, with a term describing external interference that can reflect a treatment, e.g. chemotherapy. We mainly consider two types of delayed models, the one with the delay introduced in the per capita growth rate (we cal...
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| Format: | Article |
| Language: | English |
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AIMS Press
2013-03-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.551 |
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| _version_ | 1832590137096142848 |
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| author | Marek Bodnar Monika Joanna Piotrowska Urszula Foryś |
| author_facet | Marek Bodnar Monika Joanna Piotrowska Urszula Foryś |
| author_sort | Marek Bodnar |
| collection | DOAJ |
| description | In this paper we study the delayed Gompertz model, as a typical model of tumor growth, with a term describing external interference that can reflect a treatment, e.g. chemotherapy. We mainly consider two types of delayed models, the one with the delay introduced in the per capita growth rate (we call it the single delayed model) and the other with the delay introduced in the net growth rate (the double delayed model).We focus on stability and possible stability switches with increasing delay for the positive steady state. Moreover, we study a Hopf bifurcation, including stability of arising periodic solutions for a constant treatment. The analytical results are extended by numerical simulations for a pharmacokinetic treatment function. |
| format | Article |
| id | doaj-art-a4e6b804c14f4930a6d8361f0c1f7f20 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2013-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-a4e6b804c14f4930a6d8361f0c1f7f202025-01-24T02:26:12ZengAIMS PressMathematical Biosciences and Engineering1551-00182013-03-0110355156310.3934/mbe.2013.10.551Gompertz model with delays and treatment: Mathematical analysisMarek Bodnar0Monika Joanna Piotrowska1Urszula Foryś2Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 WarsawInstitute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 WarsawInstitute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 WarsawIn this paper we study the delayed Gompertz model, as a typical model of tumor growth, with a term describing external interference that can reflect a treatment, e.g. chemotherapy. We mainly consider two types of delayed models, the one with the delay introduced in the per capita growth rate (we call it the single delayed model) and the other with the delay introduced in the net growth rate (the double delayed model).We focus on stability and possible stability switches with increasing delay for the positive steady state. Moreover, we study a Hopf bifurcation, including stability of arising periodic solutions for a constant treatment. The analytical results are extended by numerical simulations for a pharmacokinetic treatment function.https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.551stabilitydelay equationhopf bifurcationgompertz modelstability switches. |
| spellingShingle | Marek Bodnar Monika Joanna Piotrowska Urszula Foryś Gompertz model with delays and treatment: Mathematical analysis Mathematical Biosciences and Engineering stability delay equation hopf bifurcation gompertz model stability switches. |
| title | Gompertz model with delays and treatment: Mathematical analysis |
| title_full | Gompertz model with delays and treatment: Mathematical analysis |
| title_fullStr | Gompertz model with delays and treatment: Mathematical analysis |
| title_full_unstemmed | Gompertz model with delays and treatment: Mathematical analysis |
| title_short | Gompertz model with delays and treatment: Mathematical analysis |
| title_sort | gompertz model with delays and treatment mathematical analysis |
| topic | stability delay equation hopf bifurcation gompertz model stability switches. |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.551 |
| work_keys_str_mv | AT marekbodnar gompertzmodelwithdelaysandtreatmentmathematicalanalysis AT monikajoannapiotrowska gompertzmodelwithdelaysandtreatmentmathematicalanalysis AT urszulaforys gompertzmodelwithdelaysandtreatmentmathematicalanalysis |