Analytical Study of the Existence of a Hopf Bifurcation in the Tumor Cell Growth Model with Time Delay
Abstract In this paper, we study a mathematical model of an immune response system consisting of a number of immune cells that work together to protect the human body from invading tumor cells. The delay differential equation is used to model the immune system caused by a natural delay in the activa...
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| Format: | Article |
| Language: | English |
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UIN Syarif Hidayatullah
2021-05-01
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| Series: | InPrime |
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| Online Access: | https://journal.uinjkt.ac.id/index.php/inprime/article/view/19515 |
| _version_ | 1797293181229858816 |
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| author | A. Yusnaeni Kasbawati Kasbawati Toaha Syamsuddin |
| author_facet | A. Yusnaeni Kasbawati Kasbawati Toaha Syamsuddin |
| author_sort | A. Yusnaeni |
| collection | DOAJ |
| description | Abstract
In this paper, we study a mathematical model of an immune response system consisting of a number of immune cells that work together to protect the human body from invading tumor cells. The delay differential equation is used to model the immune system caused by a natural delay in the activation process of immune cells. Analytical studies are focused on finding conditions in which the system undergoes changes in stability near a tumor-free steady-state solution. We found that the existence of a tumor-free steady-state solution was warranted when the number of activated effector cells was sufficiently high. By considering the lag of stimulation of helper cell production as the bifurcation parameter, a critical lag is obtained that determines the threshold of the stability change of the tumor-free steady state. It is also leading the system undergoes a Hopf bifurcation to periodic solutions at the tumor-free steady-state solution.
Keywords: tumor–immune system; delay differential equation; transcendental function; Hopf bifurcation.
Abstrak
Dalam makalah ini, dikaji model matematika dari sistem respon imun yang terdiri dari sejumlah sel imun yang bekerja sama untuk melindungi tubuh manusia dari invasi sel tumor. Persamaan diferensial tunda digunakan untuk memodelkan sistem kekebalan yang disebabkan oleh keterlambatan alami dalam proses aktivasi sel-sel imun. Studi analitik difokuskan untuk menemukan kondisi di mana sistem mengalami perubahan stabilitas di sekitar solusi kesetimbangan bebas tumor. Diperoleh bahwa solusi kesetimbangan bebas tumor dijamin ada ketika jumlah sel efektor yang diaktifkan cukup tinggi. Dengan mempertimbangkan tundaan stimulasi produksi sel helper sebagai parameter bifurkasi, didapatkan lag kritis yang menentukan ambang batas perubahan stabilitas dari solusi kesetimbangan bebas tumor. Parameter tersebut juga mengakibatkan sistem mengalami percabangan Hopf untuk solusi periodik pada solusi kesetimbangan bebas tumor.
Kata kunci: sistem tumor–imun; persamaan differensial tundaan; fungsi transedental; bifurkasi Hopf. |
| first_indexed | 2024-03-07T20:09:09Z |
| format | Article |
| id | doaj.art-a00828721d2a4e9ba8a3cedaa5f93ebd |
| institution | Directory Open Access Journal |
| issn | 2686-5335 2716-2478 |
| language | English |
| last_indexed | 2024-03-07T20:09:09Z |
| publishDate | 2021-05-01 |
| publisher | UIN Syarif Hidayatullah |
| record_format | Article |
| series | InPrime |
| spelling | doaj.art-a00828721d2a4e9ba8a3cedaa5f93ebd2024-02-28T02:16:03ZengUIN Syarif HidayatullahInPrime2686-53352716-24782021-05-0131202810.15408/inprime.v3i1.195157927Analytical Study of the Existence of a Hopf Bifurcation in the Tumor Cell Growth Model with Time DelayA. Yusnaeni0Kasbawati Kasbawati1Toaha Syamsuddin2Hasanuddin UniversityHasanuddin UniversityHasanuddin UniversityAbstract In this paper, we study a mathematical model of an immune response system consisting of a number of immune cells that work together to protect the human body from invading tumor cells. The delay differential equation is used to model the immune system caused by a natural delay in the activation process of immune cells. Analytical studies are focused on finding conditions in which the system undergoes changes in stability near a tumor-free steady-state solution. We found that the existence of a tumor-free steady-state solution was warranted when the number of activated effector cells was sufficiently high. By considering the lag of stimulation of helper cell production as the bifurcation parameter, a critical lag is obtained that determines the threshold of the stability change of the tumor-free steady state. It is also leading the system undergoes a Hopf bifurcation to periodic solutions at the tumor-free steady-state solution. Keywords: tumor–immune system; delay differential equation; transcendental function; Hopf bifurcation. Abstrak Dalam makalah ini, dikaji model matematika dari sistem respon imun yang terdiri dari sejumlah sel imun yang bekerja sama untuk melindungi tubuh manusia dari invasi sel tumor. Persamaan diferensial tunda digunakan untuk memodelkan sistem kekebalan yang disebabkan oleh keterlambatan alami dalam proses aktivasi sel-sel imun. Studi analitik difokuskan untuk menemukan kondisi di mana sistem mengalami perubahan stabilitas di sekitar solusi kesetimbangan bebas tumor. Diperoleh bahwa solusi kesetimbangan bebas tumor dijamin ada ketika jumlah sel efektor yang diaktifkan cukup tinggi. Dengan mempertimbangkan tundaan stimulasi produksi sel helper sebagai parameter bifurkasi, didapatkan lag kritis yang menentukan ambang batas perubahan stabilitas dari solusi kesetimbangan bebas tumor. Parameter tersebut juga mengakibatkan sistem mengalami percabangan Hopf untuk solusi periodik pada solusi kesetimbangan bebas tumor. Kata kunci: sistem tumor–imun; persamaan differensial tundaan; fungsi transedental; bifurkasi Hopf.https://journal.uinjkt.ac.id/index.php/inprime/article/view/19515tumor–immune systemdelay differential equationtranscendental functionhopf bifurcation |
| spellingShingle | A. Yusnaeni Kasbawati Kasbawati Toaha Syamsuddin Analytical Study of the Existence of a Hopf Bifurcation in the Tumor Cell Growth Model with Time Delay InPrime tumor–immune system delay differential equation transcendental function hopf bifurcation |
| title | Analytical Study of the Existence of a Hopf Bifurcation in the Tumor Cell Growth Model with Time Delay |
| title_full | Analytical Study of the Existence of a Hopf Bifurcation in the Tumor Cell Growth Model with Time Delay |
| title_fullStr | Analytical Study of the Existence of a Hopf Bifurcation in the Tumor Cell Growth Model with Time Delay |
| title_full_unstemmed | Analytical Study of the Existence of a Hopf Bifurcation in the Tumor Cell Growth Model with Time Delay |
| title_short | Analytical Study of the Existence of a Hopf Bifurcation in the Tumor Cell Growth Model with Time Delay |
| title_sort | analytical study of the existence of a hopf bifurcation in the tumor cell growth model with time delay |
| topic | tumor–immune system delay differential equation transcendental function hopf bifurcation |
| url | https://journal.uinjkt.ac.id/index.php/inprime/article/view/19515 |
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