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Mathematical analysis of a B-cell chronic lymphocytic leukemia model with immune response

Youcef Belgaid

Youcef Belgaid

Biomathematics Laboratory, Univ Sidi-Bel-AbbésSidi-Bel-Abbès, Algeria
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Belgaid, Youcef
, 
Mohamed Helal

Mohamed Helal

Biomathematics Laboratory, Univ Sidi-Bel-AbbésSidi-Bel-Abbès, Algeria
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Helal, Mohamed
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Ezio Venturino

Ezio Venturino

Dipartimento di Matematica “Giuseppe Peano”, Università; di TorinoTurin, Italy
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Venturino, Ezio
  
Dec 26, 2019
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Published Online: Dec 26, 2019
Page range: 551 - 558
Received: Jun 23, 2019
Accepted: Sep 26, 2019
DOI: https://doi.org/10.2478/AMNS.2019.2.00052
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© 2019 Youcef Belgaid et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 Public License.

Model parameters and their meaning

bB constant source rate of B-CLL produced by bone marrow
r replication rate of the leukemic B cells
dB natural mortality rate of the leukemic B
dBN killing rate of B-CLL cells by N cells
dBT killing rate of B-CLL cells by T cells
bN constant source rate of N cells
dN mortality rate of N cells
dNB deactivation rate of N cells by contact with B-CLL cells
bT constant production rate of T cells
dT natural mortality rate of T cells
dTB T cells activity suppression rate by contact with B-CLL cells
k fraction of TH cell activation that results in T cells recruitment
aTH maximal TH cells activation rate by contact with B-CLL cells
bTH constant production rate of TH cells
dTH natural mortality rate of TH cells
s half saturation constant
L parameter shaping the saturating sigmoid response

Signs of the coefficients of equation (12) for the case L = 3

a6 a5 a4 a3 a2 a1 a0 sign variations positive roots
- - + + + + + 1 1
- - + + + - + 3 1 or 3
- - + - + + + 3 1 or 3
- - + - + - + 5 1 or 3 or 5
- - - + + + + 1 1
- - - + + - + 3 1 or 3
- - - - + + + 1 1
- - - - + - + 3 1 or 3

Signs of the coefficients of equation (7) for a general value of L

aL+3 aL+2 aL+1 aL aL–1 … a4 a3 a2 a1 a0 sign variations positive roots
- - + + 0 0 0 + + + + 1 1
- - + + 0 0 0 + + - + 3 1 or 3
- - + - 0 0 0 + + + + 3 1 or 3
- - + - 0 0 0 + + - + 5 1 or 3 or 5
- - - + 0 0 0 + + + + 1 1
- - - + 0 0 0 + + - + 3 1 or 3
- - - - 0 0 0 + + + + 1 1
- - - - 0 0 0 + + - + 3 1 or 3
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Life Sciences, Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics, Physics, other
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