The mathematical model of stem cells is discussed with its motivation to describe the tissue relationship by technically introducing a two compartments model. The clear link between the proliferation phase of stem cells and the circulating neutrophil phase is set forth after delay feedback control of the state variable of stem cells. Hopf bifurcation is discussed with varying free parameters and time delays. Based on the center manifold theory, the normal form near the critical point is computed and the stability of bifurcating periodical solution is rigorously discussed. With the aids of the artificial tool on-hand which implies how much tedious work doing by DDE-Biftool software, the bifurcating periodic solution after Hopf point is continued by varying time delay.
References
[1]
Adimy, M., Crauste, F. and Ruan, S. (2005) A Mathematical Study of the Hematopoiesis Process with Applications to Chronic Myelogenous Leukemia. SIAM Journal on Apllied Mathematics, 65, 1328-1352. https://doi.org/10.1137/040604698
[2]
Adimy, M., Crauste, F. and Ruan, S. (2005) Stability and Hopf Bifurcation in a Mathematical Model of Pluripotent Stem Cell Dynamics. Nonlinear Analysis: Real World Applications, 6, 651-670. https://doi.org/10.1016/j.nonrwa.2004.12.010
[3]
Daniel C. and Humphries, A.R. (2017) Dynamics of a Mathematical Hematopoietic Stem-Cell Population Model. ArXiv: 1712.08308. https://arxiv.org/abs/1712.08308
[4]
Fortin, P. and Mackey, M.C. (1999) Periodic Chronic Myelogenous Leukemia: Spectral Analysis of Blood Cell Counts and Aetiological Implications. British Journal of Haematology, 104, 336-345. https://doi.org/10.1046/j.1365-2141.1999.01168.x
[5]
Mackey, M.C. (1978) United Hypothesis for the Origin of Aplasric Anemia and Periodic Hematopoiesis. Blood, 51, 941-956.
[6]
Haurie, C., Dale, D.C. and Mackey, M.C. (1998) Cyclical Neutropenia and Other Periodic Hematological Disorders: A Review of Mechanisms and Mathematical Models. Blood, 92, 2629-2640.
[7]
Alaoui, T.H., Yafia, R. and Aziz-Alaoui, M.A. (2007) Hopf Bifurcation Direction in a Delayed Hematopoietic Stem Cells Model. Arab Journal of Mathematics and Mathematical Sciences, 1, 35-50.
[8]
Zhuge, C.J., Lei, J.Z. and Mackey, M.C. (2012) Neutrophil Dynamics in Response to Chemotherapy and G-SCF. Journal of Theory Biology, 293, 111-120. https://doi.org/10.1016/j.jtbi.2011.10.017
[9]
Brooks, G., Langlois, G.P., Lei, J.Z. and Mackey, M.C. (2012) Neutrophil Dynamics after Chemotherapy and G-CSF: The Role of Pharmacokinetics in Shaping the Response. Journal of Theory Biology, 315, 97-109. https://doi.org/10.1016/j.jtbi.2012.08.028
[10]
Lei, J.Z. and Mackey, M.C. (2014) Understanding and Treating Cytopenia through Mathematical Modeling. In: Corey, S., Kimmel, M. and Leonard, J., Eds., A Systems Biology Approach to Blood, Advances in Experimental Medicine and Biology, Vol. 844, Springer, New York, 279-302. https://doi.org/10.1007/978-1-4939-2095-2_14
[11]
Ma, S.Q. (2014) Stability and Bifurcation Analysis of a Type of Hematopoietic Stem Cell Model. International Journal of Modern Nonlinear Theory and Application, 10, 13-27. https://doi.org/10.4236/ijmnta.2021.101002
[12]
Ma, S.Q. (2017) Periodic Oscillation in Neutrophil Models with Time Delays. International Journal of Modern Nonlinear Theory and Application, 6, 119-133. https://doi.org/10.4236/ijmnta.2017.64011
[13]
Ma, S.Q. (2019) Hopf Bifurcation of a Type of Neuron Model with Multiple Time Delays. International Journal of Bifurcation and Chaos, 29, Article ID: 1950163. https://doi.org/10.1142/S0218127419501633
[14]
Ma, S.Q. (2022) Bifurcation Analysis of Periodic Oscillation in a Hematopoietic Stem Cells Model with Time Delay Control. Mathematical Problems in Engineering, 2022, Article ID: 7304280. https://doi.org/10.1155/2022/7304280
[15]
Sieber, J., Engelborghs, K., Luzyanina, T., Samaey, G. and Roose, D. (2016) DDE-BIFTOOL v. 3.1.1 Manual—Bifurcation Analysis of Delay Differential Equations. Dynamical Systems. ArXiv: 1406.7144. https://arxiv.org/abs/1406.7144
[16]
Engelborghs, K., Luzyanina, T. and Samae, G. (2001) DDE-BIFTOOL v. 2.00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations. Technical Report TW330.
[17]
Verheyden, K., Luzyanina, T. and Roose, D. (2004) Location and Numerical Preservation of Characteristic Roots of Delay Differential Equations by LMS Methods. Technical Report TW-382.
[18]
Kuzenetsov, Y.A. (2004) Elements of Applied Bifurcation Theory Applied Mathematical Sciences. Vol. 112, Springer, New York.
[19]
Hale, J.K. and Lunel, S.M.V. (1993) Introduction of Functional Differential Equations. In: Bloch, A., Charles, L., Epstein, A.G. and Greengard, L., Eds., Applied Mathematical Sciences, Vol. 99, Springer, New York, 1-10. https://doi.org/10.1007/978-1-4612-4342-7
[20]
Hassard, B.D., Kazarinoff, N.D. and Wan, Y.H. (1981) Theory and Applications of Hopf Bifurcation. Combridge University Press, Cambridge.
[21]
Faria, T. and Magalhaes, M.L. (1995) Normal Form for Retarded Functional Differential Equations with Parameters and Applications to Hopf Bifurcation. Journal of Differential Equations, 122, 181-200. https://doi.org/10.1007/978-1-4612-4342-7
