Geodesic
A stability theorem for equilibria of delay differential equations in a critical case with application to a model of cell evolution
Karim Amin1, 2 ; Irina Badralexi1 ; Andrei Halanay1 ; Ragheb Mghames1, 2
1 Department of Mathematics and Informatics, University Politehnica of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania.
2 School of Arts and Sciences, Department of Mathematics and Physics, Lebanese International University, Bekaa, Lebanon.
Mathematical modelling of natural phenomena, Tome 16 (2021), article no. 36 Cet article a éte moissonné depuis la source EDP Sciences

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In this paper the stability of the zero equilibrium of a system with time delay is studied. The critical case of a multiple zero root of the characteristic equation of the linearized system is treated by applying a Malkin type theorem and using a complete Lyapunov-Krasovskii functional. An application to a model for malaria under treatment considering the action of the immune system is presented.
DOI : 10.1051/mmnp/2021021

Karim Amin 1, 2 ; Irina Badralexi 1 ; Andrei Halanay 1 ; Ragheb Mghames 1, 2

1 Department of Mathematics and Informatics, University Politehnica of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania.
2 School of Arts and Sciences, Department of Mathematics and Physics, Lebanese International University, Bekaa, Lebanon. 
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Karim Amin; Irina Badralexi; Andrei Halanay; Ragheb Mghames. A stability theorem for equilibria of delay differential equations in a critical case with application to a model of cell evolution. Mathematical modelling of natural phenomena, Tome 16 (2021), article  no. 36. doi: 10.1051/mmnp/2021021

[1] M. Adimy, Y. Bourfia, M.L. Hbid, C. Marquet Age-structured model of hematopoiesis dynamics with growth factor-dependent coefficients Electr. J. Differ. Equ 2016 1 20

[2] M. Adimy, F. Crauste, S. Ruan Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases Bull. Math. Biol 2006 2321 2351

[3] I. Badralexi, A. Halanay, R. Mghames A delay differential equations model for maintenance therapy in acute lymphoblastic leukemia U.P.B. Sci. Bull. Series A 2020 13 24

[4] S. Balea, A. Halanay, D. Jardan, M. Neamtu, C.A. Safta Stability analysis of a feedback model for the action of the immune system in leukemia Math. Model. Nat. Phenom 2014 108 132

[5] R. Bellman and K.L. Cooke, Differential-Difference Equations. Academic Press, New York (1963).

[6] P. Birget, M. Greischar, S. Reece, N. Mideo Altered life history strategies protect malaria parasites against drugs Evolut. Appl 2017 1 14

[7] C. Colijn, M.C. Mackey A mathematical model for hematopoiesis: I. Periodic chronic myelogenous leukemia J. Theor. Biol 2005 117 132

[8] L.E. El’sgol’ts and S.B. Norkin, Introduction to the theory of differential equations with deviating arguments (in Russian). Nauka, Moscow (1971).

[9] A. Halanay, Differential Equations. Stability, Oscillations, Time Lags. Academic Press, New York (1966).

[10] A. Halanay, D. Candea, R. Radulescu Existence and stability of limit cycles in a two-delays model of hematopoiesis including asymmetric division Math. Model. Nat. Phenom 2014 58 78

[11] J. Hale and S.M. Verduyn-Lunel, Introduction to Functional Differential Equations. Springer, New York (1993).

[12] D.H. Kerlin, M.L. Gatton Preferential Invasion by Plasmodium Merozoites and the Self-Regulation of Parasite Burden PLoS ONE 2013 e57434

[13] V.L. Kharitonov, A.P. Zhabko Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems Automatica 2003 15 20

[14] V.L. Kharitonov Time Delay Systems: Lyapunov Functionals and Matrices 2013

[15] N. Khodzhaeva, A. Baranova, A. Tokmalaev The immunological Plasmodium falciparum malaria characteristics of children in Tajikistan Republic Hindawi J. Tropical Med. 2019

[16] P. Kim, P. Lee, D. Levy A theory of immunodominance and adaptive regulation Bull. Math. Biol 2011 1645 1665

[17] B. Ma, C. Li, J. Warner Structured mathematical models to investigate the interactions between Plasmodium falciparum malaria parasites and host immune response. Math. Biosci. 2019 65 75

[18] I.G. Malkin, Theory of stability of motion (in Russian), Nauka, Moskow (1966) English translation: Atomic Energy Comm. Translation AEC-TR-3352.

[19] G. Molineux, M. Foote and S. Elliott, Erythropoiesis and Eythropoietins. Second Edition Birkhauser (2009).

[20] C. Tomasetti, D. Levi Role of symmetric and asymmetric division of stem cells in developing drug resistance PNAS 2010 16766 16771

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