Leukemia mathematical model

In this paper, we study a mathematical model of leukemia diseases. We find sufficient conditions for existence and local stability of steady states.


Introduction
Hematopoietic stem cells are found in the bone marrow and are able to renew themselves through cell division producing blood cells, a significant increase in the number of white blood cells disease causes chronic myeloid leukemia which is characterized by a chromosomal anomaly acquired, that is the translocation between chromosome 9 and 22 giving birth to an abnormal chromosome called the Philadelphia chromosome, this translocation generates a protein from the merger of the Bcr gene on chromosome 22 and Abl genes on chromosome 9, this protein is a tyrosine kinase Bcr-Abl. Generally, tyrosine kinases are components that control cell proliferation, differentiation and apoptosis, they have a very important role in signal transduction because the tyrosine kinase can transfer a phosphate group from adenosine triphosphate to another protein in a cell but the tyrosine kinase Bcr-Abl instructs cells to grow out of control and prevents them from undergoing apoptosis, resulting in the formation of a tumor. Among the treatments can produce a significant chance of cure is IMATINIB [5] which is a competitive inhibitor of the tyrosine kinase activity because it will bind to quote binding of adenosine tri-phosphate prevents tyrosine to give these orders cancerous proliferation. The first mathematical models describing the dynamics of hematopoietic stem cells have been proposed by Mackey [3]. In recent works Adimy & Crauste [1], Dingli & Michor [2] and Michor et al. [4] have studied the dynamics of normal and cancer stem cells in chronic myeloid leukemia.
In our work we focus on the development of normal, cancerous and resistant hematopoietic stem cells in chronic myeloid leukemia. We study the existence of equilibrium points, their local stability and we give some numerical simulations.

The model
We consider the following system with ,

Existence and stability of equilibrium
In this section, we analyze the existence and local stability of feasible equilibrium of (1).

Existence of equilibrium
The equilibrium points of (1) are Theorem 3.1: The system (1) admits feasible equilibrium points according to the following conditions.
, g 0r < m and c y g 0r n c y g 0r + c x (m − g 0r ) < d 0 < c y g 0r n c y g 0r + c x (m − g 0r ) the equilibrium point E 3 exists. 5. If g 0 < m and g 2 < g + h 2 (u) the equilibrium point E 4 exists. 6. If g 0r < m and d 2r < d r + h 2r (u) the equilibrium point E 5 exists.

Local stability of equilibrium
The general form of the Jacobian with respect to each equilibrium point is given by Local Stability of equilibrium points 01006-p.3

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We have

Theorem 3.2:
1. The equilibrium point E 0 is asymptotically stable if d 2r < d r + h 2r (u), g 2 < g + h 2 (u), d 2 < d, n < d 0 and m < g 0r . 2. The equilibrium point E 1 is asymptotically stable if d 2r < d r + h 2r (u), g 2 < g + h 2 (u), d 2 < d, d 0 < n and mc x d 0 c x d 0 +c y (n−d 0 ) < g 0r . 3. The equilibrium point E 2 is unstable. 4. The equilibrium point E 3 is unstable. 5. The equilibrium point E 4 is unstable. 6. The equilibrium point E 5 is asymptotically stable if d 2r < d r + h 2r (u),g 2 < g + h 2 (u), d 2 < d, g 0r < m and n c y g 0r For the equilibrium point E 0 we have We have Then, the eigenvalues are 1 For the equilibrium point E 1 we have We have Then, the eigenvalues are 1 − g 0r and 6 = d 0 n d 0 − 1 .
We note that a 55 is a positive eigenvalue, then E 2 is unstable.

4.
For the equilibrium point E 3 we have WMLS 2014 such that According to the conditions of existence of the equilibrium point E 3 we can prove that one of eigenvalues is positive that is E 3 is unstable.

5.
For the equilibrium point E 4 we have and

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We have Note that = g 0 − g 0r is a positive eigenvalue which implies that E 4 is unstable. 6.
For the equilibrium point E 5 we have We have Then the eigenvalues are n c y g 0r c y g 0r + c x (m − g 0r ) − d 0 , 5 = g 0r − g 0 and 6 = g 0r g 0r m − 1 .

Numerical Simulation for the local stability
In this section, we give some numerical simulations for the points E 1 and E 5 to illustrate our results.