As an applied mathematician, I work in a wide variety of fields. I have contributed to theoretical and applied mathematics in mathematical biology. Although I enjoy assisting many different disciplines, I also realize the importance of having areas of focus and expertise. Therefore, I have three main research interests: · Study of Hopf bifurcation in differential equation. · Mathematical biology. Heamatopoiesis model Predator prey model Tumor-immune system competition model · Recently, reaction diffusion systems and slow-fast dynamical systems · Numerical simulations via Matlab software
The first part of my research focuses on the stability of the branch of bifurcating periodic solutions (Hopf bifurcation) via the variation of constant formula developed by J. Hale in 1997, the reduction of the integral equation to a two dimensional system via the center manifold and Linsted Poincaré expansion of the elements of bifurcation given by many authors Friedmann et al. and Hassard et al. An estimation of the region of stability of bifurcating periodic solutions by estimating the distance between the solution of the original equation and the bifurcated periodic solution.
1- The second part is devoted to analyzing the stability of steady states of a system of delay differential equations modeling the production of stem cells in bone marrow which called the heamatopoieisis phenomena. The model describes the two main phases of the cycle of a stem cell : proliferating phase and resting (quiescent) phase This model is introduced by M. C. Mackey and is extensively studied by M. Adimy in the recent years. I prove the occurrence of Hopf bifurcation around the non trivial steady state when the the delay crosses a critical value by taking the delay (necessary time of the division of a stem cell) as a parameter of bifurcation. The stability of bifurcating periodic solutions is demonstrated via the method of O. Dieckmann and the adjoint operator and the duality product introduced in J. Hale. A one of my recent work discuss the mathematical modeling and existence of oscillating solutions of a system of delay differential equations. This system models the interaction between the protein and three different types of cells : Precursor cells, stem cells and blood cells. The first model modeling the production and developpement of precursor cells is introduced by A. Grabosh, the model is given by system of four type of differential equations : ordinary differential equation, partial differential equation, delay differential equation and integro-differential equation. The author assume that the number of stem cells is constant and the protein is an equilibrium (dE/dt=0) and she obtain a structured model with maturity. The author studied the existence and positivity and stability of the equilibrium points via the dual semi-group theory. My model is an unstructured one which extracted from the model of A. Grabosch, I assume that the number of stem cells is not constant and the protein is an equilibrium (dE/dt=0) and the rate of production of precursor cells and the velocity of maturation and the death rate are constants. I obtain a model governed by a system of three delay differential equations modeling the interaction between stem cells, precursor cells and blood cells. The mathematical study of this model has been made with respect to the time delay. This part of my research is focused on the mathematical study of system of two differential equations with discrete delay. The model was initially introduced by A. Alaoui et al which describes the interaction between two types of population the first is prey and the second is the predator one. The authors take into account a modified functional response of Leslie-Gower and Holling-type III. The stability of possible steady states and the permanence results are demonstrated by the same authors. In my work, I prove the existence of bifurcating periodic solutions via Hopf bifurcation theorem (J. Hale et al. and Hassard et al.) . The stability of bifurcation periodic solutions is proved via normal form theory (T. Faria et al).
This part is devoted to the mathematical analysis of an unstructured model given by a system of two differential equation with discrete delay, which describes the interaction between two types of tumorale cells : proliferating cells and quiescent cells. The model without delay is given by M. Gyllenberg et al.. To made the model more realistic, I introduce the delay, which describes the necessary time of cell division. I prove the existence of steady states and the existence of periodic oscillations via Hopf bifurcation theorem.
In this part, I have extensively studied a model which describes the interaction between immune system and tumor site. The first model was given by Kuznetsov et al. by five ordinary differential equations modeling five types of cells : tumor cells, inactive tumor cells, effectors cells, inactive effectors cells and the complex tumor-effectors cells. The model was studied numerically by D. Kirschner with two ordinary differential equations by taking into account the effect of immunotherapy and with functional immune response of Michaelis-Menten type and the interaction with the immune cells. From experimental study, M. galach consider that the inactive effectors and tumor cells and complex cells are equilibriums and the functional immune response takes the form of Lotka-Volterra, she study the obtained model numerically. In my research works, I reconsider the model given by M. Galach without delay and by introducing the time delay. I prove that the existence of the possible steady states, in the first step i consider the model without delay and by taking the source of the immune cells as a parameter of bifurcation , I prove the occurrence of Hopf bifurcation for some critical value of this source. The stability of the existing periodic solutions is showed. In the second step, I consider the model with delay and I prove the existence of periodic solutions around each non trivial steady state (which the meaningful equilibrium points : correspond to the existence of tumor site) via Hopf bifurcation theorem in the two cases of the immune response parameter (negative and positive). The conditions of stability of the steady states and the occurrence of Hopf bifurcation are given in each case by the expression of the parameters of the model. The stability of bifurcating branch via the method of O. Dieckmann is demonstrated in each case with respect to the values of the parameters of the model.
Numerical applications and numerical simulations with Matlab software are given in order to illustrate the theoretical results proved in my research works. From the left to the right : Aziz Alaoui M. A.(Le Havre Univ); C. Bertelles (Le Havre Univ); P. Bourgine (Institut Polytech Paris); R. Yafia(Ibn Zohr Univ); F. Ghadi(Ibn Zohr Univ), A. El Mossadik(Ibn Zohr Univ). |