Mohamed Helal

University of Sidi-Bel-Abbes · Department of Mathematics

The work deals with the basic results of a fractional order predator-prey system with harvesting. More precisely, crop-pest interactions were studied. Mathematical results like the existence, uniqueness, and positivity of the solutions are derived. It is shown that the fractional order system undergoes a possible Hopf-bifurcation at the interior eq...

In this study, the Z-type control method is applied to an intraguild crop-pest-natural enemy model, assuming that the natural enemy can predate on both crop and pest populations. For this purpose, the indirect Z-type controller is considered in the natural enemy population. After providing the design function for the crop-pest-natural enemy model w...

In this paper, we investigate a time-delayed model describing the evolution of Alzheimer disease (AD). A necessary and sufficient conditions for the existence of steady states are given. After that, we analyze the asymptotic behavior of the model, and study the local asymptotic stability of each equilibrium.

This work is devoted to the analysis of a mathematical model of chronic myeloid leukemia disease with treatment of stem and differentiated cancerous cells. The existence of a Bang-bang optimal control is studied, and the sufficient conditions for existence of switching times are obtained, using two strategies of optimal control.

In this work, we introduce a minimal model for the current pandemic. It incorporates the basic compartments: exposed, and both symptomatic and asymptomatic infected. The dynamical system is formulated by means of fractional operators. The model equilibria are analyzed. The theoretical results indicate that their stability behavior is the same as fo...

The main objective is to study the stability of the periodic solutions of an impulsive system arising in the modeling of the use of spiders as biological agents against pest crops. The abstract functional analytic mathematical framework is suitably set up and used for the scope.

The main objective is to study the stability of the periodic solutions of an impulsive system arising in the modeling of the use of spiders as biological agents against pest crops. The abstract functional analytic mathematical framework is suitably set up and used for the scope.

We prove the existence of nontrivial periodic solutions of a pulsed chemotherapeutic treatment model, by the mean of Lyapunov-Schmidt bifurcation method of a cancer model. The results obtained are applied to the model with competition between normal, sensitive tumor andresistant tumor cells. The existence of bifurcated nontrivial periodic solutions...

We study the existence of positive solutions for a model of Prion disease with impulse effects.

In this paper, we investigate a time-delayed model describing the dynamics of the hematopoietic stem cell population with treatment. First, we give some property results of the solutions. Second, we analyze the asymptotic behavior of the model, and study the local asymptotic stability of each equilibrium: trivial and positive ones. Next, a necessar...

In this work, we consider a mathematical model containing three partial differential equations describing the dynamics of hematopoietic stem cell population in the chronic myeloid leukemia. The model describes the evolution of normal, leukemic and resistant leukemic hematopoietic stem cells. We investigate conditions for existence and stability of...

The spread of epidemics has always threatened humanity. In the present circumstance of the Coronavirus pandemic, a mathematical model is considered. It is formulated via a compartmental dynamical system. Its equilibria are investigated for local stability. Global stability is established for the disease-free point. The allowed steady states are an...

We investigate a mathematical model for leukemia, described by delay differential equations. We determine the conditions of existence of trivial, nonpathologic, blast and chronic equilibria. After that, we study their local stability and the global stability of the trivial equilibrium. AMS Subject Classification: 34K20, 34K60, 92B05, 92C37, 92D25

A B-cell chronic lymphocytic leukemia has been modeled via a highly nonlinear system of ordinary differential equations. We consider the rather important theoretical question of the equilibria existence. Under suitable assumptions all model populations are shown to coexist.

In this work we develop a mathematical model of chronic myeloid leukemia including treatment with instantaneous effects. Our analysis focuses on the values of growth rate which give either stability or instability of the disease free equilibrium. If the growth rate of sensitive leukemic stem cells is less than some threshold , we obtain the stabili...

Alzheimer’s disease (AD) is a neuro-degenerative disease affecting more than 46 million people worldwide in 2015. AD is in part caused by the accumulation of A\(\beta \) peptides inside the brain. These can aggregate to form insoluble oligomers or fibrils. Oligomers have the capacity to interact with neurons via membrane receptors such as prion pro...

A mathematical model with delay for stem cells in a leukemia diseases is analyzed. We prove the existence of nontrivial steady states of the system describing the diseases. We determine the values of the delay parameter to have stability or instability of the steady states.

We develop an impulsive model for zoonotic visceral leishmaniasis disease on a population of dogs. The disease infects a population D of dogs. We determine the basic reproduction number R0, which depends on the vectorial capacity C. Our analysis focuses on the values of C which give either stability or instability of the disease-free equilibrium (D...

In this work we investigate an impulsive chemotherapy model for a population of cells containing normal cells, sensitive and resistant tumor cells. First we study the exponential stability of trivial periodic positive solution corresponding to the healthy case, after that we study the possibility to have bifurcation of nontrivial periodic positive...

In this work, we consider a mathematical model describing the dynamics of visceral leishmaniasis in a population of dogs í µí°·. First, we consider the case of constant total population í µí°·, this is the case where birth and death rates are equal, in this case transcritical bifurcation occurs when the basic reproduction number ℛ 0 is equal to one...

A chemotherapeutic treatment model for cell population with resistant tumor is considered. We consider the case of two drugs one with pulsed effect and the other one with continuous effect. We investigate stability of the trivial periodic solutions and the onset of nontrivial periodic solutions by the mean of Lyapunov-Schmidt bifurcation. Nous cons...

In this paper we considered a predator-prey model with state-dependent impulse. We determine periodic solutions for the model without impulses and we prove the existence of nontrivial periodic solution in the case of impulse depending on the state of the model.

A pulsed chemotherapeutic treatment model is investigated in this work. We prove the existence of nontrivial periodic solutions by the mean of Lyapunov-Schmidt bifurcation method of a cancer model. In this model we consider the case of application of two drugs, the first one P with continuous effect, it appears in the differential equations, and th...

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

In this work we consider a mathematical model based on a system of ordinary differential equations describing the evolution of population of dogs infected by leishmania diseases. By analyzing the corresponding characteristic equations, the local stability of infection free equilibrium point and infection equilibrium point are discussed. It is shown...

A model of prion diseases with impulse effects is studied in this work. First we transform the model to a system of three differential equations with impulse effects in order to study the stability of periodic solution. After that we study the general model by the mean of evolution semi group in order to find conditions of existence of mild solutio...

In this paper, a model describing the dynamic of chronic myeloid leukemia is studied. By analyzing the corresponding characteristic equations, the local stability of trivial and nontrivial equilibria are discussed. By establishing appropriate Lyapunov functions, we prove the global stability of the positive constant equilibrium solutions.

We introduce a mathematical model of the in vivo progression of Alzheimer's disease with focus on the role of prions in memory impairment. Our model consists of differential equations that describe the dynamic formation of {\beta}-amyloid plaques based on the concentrations of A{\beta} oligomers, PrPC proteins, and the A{\beta}-x-PrPC complex, whic...

This thesis deals with a different mathematical models deriving from natural phenomena. The essential tools in this study are differential inclusions, differential or partial differential equations (or systems of equations) and bifurcation theory. The nature of these equations depends on the problem being addressed: it may be transport equations, r...

In this paper, we present some existence results of solutions and study the topological structure of solution sets for the following first-order impulsive neutral functional differential inclusions with initial condition. Mathematical equation representative. where J := [0; b] and 0 = t 0 < t 1 < ⋯ < t m < t m+1 = b (mεℕ*), F is a set-valued map an...