In this paper, we develop a business cycle model with general investment, variable depreciation r... more In this paper, we develop a business cycle model with general investment, variable depreciation rate of capital stock and two delays. The first delay describes the time lag between the decision of investment and its implementation, while the second one models the time lag for investment to be produced. The well-posedness and the existence of economic equilibrium are carefully investigated. Moreover, the stability of the economic equilibrium and the existence of Hopf bifurcation are established. The case when the two delays are equal is rigourously studied.
The Cochlea plays a crucial role in the hearing of mammalian species including man. The basic fun... more The Cochlea plays a crucial role in the hearing of mammalian species including man. The basic function of the cochlea is to map sounds of different frequencies into corresponding characteristic positions on the basilar membrane. Many disciplines meet in the study of the auditory system to understand the truth function of the cochlea. An abnormality or small perturbation in the physical parameters of the cochlea may result a malfunction in the auditory system. In this paper, we developed a mathematical model in the order to show numerically the effect of stiffness and damping on the amplitude displacement in the case of a passive cochlea with the objective to study the ear dysfunction.
An increase of pressure in the structure of cochlea may cause a hearing loss. In this paper, we e... more An increase of pressure in the structure of cochlea may cause a hearing loss. In this paper, we established the relationship between the fluid pressure and the amplitude of displacement of Basilar Membrane to clarify the mechanisms of hearing loss caused by increasing of this pressure. So, a mathematical cochlear model was formulated using finite difference method in order to explain and demonstrate this malfunction in passive model. Numerical simulations may be considered as helpful tools which may extend and complete the understanding of a cochlea dysfunction.
In HIV infection, the latent cells represent a reservoir that contributes to the failure of the H... more In HIV infection, the latent cells represent a reservoir that contributes to the failure of the Highly Active Anti-Retroviral Therapy (HAART). This fact requires investigating the possible strategy to improve the administration of the HAART therapy, in order to guarantee the control of the virus load to the lost level as long as possible. In this work, we aim to study the possibility of reducing the latent infected CD4 + reservoir in the HIV infection by considering a mathematical model of two types of latently infected CD4 + : fast and slow, and eight virus strains: wild-type, three single mutants, three double mutants and a fully resistant triple mutant. The HAART therapy is considered as an optimal control problem that aimes to reduce the virus load and the infected cells. Our optimal control approach shows the impact of the optimal HAART therapy on reducing two different types of the reservoirs of the latent infected CD4 + cells.
ABSTRACT Résumé Les cartes thématiques, particulièrement la carte pluviométrique du Maroc datent ... more ABSTRACT Résumé Les cartes thématiques, particulièrement la carte pluviométrique du Maroc datent d'une quarantaine d'année. Il est indéniable que leur réactualisation permettra de mettre à la disposition des chercheurs de nouvelles données pertinentes sur la pluviométrie au Maroc, et servir comme un outil d'aide à la décision pour les décideurs politiques.
In this paper, we are interested in control of a substance which cir- culate among organs in leav... more In this paper, we are interested in control of a substance which cir- culate among organs in leaving being. We determine, under certain con- ditions, the optimal control which steers such systems from an initial state to a desired one.The linear quadratic optimal control problem of such systems is analyzed using the Hilbert Uniqueness Method (HUM). To illustrate our approach,
In this paper, we develop a mathematical model that describes the spatiotemporal dynamics of busi... more In this paper, we develop a mathematical model that describes the spatiotemporal dynamics of business cycle under the goods and services market as well as the money market. We first prove that the developed model is mathematically and economically well-posed. The conditions for the existence of economic equilibrium are rigorously established. Moreover, the stability analysis and the existence of Hopf bifurcation are carefully investigated. Finally, our theoretical results are illustrated with some numerical simulations.
In this chapter, we propose and analyze a class of three spatiotemporal models describing infecti... more In this chapter, we propose and analyze a class of three spatiotemporal models describing infectious diseases caused by viruses such as the human immunodeficiency virus (HIV) and the hepatitis B virus (HBV). The first model with cellular immunity, the second with humoral immunity and the third with cellular and humoral immune responses. In the three proposed models, the disease transmission process is modeled by a general incidence function which includes several forms existing in the literature. In addition, the global analysis of the proposed models is rigorously investigated. Furthermore, biological findings of our analytical results are presented. Moreover, mathematical virus models and results presented in many previous studies are extended and generalized.
Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, 2019
In this work, we propose and analyze a fractional order model for hepatitis B virus (HBV) infecti... more In this work, we propose and analyze a fractional order model for hepatitis B virus (HBV) infection with capsids and cure of infected cells. We first prove the existence, positivity, and boundedness of solutions in order to ensure the well-posedness of our proposed model. By constructing appropriate Lyapunov functionals, the global stability of the steady states is established. Numerical simulations are presented in order to validate our theoretical results.
Communications in Mathematical Biology and Neuroscience, 2019
Actually, cancer is considered one of the leading causes of death in the world. Various therapeut... more Actually, cancer is considered one of the leading causes of death in the world. Various therapeutic strategies have been developed to combat this dangerous disease. This article investigates a promising therapeutic strategy by proposing a mathematical model that describes the dynamics of cancer treatment with oncolytic viruses. The proposed model integrates the time needed for infected tumor cells to produce new virions after viral entry, the probability of surviving during the latent period, and the saturation effect. We first prove the well-posedness of model and the existence of three equilibria that represent the desired outcome of therapy, the complete failure of therapy and the partial success of therapy. Furthermore, the stability analysis of equilibria and the existence of Hopf bifurcation are rigourously investigated.
Nonlinear Analysis and Differential Equations, 2016
In this paper, we propose a stochastic viral infection model with general functional response. Th... more In this paper, we propose a stochastic viral infection model with general functional response. The well posedness of the proposed model is investigated. Also, the extinction of the infection is fully determined by the basic reproduction number R 0. Furthermore, the dynamical behavior around the chronic infection equilibrium is established. Finally, Numerical simulations are given to illustrate our theoretical results.
A new stochastic SIRS epidemic model with specific functional response is proposed and analyzed. ... more A new stochastic SIRS epidemic model with specific functional response is proposed and analyzed. First, we show that the model is biologically well-posed by proving the global existence, positivity and boundedness of solutions. Moreover, sufficient conditions for the extinction and persistence of the disease are also obtained. In the end, some numerical simulations are presented to illustrate our analytical results.
International Journal of Tomography and Statistics
The aim of this work is to investigate a new mathematical model that describe HIV infection of CD... more The aim of this work is to investigate a new mathematical model that describe HIV infection of CD4+ T-cells during therapy. A novel feature is that the both therapy and the cure rate are incorporated into the model. We prove that the infection will die out if the basic reproduction number R0 lesss than 1 while the HIV infection may become endemic if R0 greater than 1. Stability analysis of the both endemic and free steady state are also studied. To illustrate our results, numerical simulations are also presented.
Communications in Mathematical Biology and Neuroscience, 2020
In this paper, we develop a mathematical model to describe the interactions between Chikungunya v... more In this paper, we develop a mathematical model to describe the interactions between Chikungunya virus (CHIKV), host cells and antibodies. The proposed model considers two types of infected cells and incorporates two modes of transmission, the classical virus-to-cell infection and the direct cell-to-cell transmission. These both modes are modeled by two general incidence functions that include many special cases existing in the literature. We first prove the well-posedness of the model, including the positivity and boundedness of solutions. The stability and instability of equilibria are established by means of direct and indirect Lyapunov methods. Furthermore, numerical simulations are presented in order to support our analytical results.
Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of nec... more Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann–Liouville sense with its associated integral for the recently introduced weighted generalized fractional derivative with Mittag–Leffler kernel. We rewrite these operators equivalently in effective series, proving some interesting properties relating to the left and the right fractional operators. These results permit us to obtain the corresponding integration by parts formula. With the new general formula, we obtain an appropriate weighted Euler–Lagrange equation for dynamic optimization, extending those existing in the literature. We end with the application of an optimization variational problem to the quantum mechanics framework.
Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of nec... more Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann–Liouville sense with its associated integral for the recently introduced weighted generalized fractional derivative with Mittag–Leffler kernel. We rewrite these operators equivalently in effective series, proving some interesting properties relating to the left and the right fractional operators. These results permit us to obtain the corresponding integration by parts formula. With the new general formula, we obtain an appropriate weighted Euler–Lagrange equation for dynamic optimization, extending those existing in the literature. We end with the application of an optimization variational problem to the quantum mechanics framework.
Communications in Mathematical Biology and Neuroscience, 2018
The purpose of this work is to investigate the almost surely exponentially stable of a stochastic... more The purpose of this work is to investigate the almost surely exponentially stable of a stochastic SIS model with double epidemic hypothesis and specific nonlinear incidence rate. We establish the global existence and positivity of solution. Furthermore, the stability of the disease-free equilibrium of the model are showed. The analytical results are illustrated by computer simulations.
International Journal for Computational Biology, 2014
Hepatitis B is considered as the most common hepatic in the world and may lead to cirrhosis and l... more Hepatitis B is considered as the most common hepatic in the world and may lead to cirrhosis and liver cancer. It is caused by the hepatitis B virus, which attacks and can damage the liver. In this paper we investigate a new mathematical model to study the dynamic process of HBV infection on the liver. This model is based on a three dimensional cellular automaton, which is composed of four state variables. The model takes into account the heterogeneous feature and the spatial localization of the population studied. Furthemore, since the virus doesn't remain only on the liver surface but penetrates into the organ, our model describes better the behavior of interactions between cells and hepatitis B virus in the liver than the previous works found in the literature, which have used only two cellular automata in their models.
In this paper, we develop a business cycle model with general investment, variable depreciation r... more In this paper, we develop a business cycle model with general investment, variable depreciation rate of capital stock and two delays. The first delay describes the time lag between the decision of investment and its implementation, while the second one models the time lag for investment to be produced. The well-posedness and the existence of economic equilibrium are carefully investigated. Moreover, the stability of the economic equilibrium and the existence of Hopf bifurcation are established. The case when the two delays are equal is rigourously studied.
The Cochlea plays a crucial role in the hearing of mammalian species including man. The basic fun... more The Cochlea plays a crucial role in the hearing of mammalian species including man. The basic function of the cochlea is to map sounds of different frequencies into corresponding characteristic positions on the basilar membrane. Many disciplines meet in the study of the auditory system to understand the truth function of the cochlea. An abnormality or small perturbation in the physical parameters of the cochlea may result a malfunction in the auditory system. In this paper, we developed a mathematical model in the order to show numerically the effect of stiffness and damping on the amplitude displacement in the case of a passive cochlea with the objective to study the ear dysfunction.
An increase of pressure in the structure of cochlea may cause a hearing loss. In this paper, we e... more An increase of pressure in the structure of cochlea may cause a hearing loss. In this paper, we established the relationship between the fluid pressure and the amplitude of displacement of Basilar Membrane to clarify the mechanisms of hearing loss caused by increasing of this pressure. So, a mathematical cochlear model was formulated using finite difference method in order to explain and demonstrate this malfunction in passive model. Numerical simulations may be considered as helpful tools which may extend and complete the understanding of a cochlea dysfunction.
In HIV infection, the latent cells represent a reservoir that contributes to the failure of the H... more In HIV infection, the latent cells represent a reservoir that contributes to the failure of the Highly Active Anti-Retroviral Therapy (HAART). This fact requires investigating the possible strategy to improve the administration of the HAART therapy, in order to guarantee the control of the virus load to the lost level as long as possible. In this work, we aim to study the possibility of reducing the latent infected CD4 + reservoir in the HIV infection by considering a mathematical model of two types of latently infected CD4 + : fast and slow, and eight virus strains: wild-type, three single mutants, three double mutants and a fully resistant triple mutant. The HAART therapy is considered as an optimal control problem that aimes to reduce the virus load and the infected cells. Our optimal control approach shows the impact of the optimal HAART therapy on reducing two different types of the reservoirs of the latent infected CD4 + cells.
ABSTRACT Résumé Les cartes thématiques, particulièrement la carte pluviométrique du Maroc datent ... more ABSTRACT Résumé Les cartes thématiques, particulièrement la carte pluviométrique du Maroc datent d'une quarantaine d'année. Il est indéniable que leur réactualisation permettra de mettre à la disposition des chercheurs de nouvelles données pertinentes sur la pluviométrie au Maroc, et servir comme un outil d'aide à la décision pour les décideurs politiques.
In this paper, we are interested in control of a substance which cir- culate among organs in leav... more In this paper, we are interested in control of a substance which cir- culate among organs in leaving being. We determine, under certain con- ditions, the optimal control which steers such systems from an initial state to a desired one.The linear quadratic optimal control problem of such systems is analyzed using the Hilbert Uniqueness Method (HUM). To illustrate our approach,
In this paper, we develop a mathematical model that describes the spatiotemporal dynamics of busi... more In this paper, we develop a mathematical model that describes the spatiotemporal dynamics of business cycle under the goods and services market as well as the money market. We first prove that the developed model is mathematically and economically well-posed. The conditions for the existence of economic equilibrium are rigorously established. Moreover, the stability analysis and the existence of Hopf bifurcation are carefully investigated. Finally, our theoretical results are illustrated with some numerical simulations.
In this chapter, we propose and analyze a class of three spatiotemporal models describing infecti... more In this chapter, we propose and analyze a class of three spatiotemporal models describing infectious diseases caused by viruses such as the human immunodeficiency virus (HIV) and the hepatitis B virus (HBV). The first model with cellular immunity, the second with humoral immunity and the third with cellular and humoral immune responses. In the three proposed models, the disease transmission process is modeled by a general incidence function which includes several forms existing in the literature. In addition, the global analysis of the proposed models is rigorously investigated. Furthermore, biological findings of our analytical results are presented. Moreover, mathematical virus models and results presented in many previous studies are extended and generalized.
Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, 2019
In this work, we propose and analyze a fractional order model for hepatitis B virus (HBV) infecti... more In this work, we propose and analyze a fractional order model for hepatitis B virus (HBV) infection with capsids and cure of infected cells. We first prove the existence, positivity, and boundedness of solutions in order to ensure the well-posedness of our proposed model. By constructing appropriate Lyapunov functionals, the global stability of the steady states is established. Numerical simulations are presented in order to validate our theoretical results.
Communications in Mathematical Biology and Neuroscience, 2019
Actually, cancer is considered one of the leading causes of death in the world. Various therapeut... more Actually, cancer is considered one of the leading causes of death in the world. Various therapeutic strategies have been developed to combat this dangerous disease. This article investigates a promising therapeutic strategy by proposing a mathematical model that describes the dynamics of cancer treatment with oncolytic viruses. The proposed model integrates the time needed for infected tumor cells to produce new virions after viral entry, the probability of surviving during the latent period, and the saturation effect. We first prove the well-posedness of model and the existence of three equilibria that represent the desired outcome of therapy, the complete failure of therapy and the partial success of therapy. Furthermore, the stability analysis of equilibria and the existence of Hopf bifurcation are rigourously investigated.
Nonlinear Analysis and Differential Equations, 2016
In this paper, we propose a stochastic viral infection model with general functional response. Th... more In this paper, we propose a stochastic viral infection model with general functional response. The well posedness of the proposed model is investigated. Also, the extinction of the infection is fully determined by the basic reproduction number R 0. Furthermore, the dynamical behavior around the chronic infection equilibrium is established. Finally, Numerical simulations are given to illustrate our theoretical results.
A new stochastic SIRS epidemic model with specific functional response is proposed and analyzed. ... more A new stochastic SIRS epidemic model with specific functional response is proposed and analyzed. First, we show that the model is biologically well-posed by proving the global existence, positivity and boundedness of solutions. Moreover, sufficient conditions for the extinction and persistence of the disease are also obtained. In the end, some numerical simulations are presented to illustrate our analytical results.
International Journal of Tomography and Statistics
The aim of this work is to investigate a new mathematical model that describe HIV infection of CD... more The aim of this work is to investigate a new mathematical model that describe HIV infection of CD4+ T-cells during therapy. A novel feature is that the both therapy and the cure rate are incorporated into the model. We prove that the infection will die out if the basic reproduction number R0 lesss than 1 while the HIV infection may become endemic if R0 greater than 1. Stability analysis of the both endemic and free steady state are also studied. To illustrate our results, numerical simulations are also presented.
Communications in Mathematical Biology and Neuroscience, 2020
In this paper, we develop a mathematical model to describe the interactions between Chikungunya v... more In this paper, we develop a mathematical model to describe the interactions between Chikungunya virus (CHIKV), host cells and antibodies. The proposed model considers two types of infected cells and incorporates two modes of transmission, the classical virus-to-cell infection and the direct cell-to-cell transmission. These both modes are modeled by two general incidence functions that include many special cases existing in the literature. We first prove the well-posedness of the model, including the positivity and boundedness of solutions. The stability and instability of equilibria are established by means of direct and indirect Lyapunov methods. Furthermore, numerical simulations are presented in order to support our analytical results.
Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of nec... more Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann–Liouville sense with its associated integral for the recently introduced weighted generalized fractional derivative with Mittag–Leffler kernel. We rewrite these operators equivalently in effective series, proving some interesting properties relating to the left and the right fractional operators. These results permit us to obtain the corresponding integration by parts formula. With the new general formula, we obtain an appropriate weighted Euler–Lagrange equation for dynamic optimization, extending those existing in the literature. We end with the application of an optimization variational problem to the quantum mechanics framework.
Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of nec... more Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann–Liouville sense with its associated integral for the recently introduced weighted generalized fractional derivative with Mittag–Leffler kernel. We rewrite these operators equivalently in effective series, proving some interesting properties relating to the left and the right fractional operators. These results permit us to obtain the corresponding integration by parts formula. With the new general formula, we obtain an appropriate weighted Euler–Lagrange equation for dynamic optimization, extending those existing in the literature. We end with the application of an optimization variational problem to the quantum mechanics framework.
Communications in Mathematical Biology and Neuroscience, 2018
The purpose of this work is to investigate the almost surely exponentially stable of a stochastic... more The purpose of this work is to investigate the almost surely exponentially stable of a stochastic SIS model with double epidemic hypothesis and specific nonlinear incidence rate. We establish the global existence and positivity of solution. Furthermore, the stability of the disease-free equilibrium of the model are showed. The analytical results are illustrated by computer simulations.
International Journal for Computational Biology, 2014
Hepatitis B is considered as the most common hepatic in the world and may lead to cirrhosis and l... more Hepatitis B is considered as the most common hepatic in the world and may lead to cirrhosis and liver cancer. It is caused by the hepatitis B virus, which attacks and can damage the liver. In this paper we investigate a new mathematical model to study the dynamic process of HBV infection on the liver. This model is based on a three dimensional cellular automaton, which is composed of four state variables. The model takes into account the heterogeneous feature and the spatial localization of the population studied. Furthemore, since the virus doesn't remain only on the liver surface but penetrates into the organ, our model describes better the behavior of interactions between cells and hepatitis B virus in the liver than the previous works found in the literature, which have used only two cellular automata in their models.
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