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Abdelhadi Abta

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Armando Marques-Guedes

UNL - New University of Lisbon

Gwen Robbins Schug

University of North Carolina at Greensboro

Gabriel Gutierrez-Alonso

University of Salamanca

Macquarie University

Swansea University

Graduate Center of the City University of New York

Richard Cloutier

Université du Québec à Rimouski

Moscow State University

Professor Dr. Loutfy H . Madkour

Al-Baha University, Kingdom of Saudi Arabia

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Papers by Abdelhadi Abta

Mathematical modeling and optimal control of multi-strain COVID-19 spread in discrete time

Frontiers in applied mathematics and statistics, Apr 15, 2024

This research article presents a mathematical model that tracks and monitors the spread of COVID-... more This research article presents a mathematical model that tracks and monitors the spread of COVID-strains in a discrete time frame. The study incorporates two control strategies to reduce the transmission of these strains: vaccination and providing appropriate treatment and medication for each strain separately. Optimal controls were established using Pontryagin's maximum principle in discrete time, and the optimality system was solved using an iterative method. To validate the e ectiveness of the theoretical findings, numerical simulations were conducted to demonstrate the impact of the implemented strategies in limiting the spread of COVID-mutant strains.

Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates

Applied Mathematics and Computation, 2012

We analyze stability of equilibria for a delayed SIR epidemic model, in which population growth i... more We analyze stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic reproduction number R 0 exceeds one, while the trivial equilibrium and the disease-free equilibrium always exist. First we show that the disease-free equilibrium is globally asymptotically stable if and only if R 0 ≤ 1. Second we show that the model is permanent and it has a unique endemic equilibrium if and only if R 0 > 1. Moreover, using a threshold parameter R 0 characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for 1 < R 0 ≤ R 0 and it loses stability as the length of the delay increases past a critical value for 1 < R 0 < R 0. Our result is an extension of the stability results in [

Optimal Control Strategy for SEIR with Latent Period and a Saturated Incidence Rate

ISRN Applied Mathematics, 2013

We propose an SEIR epidemic model with latent period and a modified saturated incidence rate. Thi... more We propose an SEIR epidemic model with latent period and a modified saturated incidence rate. This work investigates the fundamental role of the vaccination strategies to reduce the number of susceptible, exposed, and infected individuals and increase the number of recovered individuals. The existence of the optimal control of the nonlinear model is also proved. The optimality system is derived and then solved numerically using a competitive Gauss-Seidel-like implicit difference method.

An automated optimal vaccination control with a multi-region SIR epidemic model

Journal of Mathematical and Computational Science, 2021

Many mathematical models describing the evolution of infectious diseases underestimate the effect... more Many mathematical models describing the evolution of infectious diseases underestimate the effect of the Spatio-temporal spread of epidemics. Currently, the COVID-19 epidemic shows the importance of taking into account the spatial dynamic of epidemics and pandemics. In this contribution, we consider a multi-region discretetime epidemic model that describes the spatial spread of an epidemic within different geographical zones assumed to be connected with the movements of their populations. Based on the fact that there are several limitations in medical resources, the authorities and health decision-makers must define a threshold of infections in order to determine if a zone is epidemic or not yet. We propose a new approach of optimal control by defining new importance functions to identify affected zones and then the need for the control intervention there. Numerical results are provided to illustrate our findings by applying this new approach in two adjacent regions of Morocco,

Optimal control therapy and vaccination for a new coronavirus model

International Conference on Research in Applied Mathematics and Computer Science, 2021

Global Stability for Delay Sir and Seir Epidemic Models with Saturated Incidence Rates

In this article we propose a comparison of a delayed SIR model and its corresponding SEIR model i... more In this article we propose a comparison of a delayed SIR model and its corresponding SEIR model in terms of global stability. We consider a saturated incidence rate and we determine, using Lyapunov functionals, conditions by which the disease-free equilibrium and the endemic equilibrium are globally asymptotically stable. Also some numerical simulations are given to compare a global behaviour of a delayed SIR model and its corresponding SEIR model.

Optimal Control of a Delayed Alcoholism Model with Saturated Treatment

Differential Equations and Dynamical Systems, 2021

In the present work, we have proposed a new mathematical model of alcohol abuse with delay, satur... more In the present work, we have proposed a new mathematical model of alcohol abuse with delay, saturated incidence function and logistic recruitment. The model is made up of the following four population classes: occasional drinkers, heavy drinkers, drinkers during treatment and drinkers who are temporarily recovered. In particular, we incorporate time delay because the non consumer population will take a period of time to become an alcohol consumer. We have studied the optimal control problem by considering a saturated control function and an objective function of type $$L^{1}$$ . The delay is incorporated in our model to make it more realistic and to describe the latency period. The existence of the optimal control is also proved. Pontryagin’s maximum principle with delay is used to characterize these optimal controls. The optimality system is derived and then solved numerically using an algorithm based on the forward and backward difference approximation.

A Multiregion Discrete-Time Epidemic Model of Mycobacterium tuberculosis Infections: An Optimal Control Approach

Discrete Dynamics in Nature and Society, 2021

The main goal of this article is to devise the spatial-temporal spread of TB, in multiple neighbo... more The main goal of this article is to devise the spatial-temporal spread of TB, in multiple neighboring domains, taking into account the epidemiological diversity of their populations. However, since both the environment and any population are spatially heterogeneous, it is obviously desirable to include spatial structure into an epidemic model. Individuals with tuberculosis can spread the disease by moving from one area to another. In addition, people travel by air between cities, so diseases can be spread quickly between very distant places (as was the case with the COVID-19). In our model, each region's studied population is divided into five compartments S, L1, I, L2, and R. Further, we introduce in our discrete systems three control variables which represent the effectiveness rates of vaccination, travel-blocking operation, and treatment. We focus in our study to control the outbreaks of an epidemic that affects a hypothetical population belonging to a specific region. Firstl...

Stability Analysis and Optimal Vaccination Strategies for an SIR Epidemic Model with a Nonlinear Incidence Rate

An SIR epidemic model with saturated incidence rate and a vaccination program is formulated, wher... more An SIR epidemic model with saturated incidence rate and a vaccination program is formulated, where the susceptibles are assumed to satisfy the logistic equation. The incidence term is of saturated form with the infected individuals. First, we have discussed the existence and the stability of both the disease free and endemic equilibrium. Second, the impact of vaccination in reducing R(u) is tackled. Then, to achieve control of the disease, a control problem is formulated and it is shown that an optimal control exists for our model. The optimality system is derived and solved. Finally, numerical simulations are performed to illus- trate and verify the analytical results.

Boundary Exponential Stabilization for a Class of Coupled Reaction-Diffusion Equations with State Delay

Journal of Dynamical and Control Systems, 2021

A class of coupled parabolic PDEs with time delay is considered. We treat the problem of boundary... more A class of coupled parabolic PDEs with time delay is considered. We treat the problem of boundary exponential stabilization where the goal is to present a state feedback law with actuation on only one end of the domain which provide the exponential stability of the closed-loop system. We consider both the Dirichlet and Neumann boundary conditions. The corresponding proposed control law is given in explicit form. Numerical simulations are carried out to show the effectiveness of the obtained results.

Hopf Bifurcation Analysis In a Delayed SIR Model

In this paper, we analyze a delayed SIR epidemic model with saturated incidence rate. Our goal in... more In this paper, we analyze a delayed SIR epidemic model with saturated incidence rate. Our goal in this model is to introduce the incubation period into susceptible and infectious individuals in the incidence rate. The dynamics are studied in terms of local stability and of the description of the Hopf bifurcation, that is proven to exist as the delay (taken as a parameter of bifurcation) cross some critical value. Additionally we conclude with a numerical illustrations.

Optimal control of a delayed rumor propagation model with saturated control functions and $$L^1$$-type objectives

Social Network Analysis and Mining, 2020

Rumor is an important form of social interaction, and its spreading has a significant impact on h... more Rumor is an important form of social interaction, and its spreading has a significant impact on human lives. The optimal control theory is an important tool to better manage the spread of rumors. Most of the literature on rumor propagation models deals with quadratic cost functions relative to the control variable. In this paper, we have considered a time-delay rumor propagation model with saturated control functions and an objective function of $$L^1$$ -type linear with respect to the control variables. In the general case, the introduction of the delay in the dynamic systems represents the time lag between the action on the system and the response of the system to this action. The delay is incorporated in our model to make it more realistic and to describe the latency period. The existence of the optimal control pair is also proved. Pontryagin’s maximum principle with delay is used to characterize these optimal controls. The optimality system is derived and then solved numerically using an algorithm based on the forward and backward difference approximation.

A comparison of delayed SIR and SEIR epidemic models

Nonlinear Analysis: Modelling and Control, 2011

In epidemiological research literatures, a latent or incubation period can be medelled by incorpo... more In epidemiological research literatures, a latent or incubation period can be medelled by incorporating it as a delay effect (delayed SIR models), or by introducing an exposed class (SEIR models). In this paper we propose a comparison of a delayed SIR model and its corresponding SEIR model in terms of local stability. Also some numerical simulations are given to illustrate the theoretical results.

Stability Analysis of a Delayed Rumor Propagation Model

Differential Equations and Dynamical Systems, 2015

In this paper, we investigated a novel rumor spreading model with latent and constant recruitment... more In this paper, we investigated a novel rumor spreading model with latent and constant recruitment. The delay is introduced into the model in order to modeled the latent period. The local dynamics of the model is completely determined by using concepts from the Dynamical Systems Theory. The stability of equilibrium points is established, according to a threshold number R 0. Finally, some numerical simulations are performed to verify the theoretical analysis.

Optimal Antiviral Treatment Strategies of HBV Infection Model with Logistic Hepatocyte Growth

ISRN Biomathematics, 2013

This study considers an optimal therapy strategy for HBV infection by incorporating two controls ... more This study considers an optimal therapy strategy for HBV infection by incorporating two controls laws into a previous hepatitis B viral infection model with logistic hepatocyte growth. Our goal is to maximize the number of healthy cells and to minimize the cost of the therapy. In this context, the existence of an optimal control is proved. The optimal control is obtained by solving the optimality system which was composed of three nonlinear ODEs with initial conditions and three nonlinear adjoint ODEs with transversality conditions. The results were analysed and interpreted numerically using MATLAB.

Optimal Control of a Delayed SIRS Epidemic Model with Vaccination and Treatment

Acta Biotheoretica, 2015

This article deals with optimal control applied to vaccination and treatment strategies for an SI... more This article deals with optimal control applied to vaccination and treatment strategies for an SIRS epidemic model with logistic growth and delay. The delay is incorporated into the model in order to modeled the latent period or incubation period. The existence for the optimal control pair is also proved. Pontryagin's maximum principle with delay is used to characterize these optimal controls. The optimality system is derived and then solved numerically using an algorithm based on the forward and backward difference approximation.

Global Analysis for a Delay-Distributed SIR Epidemic Model

ESAIM: Proceedings, 2013

En épidémiologie le retard joue un rôle très important dans la représentation de la période d'inc... more En épidémiologie le retard joue un rôle très important dans la représentation de la période d'incubation. Dans ce travail, nous étudions un modèle SIR à retard distribué, avec une fonction d'incidence non linéaire. Pour analyser la stabilité globale du point d'équilibre sans maladie et du point d'équilibre endémique, nous utilisons la théorie de Lyapunov et le principe d'invariance de LaSalle.

The Hopf Bifurcation Analysis and Optimal Control of a Delayed SIR Epidemic Model

International Journal of Analysis, 2014

We propose a delayed SIR model with saturated incidence rate. The delay is incorporated into the ... more We propose a delayed SIR model with saturated incidence rate. The delay is incorporated into the model in order to model the latent period. The basic reproductive number R0 is obtained. Furthermore, using time delay as a bifurcation parameter, it is proven that there exists a critical value of delay for the stability of diseases prevalence. When the delay exceeds the critical value, the system loses its stability and a Hopf bifurcation occurs. The model is extended to assess the impact of some control measures, by reformulating the model as an optimal control problem with vaccination and treatment. The existence of the optimal control is also proved. Finally, some numerical simulations are performed to verify the theoretical analysis.

Mathematical modeling and optimal control of multi-strain COVID-19 spread in discrete time

Frontiers in applied mathematics and statistics, Apr 15, 2024

This research article presents a mathematical model that tracks and monitors the spread of COVID-... more This research article presents a mathematical model that tracks and monitors the spread of COVID-strains in a discrete time frame. The study incorporates two control strategies to reduce the transmission of these strains: vaccination and providing appropriate treatment and medication for each strain separately. Optimal controls were established using Pontryagin's maximum principle in discrete time, and the optimality system was solved using an iterative method. To validate the e ectiveness of the theoretical findings, numerical simulations were conducted to demonstrate the impact of the implemented strategies in limiting the spread of COVID-mutant strains.

Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates

Applied Mathematics and Computation, 2012

We analyze stability of equilibria for a delayed SIR epidemic model, in which population growth i... more We analyze stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic reproduction number R 0 exceeds one, while the trivial equilibrium and the disease-free equilibrium always exist. First we show that the disease-free equilibrium is globally asymptotically stable if and only if R 0 ≤ 1. Second we show that the model is permanent and it has a unique endemic equilibrium if and only if R 0 > 1. Moreover, using a threshold parameter R 0 characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for 1 < R 0 ≤ R 0 and it loses stability as the length of the delay increases past a critical value for 1 < R 0 < R 0. Our result is an extension of the stability results in [

Optimal Control Strategy for SEIR with Latent Period and a Saturated Incidence Rate

ISRN Applied Mathematics, 2013

We propose an SEIR epidemic model with latent period and a modified saturated incidence rate. Thi... more We propose an SEIR epidemic model with latent period and a modified saturated incidence rate. This work investigates the fundamental role of the vaccination strategies to reduce the number of susceptible, exposed, and infected individuals and increase the number of recovered individuals. The existence of the optimal control of the nonlinear model is also proved. The optimality system is derived and then solved numerically using a competitive Gauss-Seidel-like implicit difference method.

An automated optimal vaccination control with a multi-region SIR epidemic model

Journal of Mathematical and Computational Science, 2021

Many mathematical models describing the evolution of infectious diseases underestimate the effect... more Many mathematical models describing the evolution of infectious diseases underestimate the effect of the Spatio-temporal spread of epidemics. Currently, the COVID-19 epidemic shows the importance of taking into account the spatial dynamic of epidemics and pandemics. In this contribution, we consider a multi-region discretetime epidemic model that describes the spatial spread of an epidemic within different geographical zones assumed to be connected with the movements of their populations. Based on the fact that there are several limitations in medical resources, the authorities and health decision-makers must define a threshold of infections in order to determine if a zone is epidemic or not yet. We propose a new approach of optimal control by defining new importance functions to identify affected zones and then the need for the control intervention there. Numerical results are provided to illustrate our findings by applying this new approach in two adjacent regions of Morocco,

Optimal control therapy and vaccination for a new coronavirus model

International Conference on Research in Applied Mathematics and Computer Science, 2021

Global Stability for Delay Sir and Seir Epidemic Models with Saturated Incidence Rates

In this article we propose a comparison of a delayed SIR model and its corresponding SEIR model i... more In this article we propose a comparison of a delayed SIR model and its corresponding SEIR model in terms of global stability. We consider a saturated incidence rate and we determine, using Lyapunov functionals, conditions by which the disease-free equilibrium and the endemic equilibrium are globally asymptotically stable. Also some numerical simulations are given to compare a global behaviour of a delayed SIR model and its corresponding SEIR model.

Optimal Control of a Delayed Alcoholism Model with Saturated Treatment

Differential Equations and Dynamical Systems, 2021

In the present work, we have proposed a new mathematical model of alcohol abuse with delay, satur... more In the present work, we have proposed a new mathematical model of alcohol abuse with delay, saturated incidence function and logistic recruitment. The model is made up of the following four population classes: occasional drinkers, heavy drinkers, drinkers during treatment and drinkers who are temporarily recovered. In particular, we incorporate time delay because the non consumer population will take a period of time to become an alcohol consumer. We have studied the optimal control problem by considering a saturated control function and an objective function of type $$L^{1}$$ . The delay is incorporated in our model to make it more realistic and to describe the latency period. The existence of the optimal control is also proved. Pontryagin’s maximum principle with delay is used to characterize these optimal controls. The optimality system is derived and then solved numerically using an algorithm based on the forward and backward difference approximation.

A Multiregion Discrete-Time Epidemic Model of Mycobacterium tuberculosis Infections: An Optimal Control Approach

Discrete Dynamics in Nature and Society, 2021

The main goal of this article is to devise the spatial-temporal spread of TB, in multiple neighbo... more The main goal of this article is to devise the spatial-temporal spread of TB, in multiple neighboring domains, taking into account the epidemiological diversity of their populations. However, since both the environment and any population are spatially heterogeneous, it is obviously desirable to include spatial structure into an epidemic model. Individuals with tuberculosis can spread the disease by moving from one area to another. In addition, people travel by air between cities, so diseases can be spread quickly between very distant places (as was the case with the COVID-19). In our model, each region's studied population is divided into five compartments S, L1, I, L2, and R. Further, we introduce in our discrete systems three control variables which represent the effectiveness rates of vaccination, travel-blocking operation, and treatment. We focus in our study to control the outbreaks of an epidemic that affects a hypothetical population belonging to a specific region. Firstl...

Stability Analysis and Optimal Vaccination Strategies for an SIR Epidemic Model with a Nonlinear Incidence Rate

An SIR epidemic model with saturated incidence rate and a vaccination program is formulated, wher... more An SIR epidemic model with saturated incidence rate and a vaccination program is formulated, where the susceptibles are assumed to satisfy the logistic equation. The incidence term is of saturated form with the infected individuals. First, we have discussed the existence and the stability of both the disease free and endemic equilibrium. Second, the impact of vaccination in reducing R(u) is tackled. Then, to achieve control of the disease, a control problem is formulated and it is shown that an optimal control exists for our model. The optimality system is derived and solved. Finally, numerical simulations are performed to illus- trate and verify the analytical results.

Boundary Exponential Stabilization for a Class of Coupled Reaction-Diffusion Equations with State Delay

Journal of Dynamical and Control Systems, 2021

A class of coupled parabolic PDEs with time delay is considered. We treat the problem of boundary... more A class of coupled parabolic PDEs with time delay is considered. We treat the problem of boundary exponential stabilization where the goal is to present a state feedback law with actuation on only one end of the domain which provide the exponential stability of the closed-loop system. We consider both the Dirichlet and Neumann boundary conditions. The corresponding proposed control law is given in explicit form. Numerical simulations are carried out to show the effectiveness of the obtained results.

Hopf Bifurcation Analysis In a Delayed SIR Model

In this paper, we analyze a delayed SIR epidemic model with saturated incidence rate. Our goal in... more In this paper, we analyze a delayed SIR epidemic model with saturated incidence rate. Our goal in this model is to introduce the incubation period into susceptible and infectious individuals in the incidence rate. The dynamics are studied in terms of local stability and of the description of the Hopf bifurcation, that is proven to exist as the delay (taken as a parameter of bifurcation) cross some critical value. Additionally we conclude with a numerical illustrations.

Optimal control of a delayed rumor propagation model with saturated control functions and $$L^1$$-type objectives

Social Network Analysis and Mining, 2020

Rumor is an important form of social interaction, and its spreading has a significant impact on h... more Rumor is an important form of social interaction, and its spreading has a significant impact on human lives. The optimal control theory is an important tool to better manage the spread of rumors. Most of the literature on rumor propagation models deals with quadratic cost functions relative to the control variable. In this paper, we have considered a time-delay rumor propagation model with saturated control functions and an objective function of $$L^1$$ -type linear with respect to the control variables. In the general case, the introduction of the delay in the dynamic systems represents the time lag between the action on the system and the response of the system to this action. The delay is incorporated in our model to make it more realistic and to describe the latency period. The existence of the optimal control pair is also proved. Pontryagin’s maximum principle with delay is used to characterize these optimal controls. The optimality system is derived and then solved numerically using an algorithm based on the forward and backward difference approximation.

A comparison of delayed SIR and SEIR epidemic models

Nonlinear Analysis: Modelling and Control, 2011

In epidemiological research literatures, a latent or incubation period can be medelled by incorpo... more In epidemiological research literatures, a latent or incubation period can be medelled by incorporating it as a delay effect (delayed SIR models), or by introducing an exposed class (SEIR models). In this paper we propose a comparison of a delayed SIR model and its corresponding SEIR model in terms of local stability. Also some numerical simulations are given to illustrate the theoretical results.

Stability Analysis of a Delayed Rumor Propagation Model

Differential Equations and Dynamical Systems, 2015

In this paper, we investigated a novel rumor spreading model with latent and constant recruitment... more In this paper, we investigated a novel rumor spreading model with latent and constant recruitment. The delay is introduced into the model in order to modeled the latent period. The local dynamics of the model is completely determined by using concepts from the Dynamical Systems Theory. The stability of equilibrium points is established, according to a threshold number R 0. Finally, some numerical simulations are performed to verify the theoretical analysis.

Optimal Antiviral Treatment Strategies of HBV Infection Model with Logistic Hepatocyte Growth

ISRN Biomathematics, 2013

This study considers an optimal therapy strategy for HBV infection by incorporating two controls ... more This study considers an optimal therapy strategy for HBV infection by incorporating two controls laws into a previous hepatitis B viral infection model with logistic hepatocyte growth. Our goal is to maximize the number of healthy cells and to minimize the cost of the therapy. In this context, the existence of an optimal control is proved. The optimal control is obtained by solving the optimality system which was composed of three nonlinear ODEs with initial conditions and three nonlinear adjoint ODEs with transversality conditions. The results were analysed and interpreted numerically using MATLAB.

Optimal Control of a Delayed SIRS Epidemic Model with Vaccination and Treatment

Acta Biotheoretica, 2015

This article deals with optimal control applied to vaccination and treatment strategies for an SI... more This article deals with optimal control applied to vaccination and treatment strategies for an SIRS epidemic model with logistic growth and delay. The delay is incorporated into the model in order to modeled the latent period or incubation period. The existence for the optimal control pair is also proved. Pontryagin's maximum principle with delay is used to characterize these optimal controls. The optimality system is derived and then solved numerically using an algorithm based on the forward and backward difference approximation.

Global Analysis for a Delay-Distributed SIR Epidemic Model

ESAIM: Proceedings, 2013

En épidémiologie le retard joue un rôle très important dans la représentation de la période d'inc... more En épidémiologie le retard joue un rôle très important dans la représentation de la période d'incubation. Dans ce travail, nous étudions un modèle SIR à retard distribué, avec une fonction d'incidence non linéaire. Pour analyser la stabilité globale du point d'équilibre sans maladie et du point d'équilibre endémique, nous utilisons la théorie de Lyapunov et le principe d'invariance de LaSalle.

The Hopf Bifurcation Analysis and Optimal Control of a Delayed SIR Epidemic Model

International Journal of Analysis, 2014

We propose a delayed SIR model with saturated incidence rate. The delay is incorporated into the ... more We propose a delayed SIR model with saturated incidence rate. The delay is incorporated into the model in order to model the latent period. The basic reproductive number R0 is obtained. Furthermore, using time delay as a bifurcation parameter, it is proven that there exists a critical value of delay for the stability of diseases prevalence. When the delay exceeds the critical value, the system loses its stability and a Hopf bifurcation occurs. The model is extended to assess the impact of some control measures, by reformulating the model as an optimal control problem with vaccination and treatment. The existence of the optimal control is also proved. Finally, some numerical simulations are performed to verify the theoretical analysis.