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This research article presents a mathematical model that tracks and monitors the spread of COVID-19 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 effectiveness of the theoretical findings, numerical simulations were conducted to demonstrate the impact of the implemented strategies in limiting the spread of COVID-19 mutant strains. [ABSTRACT FROM AUTHOR]

We highlight and study in this paper the phenomenon of the spread of addiction to electronic games, where the addict goes through stages before reaching the degree of addiction. In order to model this phenomenon, we have divided people into four groups, which are potential gamers, engaged gamers, addicted gamers, and gamers who have recovered from addiction. We propose a discrete mathematical model with control strategies using three controls that represent, respectively, u k , which represents awareness of the dangers of electronic games through written and visual media; v k , which represents the effort to direct children and adolescents to educational and entertaining alternative means; and w k , which represents creating rehabilitation centers for addicts to quit electronic game addiction. To characterize optimal controls, we use Pontryagin's maximum principle and the system of optimality solved by an iterative method. Finally, numerical simulations are presented with and without controls. Using a cost-effectiveness analysis, we will show that the control that represents the creation of rehabilitation centers for gaming addicts is the most cost-effective strategy to control the spread of gaming addiction. [ABSTRACT FROM AUTHOR]

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. Firstly, we analyze the epidemic model when the control strategy is based on the vaccination control only, and secondly, when the travel-blocking control is added, we finish with the introduction of the treatment control. The optimal control theory, based on Pontryagin's maximum principle, is applied thrice in this paper, for the characterizations of the vaccination, travel-blocking, and treatment controls. The numerical results associated with the multipoint boundary value problems are obtained based on the forward-backward sweep method combined with progressive-regressive Runge–Kutta fourth-order schemes. [ABSTRACT FROM AUTHOR]

n this paper, we formulate an optimal control problem based on a tubercu- losis model with multiple infectious compartments and time delays. In order to have a more realistic model that allows highlighting the role of detection, loss to follow-up and treatment in TB transmission, we propose an extension of the classical SEIR model by dividing infectious patients in the compart- ment (I) into three categories: undiagnosed infected (I), diagnosed patients who are under treatment (T) and diagnosed patients who are lost to follow-up (L). We incorporate in our model delays representing the incubation period and the time needed for treatment. We also introduce three control variables in our delayed system which represent prevention, detection and the efforts that prevent the failure of treatment. The purpose of our control strategies is to minimize the number of infected individuals and the cost of intervention. The existence of the optimal controls is investigated, and a characterization of the three controls is given using the Pontryagin's maximum principle with delays. To solve numerically the optimality system with delays, we present an adapted iterative method based on the iterative Forward-Backward Sweep Method (FBSM). Numerical simulations performed using Matlab are also pro- vided. They indicate that the prevention control is the most effective one. To the best of our knowledge, it is the first work to apply optimal control theory to a TB model which considers infectious patients diagnosis, loss to follow-up phenomenon and multiple time delays. [ABSTRACT FROM AUTHOR]

In this paper, we aim to investigate optimal control to a new mathematical model that describes agree-disagree opinions during polls, which we presented and analyzed in Bidah et al., 2020. We first present the model and recall its different compartments. We formulate the optimal control problem by supplementing our model with a objective functional. Optimal control strategies are proposed to reduce the number of disagreeing people and the cost of interventions. We prove the existence of solutions to the control problem, we employ Pontryagin's maximum principle to find the necessary conditions for the existence of the optimal controls, and Runge–Kutta forward-backward sweep numerical approximation method is used to solve the optimal control system, and we perform numerical simulations using various initial conditions and parameters to investigate several scenarios. Finally, a global sensitivity analysis is carried out based on the partial rank correlation coefficient method and the Latin hypercube sampling to study the influence of various parameters on the objective functional and to identify the most influential parameters. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study and investigate the optimal control strategy of a discrete mathematical model of drug consumption. The population that we are going to study is divided into six compartments: potential drug users, light drug users, heavy drug users, heavy drug users-dealers and providers, temporary quitters of drug consumption, and permanent quitters of drug consumption. Our objective is to find the best strategy to reduce the number of light drug users, heavy drug users, heavy drug users-dealers and providers, and temporary quitters of drug consumption. We use four control strategies which are awareness programs through media and education, preventing contact through security campaigns, treatment, and psychological support along with follow-up. Pontryagin's maximum principle in discrete time is used to characterize the optimal controls. The numerical simulation is carried out using MATLAB. Consequently, the obtained results confirm the effectiveness of the optimization strategy. [ABSTRACT FROM AUTHOR]

We propose an optimal control strategy by conducting awareness campaigns for diabetics about the severity of complications of diabetes and the negative impact of an unbalanced lifestyle and the surrounding environment, as well as treatment and psychological follow-up. Pontryagin's maximum principle is used to characterize the optimal controls, and the optimality system is solved by an iterative method. Finally, some numerical simulations are performed to verify the theoretical analysis using MATLAB. [ABSTRACT FROM AUTHOR]

In this paper, a new rumor spreading model in social networks has been investigated. We propose a new version primarily based on the cholera model in order to take into account the expert pages specialized in the dissemination of rumors from an existing IRCSS model. In the second part, we recommend an optimal control strategy to fight against the spread of the rumor, and the study aims at characterizing the three optimal controls which minimize the number of spreader users, fake pages, and corresponding costs; theoretically, we have proved the existence of optimal controls, and we have given a characterization of controls in terms of states and adjoint functions based on a discrete version of Pontryagin's maximum principle. To illustrate the theoretical results obtained, we propose numerical simulations for several scenarios applying the forward-backward sweep method (FBSM) to solve our optimality system in an iterative process. [ABSTRACT FROM AUTHOR]

In this work, we propose a multifishing area prey-predator discrete-time model which describes the interaction between the prey and middle and top predators in various areas, which are connected by their movements to their neighbors, to provide realistic description prey effects of two predators. A grid of colored cells is presented to illustrate the entire domain; each cell may represent a subdomain or area. Next, we propose two harvesting control strategies that focus on maximizing the biomass of prey, in the targeted area, and minimizing the biomass of middle and top predators coming from the neighborhood of this targeted area to ensure sustainability and maintain a differential chain system. Theoretically, we have proved the existence of optimal controls, and we have given a characterization of controls in terms of states and adjoint functions based on a discrete version of Pontryagin's maximum principle. To illustrate the theoretical results obtained, we propose numerical simulations for several scenarios applying the forward-backward sweep method (FBSM) to solve our optimality system in an iterative process. [ABSTRACT FROM AUTHOR]

In this paper, we propose a discrete mathematical model that describes the interaction between the classes of drinkers, namely, potential drinkers P , moderate drinkers M , heavy drinkers H , poor heavy drinkers T p , rich heavy drinkers T r , and quitters of drinking Q . We also focus on the importance of treatment within addiction treatment centers aiming to find the optimal strategies to minimize the number of drinkers and maximize the number of heavy drinkers who join addiction treatment centers. We use three controls which represent awareness programs through media and education for the potential drinkers, efforts to encourage the heavy drinkers to join addiction treatment centers, and psychological support with follow-up for the individuals who quit drinking. We use Pontryagin's maximum principle in discrete time to characterize these optimal controls. The resulting optimality system is solved numerically by Matlab. Consequently, the obtained results confirm the performance of the optimization strategy. [ABSTRACT FROM AUTHOR]

We propose a mathematical model that describes the dynamics of citizens who have the right to register on the electoral lists and participate in the political process and the negative influence of abstainers, who abstain from registration on the electoral lists, on the potential electors. By using Routh–Hurwitz criteria and constructing Lyapunov functions, the local stability and the global stability of abstaining-free equilibrium and abstaining equilibrium are obtained. We also study the sensitivity analysis of the model parameters to know the parameters that have a high impact on the reproduction number ℜ 0 . In addition, we propose an optimal strategy for an awareness program that helps politicians and officials to increase the rate of citizens registered on the electoral lists with an optimal effort. Pontryagin's maximum principle is used to characterize the optimal controls, and the optimality system is solved by an iterative method. Finally, some numerical simulations are performed to verify the theoretical analysis using Matlab. [ABSTRACT FROM AUTHOR]

In this paper, we propose a free final time optimal control approach applied to 4 ordinary differential equations which describe the tumor‐immune interactions after the injection of the bacillus Calmette‐Guérin (BCG) in the bladder of a hypothetical patient. The main goal of this optimal control strategy is to find the optimal dosage amount needed in each instillation of BCG for stimulating the immune‐system and then killing superficial bladder tumors and to determine the optimal duration of treatment, adequate for stopping the intravesical therapy with lesser side‐effects. For this, we introduce into the model of interest, a control function which represents the dose of BCG immunotherapy procedure and we formulate a minimization problem where the final time is considered free (nonfixed). The characterization of the sought optimal control noted u∗ is derived based on Pontryagin's maximum principle, while the formulation of the sought optimal final time noted t f i n a l ∗ is based on formulae of sensitivity which are obtained conditions from the derivative of the objective function with respect to t f i n a l ∗. We investigate the resolution of the free final time optimal control problem in 3 possible cases: (a) when t f i n a l ∗ is quadratic in the final gain function, (b) when the final gain function does not depend on t f i n a l ∗, and (c) when t f i n a l ∗ is linear in the final gain function. Finally, we obtain the sought optimal dose of BCG, and we conclude that in case (a), we obtain an optimal duration which is more beneficial regarding the activation of immune cells while cases (b) and (c) both provide an optimal duration which is more adequate for the minimization of the tumoral population. [ABSTRACT FROM AUTHOR]

We study, in this paper, infection dynamics when an epidemic emerges to many regions which are connected with their neighbors by any kind of anthropological movement. For this, we devise a multi-regions discrete-time model with the three classical SIR compartments, describing the spatial-temporal behaviors of homogenous susceptible, infected and removed populations. We suppose a large geographical domain, presented by a grid of colored cells, to exhibit at each instant i the spatial propagation of an epidemic which affects its different parts or sub-domains that we call here cells or regions. In order to minimize the number of infected individuals in some regions, we suggest an optimal control approach based on a travel-blocking vicinity strategy which aims to control a group of cells, or a patch, by restricting movements of infected people coming from its neighboring cells. We apply a discrete version of Pontryagin's maximum principle to state the necessary conditions and characterization of the travel-blocking optimal controls. We provide cellular simulations based on discrete progressive-regressive iterative schemes associated with the obtained multi-points boundary value problems. For illustrating the modeling and optimal control approaches, we consider an example of 100 regions. [ABSTRACT FROM AUTHOR]

The main goal of this article is to exhibit the importance of awareness programs and travel-blocking operations, in the prevention of HIV/AIDS outbreaks, based on a multi-domains SIR epidemic model. The model devised describes the spatial-temporal spread of HIV/AIDS, in p neighboring domains, taking into account the epidemiological diversity of their populations. For a first case of the study, we focus on discussing the benefits of awareness campaigns that aim to raise public consciousness among susceptible people, about the danger of HIV/AIDS. Thus, for the reason that intra-domains interventions are not always sufficient to prevent people belonging to a domain (neighborhood, town or city for example), from a rapid expansion of HIV/AIDS, we propose the travel-blocking strategy, as a second approach presenting inter-domains interventions that health-policy makers could follow, by restricting the number of suspicious people that can participate in spreading HIV/AIDS via travel, coming from domains at high-risk of infection to enter domains with lower risk. The optimal control theory, based on Pontryagin's maximum principle, is applied twice in this paper, for the characterizations of the awareness and travel-blocking controls. The numerical results associated to the multi-points boundary value problems are obtained based on the Forward-Backward Sweep Method combined with progressive-regressive Runge-Kutta fourth-order schemes. [ABSTRACT FROM AUTHOR]

Vaccines are not administered on a continuous basis, but injections are practically introduced at discrete times often separated by an important number of time units, and this differs depending on the nature of the epidemic and its associated vaccine. In addition, especially when it comes to vaccination, most optimization approaches in the literature and those that have been subject to epidemic models have focused on treating problems that led to continuous vaccination schedules but their applicability remains debatable. In search of a more realistic methodology to resolve this issue, a control modeling design, where the control can be characterized analytically and then optimized, can definitely help to find an optimal regimen of pulsed vaccinations. Therefore, we propose a susceptible-infected-removed (SIR) hybrid epidemic model with impulse vaccination control and a compartment that represents the number of vaccinated individuals supposed to not acquire sufficient immunity to become permanently recovered due to the short-term effect of vaccines. A basic reproduction number, when the control is defined as a constant parameter, is calculated. Since we also need to find the optimal values of this impulse control when it is defined as a function of time, we start by stating a general form of an impulse version of Pontryagin's maximum principle that can be adapted to our case, and then we apply it to our model. Finally, we provide our numerical simulations that are obtained via an impulse progressive-regressive iterative scheme with fixed intervals between impulse times (theoretical example of an impulse at each week), and we conclude with a discussion of our results. [ABSTRACT FROM AUTHOR]