Mostafa Rachik

Optimal control and infectiology : Application to an HIV/AIDS model

An existing infectious model describing the interaction of HIV virus and the immune system of the human body is utilized to determine the optimal methodology for administering anti-viral medication therapies to fight HIV infection. This work investigates the fundamental role of chemotherapy treatment in controlling the virus reproduction. We work in the nonlinear optimal control framework. The existence and the uniqueness results of the solution are discussed. A characterization of the optimal control via adjoint variables is established. We obtain an optimality system that we seek to solve numerically by a competitive Gauss–Seidel like implicit difference method.

On the analysis of a multi-regions discrete SIR epidemic model: an optimal control approach

In this paper, we devise a discrete time SIR model depicting the spread of infectious diseases in various geographical regions that are connected by any kind of anthropological movement, which suggests disease-affected people can propagate the disease from one region to another via travel. In fact, health policy-makers could manage the problem of the regional spread of an epidemic, by organizing many vaccination campaigns, or by suggesting other defensive strategies such as blocking movement of people coming from borders of regions at high-risk of infection and entering very controlled regions or with insignificant infection rate. Further, we introduce in the discrete SIR systems, two control variables which represent the effectiveness rates of vaccination and travel-blocking operation. 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. The multi-points boundary value problems, associated to the optimal control problems studied here, are obtained based on a discrete version of Pontryagin’s maximum principle, and resolved numerically using a progressive-regressive discrete scheme that converges following an appropriate test related to the Forward-Backward Sweep Method on optimal control.

Exact controllability and optimal control for distributed systems with a discrete delayed control

Abstract This paper deals with the exact controllability and the quadratic cost control problems for distributed systems with a discrete delayed control. The Hilbert uniqueness method is developed to express the solution of both problems. To illustrate the theory an example of diffusion system is treated.

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

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.

A Multi-Regions SIRS Discrete Epidemic Model With a Travel-Blocking Vicinity Optimal Control Approach on Cells

In Susceptible-Exposed-Infected-Susceptible (SEIS) compartmental models, an infected population is divided into two categories; symptomatic infected individuals represented by the I variable, and asymptomatic infected individuals, people who are not yet infectious, or those who are just exposed to infection, represented by the variable E. In these models, an infected population recovers with no immunity, and then, it moves immediately to the susceptible compartment once it becomes recovered, For this, we devise a multi-regions SEIS discrete-time model which describes infection dynamics when an epidemic is emerging in regions that are connected with their neighbors by movement. The main goal from this kind of modeling, is to introduce after, controls variables which restrict movements of the infected individuals coming from the vicinity of the region targeted by our control strategy, called here; the travel-blocking vicinity optimal control approach. A grid of colored cells is presented to illustrate the whole domain affected by the epidemic while each cell represents a sub-domain or region. The infection is supposed starting from only one cell located in one of the corners of the grid, while the region aiming to control, is supposed to be located in the 4th line and 7th column of the grid, as an example to show the effectiveness of the proposed control strategy when it is applied to a cell with 8 neighboring cells.

Modeling the impact of immigration and climatic conditions on the epidemic spreading based on cellular automata approach

Abstract This study has three objectives. First, it aims to investigate epidemic immigration by developing spatiotemporal models based on cellular automata approach. We have proposed delayed and nonlocal transition functions which depend not only on the previous states, but also on several past states. Furthermore, the cell state depends on the non-neighbor cells in addition to the neighbor ones. Second, this study aims to explore the impact of the neighborhood architecture on the epidemic diffusion. Third, this study also endeavors to examine the impact of the virulence variation on the epidemic spreading. In this context, we have considered the virulence variation from one region to another, and from one season to another and we have emphasized its role as a decisive parameter. On top of that, we have developed basic and useful online application called the “Epidemic Application” for predicting the spread of diseases over time and space, assessing the effect of different spatial parameters, and providing adequate spatiotemporal simulations in order to help health organizations. Moreover, the suggested approach can be implemented in a variety of scientific and ecological fields.

Advanced Approach for Observability of Distributed Systems Using Internal Pointwise Sensor

This paper will serve as a basic introduction to the observability of diffusion process as an example of distributed parameter systems. The aim of this research is to reconstruct initial state not well known \(x_{0}\), which is known in certain subregions and unknown in others, and to give important results related to internal pointwise sensor in different geometrical situations. Many applications are investigated whether in one-dimensional case or two-dimensional one.

The

Discrete-time linear systems with perturbed initial state are considered. A disturbance that infects the initial state is said to be

ε

\epsilon

-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems.

-tolerable if the corresponding output signal is relatively insensitive to their effects. In this paper, we will define a new set that characterize each gain matrix K and the associated feedback control law ui=Kxi, this set will be called the