A NEW DISCRETE-TIME EPIDEMIC MODEL DESCRIBING INFORMATION SPREAD AND ITS IMPACT ON THE AGREE-DISAGREE MODEL

. In this study, we propose a new modiﬁed discrete-time epidemic model that characterizes the diffusion of information and its impact on the agreement/disagreement model. So the goal is to increase the amount of information to inﬂuence people’s opinions. To do this, we proposed a control strategy based on the increase in the number of posts that inﬂuence people to agree with the subject studied (election, vaccine against COVID 19). The Pontryagin maximum principle is used to describe optimal control. Finally, numerical simulations are performed to verify the theoretical analysis using MATLAB


INTRODUCTION
The internet presents an enormous network of growth which leads to the production of unsecured information, everyone could post and share anything online [1]. The anarchic nature of social media is desirable for maintaining open debates without control, this dilemma creates issues about the quality of information circulated in different kinds of network websites [2]. F A which denotes the amount of information that supports the topic under examination, that exists on websites and social media, to illustrate the power of its information in changing the opinions of people to be in agreement [13].
The paper is structured as follows. In Section 2, we briefly describe our proposed model, and we give some basic properties of the model. In Section 3, we introduce the control problem and provide some results. The Existence and Characterization of Optimal Control Using Pontryagin's Maximum Principle. Numerical simulation on MATLAB supports the theoretical results in Section 4. Lastly, we conclude our work in Section 5.

PRESENTATION OF THE MODEL
Many models consider that the propagation of information is analogous to an epidemic [14,11]. Moreover, in the modeling of the transmission of information, the population is supposed to be divided into three compartments Ignored, Agreed, and Aisagreed similar to the SIR models [11,15,16]. This type of model can then be used to describe the impact of social media on the human population through their publication [11].
In our paper, we put ourselves in the position in front of a problem of public opinion that divided the population into three categories: Ignorant (I), Agree (A), Disagree (D), Example (elections, Corona vaccine), without losing generalities we suppose that we want to increase the number of people who agree, for this, we consider the model of information transmission IAD [11], where the compartment "Ignorant" is used to indicate the people who have no idea of the subject of the study or who are not interested in the subject, the compartment "Agree" is used to indicate that a person agrees with a studied subject. The compartment "Disagree" is used to indicate people do not agree with the topic studied.
In the innovation, we introduced a new compartment called Quantity of information F A that denotes the amount of information that supports to be with the subject examined, that exists on the websites and social media.
The model resulting from these arguments is governed by the following system: and α is the proportion in which an ignorant person meets/contacts an agreeing person and also becomes agreed, (β ) is the probability that an ignorant person will contact a person who disagrees and also become disagreed. The probability that an ignorant person will encounter the amount of information that influences him to agree is θ . Users can delete messages, videos, and images for any reason at a rate of µ. Agree and disagree individuals can change their opinions and become ignorant people with respective rates δ 1 , and δ 2 , respectively. The meaning of each parameter is given in the table (1).  Parameter Description α the rate at which an ignorant person will contact someone who disagrees and who also disagrees β the rate at which an ignorant person will contact someone who disagrees and who also disagrees The rate at which a person agrees becomes ignorant The rate at which a person disagrees becomes ignorant θ The rate at which an ignorant person will encounter the amount of information that will influence them to agree.

σ 1 Factor of loss of interest of individuals in agreement
µ Probability that the agreed information is deleted P A A novel information (agree) posting rate

THE OPTIMAL CONTROL PROBLEM
We study the impact of information on people's opinions, for example, an influencer with thousands of fans who sells cosmetics, by publishing a video in which she shows how a product gives very satisfactory results. She thus changes the opinion of some girls on this product.
Another example: during elections, a group of candidates chooses social media sites to persuade a large group of young people to vote through a series of publications. The World Health Organization and country governments have used the media to convince people to take the covid 19 vaccine. So our noted control strategy (u) is the new posts (advertisement, videos, images) which consist of increasing the amount of information to influence people's opinions to agree with the topic under study. We present the new control variable u as follows: 3.1. Objective functional. The goal of the optimal control problem is to minimize the objective function given by where A > 0, α D > 0, α A > 0 are the constants of weights of the controls, the sharers and the withdrawn, separately, u = (u 0 , · · · , u N −1 ), and N is the final moment of our control strategize.
The goal is to reduce the number of people who disagree and maximize the number of people who agree with an optimal cost. In other words, we are seeking the control u * such that where U is the control defined by Theorem 3.1. There exists an optimal control u * ∈ U such that Since there is a finite number of uniformly bounded sequences, there exists u * ∈ U and I * , A * , D * F A * such that, on a sequence, Lastly, due to the finite dimension structure of the system (2)-(4) and the goal function J (u), we obtain that (u * ) is an optimal control with corresponding states I * , A * , D * , F A * . Which completes the proof.
For i = 0, · · · , N − 1, the optimal control (u * ) can be determined from the optimality conditions we obtain the optimal control as follows By the limits in U of the control in the definition (5) , it is easy to get u * i in the following form This achieve the demonstration.

DISCUSSION
We give numerical simulations for the aforementioned optimization problem in this part. We use several types of data to model our work when writing the program in MATLAB. With a discrete iterative method that converges after a sufficient test akin to the FBSM, the optimality systems are solved. The adjoint system is then solved backward in time due to the transversality conditions after the state's system has first been solved with the starting hypothesis forward in time. The state and co-state resources obtained in the previous steps are used to update our optimal control settings. Lastly, we carry out the previous procedures up until the desired tolerance is reached.
We put in the spot where we want to raise the number of people who agree given a phenomenon, social discussion, or social opinion, where there are individuals who agree and people who are uninformed or uninterested (example: the number of people who agree with the vaccination against COVID 19, or the number of people who support a political party). In order to do that, we suggested a control approach based on the volume of data that supports the examined discussion and is noted F A to the mathematical model IAD proposed by BIDAH et al [9]. The goals of this method of control are to illustrate how information is spread and its impact on social opinion, as well as how to influence people's opinions.
Numerical simulations of our model with MATLAB using the parameters in the table have demonstrated the effectiveness of our control strategy. It can be seen from Figure (7) that the amount of information that influences people to agree with the subject under study is slightly increased, which justifies that the number of individuals disagreeing has become greater than the number of people agreeing after 40 days (see Figure   (2)).
We are interested in this case, and without losing generality we want to increase the number of people who agree, for this reason, we have proposed the u control, which is the new posts (advertisement, videos, images) that consist in increasing the amount of information in order to influence people's opinions so that they agree with the studied subject.
Because of the additional information provided by the u control, we notice from figures 1, 2 and 3 that the number of F A information is increased in a remarkable way. Moreover, from figures (3) and (4)

CONCLUSION
In this work, a new simple discrete-time epidemic model describing the spread of information in certain types of online environments such as Facebook, WhatsApp, and Twitter is examined by adding a new compartment to the IAD model [10]. We additionally propose optimal control by increasing the amount of information via new publications in order to influence people's opinions on a topic under study (elections, the covid 19 vaccine debate). A discrete version of Pontryagin's maximum theorem was applied to define the necessary conditions and the description of our optimal controls. Finally, a simulation illustrates the effectiveness of our control strategy.