OPTIMAL NEIGHBORING INFECTION CONTROL STRATEGY ON A MULTI-REGIONS-BASED SIR EPIDEMIC MODEL

1Laboratory of Analysis, Modeling and Simulation, Department of Mathematics and Computer Science, Hassan II University of Casablanca, Faculty of Sciences Ben M’Sik, Sidi Othman, BP 7955, Casablanca, Morocco 2Laboratory of Modeling, Analysis, Control and Statistics, Department of Mathematics and Computer Science, Faculty of Sciences Ain Chock, Hassan II University of Casablanca, B.P 5366 Maarif Casablanca, Morocco


INTRODUCTION
The mathematical modeling of the spread of infectious diseases has become part of epidemiology strategies decision-making in many countries (see [1,2,3,4,5,6]). They can supply an efficient tool to study the dynamics and control of infectious diseases. In the history of all these diseases, we can notice their spread from one region to another, and recently the COVID-19 pandemic from its epicenter of Wuhan in China has spread to all parts of the world, which makes taking into account the spatial spread of diseases more important during modeling processes.
In a wide geographical area, the disease becomes mobile because of the people's movements between regions, and the transmission network of the disease becomes more complex when people use inter-continental transport means.
Before the identification of the disease, the infected people would move from one region to another. After diagnosing some infected cases and identifying the disease in some areas, It may take time for a responsive decision to restrict all travel to and from the affected areas. At that time, the affected cases had already been distributed in many other regions, making the problem even more complicated. Different epidemic diseases have been recognized in recent decades and have spread to a large geographic area, such as HIV/AIDS [7], Ebola [8,9,10], Cholera [11,12,13], Malaria [14,15] and Influenza [16,17,18] and recently, the COVID-19 [19,20,21]. To control the spatial spread of the disease, it became necessary to consider all the parts that people can visit. Hence, the utility of spatial dynamics in the mathematical models of the propagation of infectious diseases.
Optimal control theory is well used as an available and effective option for decision-makers to develop and simulate control strategies, see [16,18,22,23], but there are very few applications with both space and time as discrete variables. Some authors have addressed the case of the epidemic model in discrete-time in the last decades, (see [24], [25], [26] and [27]). One reason for choosing discrete epidemic models is that the discrete model has the advantage of describing an infectious disease from the epidemic data that are usually collected in discrete time units.
In this work, we propose a new modeling and control approach based on multi-region discretetime SIR (Susceptible -Infected -Removed) model to investigate the control of the spatiotemporal spread of an epidemic that emerges in several geographical regions, consequently to show the influence existing between regions via infection connections, and to seek a reasonable control strategy which could be effective for the prevention of infectious diseases such as HIV/AIDS, Ebola, and COVID-19, or pandemics in general. We suppose that all regions are connected, and the infected people have access to all these regions. Generally, in the case of distant regions, the infection travels to a targeted region if there exists a direct or indirect mode of transport between it and the regions from where the epidemic starts. In this direction, we present a new epidemic modeling approach in where the studied area is represented by a grid of cells uniform in size, and each cell represents a region (towns, neighboring, ...).
We assume that there is an important part (patch: a set of regions) within the domain studied, that needs to be protected against the disease, and controlling its number of infected people is a higher priority, this part is called the target patch. It is assumed also that there is no way to introduce the control campaigns directly in such places, due to the sensitivity of the social group of people residing in these locations, for example, disseminating information about the infection in the district of doctors and nurses, can lead to a serious panic in the society.
To show the influence of each region on the other, we study here the effects of the vaccination on the neighborhood of a control source patch (important and/or sensitive regions). To do this, we propose this new approach that determines an optimal control based on the multi-region discrete SIR model which hence allows implementing the vaccination campaign into the control source patch and thus allows reducing the infectious groups and increasing the number of recovered individuals in the target patch with an optimal cost. The optimal control problem was the subject of an optimization criterion represented by the minimization of an objective function. The optimality systems are solved based on an iterative discrete scheme that converges following an appropriate test similar to the one related to the Forward-Backward Sweep Method (FBSM).
The paper is organized as follows: In section 2, the model is described. In section 3, we give some results concerning the existence of the optimal control, and we use a discrete version of Pontryagin's Maximum Principle, to investigate the necessary condition for the optimal control.
The analysis of the optimal control problem and numerical simulations are given in Section 4.
Finally, we conclude the paper in Section 5.

MODEL DESCRIPTION
Consider a discrete-time SIR model modeling the spread of an epidemic within a domain Ω, occupied by an homogeneous population, that is divided to M 2 cells uniform in size, i.e. Ω = p,q=1,..,M C pq , where C pq denoting a spatial location in Ω. The spread dynamics of an infectious disease is characterized by modeling population movements between those cells (neighborhoods, towns, cities, countries...).
According to the disease transmission mechanism, the host population of each cell C pq is grouped into three epidemiological compartments, S with infective of C pq and its neighborhood V pq , where V pq = {C rs ∈ Ω/r = p + k, s = q + k , k, k ∈ {−1, 0, 1}}, which is a Moor neighborhood type, see Figure 1 (a).
Thus, the infection transmission at time i is assumed to be occur between individuals present in a given cell C pq at rate: where β rs is the proportion of adequate contacts between a susceptible from a cell C pq and an infective from its neighbor cell C rs ∈ V pq , which is a constant.
At time i, movements of susceptible, infected and removed people are also considered and occurred , respectively, at rate: where V pq = V pq \ C pq , ν S rs , ν I rs and ν R rs are the proportions of susceptible, infected and removed people coming from the neighboring cell C rs ∈ V pq , respectively.
The following system describes the multi-cells discrete SIR model corresponding to a cell C pq in Ω

AN OPTIMAL CONTROL PROBLEM
We are interested in controlling a sensitive population (political, rich or diplomatic people or any kind of population that can be considered as an important part of society ) located in a target patch denoted W O , by following controls strategies in its neighboring cells which belong to the control source patch denoted W S (see Figure 1 (b), for instance). For that, we introduce a control variable u C rs i (C rs ∈ W S ) that characterizes the effectiveness of treatment (vaccination) in the above mentioned model (1-3), corresponding to cells in the control source patch W S . Then, for a given region C pq ∈ Ω the model is given by the following equations: Then, for an initial state (S the problem is to minimize the objective functional given by , A rs > 0, A 1 > 0 and A 2 > 0 are the weight constants of controls, the infected and the removed group respectively. This optimization is taken over a finite time horizon N because the treatment (vaccination) period is usually restricted to a limited time window.
To clarify, that minimization is taken over a controls set, In other words, we are seeking an optimal control u * such that The sufficient condition for existence of an optimal control for such problems follows from theorem 1 in [25]. At the same time by using a discrete version of Pontryagin's Maximum Principle [28,29] we derive necessary conditions for our optimal control. For this purpose we define the Hamiltonian as: N) are the adjoint functions to be determined suitably.
Theorem 1. Given an optimal control u * and solutions S C * pq , I C * pq and R C * pq , there exists ζ C pq k,i , i = 0...N, k = 1, 2, 3, the adjoint variables satisfying the following equations where ζ C pq where ζ C pq Proof. To proof the previous statements, we use a discrete version of Pontryagin's Maximum Principle [28,29] and setting S C pq * , I C pq * and R C pq * and u * we obtain the following adjoint equations: with transversality conditions To obtain the optimality conditions we take the variation with respect to control u C pq i * and set it Then we obtain the optimal control By the bounds in U ad of the control, it is easy to obtain u C pq i * in the following form

NUMERICAL SIMULATIONS AND DISCUSSIONS
In this section, we provide numerical simulations to demonstrate our theoretical results in the case when the studied domain represents the assembly of M 2 regions or cells (cities, towns, ...).
The code is written and compiled in MATLAB using data cited in Table 1 In all of the figures below, the redder part of the colorbars contains larger numbers of individuals while the bluer part contains the smaller numbers. In the following, we discuss with more details, the simulations we obtain, in the case when there is yet no control.    As we can observe in Figure 4. that the number R C pq in all cells of Ω are close/or equal to only 1 or two removed people, some people acquire immune responses that help them to cure naturally from the disease. Figures 5., 6., 7., 8., 9. and 10. depict dynamics of the S, I and R populations when the vaccination optimal control strategy is followed. In order to show the importance of the optimal control approach suggested in this paper, we take the example of a target cell which has 8 neighboring cells, and as done in the previous part, we investigate also here, the results obtained when the disease starts from the lower right corner. As an example, we suppose that the target patch we aim to control is W O = {C 23 }, the control source patch is W S = {C 32 ,C 34 } and we present simulations when the epidemic is more important at the corner cell C 55 .  In the following, we note that the vicinity of the patch W O is defined by In Figure 5., as supposed also above, there are 90 susceptible individuals in cell C 55 , and 100 in each other cell. We can see that at instant i = 100, the numbers S C 55 and S C pq are at most the same as in the case when there was no control strategy, while in the control source patch, S C 32 and S C 34 take almost zero values. At instant i=700, susceptible population start decreasing from the lower right corner where the disease has started, even at this instant, the number of susceptible people in the source control patch conserved its values.

Simulations with controls.
In Figure 6., and as done in the case without controls, we can also observe that the disease spreads towards the corners and borders of Ω. For instance, at instant i = 550, the number of infected people has increased in the vicinity of the patch W S and in V W O , and as more, we move away to the corners and borders, infection is still low. At instant i = 700, I C pq takes more important values in most cells except at the upper corners, noting that the number of infected people in the control source patch W S , and I C pq in the target patch W O , are more reduced, due to the effectiveness of optimal controls, than the case when there are no controls in Figure 3.
As regards the removed population, in Figure 7, at instant i=100, R C pq in the control source patch becomes more important taking values between 100 and 150 individuals, even in the target patch W O , compared with the case when there are no controls. At instants i=300, i=350 and i=700, the number of removed individuals becomes more and more important than before, and as we move away from V W S , R C pq is still zero.
Thus, we can deduce that the vaccination optimal control strategy has proved its effectiveness earlier.
We can observe in Figure 8, the different curves of states of the target cell and the control source cells, without controls, we can see more precisely the values of each state of cells in W O and W S .
In Figure 9., we see that the number of the infected population has decreased importantly in the control source patch even in the target cell and the number of the removed individuals becomes more important after the introduction of the vaccination optimal control strategy, compared with results in Figure 8. We provide Figure 10 here to show the difference, after and before introducing the optimal control strategy, between states of the target cell.
The susceptible population in that figure (a), still increasing in time after using the optimal controls, while start decreasing at the instant i=500 before. Figure 10 (b) shows that the number of infected individuals reached a big value equal to 60 individuals at instant i=600, but in the presence of controls, it takes a maximum value not exceeding 20 individuals.
While the removed population retains almost zero value in the absence of controls, the number of removed individuals increases in time to a greater value towards 48 individuals at the time i = 600 after the introduction of the optimal vaccination control strategy, as shown in Figure 10 (c).
In Figures 5., 6. and 7., we investigate the effectiveness of the optimal vaccination control approach on the SIR populations of Ω, when it is applied to the control source patches W S for the purpose of controlling a sensitive target patch W O . We can see that at instant i = 700, the number of SIR people in the two patches satisfying the control objective, that is the minimization of the infected population and the increase of the removed one.
However, the most interesting idea we can extract from these figures, is that regardless of the position of the control source patch W S concerning the target patch W O , by respecting the fact V W O ∩ V W S = / 0, the control objective is well done, and the number of susceptible people in the target patch has not decreased significantly and the number of the infected population becomes less important, and the removed one becomes huge in time.
Since when we aim to control only one cell as in the previous simulations, the vicinity set associated with this cell contains 8 cells including cells of the control source patch. Thus, for instance in the example above, it remains 6 cells, in the vicinity set of the target cell, not controlled, and then the movements of infected travelers entering from these 6 cells can increase efficiently the number of the infected individuals in that cell.

CONCLUSIONS
In this paper, we presented a multi-region SIR epidemic model in the discrete-time, and we investigated a neighboring infection optimal control strategies.
We presented a new epidemic modeling approach, based on dividing the studied domain to several parts called cells or regions, these cells are assembled in one grid represents the studied domain. Two cells are neighboring in the grid, meaning that there are means of transport between them, whatever their geographical locations. In order to contribute to the improvement of the epidemic modeling and control, we considered a discrete-time multi-regions SIR epidemic model in which we have introduced also movements of susceptibles and removed people.
A new optimal control approach is investigated to characterize the treatment injected in a control source patch and which allow reducing the number of infected population and to increase the removed ones in a target patch which is considered an important class of society. A discrete version of Pontryagin's maximum principle is done to analyze the optimal control problem and numerical simulation is given to illustrate the obtained results.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.