AUTOMATED OPTIMAL VACCINATION AND TRAVEL-RESTRICTION CONTROLS WITH A DISCRETE MULTI-REGION SIR EPIDEMIC MODEL

1Laboratory of Analysis, Modeling and Simulation (LAMS), 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
Among the fields of mathematical biology [1], mathematical epidemiology (ME) [2], that is to say, the construction and analysis of mathematical models describing the spread and control of infectious diseases, epidemics and pandemics, which has great utility influence. In fact, mathematical models of epidemics are currently commonly used in the evaluation of public health policies by national and international health agencies. In addition, an increasing number of articles in ME are published by reputable medical journals [4,5,6,7,8]. Epidemiological modeling and its applications have undergone a great evolution in recent decades, which has made it possible to analyze the evolution of epidemics from their population systems [12] and for different geographic areas [13,14,15].
A major assumption in many mathematical models of epidemics is that the population can be divided into a set of separate states. These states are defined according to the state of the disease.
The simplest model, which calculates the theoretical number of people infected with a contagious disease in a closed population over time, described by Kermack and Mckendrick (1927), consists of three components: susceptible (S), infected (I) and recovered (R). The disease states are defined as follows: • Susceptible: Individuals that have never been infected and can thus contract the disease.
Once infected, they move to the infected state.
• Infected: Individuals that can transmit the disease to susceptible individuals. The time that individuals spend in the infected state is the infectious period; then they enter the recovered state.
• Recovered: Individuals in recovery are assumed to be immune for life [21].
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.
The authors in [13] present the first work in the modeling and control of spatio-temporal spread of an epidemic using a multi-region SIR discrete-time model, as a generalization of the concept of classical models and aiming at a description of the evolution of pandemics, Zakary et al proposed a new approach of modeling of the spread of epidemics from one area to another using finite-dimensional models for the Spatio-temporal propagation of epidemics as an alternative of the partial derivatives models which are of infinite dimension. The authors also suggested some control strategies such as awareness-raising, vaccination, and travel-restriction approaches that could prevent specific infectious diseases such as HIV / AIDS, Ebola, or other epidemics in general [13,15,22,14,35], other researchers have shown the power and effectiveness of educational workshops and awareness programs in reducing the number of infected individuals [36,37,23].
In this paper, we propose a new optimal control approach mainly based on a multi-regions discrete-time model and a new form of multi-objective optimization criteria with importance functions and which is subject to multi-points boundary value optimal control problems. With more clarifications and essential details, we devise here a multi-regions discrete model for the study of the spread of an epidemic in M different zones, and analyze the effectiveness of vaccination (or awareness) and travel-restriction optimal control strategies when vaccination campaigns and/or travel restriction are applied in infected zones based on the number of infections.
Here, we study the case when controls are applied to people who belong to all those regions and which are supposed to be reachable for every agent (nurse, doctor or media) who is responsible for the accomplishment of control strategies followed against the disease.
We consider an area as an infected zone if its number of infected individuals exceeds a threshold defined by the health decision-makers. Therefore, varying the values of this threshold and then simulating the infection situation for different values of these thresholds shows that it is necessary to think about reducing the time between the first infection and the implementation of In our modeling approach we divided the studied area Ω into different zones that we call cells (or regions or zones). A cell C j ∈ Ω can represent a city, a country or a larger domain. These cells are supposed to be connected by movements of their populations within the domain Ω. We define also a neighboring cells C k of the cell C j all zones connected with C j via every transport mean, thus a cell C j ∈ Ω can have more than one neighboring cell. Here, we suppose that a cell can be infected due to movements of infected people which enter from its neighboring zones.
We carry out the map of the studied area and then we use different threshold values in the controlled multi-region SIR model to simulate the epidemic spread within the Casablanca-Settat region and Rabat-Sale-Kenitra region illustrated in the Fig.1, by combining the ArcGIS and Matlab programs.
The paper is organized as follows: Section 2. presents the discrete-time multi-region SIR epidemic system. In Section 3., we announce theorems of the existence and characterization of the sought optimal controls functions related to the optimal control approach we propose.
Finally, in section 4., we provide simulations of the numerical results applied to the Casablanca-Settat region and Rabat-Sale-Kenitra region as domain of interest. Section 5 concludes.

MODEL DESCRIPTION AND DEFINITIONS
Based on modeling assumptions of the reference [13], we assume that there are M geographical zones denoted C j (sub-domains, regions, cities, towns ...) of the studied domain Ω Where C j can represent a city, a country or a larger domain. For instance, in the Fig.1 we show an example of geographical discretization of tow regions of Morocco, that is, Casablanca-Settat and Rabat-Sale-Kenitra with 16 zones. We define V (C j ), the vicinity set, composed by all neighboring cells of C j given by Where C j ∩ C k = / 0 means that there exists at least one mean of transport between C j and C k . Note that this definition of V (C j ) is more general where it defines a more general form of vicinity regardless the geographical location of zones.
The multi-regional discrete-time SIR model associated to C j when there is no control introduced yet is then where the disease transmission coefficient β jk > 0 is the proportion of adequate contacts in domain C j between a susceptible from C j ( j = 1, ..., M) and an infective from another domain C k , d j is the birth and death rate and γ j is the recovery rate ans α j is the proportion of mortality due to the disease. The biological background requires that all parameters be non-negative.

TRAVEL-RESTRICTION AND VACCINATION CONTROLS
3.1. Presentation of the model with controls. In this section, we introduce a control variable u C j i that characterizes the effectiveness of the vaccination in the above mentioned model (1)(2)(3). This control in some situations can represent the effect of the awareness and media programs [14,22].
In almost all infectious diseases, the authorities determine the threshold of risk based on many factors, such as availability of medical equipment, budgets, and medical personnel ... Thus, they can wait some time to see the course of events before the intervention. If the number of casualties exceeds a predefined limit, decision-makers have no choice but to start trying to control the situation. This motivate us to define a Boolean function ε associated to the domain C j , that will be called the importance function of C j . Where ε C j i is either equaling to 1, in the case when the number of infections in the cell C j at instant i is greater than or equal to the threshold of vaccination I C j V defined by the authorities and health decision-makers, or ε C j i = 0 otherwise. Therefore, we define the importance function associated to the vaccination control ε C j v,i by the Heaviside step function H as follows In the case of epidemics and pandemics, and in the absence of effective treatment, governments tend to take non-drug measures to reduce the number of victims. Travel restrictions, selfisolation and social isolation are the most commonly used non-drug measures in such situations.
Therefore, we introduce a second control variable v C j i that characterizes the travel-restriction operation which aims to restrict movements of people coming from neighboring zones C k ∈ V (C j ) to the affected zone C j , in order to facilitate the categorization of people depending on their cases. Thus, we define a threshold based on the number of infections to determine affected zones. Then, I C j T is the tolerable number of infections in the zone C j before the closure of this zone, i.e. if the number of infections I C j i in C j exceeds I C j T , C j is called an affected zone and then the travel-restriction will be applied. Therefore, the importance function associated to the travel-restriction control ε C j T,i is also defined by the Heaviside step function H as follows Based on all these considerations, for a given zone C j ∈ Ω, the model is given by the following Our goal is obviously to try to minimize the population of the susceptible and infected groups and the cost of vaccinations and travel-restriction in all affected zones.

3.
2. An optimal control approach. We devise in this paper an automated optimal control approach for each region with different importance functions ε C j * ,i , j = 1, ..., M. We characterize optimal controls that minimize the number of the infected people and maximize the ones in the recovered category for all affected regions. Then, we are interested by minimizing the functional where A j > 0, B j > 0, α C j I > 0, α C j R > 0 are the weight constants of controls, the infected and the recovered populations in the region C j respectively, and u = u C 1 , ...., Our goal is to minimize the infected individuals, minimize the systemic costs of vaccinations and travel-restriction attempting to increase the number of recovered people in each zone C j ∈ Ω. In other words, we are seeking optimal controls u * and v * such that where U and V are the control sets defined by [38] we derive necessary conditions for our optimal controls. For this purpose we define the Hamiltonian as Given optimal controls u C j * , v C j * and solutions S C j * , I C j * and R C j * , there exists ζ j k,i , i = 1, ..., N, k = 1, 2, 3, the adjoint variables satisfying the following equations Proof. Using Pontryagin's Maximum Principle [38], we obtain the following adjoint equations To obtain the optimality conditions we take the variation with respect to control u C j i and v C j i and solve for u C j and v C j respectively we get  Fig.38, with the initial populations given in Table 1.
By taking bounds of controls from U and V we get the result.

NUMERICAL RESULTS
In this section, we present numerical simulations associated to the above mentioned optimal control problem. We write a code in MAT LAB T M and simulated our results for several scenarios. The optimality systems is solved based on an iterative discrete scheme that converges following an appropriate test similar the one related to the Forward-Backward Sweep Method (FBSM). The state system with an initial guess is solved forward in time and then the adjoint system is solved backward in time because of the transversality conditions. Afterwards, we update the optimal control values using the values of state and co-state variables obtained at the previous steps. Finally, we execute the previous steps till a tolerance criterion is reached.

Area of interest.
We chose the Casablanca-Settat region and the Rabat-Sale-Kenitra region as the studied area Ω in this paper because we are convinced that we can find some useful data to support our work. They are the most populated and dynamic regions of Morocco, which contain the Rabat city as the capital of Morocco and the seventh largest city in the country with an urban population of around 580,000 inhabitants (2014) and a metropolitan population of more than 1.2 million inhabitants. It is also the capital of the administrative region of Rabat-Sale-Kenitra. They contain also the Casablanca city as the economic and industrial capital of Morocco because with its demographic growth and continuous development of the industrial sector, and the 14 other provinces (see Fig.1), in order to illustrate the objective of our work. Susceptible: The real populations given in Table 1.
Infected: 100 infections only in the city of Casablanca, and 0 for the others.
Recovered: We assume that i = 0 represents the first appearance of the epidemic, therefore, there are no recovered individuals.
Parameters We use the parameters' values given in Table 2 for all zones.                    remain stagnant until the end of the period. We notice a slight improvement compared to the vaccination strategy for 500 infected and travel-restriction for 1000 infected and we can say that it gives almost the same values. It is noted that the infected from regions C 8 , C 12 , C 13 , C 14 , C 15 and C 16 which surround the city of Casablanca, that experienced the appearance of 100 cases infected in the initial state, experienced an increase from the start of the strategy, then the regions C 6 , C 7 , C 9 , C 11 which started to grow from the time i = 50, then the regions farthest from Casablanca which started to grow at the time i = 75. On the other hand, this strategy gives good  results compared to without control and also compared to the strategy with travel-restriction at 500 infected, but less effective than the strategy with travel-restriction at 0 infected.   values which vary between 3.10 5 and 3.10 6 . The regions closest to Casablanca begin to grow at the start, then the least close and then the farthest from Casablanca. This strategy gives better results for the recovered than that without control, whose recovered does not exceed 230 cases, or with the strategies with travel-restriction at 500 infected or at 0 infected that do not exceed the 35 recovered.

CONCLUSION
In this paper, we devised a novel optimization approach that represents a general form of optimal control approaches studied in multi-region framework. We applied this approach to a multi-region discrete epidemic model which has been firstly proposed in [13]. We suggested in this article, a new analysis of infection dynamics in M regions which we supposed to be accessible for health authorities. By defining new importance functions to identify affected areas that must be dealt with, we investigated the effectiveness of optimal vaccination and travelrestriction control approaches, we introduced into the model, control functions associated with appropriate control strategies followed in the targeted regions by mass vaccination campaigns and restrictions and considering different scenarios to compare different strategies. Based on our numerical simulations, we showed the geographical spread of the epidemic and the influence of each region on another and then we deduced the effectiveness of each strategy followed. We concluded that the last scenario of optimal control approach when I T min = 0 and I V min = 500 has given better results than the other cases regarding the maximization of the number of recovered individuals and minimization of the spread of infection in all regions studied.

DATA AVAILABILITY
Data of the actual populations of the Casablanca-Settat region from [40] and for the Rabat-Salé-Kénitra region from [41].

FUNDING STATEMENT
The author(s) received no financial support for the research, authorship, and/or publication of this article.

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