NEW AUTOMATED OPTIMAL VACCINATION CONTROL WITH A MULTI-REGION SIRS EPIDEMIC MODE

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. Many mathematical models describing the evolution of infectious diseases underestimate the effect of the Spatio-temporal spread of epidemics. Currently, the COVID-19 epidemic shows the importance of taking into account the spatial dynamic of epidemics and pandemics. In this paper, we consider a multi-region discrete-time SIRS epidemic model that describes the spatial spread of an epidemic within different geographical zones assumed to be connected with the movements of their populations (cities, towns, neighbors...). Judging by the fact that there are several restrictions in medical resources and some delay in decision-making, the authorities and health decision-makers must define a threshold of infections in order to determine if a zone is epidemic or not yet. We propose a new approach of optimal control by defining new importance functions to identify affected zones and then the need for the control intervention. This optimal control strategy allows to reduce the infectious individuals and increase the number of recovered ones in the targeted domain and this with an optimal cost. Numerical results are provided to illustrate our findings by applying this new approach in the Casablanca-Settat region of Morocco. We investigate different scenarios to show the most effective scenario, based on thresholds’ values.


INTRODUCTION
A quarter to a third of all deaths in the world are due to infectious diseases, as reported by the World Health Organization (WHO). It also noted that infectious diseases made up four to five of the ten leading causes of death in 2008 [1].
In the fields of mathematical biology [2] and mathematical epidemiology (ME) [3], it is found that mathematical models have been an important tool in the analysis of the epidemiological characteristics of infectious diseases since the pioneering work of Kermack and McKendrick [1], I.e. the construction and analysis of mathematical models describing the spread and control of infectious diseases, epidemics and pandemics, which have a considerable influence. National and international health agencies often use mathematical models of epidemics in the evaluation of public health policies. Recently, an important number of articles in ME sector are published by high-profile research medical journals [4,5,6,7,8]. The evolution analysis of the epidemics from their population systems [9] and in different geographical areas [10,11,12] has become feasible thanks to epidemiological modeling. That allowed these last to undergo a big evolution over the past decades.
One of the main assumption in many mathematical models of epidemics is that the population can be divided into an ensemble 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 [13].
Susceptible-Infected-Susceptible (SIS) epidemic models have been applied to situations in which it is supposed that an infected population could move immediately to the susceptible compartment after being recovered from an infection due to the lack of immunization. This kind of compartmental models is also useful to model the evolution of many phenomena in different situations, see as examples, subjects treated in [14,15] Susceptible-Infected-Removed-Susceptible (SIRS) epidemic models have been applied to situations in which it has been supposed that a removed population could move to the susceptible compartment after being healed from an infection due to the loss of its immunity. This kind of compartmental models is very useful to model the evolution of many phenomena, see as examples, subjects treated in [16,17,18].
Different from the SIR model, the SIRS model also considers "short-term immunity". The so-called "short-term immunity" is that the immune individual becomes susceptible individual after a while. [19] Mathematical models have become important tools in analyzing the spread and control of infectious diseases. The model formulation process clarifies assumptions, variables and parameters.
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 [10] 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 [10,12,20,11,14], other researchers have shown the power and effectiveness of educational workshops and awareness programs in reducing the number of infected individuals [38,39,21].
In this paper, we propose a new optimal control approach mainly based on a multi-regions discrete-time system and a new form of multi-objective optimization criteria with importance indices 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 regions, and analyze the effectiveness of vaccination (or awareness) optimal control strategies when vaccination (or awareness) campaigns are organized in infected zones. 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, by 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 the control strategy. Unexpected results that in some situations the neighboring regions infected and its number of infections exceeds the threshold before the number of infections of the region source. This makes the implementation of the control strategies in the neighboring zones more important.
In our modeling approach we divided the studied area Ω into different zones that we call cells. 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 only from its neighboring zones.
controlled multi-region SIRS model to simulate the epidemic spread within the Casablanca-Settat 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 SIRS 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.   where C j can represent a city, a country or a larger domain. We note by 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.
For example, in the Table 1 we can see that the studied area consists of 9 zones.
The multi-regional discrete-time SIR model associated to C j with ε where the disease transmission coefficient β C k j > 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 C j is the proportion of mortality due to the disease. The biological background requires that all parameters be non-negative. θ j is the proportion that a recovered becomes again a susceptible. S  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 [11,20].
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 this 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 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 infected of the cell C j at instant i is greater than or equal to the threshold I C j defined by the authorities and health decision-makers, or ε C j i = 0 otherwise. Therefore, we define the importance function ε C j i by the Heaviside step function H as follows Then for a given domain C j ∈ Ω, the model is given by the following equations Our goal is obviously to try to minimize the population of the susceptible group and the cost of vaccination in all affected regions. Our control functions taking values between u C j min and u 3.2. An optimal control approach. We devise in this paper an optimal control approach for each region with different importance functions ε C j i , j = 1, ..., M. We characterize an optimal control that minimize the number of the infected people and maximize the ones in the removed category for all affected regions. Then, we are interested by minimizing the functional R > 0 are the weight constants of control, the infected and the removed in region C j respectively, and u = u C 1 , ...., Here, our goal is to minimize the number of infected people, minimize the systemic costs attempting to increase the number of removed people in each C j (with ε C j i = 1). In other words, we are seeking an optimal control u * such that where U is the control set defined by The sufficient condition for existence of an optimal control for the problem follows from theorem 1 . At the same time, by using Pontryagin's Maximum Principle [42] we derive necessary conditions for our optimal control in theorem 2. For this purpose, we define the Hamiltonian as Theorem 1. (Sufficient conditions) For the optimal control problem given by (7)

Theorem 2. (Necessary Conditions)
Given the optimal control u * and solutions S C j * , I C j * and R C j * , there exists ζ C j k,i , i = 1...N, k = 1, 2, 3, the adjoint variables satisfying the following equations R are the transversality conditions. In addition, Proof. Using Pontryagin's Maximum Principle [41] and for j ∈ I C , we obtain the following adjoint equations To obtain the optimality conditions we take the variation with respect to control u C pq i and set it equal to zero and ε Then, we obtain the optimal control By the bounds in U (and U C j ) of the control, it is easy to obtain u C j * i in the following form

Parameter
Description Value  Fig.19, with the initial populations given in Table 1.
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 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 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.

Geographical vicinity.
A shape-file is a simple, non topological format for storing the geometric location and attribute information of geographic features. Geographic features in a shape-file can be represented by points, lines, or polygons (areas). The workspace containing shape-files may also contain database tables, which can store additional attributes that can be joined to a shape-file's features [44]. ArcMap is a central application used in ArcGIS software, where we can view and explore GIS database for our study area, and where we assign  In the Fig.4, we note that at the beginning, all the regions did not record any infection until moment i = 120 and from the moment i = 120, the number of infected increases exponentially, and we see it clearly in the Fig.5 which has kept the same display in the four first maps (Fig.5 (a), (b), (c) and (d)) which means that the number of infected did not exceed 9600 in all regions.
In the instant i = 160, the region of Casablanca (C 4 ) and the neighboring regions C 1 ,C 2 ,C 5 ,C 6 ,C 7 exceeded the 9600 infected. The final state i = 200 has seen strong evolution, all the regions have exceeded 19200 infected except C 9 which has just reached 10 5 , regions C 2 and C 5 have exceeded 67300 infected.       On the other hand, for C 4 it remains almost zero at the beginning until the time i = 10 to starts to grow and reaches its maximum value about 3.3 10 6 at the instant i = 35, and then remains constant until the end of the period of vaccination. Without the infected threshold, the recovered quickly grows towards its maximum value, however for the other scenario take some time to increase.

CONCLUSION
In this paper, we devised a novel optimization approach that represents an extension of the optimal control approach studied in the work of Zakary et al. in the paper [12]. We applied this new approach to a multi region discrete epidemic model SIRS which has been firstly proposed in [10]. 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 zones and then will be treated. We investigated the effectiveness of optimal vaccination control approach, we introduced into the model, control functions associated with appropriate control strategies followed in the targeted regions by mass vaccination campaigns by considering different scenarios. 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 min = 0 has given better results than the other cases regarding the maximization of the number of removed individuals and minimization of the spread of infection in all regions studied, but this is clearly the most expensive scenario. Thus, as a result, it is necessary to define small thresholds to control the situation as much as possible.

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