SYSTEM (GIS) AND THE MATHEMATICAL MODELING OF EPIDEMICS TO ESTIMATE AND CONTROL THE SPATIO-TEMPORAL SEVERITY OF INFECTION IN THE MOST ATTRACTIVE REGIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. To investigate the spatio-temporal spread of an epidemic in several regions (cities, towns, neighbors...) that are connected by population movements, we adopt a new generalized discrete-time multi-region SIR model, in which we introduce a new diffusion terms that describe the potential attractiveness of each region. In this work we study the effect of the potential attractiveness of regions, and we show the influence of one region on the others by varying attractiveness parameters associated to each zone. We determine an optimal control strategy which allows to reduce the infectious individuals and increase the number of recovered ones in a targeted region and this with an optimal cost. We investigate also the impact of these diffusion parameters on several control strategies by considering several control scenarios. As an application of our theoretical results, first, we investigate the potential attractiveness of the Grand Casablanca-Settat region of Morocco. We calculated the potential attractiveness of each sub region of the studied area by using a qualitative method based on the use of topographic data as well as observations made in the field using the ArcGIS Geoprocessing Tool. The attractiveness map can help to simplify the implementation of control strategies, the organization of cultural events and the implementation of awareness campaigns.


INTRODUCTION
Residents, businesses and visitors can be considered as the main groups in a society, and satisfaction of the needs of these three target groups in a region (city, town, ...) is the most important factor of its attractiveness. Several studies and research [1,2] , define the attractive regions as the regions with the following potentialities: an enforced security, an effective structure of economic activities, successful business and housing policy, accessibility and mobility, access to public services and institutions, efficient transport and traffic system, knowledge-based society, information tools and resources, natural and physical environment; strong and diverse cultural and tourism sector, the city's vitality, liveability, viability and the city's image.
The attractiveness of regions allows these three groups of society to move from one region to another according to their needs. An attractive region for businesses should have a good accessibility, location, built environment, sufficient and quality workforce, quality of utilities, encouraging land prices, taxes and local requirements, new and existing customers, suppliers, finance, partners. For visitors, they are looking for temporary accommodation (hotels, apartments, campsites, family houses, etc.) of high quality at an acceptable cost, as well as accessibility to tourist sites, places of relaxation and entertainment (beaches, forests, theaters and playrooms), the availability of restaurants, parking, public transport, security, the cultural sector, and other amenities. However, an attractive city for residents should have a good, accessible, clean and secure environment, places for work (factories, large yards, important ports), institutions of education and health of quality, high-quality city culture, public safety, religious sites, green spaces, leisure facilities.
The importance of these features attracts individuals of all kinds whether it is a healthy or sick person. This allows a sick person to move toward more attractive areas which lead to a rapid spread of infections and therefore an epidemic.
Mathematical modeling in epidemiology has become an important tool for analyzing the dynamics and spread of epidemics. Mathematical models provide a more in-depth insight into the spread of infectious diseases and enable authorities to make decisions to eradicate such epidemics.
Among the modeling methods of diseases is the use of the compartmental model, in which the population is divided into different groups according to the stage of the infection, with assumptions about the transfer rate of time from one compartment to the other. In the last decades, different epidemics have spread in a wide geographical area, such as HIV / AIDS: [3,4,5], Ebola [6,7], Cholera [8,9,10], Malaria [11,12] and Influenza [13,14]. Thus the need for the incorporation of spatial spread of epidemics in mathematical models.
Based on all these ideas, in this paper, we propose a new mathematical modeling of epidemic by using a multi-region SIR model in which we incorporate the potential attractiveness of regions.
To the best of the authors knowledge, this is the first work that use the potential attractiveness of regions in the spatial temporal spread of epidemics modeling.
We suggest a new discrete time model that describes the spread of an epidemic in a geographical area noted Ω, which is divided into sub-domains (cells) noted C pq , where p and q denotes coordinates of that cell. All these cells are supposed to be connected by movements of their residents. The potential attractiveness of each cell is considered as a model parameter that will be mathematically defined later. We also investigate a control strategy which allows reducing the infectious individuals and increasing the number of recovered individuals and this with an optimal cost. To do this, we incorporate a control variable in the model describing the efficiency of vaccination campaigns and / or awareness programs that could be applied in the zone targeted by this control strategy. The optimal control problems 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 (FBSM) on optimal control. Numerical simulations are performed to show the effectiveness of the optimal control strategy and to show the influence of the potential attractiveness of one region on the others.
In order to apply these results to show the influence of the potential attractiveness of regions on the spread of epidemics, we start by eliciting a geographic model to show how we can define potential attractive zones in the Grand Casablanca-Settat region, based on the use of map data using the powerful ArcGIS Geoprocessing tool. We collect data coordinates for four attractiveness factors: Hospitals, higher institutions, industrial zones and touristic places. We carry out the attractiveness map and then we use region attractiveness values in the multi-region SIR model to simulate the epidemic spread within the Grand Casablanca-Settat region by combining the ArcGIS and Matlab programs.
The paper is organized as follows: Section 2. presents the new discrete-time multi-region SIR epidemic model. Attractiveness parameters are mathematically defined. We present in the second subsection the optimal control problem. Simulations are carried out in the third subsection. In Section 3., we present a case study of the previous theoretical results by identifying the attractiveness map of the Grand Casablanca-Settat region, and then simulations of the numerical results are provided. Finally, we conclude the paper in Section 4.

MATHEMATICAL MODEL
2.1. Model description. We consider a multi-regions discrete-time epidemic model which describes SIR dynamics within a global domain of interest Ω which in turn is divided to M 2 regions or cells. In other words, Ω = p,q=1,..,M C pq with C pq denoting a spatial location or region. We note that (C pq ) p,q=1,...,M could represent a country, a city or town, or a small domain, which belong respectively to the domain Ω which could represent a part of country or a whole country.
According to the disease transmission mechanism, the host population of each cell C pq is grouped into three epidemiological compartments, S i (C pq ) susceptible individuals, I i (C pq ) infected individuals and R i (C pq ) removed individuals of C pq at time i.
We note that there are population movements among these three epidemiological compartments, from time unit i to time i + 1. We assume that the susceptible individuals of C pq not yet infected but can be infected only through contacts with infectives of C pq and C rs ∈ V pq , where V pq is the vicinity set, composed by C pq and all neighboring cells of C pq which are denoted by (C rs ) r=p+k,s=q+k with (k, k ) ∈ {−1, 0, 1} 2 except when k = k = 0, thus Therefore, the infection transmission is assumed to occur between individuals present in a given cell C pq , and is given by ∑ C rs ∈V pq γ(C rs )S i (C pq )I i (C rs ), where γ(C rs ) is the proportion of adequate contacts between a susceptible from a cell C pq and an infective from cells C rs ∈ V pq .
We define the spatio-temporal distribution of the population based on regions attractiveness by a diffusion term which is parameterized by the potential attractiveness of each cell C pq in the domain Ω as pot(i,C pq ) and it is given by where α j (C pq , i) is the number of some attractiveness factors of C pq at time i, and F is the total number of the attractiveness factors. For instance: α 1 (C pq , i) = Number of hospitals in C pq at time i Capacity of hospitals in C pq < 1 α 2 (C pq , i) = Number of accommodation in C pq at time i Capacity of accommodation in C pq < 1 Number of green spaces in C pq at time i Capacity of green spaces in C pq < 1 α F (C pq , i) = Number of cultural sector in C pq at time i Capacity of cultural sector in C pq < 1 Then, we express the exit of the susceptible, infected and removed individuals from the cell C pq to the attractive neighboring cells C rs ∈ V pq by − ∑ C rs ∈V pq pot(i,C rs )S i (C pq ), − ∑ C rs ∈V pq pot(i,C rs )I i (C pq ) and − ∑ C rs ∈V pq pot(i,C rs )R i (C pq ), respectively. The entry of susceptible, infected and removed individuals to the cell C pq from the neighboring cells C rs ∈ V pq is expressed by ∑ C rs ∈V pq pot(i,C pq )S i (C rs ), ∑ C rs ∈V pq pot(i,C pq )I i (C rs ) and ∑ C rs ∈V pq pot(i,C pq )R i (C rs ), respectively. Based on all theses considerations, the SIR dynamics associated to domain or cell C pq ∈ Ω are described based on the following multi-region discrete time model: with S C pq 0 > 0, I C pq 0 > 0 and R C pq 0 > 0 are the given initial conditions. m > 0 is the natural death rate, ρ > 0 is death rate due to the infection, λ > 0 is the natural recovery rate from infection.
A summary of parameters description and values is given in Table 1. 2.2. An optimal control problem. Optimal control theory have proved to be a viable option for the optimization of some variables that influence for instance, the population dynamics of a group of individuals as in the case of epidemics [13,15,16,3,17,18].
Optimal control approach have been applied to models (1)-(3) to reduce along the control strategy period, the number of infectious individuals and increase the number of recovered one. For this, we introduce a control variable u C pq i that characterizes the effectiveness of treatment (vaccination) in the above mentioned model (1-3) associated to the cell C pq ∈ Ω. Then, the model is given by the following equations: The main goal of the control strategy is the minimization of infected individuals and the cost of applying the control in the cell C pq . Then, for an initial state (S 0 (C pq ), I 0 (C pq ), R 0 (C pq )), we consider an optimization criterion associated to cell C pq and we define it by the following objective function where A > 0 and τ > 0 are the constant severity weights associated to the number of infected individuals and controls respectively. See the Appendix for more mathematical details.

Numerical Simulations and Result.
In this subsection, we provide numerical simulations to demonstrate our theoretical results in the case when the studied domain represents the assembly of 5 2 cells (regions, cities ...). A code is written and compiled in MATLAB T M using data cited in Table 1 and  Parameters values associated to a cell C pq ∈ Ω, which utilized for the resolution of all multi-cells discrete-time systems (1)-(3) and (4)- (6), and then leading to simulations obtained from Figure 2 to Figure 10, with the initial conditions S 0 C pq , I 0 C pq and R 0 C pq associated to any cell C pq of Ω.

Parameter
Description Value S 0 C pq Initial value of susceptible 100 In order to show the importance of our work, and without loss of generality, we consider here Regarding the potential attractiveness parameters pot i,C pq for C pq ∈ Ω, without loss of generality, we chose a fixed value for all the regions of Ω, with the exception of the two regions In the following, we discuss in more detail the cellular simulations we obtain in the case when there is no control yet.
In the Figure 2 we can see that the susceptible population almost vanished after just 500 days, in all parts of that Figure    The population S C pq where C pq ∈ Ω, from i=0 to i=600.
The four lower sub-figures: The population I C pq where C pq ∈ Ω, from i=0 to i=600.    Compared with the results of [19,20], it can be seen that when considering the attractiveness of regions, more efficient results can be obtained, as can be seen from Figure 4   The four upper sub-figures: The population S C pq where C pq ∈ Ω, from i=0 to i=600. The four middle sub-figures: The population I C pq where C pq ∈ Ω, from i=0 to i=600. The four lower sub-figures: The population R C pq where C pq ∈ Ω, from i=0 to i=600.  first case (pot (i,C 43 ) = 0.1) presented above (see Figure 4 in (a)) but in Figure 6, populations are separated. This case is presented here for a better comparison.
It is clear from this Figure that there is no adjustment between the potential attractiveness parameter and the number of susceptibles and/or infectives and/or removeds, which means that if this parameter is large, this does not imply that the number of susceptible (infectives or removeds) persons is also large. As can be seen in Figure 6 (a), the blue line is over the green one, which means that if the controlled region is more attractive, the number of susceptibles may decrease and then increase, depending on the severity of attractiveness.
We can see that in the case of pot The very important idea that can be extracted from these figures is that in order to control a region connected with several other regions by any kind of anthropological movement, it is necessary to consider the impact of the potential attractiveness parameter, to ensure good control results.

Presentation of the studied region (Grand Casablanca-Settat). We chose the Grand
Casablanca-Settat region as the studied region in this paper because we are convinced that we can find some useful data to support our work. It is the most populated and dynamic region of Morocco, that contains Casablanca city as the economic and industrial capital of Morocco because with its demographic growth and continuous development of the industrial sector [25], and 167 other provinces and communes (see Fig. 7). Which allows us to illustrate the objective of our research. Figure 7 illustrates an example of discrete geographical domains of region of Casablanca-Settat (Morocco) where p = 168, that image was originally made based on information from [24].

The potential attractiveness. The concept of attractiveness has become a very interest-
ing field of research in population dynamics not only for humans, but also for animals, birds and even mosquitoes. Where billions of animals, including mammals, fish, insects and birds, travel long distances to track seasonal changes in resources and habitats [26]. Generally, fish migration is characterized by active displacement among very different structures (fresh water, salty waters, for example) [27], On the other hand, the spatial distribution of Cattle Heron is closely related to environmental factors such as the availability of food resources [28]. In most cases, hunting can influence the spatial and temporal distribution of birds and fish. Birds disturbed by hunting should therefore move from hunting sites to prohibited sites [29]. Even insects can FIGURE 7. Region of Casablanca-Settat in Morocco. This region is divided into 168 sub-regions (or communes) (Ω i ) 1≤i≤168 , each commune with its population is given in the following table based on data from [24].  be attracted to other places to escape deteriorating habitats, colonize new places and/or to find temporary shelter, such as winter sites [30].
In general, the potential attractiveness of animals does not have as many factors as humans.
Sometimes it can be the same, such as in wars, natural disasters ... . But usually human has more and more requirements, hospitals, green spaces, beaches, schools, public transportation, public directions .... • Touristic places: Hotels, Some worth seeing places.
From Google Maps, we retrieved the representative points of each entity described above manually, the points collected are all in WGS 84 coordinate system, these data were imported into Arcgis to create four layers, first layer for hospitals, the second for higher institutions, another for industrial zones and the last for touristic places.
In a second step we transformed each layer dots into a polygon layer, creating a buffer zone at a given distance from the input entities (5km for industrial zones , 3km for hospitals, higher institutions and touristic places).
The third step is the conversion of the result into a raster carrying two values: the value 0 for the non-attractive zone and 1 for the attractive zone. In order to produce the map of the most attractive places, the overlay tool was used in the fourth and last step in the ArcGIS environment.
This tool involves one of the most widely used approaches to solve multi-criteria problems, thus the calculation of the potential attractiveness in the Grand Casablanca-Settat region. The 3.4. Construction of the attractiveness map.

Higher Education Institutions: Higher Education Institutions (HEI) in Grand
Casablanca-Settat region include public and private faculties, schools, institutes and multidisciplinary vocational training centers. HEI attract a large number of students, towards a zone of influence which we have supposed of radius 3 Km. Then we created a map to visualize the attractive zones of HEI in the Grand Casablanca-Settat region (See (a) of Fig. 9 ).

3.4.2.
Industrial zones: Work displacements play an essential role in the structure of the environment [31]. A study in France showed that in 1999, three out of five workers worked outside There is no hospitals

Presence of a hospital
There is no tourist zones Presence of a tourist zone possibly pertaining to relaxation and holidays leisure. The tourist attractiveness is associated with the destination and features such as the atmosphere of the site, the services offered, the affordability ... [34]. In this part of the paper, hotels and some worth seeing places ( Hassan In Fig.12, we can see the global diffusion of susceptible people in the whole studied area, but after some time it can be seen that the susceptible population decreases especially in the most attractive regions such as in the Casablanca city. From Fig.13, it can be seen that the infected population increases and spreads out all over the most attractive regions. In Fig.14  These simulations give an idea of the probable direction of the spread of epidemics by considering the attractiveness of the regions. Thus, results of that paper can be used by the decision makers to identify and track the evolution of epidemics within a more complex system of connected zones. In Fig.15, we can see the diffusion of the susceptible population under control strategy which is applied in the source of infection that is Berrechid (Mun). The susceptible population is also decreased but it remains more important compared to the case when there is no control. From   Fig.16, it can be seen that the infected population reaches small numbers compared to the case when there is no control (Fig.13). Infected people also spread to all the nearest attractive areas, but with a small number of people.

CONCLUSION
In this paper, we have generalized a discrete time multi-regions SIR epidemic model proposed in [19,20], in which we defined and introduced a new potential attractiveness parameters of regions. Our results show that the most attractive regions get more infections than the others and there is no proportionality between these parameters and the susceptible population size.
An optimal control is also investigated to characterize the treatment (vaccination) which allows to reduce the infectious individuals and increase the number of recovered ones in a region with varying potentials and this with an optimal cost. A discrete version of Pontryagin's maximum principle is done to analyze the optimal control problem. The potential attractiveness parameters play a very interesting role in the control of connected regions, where the most attractive regions reach the control objective based only on the positive influence of regions. Numerical simulations with several scenarios are given to illustrate and discuss theoretical results. Finally, we applied our results to study the spread of infection within the Grand Casablanca-Settat region after determining the attractiveness map based on data originally collected and analyzed with the powerful ArcGis tools. Results of that paper can be used by the decision makers to identify and track the evolution of epidemics within a more complex system of connected zones.

APPENDIX
In the optimal control problem that we consider in this paper, our interest is the minimization of the objective function given by (7) subject to system (4)- (6). In the other word, we are seeking an optimal control u = (u C pq i ) i=0,...,N−1 such that Where U ad is the set of admissible controls defined by The sufficient condition for existence of an optimal control for the problem follows from theorem 1 in [16]. At the same time by using Pontryagin's Maximum Principle [35] we derive necessary conditions for our optimal control problem. For this purpose we define the Hamil- 3,i are the adjoint functions to be determined suitably. We obtain the following theorem Theorem 4.1. (Necessary Conditions) Given an optimal control u C pq * 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 Proof. Using Pontryagin's Maximum Principle [35] and setting S * (C pq ), I * (C pq ) and R * (C pq ) and u C pq * we obtain the following adjoint equations: