1. Introduction

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 [1-3].

In their papers [4-7], Zakary et al. have devised new mathematical models which are based on multi-regions discrete-time and continuous-time SIR models, aiming to describe the spatial-temporal evolution of epidemics which emerge in different geographical regions, and also, in order to show the influence of one region on an other one via infection travel. In [4, 5], the mentioned authors have proposed an optimization approach for proving its effectiveness when applied to epidemics and pandemics in general, and they treated particular infectious diseases such as HIV/AIDS and Ebola in [6] and [7] respectively. In other papers, the authors have also suggested multi-regions discrete-time systems which describe infection dynamics of an epidemic based on SIRS and SEIRS cellular compartmental models [8, 9]. In this research article, we aim to control a region that is affected by an influenza pandemic due to movements of infected people which enter only from its neighboring regions, with the hypothesis that in all regions, there are no removed individuals due to the lack of the immunity.

Grais el al. have discussed in their papers in [10, 11], concrete examples which show the impact of mobility of individuals via air travel, on the global spread of the influenza epidemic. In the same context, Mateus et al. reported different types of travel restrictions in roads, rails, air and borders, which have been followed in different regions and showed their effectiveness in the case of a human influenza [12], see also work of Epstein et al. in [13] and where the authors revealed the importance of mathematical models in exhibiting the effectiveness of such travel-restrictions control strategies in case of the pandemic flu.

For all these reasons, we suggest first here, a new modeling approach which is based on a multi-regions SIS discrete-time model describing the spatial-temporal spread of an influenza pandemic which emerges in a global domain of interest represented by a gridded surface of colored cells which are uniform in size. These cells are supposed to be connected by movements of their populations, and they represent sub-domains or regions, and noting that only one of these cells, that is targeted by our control strategy.

In [4], a region was represented by a sub-domain (Ω_j)j=1,..,p which belongs to a domain Ω, while here, a region is denoted by a cell (C_pq)p,q=1,..,M. For this, we assume that the pandemic can be transmitted and propagated by movements of people, from one spatial cell C_pq, to its neighbors or cells belonging to its vicinity. In a geographical scale relatively small, some infectious diseases such as African Swine Fever [14], Bovine Viral Diarrhoea virus [15, 16] and Foot-and-Mouth Disease [17], follow that pattern of spread, and C_pq can represent a farm, while in a large geographical scale such as in the case of SARS [18], HIV/AIDS [6, 19, 20], Ebola Virus [7], and ZIKA Virus [21], a cell C_pq can represent a city, a country or larger domain. Thus, by following a control strategy which is based on the travel-blocking vicinity optimal control approach suggested in [8, 9], movements of infected people who are intending to enter other cells, could be effectively restricted, and then, the incidence rate for which an infection could occur, becomes less important. In fact, the optimization criteria are also chosen here in a way to restrict the movement of people coming from one or more cells and entering other cells. Explicitly, we seek to minimize an objective function associated to C_pq and subject to its associated discrete-time system, with optimal control functions introduced as minimizers of infection incidence in order to show the effectiveness of the travel-blocking operations followed between C_pq and its neighbors. V_pq represents the vicinity set which is composed by all neighboring cells of C_pq and which are denoted by (C_rs)r=p+k,s=q+k′ with (k, k′) ∈ {−1, 0, 1}² except when k = k′ = 0. Note also as we have mentioned before, these cells are attached just in the grid, but in reality, they are not necessarily joined together. Thus, the travel-blocking vicinity optimal control approach will show the impact of the optimal blocker controls on reducing contacts between susceptible people of the targeted cell C_pq and infected people coming from one cell C_rs or more cells from V_pq. The paper is organized as follows: Section 2. presents the discrete-time multi-cells SIRS influenza pandemic system based on a colored cellular modeling approach. In Section 3, we announce a theorem of necessary conditions and characterization of the sought optimal controls functions related to the travel-blocking vicinity optimal control approach. Finally, in section 4, we provide simulations of the numerical results for an example of 100 cells when an infection starts from one cell of them and which has 5 neighboring cells, while aiming to control only one cell with 8 neighboring cells.

2. The Mathematical SIS Model

We consider a multi-regions discrete-time epidemic model which describes SIS dynamics within a domain Ω which is divided into M² regions or cells, uniform in size. In other words, 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 such as neighborhoods, which belong respectively to a domain Ω that could represent a part of continent or even a whole continent, a part of country or a whole country, etc.

We represent the S-I populations associated to a cell C_pq by the states and and we note that the transition between them, is probabilistic, with probabilities being determined by the observed characteristics of specific diseases. In addition to the death, there are population movements among these three epidemiological compartments, from time unit i to time i + 1. We assume that the susceptible individuals not yet infected but can be infected only through contacts with infected people from V_pq (Vicinity set or Neighborhood of a cell C_pq), thus, the infection transmission is assumed to occur between individuals present in a given cell C_pq, and is given by

where is the constant proportion of adequate contacts between a susceptible from a cell C_pq and an infective coming from its neighbor cell with

SIS dynamics associated to a cell C_pq are described based on the following multi-regions discrete model.

For p,q = 1, ..., M, we have

(1)

(2)

i = 0, ..., N − 1

with and are the given initial conditions.

d > 0 is the natural death rate while α > 0 is the death rate due to the infection, and θ > 0 denotes the recovery rate. By assuming that is all regions are occupied by homogeneous populations, α, d and θ are considered to be the same for all cells of Ω.

3. The Travel-Blocking Vicinity Optimal Control Approach

The main goal of the travel-blocking vicinity optimal control approach is to restrict movements of infected people coming from the set V_pq and aiming to reach the cell C_pq. For this, we introduce controls variables u^pqCrs which limits contacts between susceptible of the targeted cell C_pq and infected individuals from cells C_rs which belong to V_pq. Then, for a given cell C_pq in Ω, the discrete-time system (1)-(2) becomes

(3)

(4)

i = 0, ..., N − 1

Since our goal concerns the minimization of the number of the infected people and the cost of the vicinity optimal control approach, we consider an optimization criterion associated to cell C_pq and we define it by the following objective function

(5)

where A₁ > 0 and A_rs > 0 are the constant severity weights associated to the number of infected individuals and controls respectively. The controls functions are defined in the control set U_pq associated to the cell C_pq, by

(6)

Then, we seek optimal controls such that

The sufficient conditions for the existence of optimal controls in the case of discrete-time epidemic models have been announced in [4] [5], [22], and [23].

As regards to the necessary conditions and the characterization of our discrete optimal control, we use a discrete version of Pontryagin's maximum principle [4], [5], [24].

For this, we define an Hamiltonian associated to a cell by

with the adjoint variables associated to and respectively, and which are defined based on formulations of the following theorem.

Theorem 1. (Necessary Conditions \& Characterization) Given optimal controls and solutions and there exists the adjoint variables satisfying the following equations

(7)

(8)

with are the transversality conditions.

In addition

(9)

Proof. Using a discrete version of Pontryagin's Maximum Principle in [4], [5], [24], and setting and we obtain the following adjoint equations

with are the transversality conditions.

In order to obtain the optimality condition, we calculate the derivative of with respect to and we set it equal to zero

Then, we obtain

By the bounds in we finally obtain the characterization of the optimal controls as

4. Numerical Results and Discussions

4.1. Brief Presentation

In this section, we provide numerical simulations to demonstrate our theoretical results in the case when the studied domain represent the assembly of regions or cells (countries, cities, towns, ...). A code is written and compiled in MATLAB using data cited in Table 1. The optimality systems are solved using an iterative method where at instant the states and with an initial guess, are obtained based on a progressive scheme in time, and their adjoint variables are obtained based on a regressive scheme in time because of the transversality conditions. Afterwards, we update the optimal controls values (10) using the values of state and costate variables obtained in the previous steps. Finally, we execute the previous steps till a tolerance criterion is reached. In order to show the importance of our work, and without loss of generality, we consider here that and then we present our numerical simulations in a grid and which represents the global domain of interest

Table 1. Parameters values of       associated to a cell       and which are utilized for the resolution of all multi-regions discrete-time systems (1)-(2) and (3)-(4), and then leading to simulations obtained from Figure 1 to Figure 4, with the initial conditions       associated to any cell

At the initial instant, susceptible people are homogeneously distributed with 50 individuals in each cell except at the lower-left corner cell where we introduce 10 infected individuals and 40 susceptible ones.

In all of the figures below, the redder part of the color-bars contains larger numbers of individuals while the blue part contains the smaller numbers.

In the following, we discuss with more details, the cellular obtained simulations, in the case when there are yet no controls.

4.2. Cellular Simulations without Controls

In this section, Figures 1. and 2. depict dynamics of the susceptible population in the case when there is yet no control strategy, followed for the prevention of the epidemic, and we note that in all these figures presented here, simulations give us an idea about the spread of the disease in the case when the epidemic starts in a cell with (located in the lower-left corner of It represents the case when the vicinity set associated to the source cell of infection, contains 3 cells).

Figure 1. behavior in the absence of controls

Figure 2. behavior in the absence of controls

For instance, in Figure 1, if we suppose there are 40 susceptible people in cell located at the lower-left corner of and 50 in each other cell, we can see that at instant , the number becomes less important and takes values close/or equal to 35, 40 and 45, while in cells of take values close/or equal to 30, and as we move away from remains important. At instant we can observe that in most of cells, becomes less important, taking values between 0 and 10 near the source of infection, while in other cells, it takes values between 20 and 40 except which conserves its value in 50 since it is located far away from the source of infection. At instant becomes zero except at the opposite corners and the borders of because these cells have vicinity sets smaller than other cells. Finally at last instants, converge to zero in all cells.

Figure 2. illustrates the rapid propagation of the infection in and we can observe that at instant the number increases to a bigger value which is close/or equals to 25, while in cells of take values close/or equal to 15, and as we move away from remains less important. At instant we can see that in most cells, becomes more important, taking values that are bigger than 30 in cells which have/or are close to cells with 8 neighboring cells, while in few other cells, it takes values that are close to 15. From these numerical results, we can deduce that once the infection arrives to the center or to the cells with 8 cells in their vicinity sets, the infection becomes more important compared to the case of the previous instant. At instant takes values which equal/or are close to 28 or more in cells from where the epidemic has started, and 25 in and near to it, and as we move away towards the center and further regions, infection is important with the presence of 32 infected individuals in the 3 opposite corners and which becomes 25 at instant In fact, at the center of the number of infected people which has increased to 35 at the previous instant has been reduced, because once a cell becomes highly infected, it loses an important number of individuals which die, recover naturally after or become susceptible again, and this can be more deduced at further instants.

4.3. Cellular Simulations with Controls

Figures 3. and 4. depict dynamics of the S-I populations when the travel-blocking vicinity optimal control strategy is followed.

Figure 3. behavior in the presence of optimal controls (10)

Figure 4. behavior in the presence of optimal controls (10)

In order to show the importance of our optimal control approach, we take the example of a 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 cell located in the lower-left border of As an example, we suppose that the cell aiming to control is

In Figure 3., as supposed also above, there are 40 susceptible people in cell and 50 in each other cell. We can see that at instant the numbers and are at most the same as in the case when there was no control strategy. However, the controlled cell contains 45 susceptible people at instant which is not exactly the same as in the case when there was yet no control strategy since in Figure 1., has decreased more significantly. Thus, we can deduce that the travel-blocking vicinity optimal control strategy has proved its effectiveness earlier in time. At instants and is also the same as done before but fortunately again, we reach our goal in keeping the number close to its initial value despite some small decrease. Thus, this demonstrates that most of movements of infected people coming from the vicinity set have been restricted in final times.

In Figure 4., we deduce that at instant the numbers and are at most the same as in Figure 2. At instant we can see that in most cells, is also similar to the case in Figure 3. but the controlled cell is still not really infected and contains only about one infected individual, and the number remains always not important at further instants.

ACKNOWLEDGEMENTS

This work is supported by the Systems Theory Network (Réseau Théorie des Systèmes), and Hassan II Academy of Sciences and Technologies-Morocco.