A Multi-Regions SIRS Discrete Epidemic Model With a Travel-Blocking Vicinity Optimal Control Approach on Cells

Abstract

Compared to Susceptible-Infected-Removed Susceptible (SIRS) systems, where it is supposed that a removed population has lost its immunity after being healed from an infection and moves to the susceptible compartment, the S-Exposed-I-R-S (SEIRS) compartmental models, consider also, the presence of an additional compartment named by the variable E which could represent the number of asymptomatic infected individuals, people who are not yet infectious or just exposed to infection. Based on these assumptions, we devise a multi-regions SEIRS discrete-time model which describes infection dynamics due to the presence of an epidemic in regions that are connected with their neighbors by any kind of anthropological movement. The main goal from this kind of modeling, is to introduce after, controls variables which restrict movements of the infected individuals coming from the vicinity of the region targeted by our control strategy we call here by the travel-blocking vicinity optimal control approach. A grid of colored cells is presented to illustrate the whole domain affected by the epidemic while each cell represents a sub-domain or region. In order to illustrate an example of these SEIRS dynamics, we choose an example of infection which is supposed starting from only one cell located in one of the corners of the grid, while the region aiming to control, is supposed to be located in the 2nd line and 4th column of the grid. It is important to note, this optimization approach could be applied to any cell of the grid, and the source of infection could also be supposed to start from any cell. In fact, the example is presented, just to show the effectiveness of the proposed control strategy when it is applied to a cell with an important number of connections (i.e. with 8 neighboring cells in our simulations).

Role of Media and Effects of Infodemics and Escapes in the Spatial Spread of Epidemics: A Stochastic Multi-Region Model with Optimal Control Approach

“…Note that under the previous observations, (12) and (13), for (x (14) becomes linear. Therefore, the state system admits a unique strong solution [39] that is written in the following form…”

“…Recently, multi-region epidemic models as in [10,11] have been interested in the study of the spatial spread of some epidemics using metapopulation-like differential equations. Other papers as in [12][13][14][15][16][17][18][19][20][21] chose to model the same phenomenon, namely the regional spread of epidemics, using the framework of difference equations. All these last references treated dynamics of epidemics in the presence of a so-called travel-blocking control strategy.…”

Travel-blocking Optimal Control Policy on Borders of a Chain of Regions Subject to SIRS Discrete Epidemic Model

“…In order to avoid repetitions, we call readers to [11,12,13,14,17,18], to understand our programming code used in the simulations below. However, here are some important information to make our comments clearer.…”

“…In order to describe the spatial-temporal dynamics of infection which emerges in different geographical regions and to show the influence of one region on another via infection mobility, a new modeling approach using multi-regions discrete-time epidemic models has been recently presented for this subject in [9,10], and also in [11,12,13,14] using cellular simulations for the study of this type of models when they are in the form of SIR, S-Exposed-IRS (SEIRS), SEIS and SIS discrete systems respectively. The investigation of the advantages of such epidemic modeling problems has been done for special emerging diseases such as Ebola and Human Immunodeficiency Virus infection and Acquired Immune Deficiency Syndrome (HIV/AIDS) in [15] and [16] respectively.…”

“…Here, we deal with the optimal control problem of an epidemic that is described using the model devised in [17] and where the authors tried to exhibit the effectiveness of movements restrictions of the infected individuals coming from the vicinity of a region based on their so-called travel-blocking vicinity optimal control strategy. However, this mentioned reference has not discussed the case when there is an urgent need to control a group of regions or a patch and not only one region.…”