A multiregions discretetime epidemic model with a travelblocking vicinity optimal control approach on patches
 Omar Zakary^{1}Email authorView ORCID ID profile,
 Mostafa Rachik^{1},
 Ilias Elmouki^{1} and
 Samih Lazaiz^{2}
https://doi.org/10.1186/s1366201711684
© The Author(s) 2017
Received: 26 December 2016
Accepted: 4 April 2017
Published: 26 April 2017
Abstract
We study, in this paper, infection dynamics when an epidemic emerges to many regions which are connected with their neighbors by any kind of anthropological movement. For this, we devise a multiregions discretetime model with the three classical SIR compartments, describing the spatialtemporal behaviors of homogenous susceptible, infected and removed populations. We suppose a large geographical domain, presented by a grid of colored cells, to exhibit at each instant i the spatial propagation of an epidemic which affects its different parts or subdomains that we call here cells or regions. In order to minimize the number of infected individuals in some regions, we suggest an optimal control approach based on a travelblocking vicinity strategy which aims to control a group of cells, or a patch, by restricting movements of infected people coming from its neighboring cells. We apply a discrete version of Pontryagin’s maximum principle to state the necessary conditions and characterization of the travelblocking optimal controls. We provide cellular simulations based on discrete progressiveregressive iterative schemes associated with the obtained multipoints boundary value problems. For illustrating the modeling and optimal control approaches, we consider an example of 100 regions.
Keywords
1 Introduction
1.1 Main references and description of the problem
In 1927, Kermack and McKendrick devised the SusceptibleInfectedRemoved (SIR) model which has presented an interesting contribution to the mathematical theory of epidemics [1]. The mathematical SIR model is in the form of three compartments: susceptible, infected or removed. Susceptible populations are healthy and do not carry the epidemic but can contract it from infected individuals which carry the infection and can pass it to susceptible hosts, while the removed people are no longer infected and acquire immunity from future contagion.
More clearly, we propose a cellular representation of regions, assembled all together in one grid of cells, and we study the transmission dynamics of the epidemic in these regions when a travelblocking vicinity strategy is followed for controlling one region or more to show the impact of infection connections that relate it with other regions via travel. In Figure 1(b), we can see the example of all nine regions presented in (a), how they can be converted to cells, assembled in one grid which represents a part of the earth as the global domain of interest.
Based on this new kind of representations, we can discuss the spread of the epidemic and the effectiveness of a control strategy in one region, with the possibility to analyze the SIR dynamics in this region without and with control, and exhibiting the importance of the direct influence between it and its vicinity. As observed in Figure 1(a) and (b), region I is exposed to infection via travel of infected people coming from its vicinity which contains regions D, F and H, or for the same reason, region H can directly become highly infected due to connections with regions D, E, F, G and I.
In addition to all these considerations, we note that if Zakary et al. have supposed that all regions are connected by infected travelers to show the influence of SIR dynamics of one region on other regions, the cellular model we propose here has also the advantage to exhibit this kind of influence even in the absence of direct connections between regions. This can easily be understood from the example in Figure 1 where we can see that region A can also be infected by individuals coming from region G via region D. The numerical results we will provide further are more convincing to show such kind of influence.
In the following, we provide a brief presentation of the new epidemic modeling and travelblocking vicinity optimal control approaches.
1.2 The new epidemic model and the vicinity travelblocking optimal control strategy
We suggest here a new modeling approach which is based on a multiregions discretetime epidemic model describing the spatialtemporal spread of an epidemic which emerges in a global domain of interest Ω represented by a grid of colored cells which are uniform in size. These cells are supposed to be connected by movements of their populations, and they represent subdomains of Ω or regions. Note that several cells are targeted by our control strategy, which means we suggest an optimization strategy that is not limited to controlling only one cell.
In [2], each region was represented by a subdomain \((\Omega_{j})_{j=1,\ldots,p}\), while here each region or cell is denoted by \((C_{pq})_{p,q=1,\ldots,M}\).
For this, we assume that the epidemic 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 fact, in a relatively small geographical scale, some infectious diseases, such as African swine fever [9], Bovine Viral Diarrhoea virus [10, 11] and footandmouth disease [12], 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 [13], HIV/AIDS [4, 14], Ebola virus [5] and ZIKA virus [15], a cell \(C_{pq}\) can represent a city or country. Thus, the multicells model with the vicinity optimal control strategy we propose here can represent good approaches for infection dynamics studies regardless of the area size. In fact, the optimization criteria are chosen in a way to restrict the movement of people coming from several cells and entering other cells. Explicitly, we seek to minimize an objective function associated with a group of cells or patch \(P={ \bigcup_{p,q=1}^{m}C_{pq}}\) with \(m< M\), subject to the discretetime system associated with \(C_{pq}\), with optimal controls functions introduced as effectiveness rates of the travelblocking operations followed between P and its neighbors. \(V_{pq}\) is the vicinity set composed of all neighboring cells of \(C_{pq}\) which are denoted by \((C_{rs})_{r=p+k,s=q+k'}\) with \((k,k')\in \{1,0,1\}^{2}\) except when \(k=k'=0\). Note also, as we have mentioned before, that these cells are attached just in the grid, but in reality they are not necessarily joined together as seen in example of Figure 1(a). For instance, in Figure 1(b), the vicinity sets associated with regions or cells \(C_{11}=\{A\}\), \(C_{22}=\{E\}\) and \(C_{32}=\{H\}\) are defined by \(V_{11}=\{B,D,E\}\), \(V_{22}=\{A,B,C,D,F,G,H,I\}\) and \(V_{32}=\{D,E,F,G,I \}\), respectively. Thus, the travelblocking vicinity optimal control approach will show the impact of the optimal travelblocking control on reducing contacts between susceptible people of the targeted patch P and infected people coming from cells \(C_{rs}\) in \(V_{pq}\).
The paper is organized as follows. Section 2 presents the discretetime multicells epidemic system based on a colored cell modeling approach. In Section 3, we announce a theorem of necessary conditions and characterization of the sought optimal control functions related to the travelblocking vicinity optimal control approach. Finally, in Section 4, we provide simulations of the numerical results for an example of 100 hypothetical cities when an infection starts from one cell which has three neighboring cells (respectively, the case of a cell with eight neighboring cells is investigated), while aiming to control a patch of four cells, and in another example, two patches of one and four cells, respectively.
2 A discretetime multiregions epidemic model
We consider a multiregions discretetime epidemic model which describes SIR dynamics within a global domain of interest Ω, which in turn is divided to \(M^{2}\) regions, or cells, uniform in size. In other words, \(\Omega ={ \bigcup_{p,q=1}^{M}C_{pq}}\) with \(C_{pq}\) denoting a spatial location or region.
We note that \((C_{pq})_{p,q=1,\ldots,M}\) could represent a country, a city or a town, or a small domain such as neighborhoods, which belong respectively to the global domain of interest Ω, which could in turn represent a part of a continent or even a whole continent, a part of a country or a whole country, etc.
SIR dynamics associated with a domain or cell \(C_{pq}\) are described based on the following multicells discrete model.
Here, \(d>0\) is the natural death rate, while \(\alpha >0\) is the death rate due to the infection, \(\gamma >0\) denotes the natural recovery rate from infection. By assuming that all regions are occupied by homogeneous populations, α, d and γ are considered to be the same for all cells of Ω.
3 A travelblocking vicinity optimal control approach
The sufficient conditions for the existence of optimal controls in the case of discretetime epidemic models have been announced in [2, 3, 16] and [17].
Theorem 1
Necessary conditions and characterization
Proof
4 Numerical results and discussions
4.1 Brief presentation
Parameter values of α , β , γ and d associated with a cell \(\pmb{C_{pq}}\) , \(\pmb{p,q=1,\ldots,M}\) , utilized for the resolution of all multiregions discretetime systems (1)(3) and (4)(6), and then leading to simulations obtained from Figure 2 to Figure 19 , with the initial conditions \(\pmb{S_{0}^{C_{pq}}}\) , \(\pmb{I_{0}^{C_{pq}}}\) and \(\pmb{R_{0}^{C_{pq}}}\) associated with any cell \(\pmb{C_{pq}}\) of Ω
\(\boldsymbol{S_{0}^{C_{pq}}}\)  \(\boldsymbol{I_{0}^{C_{pq}}}\)  \(\boldsymbol{R_{0}^{C_{pq}}}\)  α  β  γ  d 

50  0  0  0.002  0.0001  0.003  0.0001 
At the initial instant \(i=0\), susceptible people are homogeneously distributed with 50 individuals in each cell except at the lower right corner cell \(C_{11}\), where we introduce 10 infected individuals and 40 susceptible ones. With similar values, we study the case when the epidemic starts from cell \(C_{65}\) which is near to the center of Ω.
In all of the figures, the redder part of the colorbars contains larger numbers of individuals, while the bluer part contains smaller numbers. In the following, we discuss in more detail the cellular simulations we obtain in the case when there is yet no control.
4.2 Cellular simulations without controls

when the epidemic starts in a cell \(C_{pq}\) with \(p=10\), \(q=10\) (lower right corner cell). It represents the case when the vicinity set \(V_{pq}\) associated with the source cell of infection contains three cells.

when the epidemic starts from a cell \(C_{pq}\), with \(p=6\), \(q=5\), located in the vicinity of the target patch we aim to control.
For instance, in Figure 2, if we suppose there are 40 susceptible people in cell \(C_{1010}\) located at the lower right corner of Ω, and 50 in each of the other cells, we can see that at instant \(i=150\), the number \(S^{C_{1010}}\) becomes less important and takes a value close/or equal to 20, while \(S^{C_{pq}}\) in the cells of \(V_{1010}\) takes values close/or equal to 30. As we move away from \(V_{1010}=\{C_{109},C_{910},C _{99}\}\), \(S^{C_{pq}}\) remains important. At instant \(i=300\), we can observe that in most of cells \(S^{C_{pq}}\) becomes less important, taking values between 0 and 10, while in other cells it takes values between 20 and 40 except \(S^{C_{1010}}\) which conserves its value in 50 since it is located far away from the source of infection. At instant \(i=450\), \(S^{C_{pq}}\) becomes zero except at the corners and in most cells at the borders of Ω, because these cells have vicinity sets smaller than other cells. Finally, at last instants, \(S^{C_{pq}}\) converges to zero in all cells. As regards Figure 3, when we consider \(S^{C_{65}}=40\), which is located near the center of Ω, and 50 susceptible people in each of the other cells, it is observed that the situation is more severe, because the disease reaches the corners and borders faster than in the case of Figure 2. As we can see, at instant \(i=300\), \(S^{C_{pq}}\) takes values less important in most cells except at the corners and borders since their vicinity sets contain only three to five cells respectively, but it is the result we have reached until instant \(i=450\) in Figure 2.
Figures 4 and 5 illustrate the rapid propagation of the infection when the disease starts from cell \(C_{1010}\) and from the center of Ω, respectively. In Figure 4, if we suppose there are ten infected people in cell \(C_{1010}\) and no infection in all other cells, we observe that at instant \(i=150\) the number \(I^{C_{1010}}\) increases to bigger values close/or equal to 30 in \(C_{99}\), while \(I^{C_{pq}}\) in the cells of \(V_{1010}\) takes values close/or equal to 20, and as we move away from \(V_{1010}\), \(I^{C_{pq}}\) remains less important. At instant \(i=300\), we can see that in most of cells, \(I^{C_{pq}}\) becomes more important, taking values between 30 and 35 in the cells which are close to the cells with eight neighboring cells, while in few other cells, it takes values between 0 and 20. From these numerical results, we can deduce that once the infection arrives to the center or to the cells with eight cells in their vicinity sets, the infection becomes more important compared to the case of the previous instant. At instant \(i=450\), \(I^{C_{pq}}\) takes values close/or equal to 20 in the cell from where the epidemic has started, and 25 in \(V_{1010}\) and near to it, and as we move towards the center and further regions, infection is important with the presence of more than 30 infected individuals in each cell except the ones in the three opposite corners even at instant \(i=600\). In fact, at the center of Ω, 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 or recover naturally after. All cells \(C_{pq}\) become highly infected and the number \(I^{C_{pq}}\) becomes less and less important at further instants, noting that at \(i=900\), a large number of infected individuals has decreased because many \(I^{C_{pq}}\) have died or moved to the removed compartment. In Figure 5, when we consider infection starting from near the center of Ω by considering now that \(I^{C_{65}}=10\), and no infected people in other cells, the disease spreads towards the corners and borders faster than in the first case in Figure 4. At instant \(i=150\), the number of infected people has increased in \(V_{65}\), and as we move away to the corners and borders, infection is still low. At instant \(i=300\), \(I^{C_{pq}}\) takes values more important in most cells, close/or equal to 35, except at the corners and center where \(I^{C_{pq}}\) is close/or equal to 30, which is the result we can reach until instant \(i=450\) in Figure 4, noting that \(I^{C_{65}}\) and \(I^{C_{pq}}\) in \(V^{C_{65}}\) have reduced due to death or natural recovery from the disease, while the infection has remained important in the cells which are near to the corners and borders since the infection has just arrived. The corner cells at instant \(i=450\) conserve their number of infected individuals, while cells at the borders of Ω and the ones which are close to the center lose more people due to the number of dead or recovered people, which increases more and more at further instants, leading \(I^{C_{65}}\) to decrease towards 13 and 8 at \(i=600\) and \(i=900\), respectively.
We note that in the following figure, the scale of the colorbars does not exceed ten individuals since we cannot reach a larger number of removed people when we focus only on targeting infected people which come from \(C_{1010}\). As we can observe in Figure 6, when we have supposed there are 40 susceptible people in cell \(C_{1010}\), and 50 in each of the other cells, we can see here that simultaneously, at instant \(i=150\), the numbers \(R^{C_{1010}}\) and \(R^{C_{pq}}\) in the cells of \(V_{11}\) are close/or equal to only one or two removed people, and as we move away from \(V_{1010}\), \(R^{C_{pq}}\) becomes zero. Similarly, at instant \(i=300\), the number \(R^{C_{pq}}\) is not zero and takes values between one and three, except for distant cells where it remains zero. At instant \(i=450\), \(R^{C_{pq}}\) takes values between three and five except at the opposite three corners and some cells at the borders where it does not exceed two removed people. Finally, at further instants \(R^{C_{pq}}\) converges to five in most cells at \(i=600\) and in all cells at \(i=900\) since as we go forward in time, some people acquire immune responses that help them to cure naturally from the disease. As regards Figure 7, when we consider \(S^{C_{65}}=40\) located near the center of Ω, we can see that the results at instant \(i=600\) when the disease has started from the upper left corner of Ω are at most the same as the results obtained at \(i=450\) when the disease has started near the center of Ω. At instant \(i=300\), \(R^{C_{pq}}\) has already begun to increase from the center because some infected people have disappeared as seen in the previous figure. As regards further numerical simulations, we can observe that the number of the removed people increases to five at the center at \(i=450\) until it reaches the same value in all cells of Ω except the corners at \(i=600\). It becomes more and more important at further instants reaching five removed people at the corners and six in each of the other cells at \(i=900\).
In the following, we discuss the cellular simulations we obtain in the case when the optimal controls (11) are introduced.
4.3 Cellular simulations with controls
In order to show the importance of the optimal control approach suggested in this paper, we take the example of a patch which has 12 neighboring cells. As it was done in the previous part, we investigate also here the results obtained when the disease starts from a corner and when it starts near or attached to the center. As an example, we suppose that the patch we aim to control is \(P=\{C_{33},C_{34},C_{43},C_{44} \}\), and we present simulations when the epidemic is more important at the corner cell \(C_{1010}\) and when the epidemic is more important in cell \(C_{65}\) which is attached or directly connected to P.
In Figure 10, we can see more the analogy between the number of susceptible people \(S^{C_{1010}}\) and \(S^{C_{65}}\) and infected ones \(I^{C_{11}}\) and \(I^{C_{65}}\). In fact, when the disease starts from cell \(C_{1010}\), as supposed in the section above, there are ten infected people in cell \(C_{1010}\) and no infected in each of the other cells. We can deduce that at instant \(i=150\), the numbers \(I^{C_{1010}}\) and \(I^{C_{pq}}\) are at most the same, as shown in the absence of controls. At instant \(i=300\), we can see that in most of cells, \(I^{C_{pq}}\) is similar to the case in Figure 3, and it is also more important, taking values between 20 and 30; while in other cells, it takes values between 0 and 10 as shown in the previous subsection. However, the controlled patch P is still not really infected since it does not contain yet any infected individual. At instant \(i=450\), \(I^{C_{pq}}\) takes values around 20 in neighboring cells which belong to \(V_{1010}\), and about 30 in other cells except at the three opposite corners and borders of Ω. At instant \(i=600\), most cells \(C_{pq}\) begin to lose some infected individuals due to natural recovery, and the number \(I^{C_{pq}}\) becomes less and less important at further instants, while the number of infected people in the patch P does not exceed eight infected individuals. In Figure 11, and as done in the case without controls, we can also observe that when we consider infection starting from near the center of Ω by supposing ten infected individuals in each cell of the patch P with no infected people in each of the other cells outside the patch, the disease spreads towards the corners and borders faster than in the first case in Figure 10. For instance, at instant \(i=150\), the number of infected people has increased in the vicinity of the patch P and in \(V_{65}\), and as we move away to the corners and borders, infection is still low. At instant \(i=300\), \(I^{C_{pq}}\) takes values more important in most cells except at the corners, which is the result we can reach until instant \(i=600\) in Figure 10, noting that the number of infected people in the patch P, and \(I^{C_{pq}}\) in the vicinity of the patch P, has reduced due to death or natural recovery from the disease, while the infection becomes important in cells which are near to the corners and borders. The corner cells at instant \(i=450\) conserve their number of infected individuals while some cells at the borders of Ω and the ones which are close to the center lose more people due to the number of dead or removed people, which increases more and more at further instants as observed in \(i=600\) and \(i=900\) where the number of infected people in P does not exceed 20 infected individuals, which is bigger than the number of infected people in the case when the infection has started from the corner. Thus, as deduced in Figure 9, it also shows the impact of infection which starts close to the targeted patch even if it does not start exactly in the vicinity of the patch P.
In Figure 12, when we suppose there are 40 susceptible people in cell \(C_{1010}\), and 50 others in each of the other cells, we can see that simultaneously, at instant \(i=150\), the number \(R^{C_{1010}}\) takes a value close/or equal to five, while \(R^{C_{pq}}\) in cells of \(V_{1010}\) are zero, and as we move away from \(V_{1010}\), \(R^{C_{pq}}\) is still zero. Similarly, at instant \(i=300\), the number \(R^{C_{pq}}\) is zero at the three opposite corners and borders of Ω, while it takes values between 10 and 20 in other cells, but the number of removed people in the patch P is still very close to zero due to very few people who have been infected there. At instant \(i=450\), \(R^{C_{pq}}\) takes values between 15 and 20 except at the corners and borders, while P is still not containing any individual in its removed compartment. Finally, at last instants, \(R^{C_{pq}}\) converges to 20 at \(i=600\) in all cells except in P, which does not exceed three removed people, and between 25 in all cells at \(i=900\) and a number of individuals close to five in P since not many individuals have been infected to move to the removed compartment. As regards Figure 13, when we consider \(S^{C_{65}}=40\), we can see that at instant \(i=600\), in all cells \(C_{pq}\), the number of the removed people increases to 20 in the corner cells, and 25 in most cells of Ω, and to 25 when we go forward in time as we can observe at instant \(i=900\), while the number of removed people in the patch P has not exceeded about eight removed people.
4.4 Discussions

when the travelblocking vicinity optimal control is applied to all cells \(C_{pq}\) which belong to the patch P.

when the travelblocking vicinity optimal control is applied to all cells \(C_{pq}\) which belong to the patch P, except in \(C_{33}\).
The cellular simulations on the right side are associated with the first case, while the other ones on the left side are associated with the second case.
As we can observe in Figure 17, on the left side, the number of susceptible people in \(C_{pq}\in P\) has not changed significantly compared to the initial conditions. It loses now more people as seen in the cellular simulations on the right side. Even when we consider an infection which starts from the left lower corner cell \(C_{1010}\) and the travelblocking vicinity optimal control strategy is considered to be missed in only one cell \(C_{33}\), the number of susceptible people in \(P\setminus C_{33}\) has decreased to smaller values which equal 20 individuals in each cell. Moreover, obviously, cell \(C_{33}\) loses more susceptible people towards 10, which is due to the movements of infected people that were not restricted in \(V_{33}\). As regards the cellular simulations in Figure 18, we can see in cellular simulations on the left side that when the travelblocking vicinity optimal control approach was followed in all cells \(C_{pq}\in P\), the number of infected people has not increased and conserved the zero value, while in cellular simulations on the right side, the number of infected people has increased to 20 in each cell in \(P\setminus C_{33}\) and to 25 in \(C_{33}\). Simultaneously, we can see in Figure 19 that the number of the removed people in cellular simulations on the left side has not increased since there was no real infection after applying the travelblocking vicinity optimal control strategy in all cells \(C_{pq}\in P\). However, when we do not restrict movements of infected people coming from \(V_{33}\), we can see that the number of removed people has increased to five and ten individuals in \(P\setminus C_{33}\) and \(C_{33}\), respectively. All that means that if the infection started from the upper right corner cell, the situation would be more severe. Also, this comparison shows the importance and utility of the application of the travelblocking vicinity optimal control strategy in all cells which belong to the vicinity set \(V_{P}\) since the infection which comes from only one cell could lead to undesirable results.
5 Conclusion
Some researchers have exploited the framework of compartmental modeling in epidemiology and tried to introduce the concept of networksbased models either for the description of social contagion processes as done in [19] or for the study of the propagation of electronic and computer viruses as in [20, 21]. Not very far from the main goals of this kind of epidemic models treated in the mentioned references, which aim to highlight the nature of infection connections which participate in the rapid spread of an epidemic, in this paper we have devised a multiregions discretetime model which describes infection dynamics due to the presence of an epidemic in one region and its spreading to other regions via travel. Regions have been assembled in one grid of cells, where each cell represents a region, in order to exhibit the impact of infection which comes from the vicinity of a patch. In fact, by this kind of representations, we have succeeded to show the effectiveness of the travelblocking vicinity optimal control approach when it is applied to patches. Then, we demonstrated that when we restrict movements of infected people coming from the vicinity of a targeted patch, we can keep this patch safe without or with important infection.

when an epidemic starts near the center of a global domain of interest, the situation becomes more severe in terms of the number of infected individuals compared to the case when the epidemic starts from a corner. This is due to the number of cells in the vicinity of the cell that represents the source of infection.

the optimal controls introduced in our mathematical model respond automatically to the epidemic once it is detected, and there is an analogy between their shapes and the shape of infection.

if we do not apply the travelblocking vicinity optimal control strategy to only one cell of the targeted patch, the other optimal controls are not sufficient to stop or to reduce the infection in the controlled patch.
The cellular simulations we presented in the numerical results section have illustrated the case of 100 cells threatened by infection coming from one cell first located in the corner of a global domain of interest, and then near the center of this domain, while the patch targeted for control was chosen to contain four cells located near the center.
Declarations
Acknowledgements
The authors would like to thank all the members of the editorial board who were responsible for dealing with this paper and the anonymous referees for their valuable comments and suggestions improving the content of this paper.
This work is supported by the Systems Theory Network (Réseau Théorie des Systèmes) and Hassan II Academy of Sciences and TechnologiesMorocco.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Kermack, WO, McKendrick, GA: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 115, 700721 (1927). View ArticleMATHGoogle Scholar
 Zakary, O, Rachik, M, Elmouki, I: On the analysis of a multiregions discrete SIR epidemic model: an optimal control approach. Int. J. Dyn. Control 1(14) (2016) Google Scholar
 Zakary, O, Rachik, M, Elmouki, I: A new analysis of infection dynamics: multiregions discrete epidemic model with an extended optimal control approach. Int. J. Dyn. Control 1(10) (2016) Google Scholar
 Zakary, O, Larrache, A, Rachik, M, Elmouki, I: Effect of awareness programs and travelblocking operations in the control of HIV/AIDS outbreaks: a multidomains SIR model. Adv. Differ. Equ. 2016(1), 1 (2016) MathSciNetView ArticleGoogle Scholar
 Zakary, O, Rachik, M, Elmouki, I: A multiregional epidemic model for controlling the spread of Ebola: awareness, treatment, and travelblocking optimal control approaches. Math. Methods Appl. Sci. (2016) Google Scholar
 Abouelkheir, I, Rachik, M, Zakary, O, Elmouk, I: A multiregions SIS discrete influenza pandemic model with a travelblocking vicinity optimal control approach on cells. Am. J. Comput. Appl. Math. 7(2), 3745 (2017). doi:10.5923/j.ajcam.20170702.02 Google Scholar
 Abouelkheir, I, El Kihal, F, Rachik, M, Zakary, O, Elmouki, I: A multiregions SIRS discrete epidemic model with a travelblocking vicinity optimal control approach on cells. Br. J. Math. Comput. Sci. 20(4), 116 (2017) View ArticleGoogle Scholar
 El Kihal, F, Rachik, M, Zakary, O, Elmouki, I: A multiregions SEIRS discrete epidemic model with a travelblocking vicinity optimal control approach on cells. Int. J. Adv. Appl. Math. Mech. 4(3), 6071 (2017) Google Scholar
 SánchezVizcaíno, JM, Mur, L, MartínezLópez, B: African swine fever: an epidemiological update. Transbound. Emerg. Dis. 59, 2735 (2012) View ArticleGoogle Scholar
 Fray, MD, Paton, DJ, Alenius, S: The effects of bovine viral diarrhoea virus on cattle reproduction in relation to disease control. Anim. Reprod. Sci. 60, 615627 (2000) View ArticleGoogle Scholar
 Thiaucourt, F, Yaya, A, Wesonga, H, Huebschle, OJB, Tulasne, JJ, Provost, A: Contagious bovine pleuropneumonia: a reassessment of the efficacy of vaccines used in Africa. Ann. N.Y. Acad. Sci. 916(1), 7180 (2000) View ArticleGoogle Scholar
 Grubman, MJ, Baxt, B: In: FootandMouth Disease; Clinical Microbiology Reviews, vol. 17, pp. 465493 (2004) Google Scholar
 Afia, N, Singh, M, Lucy, D: Numerical study of SARS epidemic model with the inclusion of diffusion in the system. Appl. Math. Comput. 229(2014), 480498 (2014) MathSciNetGoogle Scholar
 Zakary, O, Rachik, M, Elmouki, I: On the impact of awareness programs in HIV/AIDS prevention: an SIR model with optimal control. Int. J. Comput. Appl. 133(9), 16 (2016) Google Scholar
 Chunxiao, D, Tao, N, Zhu, Y: A mathematical model of Zika virus and its optimal control. In: Control Conference (CCC), 2016 35th Chinese, pp. 26422645. TCCT (2016) Google Scholar
 Wandi, D, Hendon, R, Cathey, B, Lancaster, E, Germick, R: Discrete time optimal control applied to pest control problems. Involve 7(4), 479489 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Dabbs, K: Optimal control in discrete pest control models. Thesis. trace.tennessee.edu (2010)
 Sethi, SP, Thompson, GL: What Is Optimal Control Theory? pp. 122. Springer, New York (2000) Google Scholar
 PastorSatorras, R, Castellano, C, Van Mieghem, P, Vespignani, A: Epidemic processes in complex networks. Rev. Mod. Phys. 87(3), 925979 (2015) MathSciNetView ArticleGoogle Scholar
 Yang, LX, Draief, M, Yang, X: The optimal dynamic immunization under a controlled heterogeneous nodebased SIRS model. Phys. A, Stat. Mech. Appl. 450, 403415 (2016) MathSciNetView ArticleGoogle Scholar
 Yang, LX, Yang, X, Wu, Y: The impact of patch forwarding on the prevalence of computer virus: a theoretical assessment approach. Appl. Math. Model. 43, 110125 (2017) MathSciNetView ArticleGoogle Scholar