# American Institute of Mathematical Sciences

doi: 10.3934/naco.2022013
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## Optimization method in counter terrorism: Min-Max zero-sum differential game approach

 1 Basic Science Department, Faculty of Computers and informatics, Suez Canal university, Ismailia, Egypt 2 Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

*Corresponding author: heba.tallah@ci.suez.edu.eg

Received  May 2021 Revised  April 2022 Early access May 2022

One of the most critical problems facing governments at present is terrorism. Most recent studies are striving to find an optimal solution to this problem that threatens the security and stability of people. To combat terrorism, government uses various means, such as improving education quality, providing labor opportunities, seeking social justice, creating religious awareness, and building security arrangements. This study aims to evaluate the optimum strategy for both government and terrorist organization using a min-max zero-sum differential game approach. In addition, it analyses the dynamics of the government's activities separately and explain the impact of the government's counter-terror measures on the activity and strength of international terror organization.

Citation: Abd El-Monem A. Megahed, Ebrahim A. Youness, Hebatallah K. Arafat. Optimization method in counter terrorism: Min-Max zero-sum differential game approach. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022013
##### References:
 [1] E. Ahmed, A. Elgazzar and A. Hegazi, On complex adaptive systems and terrorism, Physics Letters A, 337 (2005), 127-129. [2] J. Caulkins, G. Feichtinger, D. Grass and G. Tragler, Optimal control of terrorism and global reputation: A case study with novel threshold behavior, Operations Research Letters, 37 (2009), 387-391.  doi: 10.1016/j.orl.2009.07.003. [3] J. Caulkins, D. Grass, G. Feichtinger and G. Tragler, Optimizing counter-terror operations: Should one fight fire with "fire" or "water"?, Computers & Operations Research, 35 (2008), 1874-1885. [4] H.-J. Chu, J.-G. Hsieh, K.-H. Hsia and L.-W. Chen, Fuzzy differential game of guarding a movable territory, Information Sciences, 91 (1996), 113-131.  doi: 10.1016/0020-0255(95)00299-5. [5] S. Hegazy, A. Megahed, E. Youness and A. Elbann, Min-max zero-sum two persons fuzzy continuous differential games, International Journal of Applied Mathematics, 21 (2008), 1-16. [6] K.-H. Hsia and J.-G. Hsieh, A first approach to fuzzy differential game problem: guarding a territory, Fuzzy Sets and Systems, 55 (1993), 157-167.  doi: 10.1016/0165-0114(93)90128-5. [7] A. Megahed, A differential game related to terrorism: min-max zero-sum two persons differential game, Neural Computing and Applications, 30 (2016), 865-870.  doi: 10.1016/j.joems.2017.03.007. [8] A. Megahed, The development of a differential game related to terrorism: Min-max differential game, Journal of the Egyptian Mathematical Society, 25 (2017), 308-312.  doi: 10.1016/j.joems.2017.03.007. [9] A. Megahed, The stackelberg differential game for counter-terrorism, Quality & Quantity, 53 (2018), 207-220. [10] A. Megahed, A differential game related to terrorism: Stackelberg differential game of e-differentiable and e-convex function, European Journal of Pure and Applied Mathematics, 12 (2019), 654-667.  doi: 10.29020/nybg.ejpam.v12i2.3375. [11] A. Megahed and S. Hegazy, Min-max zero sum two persons continuous differential game with fuzzy controls, Asian Journal of Current Engineering and Maths, 2 (2013), 86-98. [12] A. Novak, G. Feichtinger and G. Leitmann, A differential game related to terrorism: Nash and stackelberg strategies, Journal of Optimization Theory and Applications, 144 (2010), 533-555.  doi: 10.1007/s10957-009-9643-z. [13] A. Roy and J. A. Paul, Terrorism deterrence in a two country framework: strategic interactions between r & d, defense and pre-emption, Annals of Operations Research, 211 (2013), 399-432.  doi: 10.1007/s10479-013-1431-3. [14] J. Wang and P. Wang, Counterterror measures and economic growth: A differential game, Operations Research Letters, 41 (2013), 285-289.  doi: 10.1016/j.orl.2013.02.008. [15] E. Youness, J. Hughes and N. El-Kholy, Parametric nash coalitive differential games, Mathematical and Computer Modelling, 26 (1997), 97-105.  doi: 10.1016/S0895-7177(97)00125-8. [16] E. Youness and A. Megahed, A study on fuzzy differential game, Le Matematiche, 56 (2001), 97-107. [17] E. Youness and A. Megahed, A study on large scale continuous differential games, Bull. Culcutta. Math. Soc., 94 (2002), 359-368.

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##### References:
 [1] E. Ahmed, A. Elgazzar and A. Hegazi, On complex adaptive systems and terrorism, Physics Letters A, 337 (2005), 127-129. [2] J. Caulkins, G. Feichtinger, D. Grass and G. Tragler, Optimal control of terrorism and global reputation: A case study with novel threshold behavior, Operations Research Letters, 37 (2009), 387-391.  doi: 10.1016/j.orl.2009.07.003. [3] J. Caulkins, D. Grass, G. Feichtinger and G. Tragler, Optimizing counter-terror operations: Should one fight fire with "fire" or "water"?, Computers & Operations Research, 35 (2008), 1874-1885. [4] H.-J. Chu, J.-G. Hsieh, K.-H. Hsia and L.-W. Chen, Fuzzy differential game of guarding a movable territory, Information Sciences, 91 (1996), 113-131.  doi: 10.1016/0020-0255(95)00299-5. [5] S. Hegazy, A. Megahed, E. Youness and A. Elbann, Min-max zero-sum two persons fuzzy continuous differential games, International Journal of Applied Mathematics, 21 (2008), 1-16. [6] K.-H. Hsia and J.-G. Hsieh, A first approach to fuzzy differential game problem: guarding a territory, Fuzzy Sets and Systems, 55 (1993), 157-167.  doi: 10.1016/0165-0114(93)90128-5. [7] A. Megahed, A differential game related to terrorism: min-max zero-sum two persons differential game, Neural Computing and Applications, 30 (2016), 865-870.  doi: 10.1016/j.joems.2017.03.007. [8] A. Megahed, The development of a differential game related to terrorism: Min-max differential game, Journal of the Egyptian Mathematical Society, 25 (2017), 308-312.  doi: 10.1016/j.joems.2017.03.007. [9] A. Megahed, The stackelberg differential game for counter-terrorism, Quality & Quantity, 53 (2018), 207-220. [10] A. Megahed, A differential game related to terrorism: Stackelberg differential game of e-differentiable and e-convex function, European Journal of Pure and Applied Mathematics, 12 (2019), 654-667.  doi: 10.29020/nybg.ejpam.v12i2.3375. [11] A. Megahed and S. Hegazy, Min-max zero sum two persons continuous differential game with fuzzy controls, Asian Journal of Current Engineering and Maths, 2 (2013), 86-98. [12] A. Novak, G. Feichtinger and G. Leitmann, A differential game related to terrorism: Nash and stackelberg strategies, Journal of Optimization Theory and Applications, 144 (2010), 533-555.  doi: 10.1007/s10957-009-9643-z. [13] A. Roy and J. A. Paul, Terrorism deterrence in a two country framework: strategic interactions between r & d, defense and pre-emption, Annals of Operations Research, 211 (2013), 399-432.  doi: 10.1007/s10479-013-1431-3. [14] J. Wang and P. Wang, Counterterror measures and economic growth: A differential game, Operations Research Letters, 41 (2013), 285-289.  doi: 10.1016/j.orl.2013.02.008. [15] E. Youness, J. Hughes and N. El-Kholy, Parametric nash coalitive differential games, Mathematical and Computer Modelling, 26 (1997), 97-105.  doi: 10.1016/S0895-7177(97)00125-8. [16] E. Youness and A. Megahed, A study on fuzzy differential game, Le Matematiche, 56 (2001), 97-107. [17] E. Youness and A. Megahed, A study on large scale continuous differential games, Bull. Culcutta. Math. Soc., 94 (2002), 359-368.
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