# American Institute of Mathematical Sciences

May  2019, 24(5): 2251-2280. doi: 10.3934/dcdsb.2019094

## Chattering and its approximation in control of psoriasis treatment

 1 Department of Mathematics and Computer Sciences, Texas Woman's University, Denton, TX 76204, USA 2 Faculty of Computational Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992, Russia

* Corresponding author: Ellina Grigorieva

Received  November 2017 Revised  January 2019 Published  May 2019 Early access  March 2019

We consider a nonlinear system of differential equations describing a process of the psoriasis treatment. Its phase variables are the concentrations of T-lymphocytes, keratinocytes and dendritic cells. A scalar bounded control is introduced into this system to reflect the medication dosage. For such a control system, on a given time interval the minimization problem of the Bolza type functional is stated. Its terminal term is the concentration of keratinocytes at the terminal time, and its integral term is the product of the non-negative weighting coefficient with the total cost of the psoriasis treatment. This cost is linear in the control and proportional to the concentration of keratinocytes. For the analysis of such a problem, the Pontryagin maximum principle is used. As a result of this analysis, it is shown that if the weighting coefficient is zero, then the corresponding optimal control can contain a singular arc. We establish that it is a chattering control, and therefore does not make much sense as a type of a medical treatment. If the weighting coefficient is positive, then the corresponding optimal control is bang-bang, and it can be presented as a type of psoriasis treatment. In addition, when this coefficient tends to zero, such optimal controls can be considered as chattering approximations. Therefore, the convergence of these optimal controls, the corresponding optimal solutions of the original system, and the minimum values of the functional are studied. The obtained theoretical results are illustrated by numerical calculations and the corresponding conclusions are made.

Citation: Ellina Grigorieva, Evgenii Khailov. Chattering and its approximation in control of psoriasis treatment. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2251-2280. doi: 10.3934/dcdsb.2019094
##### References:
 [1] M. H. A. Biswas, L. T. Paiva and M. R. de Pinho, A SEIR model for control of infectious diseases with constraints, Math. Biosci. Eng., 11 (2014), 761-784.  doi: 10.3934/mbe.2014.11.761. [2] F. Bonnans, P. Martinon, D. Giorgi, V. Grélard, S. Maindrault, O. Tissot and J. Liu, BOCOP 2.0.5 - User Guide, February 8, 2017, URL http://bocop.org [3] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Springer, Berlin-Heidelberg-New York, 2003. [4] X. Cao, A. Datta, F. Al Basir and P. K. Roy, Fractional-order model of the disease psoriasis: A control based mathematical approach, J. Syst. Sci. Complex., 29 (2016), 1565-1584.  doi: 10.1007/s11424-016-5198-x. [5] C. Castilho, Optimal control of an epidemic through educational compaigns, Electron. J. Differ. Eq., 2006 (2006), 1-11. [6] B. Chattopadhyay and N. Hui, Immunopathogenesis in psoriasis throuth a density-type mathematical model, WSEAS Transactions on Mathematics, 11 (2012), 440-450. [7] A. Datta and P. K. Roy, T-cell proliferation on immunopathogenic mechanism of psoriasis: a control based theoretical approach, Control Cybern., 42 (2013), 365-386. [8] A. L. Donchev, Perturbations, Approximations, and Sensitivity Analysis of Optimal Control Systems, Lecture Notes in Control and Information Sciences, vol. 52, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0043612. [9] R. V. Gamkrelidze, Sliding modes in optimal control theory, P. Steklov Inst. Math., 169 (1986), 180-193. [10] A. Gandolfi, M. Iannelli and G. Marinoschi, An age-structured model of epidermis growth, J. Math. Biol., 62 (2011), 111-141.  doi: 10.1007/s00285-010-0330-3. [11] E. V. Grigorieva and E. N. Khailov, Optimal intervention strategies for a SEIR control model of Ebola epidemics, Mathematics, 3 (2015), 961-983. [12] E. V. Grigorieva and E. N. Khailov, Estimating the number of switchings of the optimal intervention strategies for SEIR control model of Ebola epidemics, Pure and Applied Functional Analysis, 1 (2016), 541-572. [13] E.V. Grigorieva, P.B. Deignan and E.N. Khailov, Optimal control problem for a SEIR type model of Ebola epidemics, Revista de Matemática: Teoria y Aplicaciones, 24 (2017), 79-96.  doi: 10.15517/rmta.v24i1.27771. [14] E. Grigorieva, P. Deignan and E. Khailov, Optimal treatment strategies for control model of psoriasis, in Proceedings of the SIAM Conference on Control and its Applications (CT17), Pittsburgh, Pensylvania, USA, July 10-12, (2017), 86-93. doi: 10.1137/1.9781611975024.12. [15] J. E. Gudjonsson, A. Johnston, H. Sigmundsdottir and H. Valdimarsson, Immunopathogenic mechanisms in psoriasis, Clin. Exp. Immunol., 135 (2004), 1-8.  doi: 10.1111/j.1365-2249.2004.02310.x. [16] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1964. [17] H. R. Joshi, S. Lenhart, M. Y. Li and L. Wang, Optimal control methods applied to disease models, in AMS Volume on Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges, AMS Contemporary Mathematics Series, 410 (2006), 187-207. doi: 10.1090/conm/410/07728. [18] M. V. Laptev and N. K. Nikulin, Numerical modeling of mutial synchronization of auto-oscillations of epidermal proliferative activity in lesions of psoriasis skin, Biophysics, 54 (2009), 519-524. [19] U. Ledzewicz and H. Schättler, On optimal singular controls for a general SIR-model with vaccination and treatment, Discrete Cont. Dyn.-S, 2 (2011), 981-990. [20] E. B. Lee and L. Marcus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, 1967. [21] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, CRC Press, Taylor & Francis Group, London, 2007. [22] M. A. Lowes, M. Suárez-Fariñas and J. G. Krueger, Immunology of psoriasis, Annu. Rev. Immunol., 32 (2014), 227-255.  doi: 10.1146/annurev-immunol-032713-120225. [23] L. A. Lusternik and V. J. Sobolev, Elements of Functional Analysis, Gordon and Breach Publishers, Inc., 1961. [24] S. L. Mehlis and K. B. Gordon, The immunology of psoriasis and biological immunotherapy, J. Am. Acad. Dermatol., 49 (2003), 44-50. [25] R. M. Neilan and S. Lenhart, An introduction to optimal control with an application in disease modeling, in DIMACS Series in Discrete Mathematics, 75 (2010), 67-81. [26] G. Niels and N. Karsten, Simulating psoriasis by altering transit amplifying cells, Bioinformatics, 23 (2007), 1309-1312. [27] M. S. Nikol'skii, On the convergence of optimal controls in some optimization problems, Moscow University Computational Mathematics and Cybernetics, 1 (2004), 24-31. [28] H. B. Oza, R. Pandey, D. Roper, Y. Al-Nuaimi, S. K. Spurgeon and M. Goodfellow, Modelling and finite-time stability analysis of psoriasis pathogenesis, Int. J. Control, 90 (2017), 1664-1677.  doi: 10.1080/00207179.2016.1217566. [29] M. R. de Pinho, I. Kornienko and H. Maurer, Optimal control of a SEIR model with mixed constraints and $L_{1}$ cost, in Controllo'2014 - Proceedings of the 11th Portuguese conference on automatic control, Lecture Notes in Electrical Engineering, vol. 321, Springer, Switzerland, 2015, 135-145. [30] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, Mathematical Theory of Optimal Processes, The Macmillan Co., New York, 1964. [31] P. K. Roy, J. Bradra and B. Chattopadhyay, Mathematical modeling on immunopathogenesis in chronic plaque of psoriasis: A theoretical study, Lecture Notes in Engineering and Computer Science, 1 (2010), 550-555. [32] P. K. Roy and A. Datta, Negative feedback control may regulate cytokines effect during growth of keratinocytes in the chronic plaque of psoriasis: A mathematical study, International Journal of Applied Mathematics, 25 (2012), 233-254. [33] P. K. Roy and A. Datta, Impact of cytokine release in psoriasis: A control based mathematical approach, Journal of Nonlinear Evolution Equations and Applications, 2013 (2013), 23-42. [34] P. K. Roy and A. Datta, Impact of perfect drug adherence on immunopathogenic mechanism for dynamical system of psoriasis, Biomath, 2 (2013), article ID 121201, 6pp. doi: 10.11145/j.biomath.2012.12.101. [35] N. J. Savill, R. Weller and J. A. Sherratt, Mathematical modelling of nitric oxide regulation of Rete Peg formation in psoriasis, J. Theor. Biol., 214 (2002), 1-16.  doi: 10.1006/jtbi.2001.2400. [36] N. J. Savill, Mathematical models of hierarchically structured cell populations under equilibrium with application to the epidermis, Cell Proliferat., 36 (2003), 1-26.  doi: 10.1046/j.1365-2184.2003.00257.x. [37] H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer, New York-Heidelberg-Dordrecht-London, 2012. doi: 10.1007/978-1-4614-3834-2. [38] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies. An Application of Geometric Methods, Springer, New York-Heidelberg-Dordrecht-London, 2015. doi: 10.1007/978-1-4939-2972-6. [39] J. A. Sherratt, R. Weller and N. J. Savill, Modelling blood flow regulation by nitric oxide in psoriatic plaques, J. Math. Biol., 64 (2002), 623-641.  doi: 10.1006/bulm.2001.0271. [40] N.V. Valeyev, C. Hundhausen, Y. Umezawa, N.V. Kotov, G. Williams, A. Clop, C. Ainali, G. Ouzounis, S. Tsoka and F.O. Nestle, A systems model for immune cell interactions unravels the mechanism of inflammation in human skin, PLoS Computational Biology, 6 (2010), e1001024, 1-22. doi: 10.1371/journal.pcbi.1001024. [41] F. P. Vasil'ev, Optimization Methods, Factorial Press, Moscow, 2002. [42] A. Visintin, Strong convergence results related to strict convexity, Commun. Part. Diff. Eq., 9 (1984), 439-466.  doi: 10.1080/03605308408820337. [43] M. I. Zelikin and V. F. Borisov, Theory of Chattering Control: With Applications to Astronautics, Robotics, Economics and Engineering, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-2702-1. [44] M. I. Zelikin and L. F. Zelikina, The deviation of a functional from its optimal value under chattering decreases exponentially as the number of switchings grows, Diff. Equat., 35 (1999), 1489-1493. [45] M. I. Zelikin and L. F. Zelikina, Asymptotics of the deviation of a functional from its optimal value when chattering is replaced by a suboptimal regime, Russ. Math. Surv., 54 (1999), 662-664.  doi: 10.1070/rm1999v054n03ABEH000174. [46] H. Zhang, W. Hou, L. Henrot, S. Schnebert, M. Dumas, C. Heusèle and J. Yang, Modelling epidermis homoeostasis and psoriasis pathogenesis, Journal of The Royal Society Interface, 12 (2015), 1-22.  doi: 10.1098/rsif.2014.1071. [47] J. Zhu, E. Trélat and M. Cerf, Planar titling maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering, Discrete Cont. Dyn.-B, 21 (2016), 1347-1388.  doi: 10.3934/dcdsb.2016.21.1347.

show all references

##### References:
 [1] M. H. A. Biswas, L. T. Paiva and M. R. de Pinho, A SEIR model for control of infectious diseases with constraints, Math. Biosci. Eng., 11 (2014), 761-784.  doi: 10.3934/mbe.2014.11.761. [2] F. Bonnans, P. Martinon, D. Giorgi, V. Grélard, S. Maindrault, O. Tissot and J. Liu, BOCOP 2.0.5 - User Guide, February 8, 2017, URL http://bocop.org [3] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Springer, Berlin-Heidelberg-New York, 2003. [4] X. Cao, A. Datta, F. Al Basir and P. K. Roy, Fractional-order model of the disease psoriasis: A control based mathematical approach, J. Syst. Sci. Complex., 29 (2016), 1565-1584.  doi: 10.1007/s11424-016-5198-x. [5] C. Castilho, Optimal control of an epidemic through educational compaigns, Electron. J. Differ. Eq., 2006 (2006), 1-11. [6] B. Chattopadhyay and N. Hui, Immunopathogenesis in psoriasis throuth a density-type mathematical model, WSEAS Transactions on Mathematics, 11 (2012), 440-450. [7] A. Datta and P. K. Roy, T-cell proliferation on immunopathogenic mechanism of psoriasis: a control based theoretical approach, Control Cybern., 42 (2013), 365-386. [8] A. L. Donchev, Perturbations, Approximations, and Sensitivity Analysis of Optimal Control Systems, Lecture Notes in Control and Information Sciences, vol. 52, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0043612. [9] R. V. Gamkrelidze, Sliding modes in optimal control theory, P. Steklov Inst. Math., 169 (1986), 180-193. [10] A. Gandolfi, M. Iannelli and G. Marinoschi, An age-structured model of epidermis growth, J. Math. Biol., 62 (2011), 111-141.  doi: 10.1007/s00285-010-0330-3. [11] E. V. Grigorieva and E. N. Khailov, Optimal intervention strategies for a SEIR control model of Ebola epidemics, Mathematics, 3 (2015), 961-983. [12] E. V. Grigorieva and E. N. Khailov, Estimating the number of switchings of the optimal intervention strategies for SEIR control model of Ebola epidemics, Pure and Applied Functional Analysis, 1 (2016), 541-572. [13] E.V. Grigorieva, P.B. Deignan and E.N. Khailov, Optimal control problem for a SEIR type model of Ebola epidemics, Revista de Matemática: Teoria y Aplicaciones, 24 (2017), 79-96.  doi: 10.15517/rmta.v24i1.27771. [14] E. Grigorieva, P. Deignan and E. Khailov, Optimal treatment strategies for control model of psoriasis, in Proceedings of the SIAM Conference on Control and its Applications (CT17), Pittsburgh, Pensylvania, USA, July 10-12, (2017), 86-93. doi: 10.1137/1.9781611975024.12. [15] J. E. Gudjonsson, A. Johnston, H. Sigmundsdottir and H. Valdimarsson, Immunopathogenic mechanisms in psoriasis, Clin. Exp. Immunol., 135 (2004), 1-8.  doi: 10.1111/j.1365-2249.2004.02310.x. [16] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1964. [17] H. R. Joshi, S. Lenhart, M. Y. Li and L. Wang, Optimal control methods applied to disease models, in AMS Volume on Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges, AMS Contemporary Mathematics Series, 410 (2006), 187-207. doi: 10.1090/conm/410/07728. [18] M. V. Laptev and N. K. Nikulin, Numerical modeling of mutial synchronization of auto-oscillations of epidermal proliferative activity in lesions of psoriasis skin, Biophysics, 54 (2009), 519-524. [19] U. Ledzewicz and H. Schättler, On optimal singular controls for a general SIR-model with vaccination and treatment, Discrete Cont. Dyn.-S, 2 (2011), 981-990. [20] E. B. Lee and L. Marcus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, 1967. [21] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, CRC Press, Taylor & Francis Group, London, 2007. [22] M. A. Lowes, M. Suárez-Fariñas and J. G. Krueger, Immunology of psoriasis, Annu. Rev. Immunol., 32 (2014), 227-255.  doi: 10.1146/annurev-immunol-032713-120225. [23] L. A. Lusternik and V. J. Sobolev, Elements of Functional Analysis, Gordon and Breach Publishers, Inc., 1961. [24] S. L. Mehlis and K. B. Gordon, The immunology of psoriasis and biological immunotherapy, J. Am. Acad. Dermatol., 49 (2003), 44-50. [25] R. M. Neilan and S. Lenhart, An introduction to optimal control with an application in disease modeling, in DIMACS Series in Discrete Mathematics, 75 (2010), 67-81. [26] G. Niels and N. Karsten, Simulating psoriasis by altering transit amplifying cells, Bioinformatics, 23 (2007), 1309-1312. [27] M. S. Nikol'skii, On the convergence of optimal controls in some optimization problems, Moscow University Computational Mathematics and Cybernetics, 1 (2004), 24-31. [28] H. B. Oza, R. Pandey, D. Roper, Y. Al-Nuaimi, S. K. Spurgeon and M. Goodfellow, Modelling and finite-time stability analysis of psoriasis pathogenesis, Int. J. Control, 90 (2017), 1664-1677.  doi: 10.1080/00207179.2016.1217566. [29] M. R. de Pinho, I. Kornienko and H. Maurer, Optimal control of a SEIR model with mixed constraints and $L_{1}$ cost, in Controllo'2014 - Proceedings of the 11th Portuguese conference on automatic control, Lecture Notes in Electrical Engineering, vol. 321, Springer, Switzerland, 2015, 135-145. [30] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, Mathematical Theory of Optimal Processes, The Macmillan Co., New York, 1964. [31] P. K. Roy, J. Bradra and B. Chattopadhyay, Mathematical modeling on immunopathogenesis in chronic plaque of psoriasis: A theoretical study, Lecture Notes in Engineering and Computer Science, 1 (2010), 550-555. [32] P. K. Roy and A. Datta, Negative feedback control may regulate cytokines effect during growth of keratinocytes in the chronic plaque of psoriasis: A mathematical study, International Journal of Applied Mathematics, 25 (2012), 233-254. [33] P. K. Roy and A. Datta, Impact of cytokine release in psoriasis: A control based mathematical approach, Journal of Nonlinear Evolution Equations and Applications, 2013 (2013), 23-42. [34] P. K. Roy and A. Datta, Impact of perfect drug adherence on immunopathogenic mechanism for dynamical system of psoriasis, Biomath, 2 (2013), article ID 121201, 6pp. doi: 10.11145/j.biomath.2012.12.101. [35] N. J. Savill, R. Weller and J. A. Sherratt, Mathematical modelling of nitric oxide regulation of Rete Peg formation in psoriasis, J. Theor. Biol., 214 (2002), 1-16.  doi: 10.1006/jtbi.2001.2400. [36] N. J. Savill, Mathematical models of hierarchically structured cell populations under equilibrium with application to the epidermis, Cell Proliferat., 36 (2003), 1-26.  doi: 10.1046/j.1365-2184.2003.00257.x. [37] H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer, New York-Heidelberg-Dordrecht-London, 2012. doi: 10.1007/978-1-4614-3834-2. [38] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies. An Application of Geometric Methods, Springer, New York-Heidelberg-Dordrecht-London, 2015. doi: 10.1007/978-1-4939-2972-6. [39] J. A. Sherratt, R. Weller and N. J. Savill, Modelling blood flow regulation by nitric oxide in psoriatic plaques, J. Math. Biol., 64 (2002), 623-641.  doi: 10.1006/bulm.2001.0271. [40] N.V. Valeyev, C. Hundhausen, Y. Umezawa, N.V. Kotov, G. Williams, A. Clop, C. Ainali, G. Ouzounis, S. Tsoka and F.O. Nestle, A systems model for immune cell interactions unravels the mechanism of inflammation in human skin, PLoS Computational Biology, 6 (2010), e1001024, 1-22. doi: 10.1371/journal.pcbi.1001024. [41] F. P. Vasil'ev, Optimization Methods, Factorial Press, Moscow, 2002. [42] A. Visintin, Strong convergence results related to strict convexity, Commun. Part. Diff. Eq., 9 (1984), 439-466.  doi: 10.1080/03605308408820337. [43] M. I. Zelikin and V. F. Borisov, Theory of Chattering Control: With Applications to Astronautics, Robotics, Economics and Engineering, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-2702-1. [44] M. I. Zelikin and L. F. Zelikina, The deviation of a functional from its optimal value under chattering decreases exponentially as the number of switchings grows, Diff. Equat., 35 (1999), 1489-1493. [45] M. I. Zelikin and L. F. Zelikina, Asymptotics of the deviation of a functional from its optimal value when chattering is replaced by a suboptimal regime, Russ. Math. Surv., 54 (1999), 662-664.  doi: 10.1070/rm1999v054n03ABEH000174. [46] H. Zhang, W. Hou, L. Henrot, S. Schnebert, M. Dumas, C. Heusèle and J. Yang, Modelling epidermis homoeostasis and psoriasis pathogenesis, Journal of The Royal Society Interface, 12 (2015), 1-22.  doi: 10.1098/rsif.2014.1071. [47] J. Zhu, E. Trélat and M. Cerf, Planar titling maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering, Discrete Cont. Dyn.-B, 21 (2016), 1347-1388.  doi: 10.3934/dcdsb.2016.21.1347.
Optimal solutions and optimal control for $\xi = 0$ (Case B): upper row: $l_{0}^{*}(t)$, $k_{0}^{*}(t)$; lower row: $m_{0}^{*}(t)$, $u_{0}^{*}(t)$
Optimal solutions and optimal control for $\xi = 0$ (Case B): upper row: $l_{0}^{*}(t)$, $k_{0}^{*}(t)$; lower row: $m_{0}^{*}(t)$, $u_{0}^{*}(t)$
Optimal solutions and optimal control for $\xi = 0.001$: upper row: $l_{\xi}^{*}(t)$, $k_{\xi}^{*}(t)$; lower row: $m_{\xi}^{*}(t)$, $u_{\xi}^{*}(t)$
Optimal solutions and optimal control for $\xi = 0.000001$: upper row: $l_{\xi}^{*}(t)$, $k_{\xi}^{*}(t)$; lower row: $m_{\xi}^{*}(t)$, $u_{\xi}^{*}(t)$
Minimum values $J_{\xi}^{*}$ of the functional $J_{\xi}(u)$ and the corresponding numbers of iterations $M_{\xi}$
 $\xi = 10.0$ $\xi = 1.0$ $\xi = 0.1$ $\xi = 0.01$ $J_{\xi}^{*} = 3.030960$ $J_{\xi}^{*} = 3.030960$ $J_{\xi}^{*} = 3.016830$ $J_{\xi}^{*} = 2.905979$ $M_{\xi} = 69$ $M_{\xi} = 72$ $M_{\xi} = 80$ $M_{\xi} = 84$ $\xi = 0.001$ $\xi = 0.0001$ $\xi = 0.000001$ $J_{\xi}^{*} = 2.867417$ $J_{\xi}^{*} = 2.861134$ $J_{\xi}^{*} = 2.860158$ $M_{\xi} = 128$ $M_{\xi} = 206$ $M_{\xi} = 286$
 $\xi = 10.0$ $\xi = 1.0$ $\xi = 0.1$ $\xi = 0.01$ $J_{\xi}^{*} = 3.030960$ $J_{\xi}^{*} = 3.030960$ $J_{\xi}^{*} = 3.016830$ $J_{\xi}^{*} = 2.905979$ $M_{\xi} = 69$ $M_{\xi} = 72$ $M_{\xi} = 80$ $M_{\xi} = 84$ $\xi = 0.001$ $\xi = 0.0001$ $\xi = 0.000001$ $J_{\xi}^{*} = 2.867417$ $J_{\xi}^{*} = 2.861134$ $J_{\xi}^{*} = 2.860158$ $M_{\xi} = 128$ $M_{\xi} = 206$ $M_{\xi} = 286$
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