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Treatment of Psoriasis by Interleukin-10 Through Impulsive Control Strategy: A Mathematical Study

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Mathematical Modelling, Optimization, Analytic and Numerical Solutions

Part of the book series: Industrial and Applied Mathematics ((INAMA))

Abstract

Psoriasis is characterized by anomalous growth of keratinocytes (skin cells), which occurs due to abrupt signaling within immune cells and cytokines. The most significant immune cells, T cells go through differentiation with interaction of dendritic cells (DCs) to produce Type 1 T helper cell (\(\text {Th}_{1}\)) and Type 2 T helper cell (\(\text {Th}_{2}\)) subtypes. In psoriatic progression dynamics, the inflammation effect of \(\text {Th}_{1}\) mediated cytokines (pro-inflammatory) are responsible for the abnormal growth of keratinocytes. In this measure, the effect of anti-inflammatory cytokines secreted by \(\text {Th}_{2}\) subtype partially downregulate the growth of epidermal cell. In this research article, we have constructed a five-dimensional mathematical model involving T cells, dendritic cells, \(\text {Th}_{1}\), \(\text {Th}_{2}\), and keratinocyte cell populations for better understanding the development of psoriatic lesions. Moreover, we have evaluated the role of \(\text {Th}_{1}\), \(\text {Th}_{2}\), and interplay of various cytokine networks in Psoriasis through a set of nonlinear differential equations. Our analytical study reveals the preconditions for disease persistence and also validates the stability criteria of endemic equilibrium for the disease. Furthermore, we have used one-dimensional impulsive differential equation to examine the effects of different levels of biologic (\(\text{ Interleukin }\)-10) for different dosing intervals in keratinocytes cell population. We have also examined the qualitative behavior of keratinocyte by considering two different values of the parameter corresponding to the reduction of keratinocyte due to the impact of drug (IL-10). We have also found the perfect dosing intervals of biologic (\(\text{ Interleukin }\)-10) that could tolerate the keratinocytes at the desired level. Finally, our analytical and numerical computations reveal that the use of IL-10 through impulsive way is proven better treatment compared with other trivial therapeutic policies for psoriatic patients.

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References

  1. World Health Organization, Global report on psoriasis 2016, WHO Library Cataloguing-in-Publication Data (2016)

    Google Scholar 

  2. T.J. Kindt, R.A. Goldsby, B.A. Osborne, J. Kuby, Kuby Immunology (Macmillan, London, 2007)

    Google Scholar 

  3. K. Liu, M.C. Nussenzweig, Origin and development of dendritic cells. Immunol. Rev. 234(1), 45–54 (2010)

    Article  Google Scholar 

  4. A.K. Roy, P.K. Roy, E. Grigorieva, Mathematical insights on psoriasis regulation: role of Th 1 and Th 2 cells. Math. Biosci. Eng. 15(3), 717–738 (2018)

    Google Scholar 

  5. A. Coondoo, The role of cytokines in the pathomechanism of cutaneous disorders. Indian J. Derm. 57(2), 90 (2012)

    Article  Google Scholar 

  6. A. Balato, F. Ayala, M. Schiattarella, M. Megna, N. Balato, S. Lembo, Pathogenesis of Psoriasis: The Role of Pro-inflammatory Cytokines Produced by Keratinocytes (INTECH Open Access Publisher, London, 2012)

    Google Scholar 

  7. J.H. Mao, E.F. Saunier, J.P. de Koning, M.M. McKinnon, M.N. Higgins, H.T. Yang, A. Balmain, R.J. Akhurst, Genetic variants of Tgfb1 act as context-dependent modifiers of mouse skin tumor susceptibility. Proc. Natl. Acad. Sci. 103(21), 8125–8130 (2006)

    Article  Google Scholar 

  8. A. Cavani, G. Girolomoni, Interferon-?-stimulated human keratinocytes express the genes necessary for the production of peptide-loaded MHC class II molecules. J. Investig. Derm. 110(2), 138–142 (1998)

    Article  Google Scholar 

  9. A. Chiricozzi, E. Guttman-Yassky, M. Surez-Farinas, K.E. Nograles, S. Tian, I. Cardinale, S. Chimenti, J.G. Krueger, Integrative responses to IL-17 and TNF-a in human keratinocytes account for key inflammatory pathogenic circuits in psoriasis. J. Investig. Derm. 131(3), 677–687 (2011)

    Article  Google Scholar 

  10. A. Johnston, Y. Fritz, S.M. Dawes, D. Diaconu, P.M. Al-Attar, A.M. Guzman, C.S. Chen, W. Fu, J.E. Gudjonsson, T.S. McCormick, N.L. Ward, Keratinocyte overexpression of IL-17C promotes psoriasiform skin inflammation. J. Immunol. 190(5), 2252–2262 (2013)

    Article  Google Scholar 

  11. J. Baliwag, D.H. Barnes, A. Johnston, Cytokines in psoriasis. Cytokine 73(2), 342–350 (2015)

    Article  Google Scholar 

  12. A.K. Roy, F. Al Basir, P.K. Roy, A vivid cytokines interaction model on psoriasis with the effect of impulse biologic (\(\text{TNF}\)-\(\alpha \) inhibitor) therapy. J. Theor. Biol. 474, 63–77 (2019)

    Google Scholar 

  13. P. Goodwin, S. Hamilton, L. Fry, The cell cycle in psoriasis. Br. J. Derm. 90(5), 517–524 (1974)

    Article  Google Scholar 

  14. G.D. Weinstein, J.L. McCullough, P.A. Ross, Cell kinetic basis for pathophysiology of psoriasis. J. Investig. Derm. 82(6), 623–628 (1984)

    Article  Google Scholar 

  15. L.M. Johnson-Huang, M. Surez-Farias, K.C. Pierson, J. Fuentes-Duculan, I. Cueto, T. Lentini, M. Sullivan-Whalen, P. Gilleaudeau, J.G. Krueger, A.S. Haider, M.A. Lowes, A single intradermal injection of IFN-? induces an inflammatory state in both non-lesional psoriatic and healthy skin. J. Investig. Derm. 132(4), 1177–1187 (2012)

    Article  Google Scholar 

  16. A. Mussi, C. Bonifati, M. Carducci, G. D’Agosto, F. Pimpinelli, D. D’Urso, L. D’Auria, M. Fazio, F. Ameglio, Serum TNF-alpha levels correlate with disease severity and are reduced by effective therapy in plaque-type psoriasis. J. Biol. Regul. Homeost. Agents 11(3), 115–118 (1996)

    Google Scholar 

  17. A. Mussi, C. Bonifati, M. Carducci, G. D’Agosto, F. Pimpinelli, D. D’Urso, L. D’Auria, M. Fazio, F. Ameglio, Serum TNF-alpha levels correlate with disease severity and are reduced by effective therapy in plaque-type psoriasis. J. Biol. Regul. Homeost. Agents 11(3), 115–118 (1997)

    Google Scholar 

  18. P. Nockowski, J.C. Szepietowski, M. Ziarkiewicz, E. Baran, Serum concentrations of transforming growth factor beta 1 in patients with psoriasis vulgaris. Acta Derm.Venerologica Croat.: ADC 12(1), 2–6 (2003)

    Google Scholar 

  19. K.A. Papp, The long-term efficacy and safety of new biological therapies for psoriasis. Arch. Derm. Res. 298(1), 7–15 (2006)

    Article  MathSciNet  Google Scholar 

  20. K. Asadullah, W. Sterry, H.D. Volk, Interleukin-10 therapyreview of a new approach. Pharmacol. Rev. 55(2), 241–269 (2003)

    Article  Google Scholar 

  21. K. Ghoreschi, P. Thomas, S. Breit, M. Dugas, R. Mailhammer, W. van Eden, R. van der Zee, T. Biedermann, J. Prinz, M. Mack, U. Mrowietz, Interleukin-4 therapy of psoriasis induces Th2 responses and improves human autoimmune disease. Nat. Med. 9(1), 40–46 (2003)

    Article  Google Scholar 

  22. K. Reich, M. Bruck, A. Grafe, C. Vente, C. Neumann, C. Grabe, Treatment of psoriasis with interleukin-10. J. Investig. Derm. 111(6), 1235–1236 (1998)

    Article  Google Scholar 

  23. M. Friedrich, W.D. Dcke, A. Klein, S. Philipp, H.D. Volk, W. Sterry, K. Asadullah, Immunomodulation by interleukin-10 therapy decreases the incidence of relapse and prolongs the relapse-free interval in psoriasis. J. Investig. Derm. 118(4), 672–677 (2002)

    Article  Google Scholar 

  24. I.B. McInnes, G.G. Illei, C.L. Danning, C.H. Yarboro, M. Crane, T. Kuroiwa, R. Schlimgen, E. Lee, B. Foster, D. Flemming, C. Prussin, IL-10 improves skin disease and modulates endothelial activation and leukocyte effector function in patients with psoriatic arthritis. J. Immunol. 167(7), 4075–4082 (2001)

    Article  Google Scholar 

  25. A. Datta, D.K. Kesh, P.K. Roy, Effect of CD4\(+\) T-cells and CD8\(+\) T-cells on psoriasis: a mathematical study. IMHOTEP: Afr. J. Pure Appl. Math. 3(1), 1–11 (2016)

    Google Scholar 

  26. N.J. Savill, R. Weller, J.A. Sherratt, Mathematical modelling of nitric oxide regulation of rete peg formation in psoriasis. J. Theor. Biol. 214(1), 1–16 (2002)

    Article  Google Scholar 

  27. P.K. Roy, A. Datta, Impact of perfect drug adherence on immunopathogenic mechanism for dynamical system of psoriasis. BIOMATH 2(1), 1212101 (2013)

    Article  MathSciNet  Google Scholar 

  28. E. Grigorieva, E. Khailov, P. Deignan, Optimal treatment strategies for control model of psoriasis, in 2017 Proceedings of the Conference on Control and its Applications, Society for Industrial and Applied Mathematics (2017), pp. 86–93

    Google Scholar 

  29. C.A.O. **anbing, A. Datta, F.A. Basir, P.K. Roy, Fractional-order model of the disease psoriasis: a control based mathematical approach. J. Syst. Sci. Complex. 29(6), 1565–1584 (2016)

    Article  MathSciNet  Google Scholar 

  30. P.K. Roy, A. Datta, S. Rana, The Fractional-order differential equation model of psoriatic pathogenesis: a mathematical study. Afr. Diaspora J. Math., New Series 15(2), 35–46 (2013)

    Google Scholar 

  31. P.K. Roy, A. Datta, Negative feedback control may regulate cytokines effect during growth of keratinocytes in the chronic plaque of psoriasis: a mathematical study. Int. J. Appl. Math. 25(2), 233–254 (2012)

    MathSciNet  MATH  Google Scholar 

  32. A. Datta, P.K. Roy, T-cell proliferation on immunopathogenic mechanism of psoriasis: a control based theoretical approach. Control Cybern. 42 (2013)

    Google Scholar 

  33. E.J. Routh, A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion (Macmillan and Company, London, 1877)

    Google Scholar 

  34. A. Hurwitz, On the conditions under which an equation has only roots with negative real parts. Sel. Pap. Math. Trends Control Theory 65, 273–284 (1964)

    Google Scholar 

  35. V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations (World Scientific, Singapore, 1989)

    Google Scholar 

  36. R.J. Smith, P. Cloutier, J. Harrison, A. Desforges, A mathematical model for the eradication of Guinea worm disease. In Understanding the Dynamics of Emerging and Re-Emerging Infectious Diseases Using Mathematical Models, 37/661(2), 133–156 (2012)

    Google Scholar 

  37. G. Magombedze, S. Eda, V.V. Ganusov, Competition for antigen between Th1 and Th2 responses determines the timing of the immune response switch during Mycobaterium avium subspecies paratuberulosis infection in ruminants. PLoS Comput. Biol. 10(1), e1003414 (2014)

    Article  Google Scholar 

  38. Y. Kogan, Z. Agur, M. Elishmereni, A mathematical model for the immunotherapeutic control of the Th1/Th2 imbalance in melanoma. Discr. Cont. Dyn. Syst. Ser. B 18(4), 1017–1030 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Y. Kim, S. Lee, Y.S. Kim, S. Lawler, Y.S. Gho, Y.K. Kim, H.J. Hwang, Regulation of Th1/Th2 cells in asthma development: a mathematical model. Math. Biosci. Eng. 10(4), 1095–1133 (2013)

    Article  MathSciNet  Google Scholar 

  40. R. Fernandez-Botran, V.M. Sanders, T.R. Mosmann, E.S. Vitetta, Lymphokine-mediated regulation of the proliferative response of clones of T helper 1 and T helper 2 cells. J. Exp. Med. 168(2), 543–558 (1988)

    Article  Google Scholar 

  41. P.K. Denman, D.S. McElwain, D.G. Harkin, Z. Upton, Mathematical modelling of aerosolised skin grafts incorporating keratinocyte clonal subtypes. Bull. Math. Biol. 69(1), 157–179 (2007)

    Article  MathSciNet  Google Scholar 

  42. G.D. Weinstein, J.L. McCullough, P.A. Ross, Cell kinetic basis for pathophysiology of psoriasis. J. Investig. Derm. 85(6), 579–583 (1985)

    Article  Google Scholar 

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Acknowledgements

The research work is financially supported by DST-PURSE PROJECT (Phase-II), Government of India, Department of Mathematics, Jadavpur University, Kolkata-700032, India.

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Authors and Affiliations

  1. Department of Mathematics, Centre for Mathematical Biology and Ecology, Jadavpur University, Kolkata, 700032, India

    Amit Kumar Roy & Priti Kumar Roy

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  1. Amit Kumar Roy
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Correspondence to Priti Kumar Roy .

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Editors and Affiliations

  1. Department of Mathematics, Guru Nanak Dev University, Amritsar, Punjab, India

    Pammy Manchanda

  2. Laboratory Math, J.A. Dieudonné, University Côte d’Azur, Nice, France

    René Pierre Lozi

  3. School of Basic Sciences and Research, Sharda University, Greater Noida, Uttar Pradesh, India

    Abul Hasan Siddiqi

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Roy, A.K., Roy, P.K. (2020). Treatment of Psoriasis by Interleukin-10 Through Impulsive Control Strategy: A Mathematical Study. In: Manchanda, P., Lozi, R., Siddiqi, A. (eds) Mathematical Modelling, Optimization, Analytic and Numerical Solutions. Industrial and Applied Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-0928-5_15

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