×
Roy, Priti Kumar; Datta, Abhirup; Rana, Sourav

The fractional-order differential equation model of psoriatic pathogenesis: a mathematical study. (English) Zbl 1281.34081

Afr. Diaspora J. Math. 15, No. 2, 35-46 (2013).
Summary: Psoriasis is an autoimmune prevalent chronic skin disease discriminated by T-cells mediated hyperproliferation of epidermal keratinocytes. We consider a mathematical model for psoriasis, involving a set of differential equations with T-cells, dendritic cells (DCs) and epidermal keratinocytes. We introduce here the fractional-order differential equations into the mathematical model of psoriasis with effect of cytokines release to observe the impact of it on the cell-biological system. An analytical study on the basis of stability analysis with fractional derivative is furnished. Moreover, numerical simulation through nonstandard finite difference methods is applied for solving the fractional-order differential equations to support the analytical results. We establish that the effect of cytokine network through exploring the suppressed memory, the inherited property of the system dynamics, in our mathematical model that contributes a greater impact for reducing the keratinocyte cell population, causes the disease psoriasis.

Cited in 4 Documents

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34A08 Fractional ordinary differential equations
92B05 General biology and biomathematics
92C37 Cell biology
34D20 Stability of solutions to ordinary differential equations

Keywords:

T-cells; dendritic cells (DCs); keratinocytes; fractional-order differential equations; Caputo derivative; nonstandard finite difference methods
Cite Review PDF
Full Text: Euclid

References:

[1] 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 (3) (2013), 23-42. · Zbl 1330.92058
[2] R. Sabat, S. Philipp, C. H\(\ddot{\mbox o}\)flich, S. Kreutzer, E. Wallace, K. Asadullah, H. D. Volk, W. Sterry and K. Wolk, Immunopathogenesis of psoriasis. Experimental Dermatology 16 (2007), 779-798.
[3] J. G. Krueger and A. Bowcock, Psoriasis pathophysiology: Current concepts of pathogenesis. Ann Rheum Dis. 64 (2005), 30-36.
[4] A. T. Pietrzak, A. Zalewska, G. Chodorowska, D. Krasowska, A. M. Stoma, P. Nockowski, P. Osemlak, T. Paszkowski and J. M. Rolinski, Cytokines and anticytokines in psoriasis. Clinica Chimica Acta 394 (2008), 7-21.
[5] P. K. Roy and J. Bhadra, Comparative Study of the Suppression on T-Cell and Dendritic Cells in a Mathematical Model of Psoriasis. International Journal of Evolution Equations 5 (3) (2010), 309-326. · Zbl 1260.34096
[6] Y. K. Chang, M. M. Arjunan, G. M. N\('\)Gu\(\acute{\mbox e}\)r\(\acute{\mbox e}\)kata and V. Kavitha, On global solutions to fractional functional differential equations with infinite delay in Fr\(\acute{\mbox e}\)chet spaces. Computers and Mathematics with Applications 62 (2011), 1228-1237. · Zbl 1228.34015 · doi:10.1016/j.camwa.2011.03.039
[7] S. Abbas, M. Benchohra, G. M. N\('\)Gu\(\acute{\mbox e}\)r\(\acute{\mbox e}\)kata and B. A. Slimani, Darboux problem for fractional-order discontinuous hyperbolic partial differential equations in Banach algebras. Complex Variables and Elliptic Equations: An International Journal 57 (2-4) (2012), 113-120. · Zbl 1235.26006 · doi:10.1080/17476933.2011.555542
[8] M. E. Shahed and A. Alsaedi, The Fractional SIRC Model and Influenza A. Mathematical Problems in Engineering 2011 , Article ID 480378, 9 pages, doi: 10.1155/2011/480378. · Zbl 1235.92033
[9] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, 204 , Elsevier Science , 2006. · Zbl 1092.45003
[10] I. Podlubny, Fractional Differential Equations, Academic Press, 1999. · Zbl 0924.34008
[11] L. M. Petrovic, D. T. Spasic and T. M. Atanackovic, On a mathematical model of a human root dentin. Dental Materials 21 (2005), 125-128.
[12] T. J. Anastasio, The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biological Cybernetics 72 (1994), 69-79.
[13] S. Z. Rida, A. M. A. El-Sayed and A. A. M. Arafa, Effect of Bacterial Memory Dependent Growth by Using Fractional Derivatives Reaction-Diffusion Chemotactic Model. J Stat Phys. 140 (2010), 797-811. · Zbl 1197.92007 · doi:10.1007/s10955-010-0007-8
[14] E. Ahmed and A. S. Elgazzar, On fractional order differential equations model for nonlocal epidemics. Physica A 379 (2007), 607-614.
[15] Y. Ding and H. Ye, A fractional-order differential equation model of HIV infection of CD4\({+}\)T-cells. Mathematical and Computer Modelling 50 (3-4) (2009), 386-392. · Zbl 1185.34005 · doi:10.1016/j.mcm.2009.04.019
[16] S. Ganguli, D. Huh and H. Sompolinsky, Memory traces in dynamical systems. PNAS 105 (48) (2008), 18970-18975.
[17] D. R. Burrill and P. A. Silver, Making cellular memories. Cell 140 (1) (2010), 13-18.
[18] P. K. Roy, J. Bhadra and B. Chattopadhyay, Mathematical Modeling on Immunopathogenesis in Chronic Plaque of Psoriasis: A Theoretical Study. Lecture notes in Engineering and Computer Science , WCE 1 (2010), 550-555.
[19] N. J. Savill, R. Weller and J. A. Sherratt, Mathematical Modelling of Nitric Oxide Regulation of Rete Peg Formation in Psoriasis. Journal of Theoretical Biology 214 (2002), 1-16.
[20] J. A. Sherratt, R. Weller and N. J. Savill, Modelling Blood Flow Regulation by Nitric Oxide in Psoriatic Plaques. Bulletin of Mathematical Biology 64 (2002), 623-641. · Zbl 1334.92107
[21] P. K. Roy, A. Datta and A. N. Chatterjee, Saturation Effects on Immunopathogenic Mechanism of Psoriasis: A Theoretical Approach. Acta Analysis Functionalis Applicata 13 (3) (2011), 310-318. · Zbl 1245.92035
[22] 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 (2) (2012), 233-254. · Zbl 1278.93123
[23] P. K. Roy and A. Datta, Mathematical Study on T-Cell Proliferation in the Chronic Plaque of Psoriasis. American Journal of Mathematics and Sciences 2 (1) (2013), 133-140. · Zbl 1368.92085 · doi:10.11145/j.biomath.2012.12.101
[24] P. K. Roy and A. Datta, Impact of Perfect Drug Adherence on Immunopathogenic Mechanism for Dynamical System of Psoriasis. Biomath 1 (2012), 1212101. · Zbl 1368.92085 · doi:10.11145/j.biomath.2012.12.101
[25] E. Demirci, A. Unal and N. \(\ddot{\mbox O}\)zalp, A fractional-order SEIR model with density dependent death rate. Hacettepe Journal of Mathematics and Statistics 40 (2) (2011), 287-295. · Zbl 1262.92032
[26] E. H. Elbasha, C. N. Podder and A. B. Gumel, Analyzing the dynamics of an SIRS vaccination model with waning natural and vaccine-induced immunity. Nonlinear Analysis 12 (5) (2011), 2692-2705. · Zbl 1225.37104 · doi:10.1016/j.nonrwa.2011.03.015
[27] J. Lou, L. Chen and T. Ruggeri, An Impulsive Differential Model On Post Exposure Prophylaxis To HIV-1 Exposed Individual. Journal of Biological System 17 (4) (2009), 659-683. · Zbl 1342.92099 · doi:10.1142/S0218339009002934
[28] E. Ahmed, A. M. A. El-Sayed and H. A. A. El-Saka, On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, R\(\ddot{\mbox o}\)ssler, Chua and Chen systems. Physics Letters A 358 (1) (2006), 1-4. · Zbl 1142.30303 · doi:10.1016/j.physleta.2006.04.087
[29] N. \(\ddot{\mbox O}\)zalp and E. Demi\(\ddot{\mbox o}\)rci\(\ddot{\mbox o}\), A fractional order SEIR model with vertical transmission. Mathematical and Computer Modelling 54 (1-2) (2011), 1-6. · Zbl 1225.34011 · doi:10.1016/j.mcm.2010.12.051
[30] H. Ye and Y. Ding, Nonlinear dynamics and chaos in a fractional-order HIV model. Mathematical Problems in Engineering 2009 (2009) Article ID 378614, 12 pages. · Zbl 1181.37124 · doi:10.1155/2009/378614
[31] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations , World Scientific, 2005. · Zbl 1073.65552
[32] R. Anguelov and J. M. S. Lubuma, Nonstandard finite difference method by nonlocal approximation. Mathematics and Computers in Simulation 61 (3-6) (2003), 465-475. · Zbl 1015.65034 · doi:10.1016/S0378-4754(02)00106-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.