# American Institute of Mathematical Sciences

June 2018, 15(3): 717-738. doi: 10.3934/mbe.2018032

## Mathematical insights on psoriasis regulation: Role of Th1 and Th2 cells

 1 Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata-700032, India 2 Department of Mathematics and Computer Sciences, Texas Womans University, Denton, TX 76204, USA

Received  April 10, 2017 Revised  July 06, 2017 Published  December 2017

Fund Project: The research is supported by Department of Mathematics, Jadavpur University, PURSE-DST, Government of India

Psoriasis is an autoimmune disorder, characterized by hyper-proli-feration of Keratinocytes for the abnormal activation of T Cells, Dendritic Cells (DCs) and cytokine signaling. Interaction of DCs and T Cells enable T Cell to differentiate into Type 1 (Th1), Type 2 (Th2) helper T Cell depending on cytokine release. Hyper-proliferation of Keratinocytes may occur due to over expression of pro-inflammatory cytokines secreted by Th1-Cells viz. Interferon gamma ($\mbox{IFN}-{γ}$), Transforming growth factor beta ($\mbox{TGF}-β$) and Tumor necrosis factor alpha ($\mbox{TNF}-α$) etc. Deregulation of epidermal happens due to signaling of anti-inflammatory cytokines like Interleukin 10 ($\mbox{IL}-{10}$), Interleukin 4 ($\mbox{IL}-{4}$) etc., released by Th2-Cells. In this article, we have constructed a set of nonlinear differential equations involving the above cell population for better understanding the impact of cytokines on Psoriasis. System is analyzed introducing therapeutic agent (Biologic / $\mbox{IL}-{10}$) for reducing the hyper-proliferation of Keratinocytes. Effect of Biologic is used as a surrogate of control parameter to reduce the psoriatic lesions. We also studied its effect both in continuous and impulsive dosing method. Our study reveals that impulsive dosing is more applicable compare with continuous dosing to prevent Psoriasis.

Citation: Amit Kumar Roy, Priti Kumar Roy, Ellina Grigorieva. Mathematical insights on psoriasis regulation: Role of Th1 and Th2 cells. Mathematical Biosciences & Engineering, 2018, 15 (3) : 717-738. doi: 10.3934/mbe.2018032
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##### References:
Schematic of the interactions between the components of the model.
Time series plot of cell population and contour plot (A) Qualitative nature of all cells (T Cell, Dendritic Cell, $\mbox{Th}_{1}$-Cell, $\mbox{Th}_{2}$-Cell and Keratinocyte) during the disease progression. (B) Contour plot of $R_E$ as a function of $\mu_5$ and $c$.
Stability analysis using different cell population and finding the endemic equilibrium of system dynamics (A) Considering three cells Keratinocyte, T Cell, Dendritic Cell. (B) Considering three cells $\mbox{Th}_{1}$-Cell, $\mbox{Th}_{2}$-Cell and Keratinocyte.
Qualitative behaviour of $\mbox{Th}_{1}$, $\mbox{Th}_{2}$ and Keratinocyte with and with out control. Dotted line represents the cell dynamics without control and solid line represents the cell dynamics with control.
Control parameter with respect to time for better impact on psoriatic plaque.
Comparative system behaviour of continuous therapy (blue line) and impulse therapy (green line) using Biologic ($\mbox{IL}-{10}$).
System behaviour with perfect dose of Biologic ($\mbox{IL}-{10}$) in impulsive way. Different cells population are denoted by different colour line i.e Keratinocyte (red), $\mbox{Th}_{1}$-Cells (green) and $\mbox{Th}_{2}$-Cells (blue).
Parameters value using for numerical simulation.
 Parameter Assigned Value Range References $a$ 12 $\mbox{mm}^{-3}\mbox Day^{-1}$ 9 -15 $\mbox{mm}^{-3}\mbox Day^{-1}$ [33,34,54] $b$ 14 $\mbox{mm}^{-3}\mbox Day^{-1}$ 12 -14 $\mbox{mm}^{-3}\mbox Day^{-1}$ [33,35,55] $\delta_1$ 0.07 $\mbox Day^{-1}$ 0.005 -0.15 $\mbox Day^{-1}$ [34,35,56] $\delta_2$ 0.08 $\mbox Day^{-1}$ 0.00004 -0.4 $\mbox Day^{-1}$ [34,35,55] $\mu_1$ 0.02 $\mbox Day^{-1}$ 0.007 -0.1 $\mbox Day^{-1}$ [33,34,54] $\eta_1$ 0.05 $\mbox Day^{-1}$ Estimated [57] $\eta_2$ 0.0025 $\mbox Day^{-1}$ Estimated [57] $\alpha$ 0.002 $\mbox Day^{-1}$ - Assumed $\mu_2$ 0.05 $\mbox Day^{-1}$ 0.002-0.05 $\mbox Day^{-1}$ [33,35] $\beta_1$ 0.02 $\mbox Day^{-1}$ Estimated [15,58] $\beta_2$ 0.0001 $\mbox Day^{-1}$ Estimated [15,58] $\mu_3$ 0.12 $\mbox Day^{-1}$ 0.012 -0.12 $\mbox Day^{-1}$ [37] $\mu_4$ 0.24 $\mbox Day^{-1}$ 0.24 $\mbox Day^{-1}$ [58] $\gamma_1$ 0.51 $\mbox Day^{-1}$ Estimated [15,58] $\gamma_2$ 0.035 $\mbox Day^{-1}$ Estimated [15,58] $\xi_1$ 0.90 $\mbox Day^{-1}$ - Assumed $\xi_2$ 0.15 $\mbox Day^{-1}$ - Assumed $\mu_5$ 0.65 $\mbox Day^{-1}$ 0.04-0.9 $\mbox Day^{-1}$ [33] $c$ 0.50 $\mbox Day^{-1}$ Estimated [22,60]
 Parameter Assigned Value Range References $a$ 12 $\mbox{mm}^{-3}\mbox Day^{-1}$ 9 -15 $\mbox{mm}^{-3}\mbox Day^{-1}$ [33,34,54] $b$ 14 $\mbox{mm}^{-3}\mbox Day^{-1}$ 12 -14 $\mbox{mm}^{-3}\mbox Day^{-1}$ [33,35,55] $\delta_1$ 0.07 $\mbox Day^{-1}$ 0.005 -0.15 $\mbox Day^{-1}$ [34,35,56] $\delta_2$ 0.08 $\mbox Day^{-1}$ 0.00004 -0.4 $\mbox Day^{-1}$ [34,35,55] $\mu_1$ 0.02 $\mbox Day^{-1}$ 0.007 -0.1 $\mbox Day^{-1}$ [33,34,54] $\eta_1$ 0.05 $\mbox Day^{-1}$ Estimated [57] $\eta_2$ 0.0025 $\mbox Day^{-1}$ Estimated [57] $\alpha$ 0.002 $\mbox Day^{-1}$ - Assumed $\mu_2$ 0.05 $\mbox Day^{-1}$ 0.002-0.05 $\mbox Day^{-1}$ [33,35] $\beta_1$ 0.02 $\mbox Day^{-1}$ Estimated [15,58] $\beta_2$ 0.0001 $\mbox Day^{-1}$ Estimated [15,58] $\mu_3$ 0.12 $\mbox Day^{-1}$ 0.012 -0.12 $\mbox Day^{-1}$ [37] $\mu_4$ 0.24 $\mbox Day^{-1}$ 0.24 $\mbox Day^{-1}$ [58] $\gamma_1$ 0.51 $\mbox Day^{-1}$ Estimated [15,58] $\gamma_2$ 0.035 $\mbox Day^{-1}$ Estimated [15,58] $\xi_1$ 0.90 $\mbox Day^{-1}$ - Assumed $\xi_2$ 0.15 $\mbox Day^{-1}$ - Assumed $\mu_5$ 0.65 $\mbox Day^{-1}$ 0.04-0.9 $\mbox Day^{-1}$ [33] $c$ 0.50 $\mbox Day^{-1}$ Estimated [22,60]
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