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

Amit Kumar Roy; Priti Kumar Roy; Ellina Grigorieva

doi:10.3934/mbe.2018032

Article Contents

2018, Volume 15, Issue 3: 717-738. Doi: 10.3934/mbe.2018032

This issue Previous Article Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect Next Article Dynamics of a Filippov epidemic model with limited hospital beds

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 6, 2017

Published online: December 2017

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 92B05, 97M10; Secondary: 49J15.

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Figure 1. Schematic of the interactions between the components of the model.

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Figure 2. 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 $c$.

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Figure 3. 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.

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Figure 4. 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.

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Figure 5. Control parameter with respect to time for better impact on psoriatic plaque.

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Figure 6. Comparative system behaviour of continuous therapy (blue line) and impulse therapy (green line) using Biologic ($\mbox{IL}-{10}$).

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Figure 7. 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).

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Table 1. 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]

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