📂Dynamics

Dynamics in Atopic dermatitis Systems

Model

The following nonsmooth dynamic system is referred to as the Atopic dermatitis System. $$\begin{align*} {{ d P (t) } \over { d t }} =& {{ P_{\text{env}} \kappa_{P} } \over { 1 + \gamma_{B} B (t) }} - \alpha_{I} R(t) P(t) - \delta_{P} P (t) \\ {{ d B (t) } \over { d t }} =& {{ \kappa_{B} \left[ 1 - B(t) \right] } \over { \left[ 1 + \gamma_{R} R(t) \right] \left[ 1 + \gamma_{G} G(t) \right] }} - \delta_{B} K(t) B(t) \\ {{ d D (t) } \over { d t }} =& \kappa_{D} R(t) - \delta_{D} D(t) \end{align*}$$ This system is controlled by two types of switches: the reversible switch $R$ and the irreversible switch $G$. $$\begin{align*} R (t) =& \begin{cases} R_{\text{off}} & , \text{if } P(t) < P^{-} \text{ or } P^{-} \le P (t) \le P^{+} , R (t - dt) = R_{\text{off}} \\ R_{\text{on}} & , \text{if } P(t) > P^{+} \text{ or } P^{-} \le P (t) \le P^{+} , R (t - dt) = R_{\text{on}} \end{cases} \\ G (t) =& \begin{cases} G_{\text{off}} & , \text{if } D(t) < D^{+} \text{ and } G (t - dt) = G_{\text{off}} \\ G_{\text{on}} & , \text{if } D(t) \ge D^{+} \text{ or } G (t - dt) = G_{\text{on}} \end{cases} \end{align*}$$ Here, $dt \approx 0$ represents a very short time interval. According to the $R$-switch, $K(t)$ is given as follows. $$K(t) = \begin{cases} K_{\text{off}} & , \text{if } R(t) = R_{\text{off}} \\ m_{\text{on}} P(t) - \beta_{\text{on}} & , \text{if } R(t) = R_{\text{on}} \end{cases}$$

Variables

• $P(t)$: The amount of pathogen infiltrated. It is affected by the given environment $P_{\text{env}}$ and is reduced as the skin barrier strengthens.
• $B(t) \in \left[ 0, 1 \right]$: The strength of the Barrier. Closer to $1$ indicates a healthier skin condition, while closer to $0$ indicates a poorer skin condition.
• $D(t)$: The concentration of Dendritic Cells in lymph nodes. If this value exceeds a certain level, the $G$-switch is turned on.
• $R(t)$: The sensitivity of receptors characterizing the $R$-switch. It activates when pathogen $P(t)$ exceeds a certain level, contributes to reducing $P(t)$, and also damages the skin barrier. It turns off when $P(t)$ decreases below a certain level, making it a reversible switch.
• $G(t)$: Involved in immune regulation, the $G$-switch. It turns on when $D(t)$ exceeds the manageable limit $D^{+}$ and remains on permanently. Once the $G$-switch is activated, it negatively affects the skin barrier’s ability to recover permanently.
• $K(t)$: The amount of Kallikrein, an enzyme that releases kinins from the plasma. It activates when the $R$-switch is on, damaging the skin barrier.

Parameters

Description

Mathematical modeling of Atopic dermatitis or Atopic Eczema was proposed in a paper by Elisa Domínguez-Hüttinger et al., published in the Journal of Allergy and Clinical Immunology (JACI)¹, and has been a subject of ongoing research by the Tanaka group² and actively researched in Korea by Dr. Kang Yoseph and others³.

The Two Switches

While the $R$-switch and the $G$-switch may look complicated in their mathematical expressions, they can be simplified into the form shown above when drawn. Due to both the values of $P$ and $D$ and the state of the switches at that moment, the AD system is not strictly defined by a vector field but becomes a piecewise smooth system³. Depending on the on/off state of the two switches, there are four possible scenarios, each of which can be represented as four smooth subsystems.

$S_{1}$-Subsystem: $R_{\text{off}}$, $G_{\text{off}}$

$$\begin{align*} {{ d P (t) } \over { d t }} =& {{ P_{\text{env}} \kappa_{P} } \over { 1 + \gamma_{B} B (t) }} - \delta_{P} P (t) \\ {{ d B (t) } \over { d t }} =& \kappa_{B} \left[ 1 - B(t) \right] \\ {{ d D (t) } \over { d t }} =& - \delta_{D} D(t) \end{align*}$$

$S_{2}$-Subsystem: $R_{\text{off}}$, $G_{\text{on}}$

$$\begin{align*} {{ d P (t) } \over { d t }} =& {{ P_{\text{env}} \kappa_{P} } \over { 1 + \gamma_{B} B (t) }} - \delta_{P} P (t) \\ {{ d B (t) } \over { d t }} =& {{ \kappa_{B} \left[ 1 - B(t) \right] } \over { \left[ 1 + \gamma_{G} G_{\text{on}} \right] }} \\ {{ d D (t) } \over { d t }} =& - \delta_{D} D(t) \end{align*}$$

$S_{3}$-Subsystem: $R_{\text{on}}$, $G_{\text{off}}$

$$\begin{align*} {{ d P (t) } \over { d t }} =& {{ P_{\text{env}} \kappa_{P} } \over { 1 + \gamma_{B} B (t) }} - \alpha_{I} R_{\text{on}}(t) P(t) - \delta_{P} P (t) \\ {{ d B (t) } \over { d t }} =& {{ \kappa_{B} \left[ 1 - B(t) \right] } \over { \left[ 1 + \gamma_{R} R_{\text{on}}(t) \right] }} - \delta_{B} \left( m_{\text{on}} P(t) - \beta_{\text{on}} \right) B(t) \\ {{ d D (t) } \over { d t }} =& \kappa_{D} R_{\text{on}}(t) - \delta_{D} D(t) \end{align*}$$

$S_{4}$-Subsystem: $R_{\text{on}}$, $G_{\text{on}}$

$$\begin{align*} {{ d P (t) } \over { d t }} =& {{ P_{\text{env}} \kappa_{P} } \over { 1 + \gamma_{B} B (t) }} - \alpha_{I} R_{\text{on}}(t) P(t) - \delta_{P} P (t) \\ {{ d B (t) } \over { d t }} =& {{ \kappa_{B} \left[ 1 - B(t) \right] } \over { \left[ 1 + \gamma_{R} R_{\text{on}}(t) \right] \left[ 1 + \gamma_{G} G_{\text{on}}(t) \right] }} - \delta_{B} \left( m_{\text{on}} P(t) - \beta_{\text{on}} \right) B(t) \\ {{ d D (t) } \over { d t }} =& \kappa_{D} R_{\text{on}}(t) - \delta_{D} D(t) \end{align*}$$

Bifurcation

The Codimension-2 bifurcation diagram for the two parameters Barrier permeability $\kappa_{P}$ and Immune responses $\alpha_{I}$ is shown above. The time evolutions within each of the four regions of the diagram can be broadly categorized into four types.

• (1) Converging to a stable point where the skin barrier fully recovers $\displaystyle \lim_{t \to \infty} B(t) = 1$
• (4) Converging to a stable point where the skin barrier sustains chronic damage $\displaystyle \lim_{t \to \infty} B(t) = 0$
• (2) Depending on the initial conditions, either recovering or sustaining chronic damage, bistability
• (3) Oscillating oscillation between $[0,1]$ where the condition of the skin barrier $B(t)$ fluctuates between improvement and deterioration

This can be seen as a dynamical analysis of the progression of atopy based on innate conditions $\kappa_{P}$, $\alpha_{I}$, and given circumstances. Subsequent studies have drawn more detailed bifurcation diagrams or have identified oscillations subdivided into mild and severe for precision analysis.