Axisymmetric solutions for a chemotaxis model of Multiple Sclerosis

Abstract

In this paper we study radially symmetric solutions for our recently proposed reaction–diffusion–chemotaxis model of Multiple Sclerosis. Through a weakly nonlinear expansion we classify the bifurcation at the onset and derive the amplitude equations ruling the formation of concentric demyelinating patterns which reproduce the concentric layers observed in Balò sclerosis and in the early phase of Multiple Sclerosis. We present numerical simulations which illustrate and fit the analytical results.

A Derivation of the amplitude equation

In this Appendix, we give the details of the derivation of the amplitude equation (14).

At $O(\eta )$ we get a linear problem whose solution is given by (13). Taking into account this solution, at $O(\eta ^2)$ we obtain the following linear problem:

$$\begin{aligned} \mathcal {L}^{\chi _c}{} \mathbf w _2=\mathbf {F} \end{aligned}$$

(34)

where $\mathbf {F}=\frac{\partial A}{\partial T}\varvec{\gamma } J_0(k_c \varrho )+\mathbf {G}_0^{(1)}AJ_0(k_c \varrho )+(\mathbf {G}_0^{(2)}J_0(k_c \varrho )^2+\mathbf {G}_1^{(2)}J_1(k_c \varrho )^2)A^2$

and

$$\begin{aligned}&\mathbf {G}_0^{(1)}=\left( \begin{array}{c} -\frac{\chi _1}{2}k_c^2\\ 0 \\ 0 \end{array} \right) ,&\end{aligned}$$

(35)

$$\begin{aligned}&\mathbf {G}_0^{(2)}=\left( \begin{array}{c} -\frac{\chi _1}{4}k_c^2 M+M^2 \\ 0\\ \frac{3}{4}r M N \end{array} \right) ,&\end{aligned}$$

(36)

$$\begin{aligned}&\mathbf {G}_1^{(2)}=\left( \begin{array}{c} \frac{\chi _c}{4}k_c^2 M \\ 0 \\ 0 \end{array} \right) ,&\end{aligned}$$

(37)

By Fredholm alternative theorem, the Eq. (34) admits solutions if and only if $\langle \mathbf {F}, \varvec{\psi }^{*}\rangle =0$, where

$$\begin{aligned} \varvec{\psi }^{*}=\varvec{\psi }J_0(k_c \varrho ), , \end{aligned}$$

(38)

and

$$\begin{aligned} \varvec{\psi }=\left( \bar{M} , 1 , \bar{N} \right) ^T=\left( \frac{\beta }{\tau (1+k_c^2)} , 1 , \frac{2\delta }{\tau r} \right) ^T\in Ker[(K-k_c^2D^{\chi _c})^{\dag }]. \end{aligned}$$

The solvability condition $\langle \mathbf {F}, \, \varvec{\psi }^{*}\rangle =0$ for Eq. (34) leads to (14), where the expression for $\sigma$ and L are given by:

$$\begin{aligned} \sigma= & {} -\frac{\langle \mathbf {G}_0^{(1)},\varvec{\psi }\rangle }{\langle \varvec{\gamma },\varvec{\psi }\rangle }, \qquad L\nonumber \\= & {} \frac{\langle \mathbf {G}_0^{(2)},\varvec{\psi }\rangle }{\langle \varvec{\gamma },\varvec{\psi }\rangle } \frac{\int _{0}^{R} \varrho \, J_0(k_c \rho )^3 \, d \varrho }{\int _{0}^{R} \varrho \, J_0(k_c \rho )^2 \, d \varrho }+\frac{\langle \mathbf {G}_1^{(2)},\varvec{\psi }\rangle }{\langle \varvec{\gamma },\varvec{\psi }\rangle } \frac{\int _{0}^{R} \varrho \, J_0(k_c \rho ) \, J_1(k_c \rho )^2 \, d \varrho }{\int _{0}^{R} \varrho J_0(k_c \rho )^2 d \varrho } \end{aligned}$$

(39)