Mathematical modeling and its analysis for instability of the immune system induced by chemotaxis

Abstract

In this paper, we study how chemotaxis affects the immune system by proposing a minimal mathematical model, a reaction–diffusion–advection system, describing a cross-talk between antigens and immune cells via chemokines. We analyze the stability and instability arising in our chemotaxis model and find their conditions for different chemotactic strengths by using energy estimates, spectral analysis, and bootstrap argument. Numerical simulations are also performed to the model, by using the finite volume method in order to deal with the chemotaxis term, and the fractional step methods are used to solve the whole system. From the analytical and numerical results for our model, we explain not only the effective attraction of immune cells toward the site of infection but also hypersensitivity when chemotactic strength is greater than some threshold.

Appendix

In this appendix, we introduce mathematical notations used in our paper and prove the local-in-time existence of the full system (11)–(14). Let $\varOmega \subset \mathbb {R}^{d}$ be a domain with smooth boundary. The Lebesgue spaces and the Sobolev spaces on $\varOmega$ are defined as

where

and $0 \le n \in \mathbb {Z}$.

To prove the local well-posedness of (11)–(14), we need the next two lemmas. The first lemma is easily proved by using the Sobolev embedding theorem. For details, see Section 5 in Evans (2010).

Lemma 5

Let $\varOmega \subset \mathbb {R}^{d}$ be a bounded domain with smooth boundary and $u, v \in H^{n}(\varOmega )$ for , then $uv\in H^{n}(\varOmega )$ and

$$\begin{aligned} ||{uv}||_{H^{n}(\varOmega )} \le K_{1} ||{u}||_{H^{n}(\varOmega )} ||{u}||_{H^{n}(\varOmega )} , \end{aligned}$$

for some constant $K_{1}$ depending only on n and $\varOmega$.

The next lemma concerns the existence and uniqueness of a linear parabolic equation. We can easily prove the lemma by using the Galerkin approximation and we skip the proof.

Lemma 6

Let $\varOmega \subset \mathbb {R}^{d}$ ($d = 1, 2, 3$) be a bounded domain with smooth boundary. For $f \in L^{2}(0, T, H^{n - 1}(\varOmega ))$, and $u_{0}\in H^{n}(\varOmega )$, there exists a unique $u \in X_{T}^{n}$ satisfying

$$\begin{aligned} \frac{\partial u}{\partial t} - \varDelta u +\mu u&= f \quad \hbox { in}\ (0, T) \times \varOmega , \\ \frac{\partial u}{\partial \mathbf {n}}&= 0 \quad \hbox { on}\ [0,T)\times \partial \varOmega , \\ u(0, x)&= u_{0}(x) \quad \hbox { for}\ x \in \varOmega . \end{aligned}$$

where $\mu >0$ is constant. Moreover, there exists $K_{2} = K_{2}(\mu , \varOmega ) > 0$ such that

Using the previous two lemmas, we can show the local existence of a solution for Eqs. (1)–(4).

Lemma 7

(Local existence) Let $n > d / 2$ and $\varOmega \subset \mathbb {R}^{d}$ ($d = 1, 2, 3$) be a bounded domain with smooth boundary. If $(A_{0}, C_{0}, M_{0}) \in [H^{n}(\varOmega )]^{3}$ then there is $T = T(A_{0}, C_{0}, M_{0}) > 0$ such that there exists a solution $(A, C, M) \in (X_{T}^{n})^{3}$ of Eqs. (1)–(4) where

$$\begin{aligned} X_{T}^{n} = L^{\infty }([0, T), H^{n}(\varOmega )) \cap L^{2}((0, T), H^{n + 1}(\varOmega )), \end{aligned}$$

with norm $||{\cdot }||_{X_{T}^{n}}^{2} = \sup _{0 \le t < T} ||{\cdot }||_{H^{n}}^{2} + \int _{0}^{T} ||{\cdot }||_{H^{n + 1}}^{2} \;\mathrm {d}t$. Moreover, in $0\le t \le T$,

$$\begin{aligned} ||{(A,C,M)}||_{H^{n}} \le 2 ||{(A_{0}, C_{0}, M_{0})}||_{H^{n}} . \end{aligned}$$

Proof

Take $\phi \in X_{T}^{n}$ with $\phi (0, x) = M_{0}(x)$. Let $A = A(\phi )$ denote a corresponding solution for

$$\begin{aligned}&\frac{\partial A}{\partial t} - D_{A} \mathop {}\!\mathbin {\varDelta }A + \mu _{A} A = s_{A}(t, x) - \lambda _{A} \phi A, \\&\quad A(0, x) = A_{0}(x) . \end{aligned}$$

For this A, let $C = C(A(\phi ))$ be a corresponding solution for

$$\begin{aligned} \frac{\partial C}{\partial t} - D_{C} \mathop {}\!\mathbin {\varDelta }C + \mu _{C} C= & {} s_{C} \phi A, \\ C(0, x)= & {} C_{0}(x) . \end{aligned}$$

Finally for this C, let $M = M(C(A(\phi )))$ be a corresponding solution for

$$\begin{aligned} \frac{\partial M}{\partial t} - D_{M} \mathop {}\!\mathbin {\varDelta }M + \mu _{M} M= & {} -{{\mathrm{div}}}(\chi M \mathop {}\!\mathbin {\nabla }C) + s_{M} - \lambda _{M}MA, \\ M(0, x)= & {} M_{0}(x) . \end{aligned}$$

Then by the linear parabolic theory, the unique existence of A, C, and M follows. Applying Lemma 6, we can deduce

(40)

The Gronwall inequality (see Section 1.2 in Pachpatte and Ames (1997)) yields

Putting it into (40), we get $A\in X_{T}^{n}$. For C,

We also get $C \in X_{T}^{n}$. For M,

$$\begin{aligned} ||{M}|| _{X_{T}^{n}}^{2}= & {} \sup _{0\le t\le T}||{ M}|| _{H^{n}}^{2}+\int _{0}^{T}||{C}|| _{H^{n+1}}^{2} \;\mathrm {d}t\nonumber \\\le & {} K_{2}\left( ||{M_{0}}|| _{H^{n}}^{2}+\chi ^{2}\int _{0}^{T} ||{{{\mathrm{div}}}(M\nabla C)}||_{H^{n-1}}^{2} \;\mathrm {d}t \right. \nonumber \\&\quad \left. {} + s_{M}^{2} ||{\varOmega }|| T +\lambda _{M}^{2}\int _{0}^{T}||{MA}|| _{H^{n-1}}^{2} \;\mathrm {d}t \right) \nonumber \\\le & {} K_{2} \left( \phantom {\int _0^T} ||{M_{0}}|| _{H^{m}}^{2}+s_{M}^{2}||{\varOmega }|| T \right. \nonumber \\&\quad \left. {} + K_{1}^{2}\int _{0}^{T}(\chi ^{2}||{C}||_{H^{n}}^{2} +\lambda _{M}^{2}||{A}|| _{H^{n}}^{2})||{M}|| _{H^{n}}^{2} \;\mathrm {d}t \right) . \end{aligned}$$

(41)

By the Gronwall inequality,

(42)

Putting (42) into (41), we get $M\in X_{T}^{n}$. Then we can define a mapping $Q_{T}:X_{T}^{n} \rightarrow X_{T}^{n}$ with $Q_{T}(\phi ) = M$. Let $(X_{T}^{n})_{R}=\{\phi \in X_{T}^{n}:{||{\phi }||_{X_{T}^{n}}<R}\}$. Choose $R = 2||{M_{0}}||_{H^{m}}$. We will prove that $Q_{T}$ is a contraction mapping in $(X_{T}^{n})_{R}$ for sufficiently small T. It is obvious that if we take a sufficiently small T, $Q_{T}$ maps from $(X_{T}^{n})_{R}$ to itself by above a priori estimate (42). To show $Q_{T}$ is a contraction in $X_{T}^{n}$, take $\phi _{1},\phi _{2}\in X_{T}^{n}$, and let $M_{1} = Q_{T}(\phi _{1})$ and $M_{2} = Q_{T}(\phi _{2})$. Then we can also get the corresponding $A_{1}$, $A_{2}$, $C_{1}$, and $C_{2}$. Consequently, we have

$$\begin{aligned} \frac{\partial {\widetilde{A}}}{\partial t} - D_{A}\mathop {}\!\mathbin {\varDelta }{\widetilde{A}} + \mu _{A}{\widetilde{A}}&= -\lambda _{A}\phi _{1}{\widetilde{A}} - \lambda _{A}A_{2}{\widetilde{\phi }} , \\ \frac{\partial {\widetilde{C}}}{\partial t} - D_{C}\mathop {}\!\mathbin {\varDelta }{\widetilde{C}} + \mu _{C}{\widetilde{C}}&= s_{C}\phi _{1}{\widetilde{C}} + s_{C}C_{2}{\widetilde{\phi }} , \\ \frac{\partial {\widetilde{M}}}{\partial t} - D_{M}\mathop {}\!\mathbin {\varDelta }{\widetilde{M}} + \mu _{M}{\widetilde{M}}&= \chi {{\mathrm{div}}}(M_{1}\nabla {\widetilde{C}} + s_{C}\nabla C_{2}{\widetilde{M}}) \\&\quad -\lambda _{M}M_{1}{\widetilde{A}}-\lambda _{M}A_{2}{\widetilde{M}} , \end{aligned}$$

where ${\widetilde{\phi }} = \phi _{1} - \phi _{2}$, ${\widetilde{A}} = A_{1} - A_{2}$, ${\widetilde{C}} = C_{1}-C_{2}$, and ${\widetilde{M}} = M_{1} - M_{2}$. By using Lemma 6, we obtain the following estimate for ${\widetilde{A}}$:

The Gronwall inequality yields

with

$$\begin{aligned} K_{A_{1}}(T) \rightarrow 0 \quad \text {as }T \rightarrow 0 . \end{aligned}$$

Then we can compute

with

$$\begin{aligned} K_{A}(T)\rightarrow 0 \quad \text {as } T\rightarrow 0. \end{aligned}$$

We can also derive the following estimate for ${\widetilde{C}}$

with

$$\begin{aligned} K_{C}(T)\rightarrow 0 \quad \text {as } T\rightarrow 0 . \end{aligned}$$

Finally, we get an estimate for ${\widetilde{M}}$:

with

$$\begin{aligned} K_{M_{1}}(T)\rightarrow 0 \quad \text {as }T\rightarrow 0 . \end{aligned}$$

By using the Gronwall inequality,

with

$$\begin{aligned} K_{M_{2}}(T)\rightarrow 0 \quad \text {as }T\rightarrow 0 . \end{aligned}$$

Hence

with

$$\begin{aligned} K_{M}(T)\rightarrow 0 \quad \text {as } T\rightarrow 0 . \end{aligned}$$

Therefore, we can apply the contraction mapping theorem which is stated in Section 9 in Evans (2010), and we can conclude that there is the unique fixed point M of $Q_{T}$ for sufficiently small T. $\square$

Remark 2

By Lemma 7, the existence and uniqueness of (a, c, m) which is a solution of (11)–(15) is also guaranteed. Moreover, it is easy to show that for sufficiently small time $T > 0$,

$$\begin{aligned} ||{(a, c, m)}||^2_{H^n} \le 2 ||{(a_0, c_0, m_0)}||^2_{H^n} . \end{aligned}$$

(43)