# A reaction-diffusion equation with delay

Authors: - 03 April 2019

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The aim of this tutorial is to give a numerical method for solving a partial differential equation with a constant delay.

## Introduction

We consider the following one-dimentional reaction-diffusion equation with logistic production and delayed term,

this equation was suggested in  as a model of viral infection spreading in tissues. For the existence of solution and Global stability of the homogeneous in space equilibrium we refer the reader to . Here, $u(x,t)$ is the concentration of virus with respect to the space variable $x$ and time $t$. The parameters $D$ and $r$ are respectively the diffusion coefficient and replication rate constant. The function $f(u_{\tau})$ describes the concentration of immune cells as a function of the virus concentration at time $t-\tau, u_{\tau}(x, t) = u(x, t - \tau)$.

## Numerical method

We rewrite the one-dimentional reaction-diffusion equation with a constant time delay described above with Neumann boundary condition:

where the delay $\tau$ is a positive constant. we use an implicit finite difference approximation for the diffusion term and classical approach of the resolution of delay equations. let N and M denote the number of space steps and time steps with the notations $\Delta x= L/N$ and $\Delta t= T/M$, respectively.

The discretization of space $x_i$ and time $t_n$ are given by

Let $u^n_i$ be the approximation of the function $u$ at $(x,t)=(x_i,t_n)$, moreover,

We write the scheme of the equation (\ref{equa1}) at the point $u^n_i$,

Where, $u^{n-k}_i \approx u(x_i,t_n-\tau)$ is the delay variable and $k$ is determined by the equality $\tau= k \Delta t$. For simplicity of notation, we denote by $F^{n-k}_i := f(u^{n-k}_i)$ and $G^{n,k}_i:=r u^n_i( 1-u^n_i - F^{n-k}_i)$, for $i=0,..,N-1, \quad n=1,..,k$ and $n=0,..,M-1$.

The simplest form of this expression is given as follows

where $\lambda= D \Delta t /(\Delta x)^2$.

Now, we can write the following semi-linear system with Neumann boundary condition

First, we fix the parametrs as follows $T=10$, $\tau=1$, $L=1$, $D=1$, $r=1$,

clear ;
close ;
clc,
tic
T=10;
M=100;
L=1;
epsilon=10^(-6);
D=0.01;
r=1;
to=2;
N=200;
dx=L/(N);
dt=T/M;
x= (0:N)*dx ;
lambda=dt*D/(dx^2);
t=(0:M)*dt;


The tridiagonal matrix with Neumann condition

C=eye(N+1);
D=(2*lambda+1)*C-lambda*diag(ones(1,N),1)-lambda*diag(ones(1,N),-1);
B=[[-lambda;zeros(N,1)] [zeros(N+1,N-1)] [zeros(N,1);-lambda]];
A=D+B;


The initial function $u_0$

U=zeros(N+1,1);

for i=1:N+1
U(i,1)=u0(x(i),0);
end
figure (1)
plot(x,U)
xlabel('space')
title('Initial function u_0(x)') The delay function $f(u_{\tau})$

f=@( t ) 2*t ;
Q=zeros(N+1,1);
for i=1:N+1
Q(i,1)=f(x(i))   ;
end

figure (2)
plot(x,Q)
xlabel('space')
title('The function f') Ut=zeros(N+1,1);
Fretard=zeros(N+1,1);
Uf=zeros(N+1,M+1);
Uf=U;


We introduce the following test to get $u(x, t-\tau)$,

fig =        figure;
ax  = axes('Parent',fig,'XLim',[0 L],'YLim',[0 1.1]);
ax.XLabel.String = 'space';

%%gif('pdedelay4.gif','frame',fig,'DelayTime',1)
for n=1:M
if (((t(n)-to)<=0)||(abs(t(n)-to)<epsilon))
for i=1:N+1
Ut(i,1)=u0(x(i),t(n)-to);
Fretard(i,1)=f(Ut(i,1));
end
else
s=floor((t(n)-to)/dt)+1;
for i=1:N+1
Ut(i,1)=Uf(i,s);
Fretard(i,1)=f(Ut(i,1));
end
end
Uold=U;
Ur=dt*r*Uold.*(1-Uold-Fretard);
U=A\(Uold+Ur);
%%
%% We plot the evolution of solution $u(x,t)$ at
%% $t= \Delta t, \tau, 2 \tau, 3\tau, ...$,
%% when $\tau=0$, we plot the solution for $t=5, 10, 15, ...$
%%
if (((t(n+1)==dt)|| (mod(t(n+1),to)==0)) || ((to==0)&&(((mod(t(n+1),5)==0)||(n==M)))))

ll =line(x,Uold,'Color','r','Parent',ax);
ax.Title.String = ['t=',num2str(t(n+1))];
pause(1)
%%gif;
delete(ll)
end
Uf=[Uf,U];
end The solution $u(x,t)$ of PDE with delay

figure(4);
[X , Y] = meshgrid(x,[t]);
mesh (X , Y , Uf')
title('The evolution of solution u(x,t)')
xlabel('space')
ylabel('time') G.Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk, V. Volpert. Spatiotemporal dynamics of virus infection spreading in tissues. PlosOne, December 20, 2016.

 T. M. Touaoula, M. N. Frioui, N. Bessonov, V. Volpert Dynamics of solutions of a reaction-diffusion equation with delayed inhibition. Manuscript submitted for publication.