function example_steadystate
COMPILATION
[exdir,~,~]=fileparts(which('example_steadystate.m')); % compile the model amiwrap('model_steadystate','model_steadystate_syms',exdir)
Generating model struct ... Parsing model struct ... Generating C code ... headers | wrapfunctions | Compiling mex file ... amici | Building with 'Xcode with Clang'. MEX completed successfully. Building with 'Xcode with Clang'. MEX completed successfully.
SIMULATION
% time vector t = linspace(0,300,20); p = [1;0.5;0.4;2;0.1]; k = [0.1,0.4,0.7,1]; options = amioption('sensi',0,... 'maxsteps',1e4); % load mex into memory sol = simulate_model_steadystate(t,log10(p),k,[],options); tic sol = simulate_model_steadystate(t,log10(p),k,[],options); disp(['Time elapsed with cvodes: ' num2str(toc) ])
Time elapsed with cvodes: 0.0040652
ODE15S
ode_system = @(t,x,p,k) [-2*p(1)*x(1)^2 - p(2)*x(1)*x(2) + 2*p(3)*x(2) + p(4)*x(3) + p(5);
+ p(1)*x(1)^2 - p(2)*x(1)*x(2) - p(3)*x(2) + p(4)*x(3);
+ p(2)*x(1)*x(2) - p(4)*x(3) - k(4)*x(3)];
options_ode15s = odeset('RelTol',options.rtol,'AbsTol',options.atol,'MaxStep',options.maxsteps);
tic
[~, X_ode15s] = ode15s(@(t,x) ode_system(t,x,p,k),t,k(1:3),options_ode15s);
disp(['Time elapsed with ode15s: ' num2str(toc) ])
Time elapsed with ode15s: 0.10806
PLOTTING
figure
c_x = get(gca,'ColorOrder');
subplot(2,2,1)
for ix = 1:size(sol.x,2)
plot(t,sol.x(:,ix),'.-','Color',c_x(ix,:))
hold on
plot(t,X_ode15s(:,ix),'d','Color',c_x(ix,:))
end
legend('x1','x1_{ode15s}','x2','x2_{ode15s}','x3','x3_{ode15s}','Location','NorthEastOutside')
legend boxoff
xlabel('time t')
ylabel('x')
box on
subplot(2,2,2)
plot(t,abs(sol.x-X_ode15s),'--')
set(gca,'YScale','log')
legend('error x1','error x2','error x3','Location','NorthEastOutside')
legend boxoff
set(gcf,'Position',[100 300 1200 500])
FORWARD SENSITIVITY ANALYSIS
options.sensi = 1;
options.sens_ind = [3,1,2,4];
sol = simulate_model_steadystate(t,log10(p),k,[],options);
FINITE DIFFERENCES
eps = 1e-3;
xi = log10(p);
for ip = 1:4;
xip = xi;
xip(ip) = xip(ip) + eps;
solp = simulate_model_steadystate(t,xip,k,[],options);
sx_fd(:,:,ip) = (solp.x - sol.x)/eps;
sy_fd(:,:,ip) = (solp.y - sol.y)/eps;
end
PLOTTING
figure
for ip = 1:4
subplot(4,2,ip*2-1)
hold on
for ix = 1:size(sol.x,2)
plot(t,sol.sx(:,ix,ip),'.-','Color',c_x(ix,:))
plot(t,sx_fd(:,ix,options.sens_ind(ip)),'d','Color',c_x(ix,:))
end
legend('x1','x1_{fd}','x2','x2_{fd}','x3','x3_{fd}','Location','NorthEastOutside')
legend boxoff
title(['state sensitivity for p' num2str(options.sens_ind(ip))])
xlabel('time t')
ylabel('x')
box on
subplot(4,2,ip*2)
plot(t,abs(sol.sx(:,:,ip)-sx_fd(:,:,options.sens_ind(ip))),'--')
legend('error x1','error x2','error x3','Location','NorthEastOutside')
legend boxoff
title(['error of state sensitivity for p' num2str(options.sens_ind(ip))])
xlabel('time t')
ylabel('error')
set(gca,'YScale','log')
box on
end
set(gcf,'Position',[100 300 1200 500])
STEADY STATE SENSITIVITY
sssens = NaN(size(sol.sx));
for it = 2:length(t)
tt = [0,t(it)];
options.sensi_meth = 'ss';
solss = simulate_model_steadystate(tt,log10(p),k,[],options);
sssens(it,:,:) = solss.sx;
ssxdot(it,:) = solss.xdot;
end
PLOTTING
figure
for ip = 1:4
subplot(4,2,ip*2-1)
hold on
for ix = 1:size(sol.x,2)
plot(t,sol.sx(:,ix,ip),'.-','Color',c_x(ix,:))
plot(t,sssens(:,ix,ip),'d-','Color',c_x(ix,:))
end
legend('x1','x1_{ss}','x2','x2_{ss}','x3','x3_{ss}','Location','NorthEastOutside')
legend boxoff
title(['state steady sensitivity for p' num2str(ip)])
xlabel('time t')
ylabel('x')
box on
subplot(4,2,ip*2)
plot(t,abs(sol.sx(:,:,ip)-sssens(:,:,ip)),'--')
legend('error x1','error x2','error x3','Location','NorthEastOutside')
legend boxoff
title(['error of steady state sensitivity for p' num2str(ip)])
xlabel('time t')
ylabel('error')
set(gca,'YScale','log')
box on
end
set(gcf,'Position',[100 300 1200 500])
figure
scatter(sqrt(sum((ssxdot./sol.x).^2,2)),sqrt(sum(sum((sol.sx-sssens).^2,2),3)))
hold on
plot([1e-15,1e5],[1e-15,1e5],'k:')
set(gca,'YScale','log')
set(gca,'XScale','log')
box on
axis square
xlabel('||dxdt/x||_2')
ylabel('error steady state approximation')
set(gca,'FontSize',15)
set(gca,'LineWidth',1.5)
set(gcf,'Position',[100 300 1200 500])
drawnow
end