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Need help to solve 5 simultaneous first order differential equations with Initial Condition.

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HARSH ZALAVADIYA
HARSH ZALAVADIYA on 14 Apr 2021 at 12:43
Commented: HARSH ZALAVADIYA on 14 Apr 2021 at 15:31
I have to solve following first order ordinary differential equations and plot the values of Concentrations (Cmn, Cecm, Crec, Ccirc, Ccells) against time (t).
where value of r(t) is described as following.
Below is my script, but I am getting constant errors. or something i dont understand.
clc; close; clear;
%Constant Parameters
Cmn0 = 6.2E-6; %Initial conc at the MN (
tr = 1805; %release period (s)
ka = 5.01E6; %Association rate (in 1/s)
kd = 5E-4; %Dissociation rate (in 1/s)
ki = 5.05E-3; %Internalization rate (in 1/s)
kc = 5E-3; %Circulation uptake rate (in 1/s)
Rtot = 1.85E-6 ; %initial receptor concentration (in umol/mm^3)
syms r(t) C_mn(t) C_ecm(t) C_rec(t) C_circ(t) C_cells(t) %creating symbolic variable
ode1 = diff(r) == Cmn0/tr;
ode2 = diff(C_mn) == -r;
ode3 = diff(C_ecm) == r- (ka * C_ecm * (Rtot - C_rec - C_cells)) + kd * C_rec - kc * C_ecm;
ode4 = diff(C_rec) == (ka * C_ecm * (Rtot - C_rec - C_cells)) - ((kd + ki)* C_rec);
ode5 = diff(C_circ) == kc * C_ecm;
ode6 = diff(C_cells) == ki * C_rec;
odes = [ode1; ode2; ode3; ode4; ode5; ode6]
cond1 = r(0) == Cmn0/tr;
cond2 = C_mn(0) == 6.2E-6;
cond3 = C_ecm(0) == 0;
cond4 = C_rec(0) == 0;
cond5 = C_circ(0) == 0;
cond6 = C_cells(0) == 0;
conds = [cond1; cond2; cond3; cond4; cond5; cond6];
[VF,Sbs] = odeToVectorField(odes);
odsefcn = matlabFunction(VF,'File', 'Consolvefun')
[t, C] = ode45(@Consolvefun, [0 5000], conds);

Accepted Answer

Alan Stevens
Alan Stevens on 14 Apr 2021 at 14:06
You have a stiff system (ka~10^6, kd~10^-4), so use ode15s rather than ode45. The following works. I'll leave you to decide if the results make sense!
C0 = [6.2E-6, 0, 0, 0, 0];
tspan = 0:10:5000;
[t, C] = ode15s(@fn, tspan, C0);
C_mn = C(:,1);
C_ecm = C(:,2);
C_rec = C(:,3);
C_circ = C(:,4);
C_cells = C(:,5);
plot(t,C_mn,t,C_ecm,t,C_rec,t,C_circ,t,C_cells),grid
xlabel('t'),ylabel('C')
legend('Cmn','Cecm','Crec','Ccirc','Ccells')
function dCdt = fn(t, C)
%Constant Parameters
Cmn0 = 6.2E-6; %Initial conc at the MN (
tr = 1805; %release period (s)
ka = 5.01E6; %Association rate (in 1/s)
kd = 5E-4; %Dissociation rate (in 1/s)
ki = 5.05E-3; %Internalization rate (in 1/s)
kc = 5E-3; %Circulation uptake rate (in 1/s)
Rtot = 1.85E-6 ; %initial receptor concentration (in umol/mm^3)
r = Cmn0/tr*(t<=tr);
C_mn = C(1);
C_ecm = C(2);
C_rec = C(3);
C_circ = C(4);
C_cells = C(5);
dCdt = [ -r;
r- ka * C_ecm * (Rtot - C_rec - C_cells) + kd * C_rec - kc * C_ecm;
ka * C_ecm * (Rtot - C_rec - C_cells) - (kd + ki)* C_rec;
kc * C_ecm;
ki * C_rec];
end
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