# How to solve a system of ODEs using ode23s with multiple inputs.

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Ron_S on 3 Mar 2024
Edited: Ron_S on 3 Mar 2024
The code works if I use one input (input1) where it calls for the numerical integration ([t,u] = ode23s(@(t,u) gene(t,u,k,input1), tspan, init); and if I define all the inputs in the system as in1(t). This occurs in eqs(6),(16) and (37).
Is it possible though, to have 3 different intput instances, so that the second term in eq(6) will be k26*in2(t), the first term of eq(16) will be k(47)*in3(t)*u(21) and the first term in eq(37) will be k(54)*in4(t)*u(35)?
The code used is given below:
clear
clc
% Set the initial values
u(1) = 5;
u(2) = 1;
u(3) = 1;
u(4) = 1;
u(5) = 1;
u(6) = 50;
u(7) = 1;
u(8) = 100;
u(9) = 5;
u(10) = 1;
u(11) = 1;
u(12) = 1;
u(13) = 1;
u(14) = 1;
u(15) = 1;
u(16) = 1;
u(17) = 1;
u(18) = 1;
u(19) = 1;
u(20) = 1;
u(21) = 1;
u(22) = 1;
u(23) = 1;
u(24) = 1;
u(25) = 1;
u(26) = 1;
u(27) = 1;
u(28) = 1;
u(29) = 1;
u(30) = 1;
u(31) = 1;
u(32) = 1;
u(33) = 1;
u(34) = 1;
u(35) = 1;
u(36) = 1;
u(37) = 1;
u(38) = 1;
u(39) = 1;
u(40) = 1;
u(41) = 1;
%% Define anonymous function input1(t)
% Time: 0<=t<50 50<=t<100 100<=t<150 150<=t<200 200<=t
% Input: 0.5 1 1.5 1 0.5
input1 = @(t) 0.5 + 0.5*(50<=t & t<100) + 1*(100<=t & t<150) + 0.5*(150<=t & t<200);
input2 = @(t) 0.5 + 0.5*(50<=t & t<100) + 1*(100<=t & t<150) + 0.5*(150<=t & t<200);
input3 = @(t) 0.5 + 0.5*(50<=t & t<100) + 1*(100<=t & t<150) + 0.5*(150<=t & t<200);
input4 = @(t) 0.5 + 0.5*(50<=t & t<100) + 1*(100<=t & t<150) + 0.5*(150<=t & t<200);
% Set the model parameters
k1 = 3;
k2 = 2;
k3 = 5;
k4 = 4;
k5 = 5;
k6 = 1;
k7 = 4;
k8 = 3;
k9 = 1;
k10 = 1.2;
k11 = 1;
k12 = 9;
k13 = 1;
k14 = 1;
k15 = 1;
k16 = 30;
k17 = 30;
k18 = 1.5;
k19 = 25;
k20 = 1;
k21 = 10;
k22 = 20;
k23 = 1;
k24 = 0.5;
k25 = 1;
k26 = 1;
k27 = 1;
k28 = 1;
k29 = 1;
k30 = 1;
k31 = 1;
k32 = 1;
k33 = 1;
k34 = 1;
k35 = 1;
k36 = 1;
k37 = 1;
k38 = 1;
k39 = 1;
k40 = 1;
k41 = 1;
k42 = 1;
k43 = 1;
k44 = 1;
k45 = 1;
k46 = 1;
k47 = 1;
k48 = 1;
k49 = 1;
k50 = 1;
k51 = 1;
k52 = 1;
k53 = 20;
k54 = 1;
k55 = 1;
k56 = 1;
k57 = 1;
k58 = 1;
k59 = 1;
k60 = 1;
k61 = 1;
k62 = 1;
k63 = 1;
k64 = 1;
k65 = 1;
k66 = 1;
k67 = 1;
k68 = 1;
k69 = 1;
k70 = 1;
%% Perform the numerical integration
k = [k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12,k13,k14,k15,k16,k17,k18,k19,k20,k21,k22,k23,k24,k25,k26,k27,k28,k29,k30,k31,k32,k33,k34,k35,k36,k37,k38,k39,k40,k41,k42,k43,k44,k45,k46,k47,k48,k49,k50,k51,k52,k53,k54,k55,k56,k57,k58,k59,k60,k61,k62,k63,k64,k65,k66,k67,k68,k69,k70];
init = [u(1) u(2) u(3) u(4) u(5) u(6) u(7) u(8) u(9) u(10) u(11) u(12) u(13) u(14) u(15) u(16) u(17) u(18) u(19) u(20) u(21) u(22) u(23) u(24) u(25) u(26) u(27) u(28) u(29) u(30) u(31) u(32) u(33) u(34) u(35) u(36) u(37) u(38) u(39) u(40) u(41)];
tspan = [0 250];
[t,u] = ode23s(@(t,u) gene(t,u,k,input1), tspan, init);
%% Plot results
t1=0:.5:250;
tiledlayout(4,1)
nexttile
hold on
plot(t1,input1(t1),'-k', 'LineWidth',2.0)
title(''); legend('CL(Input)')
xlabel('Time t'); ylabel('Concentration')
hold off
ylim([0 2])
nexttile
hold on
plot(t,u(:,6),'-',t,u(:,7),'-',t,u(:,8),'-',t,u(:,9),'-',t,u(:,18),'-',t,u(:,39),'-',t,u(:,29),'-', 'LineWidth',2.0)
title(''); legend('Cf','Cp','Ce','E','Ce1','Ce2','B')
xlabel('Time t'); ylabel('Concentration')
ylim([0 4])
nexttile
hold on
plot(t,u(:,2),'-',t,u(:,5),'-',t,u(:,11),'-', 'LineWidth',2.0)
title(''); legend('C','R','H')
xlabel('Time t'); ylabel('Concentration')
ylim([0 60])
nexttile
hold on
plot(t,u(:,1),'-',t,u(:,3),'-',t,u(:,10),'-',t,u(:,12),'-',t,u(:,13),'-',t,u(:,4),'-', 'LineWidth',2.0)
title('','FontSize',16); legend('Sci','Sr','HR','Sp','P','Sh')
xlabel('Time t'); ylabel('Concentration')
ylim([0 1.5])
%% ODE System
function eqns = gene(t,u,k,in1)
% Using u = [Sci C Sr Sh R Cf Cp Ce E HR H Sp P Sb Rb Cf1 Sci1 C1 Sb1 Sh1 Rb1 HR1 Cp1 Ce1 E1 H1 CO Cg B Sci2 C2 Sr2 Sn2 Sp2 R2 P2 Cf2 Cp2 Ce2 H2 HR2]
% Equation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
%
%Using k17=p1, k18=p2,k19=p3, k20=mu, k21=eta, k22=alpha, k23=theta
eqns = zeros(41,1); % To start with we have 41 empty equations
eqns(1) = k(60) - k(61)*u(1)*u(2);
eqns(2) = k(70)*u(8) - k(61)*u(1)*u(2);
eqns(3) = k(1)*u(1) - k(2)*u(3);
eqns(4) = k(1)*u(1) - k(10)*u(4);
eqns(5) = k(17)*u(3) - k(11)*u(5) - k(14)*u(13)*u(5);
eqns(6) = k(3)*in1(t) + k(26)*in1(t) - k(4)*u(6);
eqns(7) = k(4)*u(6) - k(5)*u(7) + k(6)*u(8) - k(12)*u(7)*u(27);
eqns(8) = k(5)*u(7) - k(6)*u(8) + k(9)*u(10)*u(11) - k(7)*u(8) + k(8)*u(9) - k(21)*u(8) - k(22)*u(8)*u(28);
eqns(9) = k(7)*u(8) - k(8)*u(9);
eqns(10) = k(18)*u(4) - k(15)*u(10)*u(8);
eqns(11) = k(62) - k(9)*u(10)*u(11);
eqns(12) = k(1)*u(1) - k(13)*u(12);
eqns(13) = k(19)*u(12) - k(16)*u(13);
eqns(14) = k(1)*u(1) - k(27)*u(14);
eqns(15) = k(28)*u(14) - k(29)*u(15);
eqns(16) = k(47)*in1(t)*u(21) - k(30)*u(16);
eqns(17) = k(63) - k(67)*u(17)*u(18);
eqns(18) = k(64)*u(24) - k(67)*u(17)*u(18);
eqns(19) = k(20)*u(17) - k(23)*u(19);
eqns(20) = k(20)*u(17) - k(25)*u(20);
eqns(21) = k(35)*u(19) - k(36)*u(21);
eqns(22) = k(33)*u(20) - k(38)*u(22)*u(24);
eqns(23) = k(30)*u(16) - k(31)*u(23) + k(32)*u(24) - k(37)*u(23);
eqns(24) = k(31)*u(23) - k(32)*u(24) - k(41)*u(24) + k(42)*u(25) + k(34)*u(22)*u(26) - k(40)*u(24);
eqns(25) = k(41)*u(24) - k(42)*u(25);
eqns(26) = k(65) - k(34)*u(22)*u(26);
eqns(27) = k(37)*u(23) - k(12)*u(7)*u(27);
eqns(28) = k(12)*u(7)*u(27) - k(22)*u(8)*u(28);
eqns(29) = k(22)*u(8)*u(28) - k(24)*u(29);
eqns(30) = k(66) - k(68)*u(30)*u(31);
eqns(31) = k(66)*u(39) - k(68)*u(30)*u(31);
eqns(32) = k(48)*u(30) - k(52)*u(32);
eqns(33) = k(48)*u(30) - k(43)*u(33);
eqns(34) = k(48)*u(30) - k(51)*u(34);
eqns(35) = k(53)*u(32) - k(55)*u(35) - k(59)*u(36)*u(35);
eqns(36) = k(57)*u(34) - k(58)*u(36);
eqns(37) = k(54)*in1(t)*u(35) - k(56)*u(37);
eqns(38) = k(56)*u(37) - k(49)*u(38) + k(50)*u(39);
eqns(39) = k(49)*u(38) - k(50)*u(39) + k(46)*u(41)*u(40) - k(39)*u(39);
eqns(40) = k(69) - k(46)*u(41)*u(40);
eqns(41) = k(44)*u(33) - k(45)*u(41)*u(39);
end
##### 2 CommentsShow NoneHide None
Walter Roberson on 3 Mar 2024
I recommend that you set up the equations using Symbolic Toolbox, and then follow the flow shown in the first example of odeFunction() to create an anonymous function that is suitable for passing to ode23s()
Ron_S on 3 Mar 2024
Thank you @Walter Roberson, I'm slowly getting there and I think it is going to work. I did a toy system run first and it worked.

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### Answers (1)

Sam Chak on 3 Mar 2024
I simply wanted to ensure that the code was functioning correctly with the given external inputs. Therefore, I didn't make any touch-ups or beautify the display. My main focus was on getting the code to run accurately. Perhaps I'll do some brief explanations afterwards.
% Set the initial values
u(1) = 5;
u(2) = 1;
u(3) = 1;
u(4) = 1;
u(5) = 1;
u(6) = 50;
u(7) = 1;
u(8) = 100;
u(9) = 5;
u(10) = 1;
u(11) = 1;
u(12) = 1;
u(13) = 1;
u(14) = 1;
u(15) = 1;
u(16) = 1;
u(17) = 1;
u(18) = 1;
u(19) = 1;
u(20) = 1;
u(21) = 1;
u(22) = 1;
u(23) = 1;
u(24) = 1;
u(25) = 1;
u(26) = 1;
u(27) = 1;
u(28) = 1;
u(29) = 1;
u(30) = 1;
u(31) = 1;
u(32) = 1;
u(33) = 1;
u(34) = 1;
u(35) = 1;
u(36) = 1;
u(37) = 1;
u(38) = 1;
u(39) = 1;
u(40) = 1;
u(41) = 1;
init = [u(1) u(2) u(3) u(4) u(5) u(6) u(7) u(8) u(9) u(10) u(11) u(12) u(13) u(14) u(15) u(16) u(17) u(18) u(19) u(20) u(21) u(22) u(23) u(24) u(25) u(26) u(27) u(28) u(29) u(30) u(31) u(32) u(33) u(34) u(35) u(36) u(37) u(38) u(39) u(40) u(41)];
tspan = [0 250];
[t,u] = ode23s(@(t,u) gene(t,u), tspan, init);
%% Plot results
t1 = 0:.5:250;
input1 = @(t) 0.5 + 0.5*(50<=t & t<100) + 1*(100<=t & t<150) + 0.5*(150<=t & t<200);
tiledlayout(4,1)
nexttile
hold on
plot(t1,input1(t1),'-k', 'LineWidth',2.0)
title(''); legend('CL(Input)')
xlabel('Time t'); ylabel('Concentration')
hold off
ylim([0 2])
nexttile
hold on
plot(t,u(:,6),'-',t,u(:,7),'-',t,u(:,8),'-',t,u(:,9),'-',t,u(:,18),'-',t,u(:,39),'-',t,u(:,29),'-', 'LineWidth',2.0)
title(''); legend('Cf','Cp','Ce','E','Ce1','Ce2','B')
xlabel('Time t'); ylabel('Concentration')
ylim([0 4])
nexttile
hold on
plot(t,u(:,2),'-',t,u(:,5),'-',t,u(:,11),'-', 'LineWidth',2.0)
title(''); legend('C','R','H')
xlabel('Time t'); ylabel('Concentration')
ylim([0 60])
nexttile
hold on
plot(t,u(:,1),'-',t,u(:,3),'-',t,u(:,10),'-',t,u(:,12),'-',t,u(:,13),'-',t,u(:,4),'-', 'LineWidth',2.0)
title('','FontSize',16); legend('Sci','Sr','HR','Sp','P','Sh')
xlabel('Time t'); ylabel('Concentration')
ylim([0 1.5])
%% ODE System
function eqns = gene(t, u)
% Using u = [Sci C Sr Sh R Cf Cp Ce E HR H Sp P Sb Rb Cf1 Sci1 C1 Sb1 Sh1 Rb1 HR1 Cp1 Ce1 E1 H1 CO Cg B Sci2 C2 Sr2 Sn2 Sp2 R2 P2 Cf2 Cp2 Ce2 H2 HR2]
% Equation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
%
% Using k17=p1, k18=p2,k19=p3, k20=mu, k21=eta, k22=alpha, k23=theta
% Set the model parameters
k1 = 3;
k2 = 2;
k3 = 5;
k4 = 4;
k5 = 5;
k6 = 1;
k7 = 4;
k8 = 3;
k9 = 1;
k10 = 1.2;
k11 = 1;
k12 = 9;
k13 = 1;
k14 = 1;
k15 = 1;
k16 = 30;
k17 = 30;
k18 = 1.5;
k19 = 25;
k20 = 1;
k21 = 10;
k22 = 20;
k23 = 1;
k24 = 0.5;
k25 = 1;
k26 = 1;
k27 = 1;
k28 = 1;
k29 = 1;
k30 = 1;
k31 = 1;
k32 = 1;
k33 = 1;
k34 = 1;
k35 = 1;
k36 = 1;
k37 = 1;
k38 = 1;
k39 = 1;
k40 = 1;
k41 = 1;
k42 = 1;
k43 = 1;
k44 = 1;
k45 = 1;
k46 = 1;
k47 = 1;
k48 = 1;
k49 = 1;
k50 = 1;
k51 = 1;
k52 = 1;
k53 = 20;
k54 = 1;
k55 = 1;
k56 = 1;
k57 = 1;
k58 = 1;
k59 = 1;
k60 = 1;
k61 = 1;
k62 = 1;
k63 = 1;
k64 = 1;
k65 = 1;
k66 = 1;
k67 = 1;
k68 = 1;
k69 = 1;
k70 = 1;
%% Perform the numerical integration
k = [k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12,k13,k14,k15,k16,k17,k18,k19,k20,k21,k22,k23,k24,k25,k26,k27,k28,k29,k30,k31,k32,k33,k34,k35,k36,k37,k38,k39,k40,k41,k42,k43,k44,k45,k46,k47,k48,k49,k50,k51,k52,k53,k54,k55,k56,k57,k58,k59,k60,k61,k62,k63,k64,k65,k66,k67,k68,k69,k70];
%% Define anonymous function input1(t)
% Time: 0<=t<50 50<=t<100 100<=t<150 150<=t<200 200<=t
% Input: 0.5 1 1.5 1 0.5
in1 = @(t) 0.5 + 0.5*(50<=t & t<100) + 1*(100<=t & t<150) + 0.5*(150<=t & t<200);
in2 = @(t) 0.5 + 0.5*(50<=t & t<100) + 1*(100<=t & t<150) + 0.5*(150<=t & t<200);
in3 = @(t) 0.5 + 0.5*(50<=t & t<100) + 1*(100<=t & t<150) + 0.5*(150<=t & t<200);
in4 = @(t) 0.5 + 0.5*(50<=t & t<100) + 1*(100<=t & t<150) + 0.5*(150<=t & t<200);
eqns = zeros(41,1); % To start with we have 41 empty equations
eqns(1) = k(60) - k(61)*u(1)*u(2);
eqns(2) = k(70)*u(8) - k(61)*u(1)*u(2);
eqns(3) = k(1)*u(1) - k(2)*u(3);
eqns(4) = k(1)*u(1) - k(10)*u(4);
eqns(5) = k(17)*u(3) - k(11)*u(5) - k(14)*u(13)*u(5);
eqns(6) = k(3)*in1(t) + k(26)*in2(t) - k(4)*u(6);
eqns(7) = k(4)*u(6) - k(5)*u(7) + k(6)*u(8) - k(12)*u(7)*u(27);
eqns(8) = k(5)*u(7) - k(6)*u(8) + k(9)*u(10)*u(11) - k(7)*u(8) + k(8)*u(9) - k(21)*u(8) - k(22)*u(8)*u(28);
eqns(9) = k(7)*u(8) - k(8)*u(9);
eqns(10) = k(18)*u(4) - k(15)*u(10)*u(8);
eqns(11) = k(62) - k(9)*u(10)*u(11);
eqns(12) = k(1)*u(1) - k(13)*u(12);
eqns(13) = k(19)*u(12) - k(16)*u(13);
eqns(14) = k(1)*u(1) - k(27)*u(14);
eqns(15) = k(28)*u(14) - k(29)*u(15);
eqns(16) = k(47)*in3(t)*u(21) - k(30)*u(16);
eqns(17) = k(63) - k(67)*u(17)*u(18);
eqns(18) = k(64)*u(24) - k(67)*u(17)*u(18);
eqns(19) = k(20)*u(17) - k(23)*u(19);
eqns(20) = k(20)*u(17) - k(25)*u(20);
eqns(21) = k(35)*u(19) - k(36)*u(21);
eqns(22) = k(33)*u(20) - k(38)*u(22)*u(24);
eqns(23) = k(30)*u(16) - k(31)*u(23) + k(32)*u(24) - k(37)*u(23);
eqns(24) = k(31)*u(23) - k(32)*u(24) - k(41)*u(24) + k(42)*u(25) + k(34)*u(22)*u(26) - k(40)*u(24);
eqns(25) = k(41)*u(24) - k(42)*u(25);
eqns(26) = k(65) - k(34)*u(22)*u(26);
eqns(27) = k(37)*u(23) - k(12)*u(7)*u(27);
eqns(28) = k(12)*u(7)*u(27) - k(22)*u(8)*u(28);
eqns(29) = k(22)*u(8)*u(28) - k(24)*u(29);
eqns(30) = k(66) - k(68)*u(30)*u(31);
eqns(31) = k(66)*u(39) - k(68)*u(30)*u(31);
eqns(32) = k(48)*u(30) - k(52)*u(32);
eqns(33) = k(48)*u(30) - k(43)*u(33);
eqns(34) = k(48)*u(30) - k(51)*u(34);
eqns(35) = k(53)*u(32) - k(55)*u(35) - k(59)*u(36)*u(35);
eqns(36) = k(57)*u(34) - k(58)*u(36);
eqns(37) = k(54)*in4(t)*u(35) - k(56)*u(37);
eqns(38) = k(56)*u(37) - k(49)*u(38) + k(50)*u(39);
eqns(39) = k(49)*u(38) - k(50)*u(39) + k(46)*u(41)*u(40) - k(39)*u(39);
eqns(40) = k(69) - k(46)*u(41)*u(40);
eqns(41) = k(44)*u(33) - k(45)*u(41)*u(39);
end
##### 1 CommentShow -1 older commentsHide -1 older comments
Ron_S on 3 Mar 2024
Edited: Ron_S on 3 Mar 2024
Thank you so much @Sam Chak, this is exactly what I wanted to see. I'm learning how to do this in a more succinct and efficient way as suggested by @Walter Roberson, and I think I'm slowly getting there. Thank you to both contributors.

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