estimation of parameters by using lsqnonlin for system of differential equations

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time = data(:,1);
A_data= data(:,2);
beta0 = 0.5;
alpha0 = 0.004;
gamma0 = 0.1;
upsilon0 = 0.13;
epsilon0= 0.07;
lamda0= 0.1;
sigma0= 0.07;
kappa0= 0.03;
nu0 = 0.0001;
xi0 = 0.0002;
lb =[0,0,0,0,0,0,0,0,0,0]; ub = [1,1,1,1,1,1,1,1,1,1];
B0 = [beta0; alpha0; gamma0; upsilon0; epsilon0; lamda0; sigma0; kappa0; nu0; xi0 ];
options = optimoptions(@lsqnonlin,'Algorithm','trust-region-reflective');
[B,resnorm,RESIDUAL,exitflag,OUTPUT,LAMBDA,Jacobian] = lsqnonlin(@Kinetics,B0,time,A_data,lb,ub,options );
disp(B)
function Av = Kinetics(B,time);
plot(time,A_data,time,Av)
function A = Kinetics(B, t)
x0 = [1217378052,100,3,2,1,1,1,1];
[T,Av] = ode45(@DifEq, t, x0);
function dA = DifEq(t, x)
N = 1390000000;
pi = 70000;
zeta = 0.1;
eta = 0.2;
theta = 0.3;
iota = 0.3;
delta = 0.1;
rho = 0.5;
mu = 0.0000425;
beta = B(1);
alpha =B(2);
gamma = B(3);
upsilon =B(4);
epsilon = B(5);
lamda = B(6);
sigma = B(7);
kappa = B(8);
nu = B(9);
xi = B(10);
xdot = zeros(8,1);
xdot(1) = pi -beta*(zeta*x(3)+eta*x(4)+theta*x(5)+iota*x(6))*(x(1)/N) -mu*x(1);
xdot(2) = beta*(zeta*x(3)+eta*x(4)+theta*x(5)+iota*x(6))*(x(1)/N) -(delta+mu)*x(2);
xdot(3) = rho*delta*x(2)-(lamda+gamma+nu+mu)*x(3);
xdot(4) = (1-rho)*delta*x(2)-(sigma+kappa+mu)*x(4);
xdot(5) = lamda*x(3) + sigma*x(4)-(alpha+upsilon+mu)*x(5);
xdot(6) = alpha*x(6) + kappa*x(4)- (epsilon+xi+mu)*x(6);
xdot(7) = gamma*x(3) + upsilon*x(5) + epsilon*x(6);
xdot(8) = nu*x(3) + xi*x(6);
dA = xdot;
end
A = Av(:,1);
end
i got an error;
Error using lsqnonlin (line 190)
Invalid datatype. Options argument must be created with OPTIMOPTIONS.
Error in error_14bmodel (line 19)
[B,resnorm,RESIDUAL,exitflag,OUTPUT,LAMBDA,Jacobian] = lsqnonlin(@Kinetics,B0,time,A_data,lb,ub,options );
i am fresher of matlab .i requesting you sir please help anyone how to resolve this error
if possible please refer some covid 19 or malaria,dengue hepities b models sir
thank you sir

Answers (4)

Torsten
Torsten on 2 Apr 2022
[B,resnorm,RESIDUAL,exitflag,OUTPUT,LAMBDA,Jacobian] = lsqnonlin(@Kinetics,B0,time,A_data,lb,ub,options );
This is not the correct call to lsqnonlin, but to lsqcurvefit.
  3 Comments
Torsten
Torsten on 2 Apr 2022
You modify this by calling lsqcurvefit instead of lsqnonlin:
[B,resnorm,RESIDUAL,exitflag,OUTPUT,LAMBDA,Jacobian] = lsqcurvefit(@Kinetics,B0,time,A_data,lb,ub,options );
mallela ankamma rao
mallela ankamma rao on 2 Apr 2022
Thank you sir I request you sir Please modify this by Lsqnonlin .i didn't know how to modify this.i am struggling alot sir

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Star Strider
Star Strider on 2 Apr 2022
I recognise my code.
It should have the correct lsqcurvefit call, so all you need to do is substitute your differnetial equations and data into it.
An updated version of that code is:
t=[0.1
0.2
0.4
0.6
0.8
1
1.5
2
3
4
5
6];
c=[0.902 0.06997 0.02463 0.00218
0.8072 0.1353 0.0482 0.008192
0.6757 0.2123 0.0864 0.0289
0.5569 0.2789 0.1063 0.06233
0.4297 0.3292 0.1476 0.09756
0.3774 0.3457 0.1485 0.1255
0.2149 0.3486 0.1821 0.2526
0.141 0.3254 0.194 0.3401
0.04921 0.2445 0.1742 0.5277
0.0178 0.1728 0.1732 0.6323
0.006431 0.1091 0.1137 0.7702
0.002595 0.08301 0.08224 0.835];
theta0=rand(6,1);
[theta,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=lsqcurvefit(@kinetics,theta0,t,c,zeros(size(theta0)));
Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance.
fprintf(1,'\tRate Constants:\n')
Rate Constants:
for k1 = 1:length(theta)
fprintf(1, '\t\tTheta(%d) = %8.5f\n', k1, theta(k1))
end
Theta(1) = 0.89692 Theta(2) = 0.25742 Theta(3) = 0.31073 Theta(4) = 0.61683 Theta(5) = 0.62818 Theta(6) = 0.00000
tv = linspace(min(t), max(t));
Cfit = kinetics(theta, tv);
figure
hd = plot(t, c, 'p');
for k1 = 1:size(c,2)
CV(k1,:) = hd(k1).Color;
hd(k1).MarkerFaceColor = CV(k1,:);
end
hold on
hlp = plot(tv, Cfit);
for k1 = 1:size(c,2)
hlp(k1).Color = CV(k1,:);
end
hold off
grid
xlabel('Time')
ylabel('Concentration')
legend(hlp, compose('C_%d',1:size(c,2)), 'Location','N')
function C=kinetics(theta,t)
c0=[1;0;0;0];
[T,Cv]=ode45(@DifEq,t,c0);
%
function dC=DifEq(t,c)
dcdt=zeros(4,1);
dcdt(1)=-theta(1).*c(1)-theta(2).*c(1);
dcdt(2)= theta(1).*c(1)+theta(4).*c(3)-theta(3).*c(2)-theta(5).*c(2);
dcdt(3)= theta(2).*c(1)+theta(3).*c(2)-theta(4).*c(3)+theta(6).*c(4);
dcdt(4)= theta(5).*c(2)-theta(6).*c(4);
dC=dcdt;
end
C=Cv;
end
.
  37 Comments
Torsten
Torsten on 7 Apr 2022
Why do you plot in "kinetics" and not in your script when lsqnonlin has finished ?
After you got the parameters from lsqnonlin, you can call "kinetics" to get Cv for your plot:
c_data= [503,502,575,513,484,376,405,406,339,490,444,397,341,397,341,356,387,359,346];
time = [1 2 3 4 5 7 9 11 13 15 17 21 25 30 40 50 60];
beta0 = 0.5;
alpha0 = 0.004;
gamma0 = 0.1;
upsilon0 = 0.13;
epsilon0= 0.07;
lamda0= 0.1;
sigma0= 0.07;
kappa0= 0.03;
nu0 = 0.0001;
xi0 = 0.0002;
lb =[0,0,0,0,0,0,0,0,0,0]; ub = [1,1,1,1,1,1,1,1,1,1];
p0 = [beta0; alpha0; gamma0; upsilon0; epsilon0; lamda0; sigma0; kappa0; nu0; xi0 ];
options = optimoptions(@lsqnonlin,'Algorithm','trust-region-reflective');
[p,resnorm,RESIDUAL,exitflag,OUTPUT,LAMBDA,Jacobian] = lsqnonlin(@(p) kinetics(p,time,c_data),p0,lb,ub,options );
c_data_minus_Cv=kinetics(p,time,c_data);
Cv = c_data - c_data_minus_Cv;
plot(time,c_data, time,Cv)

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mallela ankamma rao
mallela ankamma rao on 13 Apr 2022
good morning sir
i got parameter values from lsqnonlin. i got figure for c_data vs fitting .
i have doubt sir .was the graph is correct or not?
if not i request you please modify sir
thank you sir
my code;
time = data(:,1);
c_data= data(:,2);
beta0 = 0.5;
alpha0 = 0.004;
gamma0 = 0.1;
upsilon0 = 0.13;
epsilon0= 0.07;
lamda0= 0.1;
sigma0= 0.07;
kappa0= 0.03;
nu0 = 0.0001;
xi0 = 0.0002;
lb =[0,0,0,0,0,0,0,0,0,0]; ub = [1,1,1,1,1,1,1,1,1,1];
p0 = [beta0; alpha0; gamma0; upsilon0; epsilon0; lamda0; sigma0; kappa0; nu0; xi0 ];
options = optimoptions(@lsqnonlin,'Algorithm','trust-region-reflective');
[p,resnorm,RESIDUAL,exitflag,OUTPUT,LAMBDA,Jacobian] = lsqnonlin(@(p) immanuel(p,time,c_data),p0,lb,ub,options );
disp(p)
function C=immanuel(p,time,c_data)
c0 = [1217378052,100,10,5,3,3,1,1];
[T,Cv]=ode45(@DifEq,time,c0);
function dC=DifEq(time,c)
N = 1390000000;
pi = 150;
zeta = 0.1;
eta = 0.2;
theta = 0.3;
iota = 0.3;
delta = 0.1;
rho = 0.5;
mu = 0.0000425;
beta = p(1);
alpha =p(2);
gamma = p(3);
upsilon =p(4);
epsilon = p(5);
lamda = p(6);
sigma = p(7);
kappa = p(8);
nu = p(9);
xi = p(10);
dcdt = zeros(8,1);
dcdt(1) = pi -beta*(zeta*c(3)+eta*c(4)+theta*c(5)+iota*c(6))*(c(1)/N) -mu*c(1);
dcdt(2) = beta*(zeta*c(3)+eta*c(4)+theta*c(5)+iota*c(6))*(c(1)/N) -(delta+mu)*c(2);
dcdt(3) = rho*delta*c(2)-(lamda+gamma+nu+mu)*c(3);
dcdt(4) = (1-rho)*delta*c(2)-(sigma+kappa+mu)*c(4);
dcdt(5) = lamda*c(3) + sigma*c(4)-(alpha+upsilon+mu)*c(5);
dcdt(6) = alpha*c(6) + kappa*c(4)- (epsilon+xi+mu)*c(6);
dcdt(7) = gamma*c(3) + upsilon*c(5) + epsilon*c(6);
dcdt(8) = nu*c(3) + xi*c(6);
dC = dcdt;
end
C=c_data-Cv(:,2);
plot(time,c_data,time,C)
end
  3 Comments
mallela ankamma rao
mallela ankamma rao on 13 Apr 2022
thanks alot for your reply dear Torsten sir
i tried sir but it showing error sir
i felt very difficult about graph sir
please resolve this sir
code:
data = [ 1 36456
2 70217
3 108526
4 145164
5 181275
6 213547
7 240654
8 272721
9 299078
10 333744
11 363778
12 394132
13 421468
14 443409
15 469810
16 487974
17 514736
18 541727
19 568561
20 593150
21 612324
22 636210
23 660446
24 684370
25 706720
26 725294
27 745627
28 761699
29 782228
30 804185
31 823231
32 842753
33 862213
34 881151
35 900118
36 918709
37 937090
38 955342
39 973350
40 991455
41 1009475
42 1026953
43 1044144
44 1061441
45 1077789
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76 1478261
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78 1500598
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81 1535790
82 1548121
83 1560820
84 1573720
85 1586918
86 1600698
87 1614958
88 1629596
89 1644646
90 1659887
91 1675382
92 1691064
93 1706817
94 1722602
95 1738639
96 1754950
97 1771695
98 1788880
99 1806476
100 1824847];
time = data(:,1);
c_data= data(:,2);
beta0 = 0.5;
alpha0 = 0.004;
gamma0 = 0.1;
upsilon0 = 0.13;
epsilon0= 0.07;
lamda0= 0.1;
sigma0= 0.07;
kappa0= 0.03;
nu0 = 0.0001;
xi0 = 0.0002;
lb =[0,0,0,0,0,0,0,0,0,0]; ub = [1,1,1,1,1,1,1,1,1,1];
p0 = [beta0; alpha0; gamma0; upsilon0; epsilon0; lamda0; sigma0; kappa0; nu0; xi0 ];
options = optimoptions(@lsqnonlin,'Algorithm','trust-region-reflective');
[p,resnorm,RESIDUAL,exitflag,OUTPUT,LAMBDA,Jacobian] = lsqnonlin(@(p) immanuel(p,time,c_data),p0,lb,ub,options );
disp(p)
c_data_minus_Cv=immanuel(p,time,c_data);
Cv = c_data - c_data_minus_Cv;
plot(time,c_data, time,Cv)
c0 = [1217378052,100,10,5,3,3,1,1];
Cv = c_data - c_data_minus_Cv;
plot(time,c_data, time,Cv)
c0 = [1217378052,100,10,5,3,3,1,1];
[T,Cv]=ode45(@DifEq,time,c0);
function dC=DifEq(time,c)
N = 1390000000;
pi = 150;
zeta = 0.1;
eta = 0.2;
theta = 0.3;
iota = 0.3;
delta = 0.1;
rho = 0.5;
mu = 0.0000425;
beta = p(1);
alpha =p(2);
gamma = p(3);
upsilon =p(4);
epsilon = p(5);
lamda = p(6);
sigma = p(7);
kappa = p(8);
nu = p(9);
xi = p(10);
dcdt = zeros(8,1);
dcdt(1) = pi -beta*(zeta*c(3)+eta*c(4)+theta*c(5)+iota*c(6))*(c(1)/N) -mu*c(1);
dcdt(2) = beta*(zeta*c(3)+eta*c(4)+theta*c(5)+iota*c(6))*(c(1)/N) -(delta+mu)*c(2);
dcdt(3) = rho*delta*c(2)-(lamda+gamma+nu+mu)*c(3);
dcdt(4) = (1-rho)*delta*c(2)-(sigma+kappa+mu)*c(4);
dcdt(5) = lamda*c(3) + sigma*c(4)-(alpha+upsilon+mu)*c(5);
dcdt(6) = alpha*c(6) + kappa*c(4)- (epsilon+xi+mu)*c(6);
dcdt(7) = gamma*c(3) + upsilon*c(5) + epsilon*c(6)-mu*c(7);
dcdt(8) = nu*c(3) + xi*c(6);
dC = dcdt;
end
ERROR:
Unrecognized function or variable 'immanuel'.
Error in errrrr>@(p)kinetics(p,time,c_data) (line 20)
[p,resnorm,RESIDUAL,exitflag,OUTPUT,LAMBDA,Jacobian] = lsqnonlin(@(p) immanuel(p,time,c_data),p0,lb,ub,options );
Error in lsqnonlin (line 218)
initVals.F = feval(funfcn{3},xCurrent,varargin{:});
Error in errrrr (line 20)
[p,resnorm,RESIDUAL,exitflag,OUTPUT,LAMBDA,Jacobian] = lsqnonlin(@(p) immanuel(p,time,c_data),p0,lb,ub,options );
mallela ankamma rao
mallela ankamma rao on 13 Apr 2022
dear torsten sir and dear star strider sir
i request you please reply anyone how to modify this code sir
how to overcome this error sir

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Torsten
Torsten on 13 Apr 2022
data = [ 1 36456
2 70217
3 108526
4 145164
5 181275
6 213547
7 240654
8 272721
9 299078
10 333744
11 363778
12 394132
13 421468
14 443409
15 469810
16 487974
17 514736
18 541727
19 568561
20 593150
21 612324
22 636210
23 660446
24 684370
25 706720
26 725294
27 745627
28 761699
29 782228
30 804185
31 823231
32 842753
33 862213
34 881151
35 900118
36 918709
37 937090
38 955342
39 973350
40 991455
41 1009475
42 1026953
43 1044144
44 1061441
45 1077789
46 1093666
47 1109004
48 1124038
49 1138716
50 1153092
51 1167220
52 1181179
53 1195019
54 1208837
55 1222551
56 1236139
57 1249576
58 1262481
59 1276016
60 1289370
61 1302464
62 1315314
63 1328090
64 1340619
65 1353343
66 1365137
67 1376739
68 1388198
69 1399694
70 1411209
71 1422686
72 1434108
73 1445094
74 1456141
75 1467207
76 1478261
77 1489368
78 1500598
79 1511874
80 1523705
81 1535790
82 1548121
83 1560820
84 1573720
85 1586918
86 1600698
87 1614958
88 1629596
89 1644646
90 1659887
91 1675382
92 1691064
93 1706817
94 1722602
95 1738639
96 1754950
97 1771695
98 1788880
99 1806476
100 1824847];
time = data(:,1);
c_data= data(:,2);
beta0 = 0.5;
alpha0 = 0.004;
gamma0 = 0.1;
upsilon0 = 0.13;
epsilon0= 0.07;
lamda0= 0.1;
sigma0= 0.07;
kappa0= 0.03;
nu0 = 0.0001;
xi0 = 0.0002;
lb =[0,0,0,0,0,0,0,0,0,0]; ub = [1,1,1,1,1,1,1,1,1,1];
p0 = [beta0; alpha0; gamma0; upsilon0; epsilon0; lamda0; sigma0; kappa0; nu0; xi0 ];
options = optimoptions(@lsqnonlin,'Algorithm','trust-region-reflective');
[p,resnorm,RESIDUAL,exitflag,OUTPUT,LAMBDA,Jacobian] = lsqnonlin(@(p) immanuel(p,time,c_data),p0,lb,ub,options );
disp(p)
Cdata_minus_Cv =immanuel(p,time,c_data);
Cv = c_data-Cdata_minus_Cv;
plot(time,[c_data,Cv])
function C=immanuel(p,time,c_data)
c0 = [1217378052,100,10,5,3,3,1,1];
[T,Cv]=ode45(@DifEq,time,c0);
function dC=DifEq(time,c)
N = 1390000000;
pi = 150;
zeta = 0.1;
eta = 0.2;
theta = 0.3;
iota = 0.3;
delta = 0.1;
rho = 0.5;
mu = 0.0000425;
beta = p(1);
alpha =p(2);
gamma = p(3);
upsilon =p(4);
epsilon = p(5);
lamda = p(6);
sigma = p(7);
kappa = p(8);
nu = p(9);
xi = p(10);
dcdt = zeros(8,1);
dcdt(1) = pi -beta*(zeta*c(3)+eta*c(4)+theta*c(5)+iota*c(6))*(c(1)/N) -mu*c(1);
dcdt(2) = beta*(zeta*c(3)+eta*c(4)+theta*c(5)+iota*c(6))*(c(1)/N) -(delta+mu)*c(2);
dcdt(3) = rho*delta*c(2)-(lamda+gamma+nu+mu)*c(3);
dcdt(4) = (1-rho)*delta*c(2)-(sigma+kappa+mu)*c(4);
dcdt(5) = lamda*c(3) + sigma*c(4)-(alpha+upsilon+mu)*c(5);
dcdt(6) = alpha*c(6) + kappa*c(4)- (epsilon+xi+mu)*c(6);
dcdt(7) = gamma*c(3) + upsilon*c(5) + epsilon*c(6);
dcdt(8) = nu*c(3) + xi*c(6);
dC = dcdt;
end
C=c_data-Cv(:,2);
end
  14 Comments
mallela ankamma rao
mallela ankamma rao on 29 Apr 2022
Good afternoon sir
sir please tell me how to draw graph for infected data vs predicted for this lscurvefit code.
i wrote code like this
plot(time,infected ,time,Cv) but it shows error sir.
code is
data = [ 1 25
2 75
3 227
4 296
5 258
6 236
7 192
8 126
9 71
10 28
11 11
12 7];
time = data(1:12,1);
infected= data(1:12,2);
beta0 = 1;
lamdaa0= 0.019;
lamdas0= 0.0715;
etas0 = 0.03;
etaq0 = 0.04;
gammaa0 = 0.2;
gammaq0 = 0.13;
gammah0 = 0.07;
mua0 = 0.0001;
muh0 = 0.0002;
lb =[0,0,0,0,0,0,0,0,0,0]; ub = [1,1,1,1,1,1,1,1,1,1];
B0 = [beta0; lamdaa0; lamdas0; etas0; etaq0; gammaa0; gammaq0; gammah0; mua0; muh0 ];
options=optimset('MaxFunEvals', 10000, 'MaxIter', 10000, 'TolFun', 0.0001, 'TolX',0.0001,'Display','on');
[B,resnorm,RESIDUAL,exitflag,OUTPUT,LAMBDA,Jacobian] = lsqcurvefit(@diff1,B0,time,infected,lb,ub,options);
disp(B)
%plot(time,infected,time,C)
function C = diff1(B,time)
x0 = [1217378052,120,3,2,1,1,0,0];
[t,Cv] = ode45(@DifEq, time, x0);
function dC = DifEq(t, x)
N = 1390000000;
pi = 700 ;
zetaa = 0.1;
zetas = 0.2;
zetaq = 0.3;
zetah = 0.3;
omega = 0.2;
theta = 0.5;
mu = 0.0000425;
beta = B(1);
lamdaa= B(2);
lamdas = B(3);
etas = B(4);
etaq = B(5);
gammaa = B(6);
gammaq = B(7);
gammah = B(8);
mua = B(9);
muh = B(10);
xdot = zeros(7,1);
xdot(1) = pi -beta*(zetaa*x(3) +zetas*x(4) +zetaq*x(5)+zetah*x(6))*(x(1)/N) -mu*x(1);
xdot(2) = beta*(zetaa*x(3) +zetas*x(4) +zetaq*x(5)+zetah*x(6))*(x(1)/N) -(omega+mu)*x(2);
xdot(3) = theta*omega*x(2)-(lamdaa+gammaa+mua+mu)*x(3);
xdot(4) = (1-theta)*omega*E-(lamdas+etas+mu)*x(4);
xdot(5) = lamdaa*Ia+lamdas*Is-(etaq+gammaq+mu)*x(5);
xdot(6) = etas*Is+etaq*Q- (gammah+muh+mu)*x(6);
xdot(7) = gammaa*x(4) + gammaq*x(5) + gammah*x(6)
end
C = Cv(:,2);
plot(time,infected ,time,Cv)
end
Error is
Unrecognized function or variable 'infected'.
Error in lscurvfit6>diff1 (line 72)
plot(time,infected ,time,Cv)
Error in lsqcurvefit (line 225)
initVals.F = feval(funfcn_x_xdata{3},xCurrent,XDATA,varargin{:});
Error in lscurvfit6 (line 19)
[B,resnorm,RESIDUAL,exitflag,OUTPUT,LAMBDA,Jacobian] = lsqcurvefit(@diff1,B0,time,infected,lb,ub,options);
Caused by:
Failure in initial objective function evaluation. LSQCURVEFIT cannot continue.
mallela ankamma rao
mallela ankamma rao on 29 Apr 2022
Mr torsten sir and Mr star strider sir please reply anyone
how can i draw graph for above code.

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