Model fitting and validation for Hubei, China

The following document presents the implemented code to fit and validate a discrete compartmental model that describes the COVID-19 transmission, specifically for the Hubei province. Note that we used the GSUA_CSB toolbox available in GitHub.

1. Load the real data

First, we load the real data that is actual, deads, and recovered cases. Note that we can define the actual cases as follow

Actual cases = accumulative - dead - recovered cases

addpath Tool/

addpath Model/

addpath Data/

load('China.mat');

Extension = length(ActualCases);

ydata = zeros(3,Extension);

ydata(1,:) = ActualCases;

ydata(2,:) = Dead;

ydata(3,:) = Recovered;

figure

subplot(1,3,1)

plot(ActualCases)

xlabel('Time(days)')

ylabel('Number of cases')

title('Active cases')

grid on

subplot(1,3,2)

plot(Dead)

xlabel('Time(days)')

title('Dead')

grid on

subplot(1,3,3)

plot(Recovered)

xlabel('Time(days)')

title('Recovered')

grid on

We take this data from Github according to [1], from January 21 to April 8, 2020.

Reference:

[1] E. Dong, H. Du, L. Gardner, An interactive web-based dashboard totrack COVID-19 in real time, The Lancet Infectious Diseases 20 (2020) 533–534.

2. Model definition

First, we define the factor (initial values and parameters) names followed by the model outputs.

VarNames={'N_0','E_0','P_0','J_1','J_{L}','D','R_j','RJ_L'};
OutNames={'S_f','S_q','E_f','E_q','L_f','L_q','H_f','H_q','J','D',...
        'R','T','RJH','RJL','RJ','Tot','JH','JL','J_{~P}','J_{P}'};
ParNames={'\beta_L','\beta_T', '\beta_P', '\phi_{EP}',...
            '\lambda_{fq}', '\vartheta_E','\gamma_L', 'k_L',...
            'k_P', '\phi_T','\lambda_{qf}', '\psi_e', '\phi_{PH}',...
            '\delta','m','\eta_L', '\vartheta_P','\gamma_H', 'z',...
            '\phi_{PL}','\eta_\vartheta','nons','a_L','b_L',...
            'a_\mu','b_\mu','\mu','a_H','b_H','\nu','time'};
FullNames = [VarNames,ParNames];
ModelName = 'ChimeraModel';

Then, we define the interval values for model factors; their values are according to the parameter nature and some studies in literature, for more information about it, see the main paper document.

Range1 = [
8e7 5.8e7;            % N_0
2000;                 % E^2_0
400;                  % P_0
    ydata(1,1) ydata(1,1);  % J_1
0;                    % J_L
    ydata(2,1) ydata(2,1);  % D
    ydata(3,1) ydata(3,1);  % R_j
0;                    % RJ_L
    ];
Range2=[
7 0.9;    % beta_L
 1;      % beta_T          
2 0.6;    % beta_P
8;      % phi_{EP}
1;      % lambda_{fq}
1;      % vartheta_E
 1;      % gamma_L
 10;     % k_L
 1;      % k_P
8;      % phi_T
1;      % lambda_{qf}
0;      % psi_e
3;      % phi_{PH}
 0.15;   % delta
 0;      % Migration input
 1;      % eta_L
 1;      % vartheta_P
 1;      % gamma_H 
 30;     % z 
 3;      % phi_{PL}
 1;      % eta_vartheta
 0;      % nons
 15;     % a_L
 15;     % b_L
 15;     % a_mu
 15;     % b_mu
 1;      % mu
 15;     % a_H
 15;     % b_H
500;     % nu
41      % m
    ];
RangeT = [Range1; Range2];

Finally, we define the real data domain

D = length(ydata(1,:));
Domain = [0 D-1];
xdata = 0:D-1;
opt = zeros(2,4)

and create the struture data that manage the model simulations, estimations, and other methodologies

[T,~] = gsua_dataprep(ModelName,RangeT,'domain',Domain,'names',FullNames,'out_names',OutNames);
Setting environment to work with user-defined function

3. Parameter estimation

Because of the nature of real data (the presence of three curves to fit the model), we perform multi-objective optimization to estimate the model factors. So, we select the input numbers to fit with real data: 9: actual cases; 10: Dead cases; 15: Identified recovered cases.

T.Properties.CustomProperties.output = [9,10,15];

Then, we define the optimization options according to the researcher’s necessities. For COVID-19 model, the estimation options are as follows

Solver='fmincon';
    
Opt = optimoptions('fmincon','UseParallel',false,'MaxFunctionEvaluations',30000, ...
    'MaxIterations', 10000, 'Display','off');

We develop a code that saves those estimations which are less than the pre-defined cost function value. Thus, we perform an unknown number of parameter estimations until obtaining N that do not beat this pre-defined value.

N = 9000;                   % Number of estimations
Best = zeros(size(T,1),N);  % Matrix that saves factor's estimated values
IpBest = Best;              % Matrix that saves optimization method initial point
Residual = zeros(1,N);      % Vector that saves estimations cost function value
ResBase = 6e-2;             % Pre-defined cost function value
parfor i = 1:N
    ResTemp = 5e8;
    while ResTemp > ResBase
        [T_est, ResTemp, IP] = gsua_pe(T,xdata,ydata,...
            'solver',Solver,'N',1,'opt',Opt,'save',false);
    end
    IpBest(:,i) = IP;
    Best(:,i) = T_est.Estfmincon;
    Residual(i) = ResTemp;
    i
end
%Sort the estimated values according its cost function value
[Residual, Idx] = sort(Residual);
T.Est = Best(:, Idx);
IpBest = IpBest(:,Idx);
gsua_save(T)
movefile("portable.mat", "Results/ChinaResults/ChinaEstimations.mat");
save('Results/ChinaResults/ChinaEstiamtionsData.mat',...
    'ydata','Residual','IpBest')

Load and plot all factor families estimated in the previous step.

load('Results/ChinaResults/ChinaEstimations.mat')
load('Results/ChinaResults/ChinaEstiamtionsData.mat')
T_est = gsua_load(T);
Setting environment to work with user-defined function

gsua_eval(T.Est,T_est,xdata,ydata);

figure

histogram(Residual)

Now, we can filter more the resulted estimations and select a defined number of families with the smallest cost function value.

Filtered = 114;

gsua_eval(T.Est(:,1:Filtered),T_est,xdata,ydata);

4. Plot nominal curve

Then, we obtain a nominal curve by estimating the median value for each factor using the filtered families in the previous step.

Median = median(T.Est(:,1:Filtered),2);

gsua_eval(Median,T_est,xdata,ydata);

5. Identification analisys

We use the following function to analyze the practical identifiability of the system for the factor families implemented to get the nominal curve (median of each factor). Also, it estimates the factors interval using a parametric methodology; for more information about it, see the main paper document.

T = gsua_ia(T_est, T.Est(:,1:Filtered));

T = 28×4 table 

	Range		Nominal	Est																																																																index
1 E^2_0	769.1663	955.1752	862.1708	119.6126	119.6126	119.6126	358.3136	358.3136	358.3136	1.0417e+03	1.0417e+03	1.0417e+03	1.0336e+03	1.0336e+03	1.0336e+03	169.4135	169.4135	169.4135	783.8378	783.8378	783.8378	635.4436	635.4436	635.4436	1.1722e+03	1.1722e+03	1.1722e+03	491.8075	491.8075	491.8075	721.0974	721.0974	721.0974	1.0036e+03	1.0036e+03	1.0036e+03	1.5241e+03	1.5241e+03	1.5241e+03	740.4483	740.4483	740.4483	872.0417	872.0417	872.0417	419.2755	419.2755	419.2755	1.3308e+03	1.3308e+03	1.3308e+03	852.2999	852.2999	852.2999	1.2517e+03	1.2517e+03	1.2517e+03	752.5699	752.5699	752.5699	1.4460e+03	1.4460e+03	1.4460e+03	1.5859e+03	1.5859e+03	1.5859e+03	1.0326e+03	1.0326e+03	1.0326e+03	813.5525	813.5525	813.5525	517.4892	517.4892	517.4892	1.2772e+03	1.2772e+03	1.2772e+03	465.9704	465.9704	465.9704	1.4588e+03	1.4588e+03	1.4588e+03	1.1271e+03	1.1271e+03	1.1271e+03	776.8274	776.8274	776.8274	1.0898e+03	1.0898e+03	1.0898e+03	1.3695e+03	1.3695e+03	1.3695e+03	566.4236	566.4236	566.4236	1.5317e+03	1.5317e+03	1.5317e+03	349.1122	349.1122	349.1122	1.2985e+03	1.2985e+03	1.2985e+03	848.3159	848.3159	848.3159	732.5444	732.5444	732.5444	1.5688e+03	1.5688e+03	1.5688e+03	0.2063
2 P_0	115.3837	147.8034	131.5936	158.3881	158.3881	158.3881	284.2355	284.2355	284.2355	43.1332	43.1332	43.1332	131.1020	131.1020	131.1020	164.8395	164.8395	164.8395	65.4614	65.4614	65.4614	130.1873	130.1873	130.1873	112.3000	112.3000	112.3000	133.9834	133.9834	133.9834	139.5791	139.5791	139.5791	229.3943	229.3943	229.3943	31.9750	31.9750	31.9750	118.8676	118.8676	118.8676	168.3362	168.3362	168.3362	154.0125	154.0125	154.0125	119.1164	119.1164	119.1164	169.4080	169.4080	169.4080	252.6324	252.6324	252.6324	353.2246	353.2246	353.2246	38.3342	38.3342	38.3342	120.4603	120.4603	120.4603	166.1618	166.1618	166.1618	90.3741	90.3741	90.3741	116.0037	116.0037	116.0037	224.4677	224.4677	224.4677	134.9129	134.9129	134.9129	47.6482	47.6482	47.6482	132.0851	132.0851	132.0851	213.1348	213.1348	213.1348	114.7985	114.7985	114.7985	196.8357	196.8357	196.8357	95.5753	95.5753	95.5753	46.4536	46.4536	46.4536	246.3735	246.3735	246.3735	186.5271	186.5271	186.5271	71.4126	71.4126	71.4126	119.9953	119.9953	119.9953	119.6773	119.6773	119.6773	0.1131
3 \beta_L	0.7972	0.8083	0.8028	0.8435	0.8435	0.8435	0.8398	0.8398	0.8398	0.8044	0.8044	0.8044	0.7884	0.7884	0.7884	0.8092	0.8092	0.8092	0.8441	0.8441	0.8441	0.7807	0.7807	0.7807	0.7919	0.7919	0.7919	0.8364	0.8364	0.8364	0.7821	0.7821	0.7821	0.7390	0.7390	0.7390	0.7826	0.7826	0.7826	0.8041	0.8041	0.8041	0.7811	0.7811	0.7811	0.7826	0.7826	0.7826	0.8233	0.8233	0.8233	0.8339	0.8339	0.8339	0.8073	0.8073	0.8073	0.7578	0.7578	0.7578	0.8101	0.8101	0.8101	0.8127	0.8127	0.8127	0.8003	0.8003	0.8003	0.8024	0.8024	0.8024	0.8005	0.8005	0.8005	0.7922	0.7922	0.7922	0.8216	0.8216	0.8216	0.7980	0.7980	0.7980	0.7969	0.7969	0.7969	0.7582	0.7582	0.7582	0.8100	0.8100	0.8100	0.8072	0.8072	0.8072	0.7914	0.7914	0.7914	0.7646	0.7646	0.7646	0.7908	0.7908	0.7908	0.8141	0.8141	0.8141	0.8274	0.8274	0.8274	0.8031	0.8031	0.8031	0.8281	0.8281	0.8281	0.1036
4 \beta_T	0.1213	0.1396	0.1304	0.1833	0.1833	0.1833	0.1538	0.1538	0.1538	0.1644	0.1644	0.1644	0.1346	0.1346	0.1346	0.1920	0.1920	0.1920	0.2630	0.2630	0.2630	0.1686	0.1686	0.1686	0.1651	0.1651	0.1651	0.1655	0.1655	0.1655	0.1180	0.1180	0.1180	0.1249	0.1249	0.1249	0.1187	0.1187	0.1187	0.1885	0.1885	0.1885	0.1159	0.1159	0.1159	0.1521	0.1521	0.1521	0.0861	0.0861	0.0861	0.1866	0.1866	0.1866	0.0947	0.0947	0.0947	0.0765	0.0765	0.0765	0.1713	0.1713	0.1713	0.1228	0.1228	0.1228	0.1193	0.1193	0.1193	0.1253	0.1253	0.1253	0.1543	0.1543	0.1543	0.1180	0.1180	0.1180	0.1287	0.1287	0.1287	0.1909	0.1909	0.1909	0.1173	0.1173	0.1173	0.0888	0.0888	0.0888	0.0996	0.0996	0.0996	0.1153	0.1153	0.1153	0.1856	0.1856	0.1856	0.1823	0.1823	0.1823	0.1162	0.1162	0.1162	0.0807	0.0807	0.0807	0.0942	0.0942	0.0942	0.1686	0.1686	0.1686	0.1322	0.1322	0.1322	0.1769
5 \beta_P	0.3738	0.3982	0.3860	0.2710	0.2710	0.2710	0.3749	0.3749	0.3749	0.4987	0.4987	0.4987	0.3891	0.3891	0.3891	0.4240	0.4240	0.4240	0.3658	0.3658	0.3658	0.4231	0.4231	0.4231	0.4057	0.4057	0.4057	0.4883	0.4883	0.4883	0.3964	0.3964	0.3964	0.4246	0.4246	0.4246	0.3379	0.3379	0.3379	0.3850	0.3850	0.3850	0.4125	0.4125	0.4125	0.3783	0.3783	0.3783	0.3628	0.3628	0.3628	0.3405	0.3405	0.3405	0.4584	0.4584	0.4584	0.4259	0.4259	0.4259	0.3949	0.3949	0.3949	0.3772	0.3772	0.3772	0.3564	0.3564	0.3564	0.3489	0.3489	0.3489	0.3870	0.3870	0.3870	0.4286	0.4286	0.4286	0.3707	0.3707	0.3707	0.3455	0.3455	0.3455	0.3848	0.3848	0.3848	0.4218	0.4218	0.4218	0.4339	0.4339	0.4339	0.2802	0.2802	0.2802	0.3726	0.3726	0.3726	0.3305	0.3305	0.3305	0.3695	0.3695	0.3695	0.4825	0.4825	0.4825	0.4371	0.4371	0.4371	0.4268	0.4268	0.4268	0.2668	0.2668	0.2668	0.0981
6 \phi_{EP}	0.6819	1.1527	0.9173	2.9684	2.9684	2.9684	1.4690	1.4690	1.4690	0.9932	0.9932	0.9932	0.9632	0.9632	0.9632	2.0582	2.0582	2.0582	0.8714	0.8714	0.8714	1.4018	1.4018	1.4018	0.5962	0.5962	0.5962	1.2017	1.2017	1.2017	2.1138	2.1138	2.1138	0.8016	0.8016	0.8016	0.6760	0.6760	0.6760	0.7311	0.7311	0.7311	1.6155	1.6155	1.6155	2.1110	2.1110	2.1110	0.8409	0.8409	0.8409	0.6897	0.6897	0.6897	0.7603	0.7603	0.7603	1.1847	1.1847	1.1847	0.5773	0.5773	0.5773	0.6191	0.6191	0.6191	0.7778	0.7778	0.7778	2.7818	2.7818	2.7818	2.2590	2.2590	2.2590	0.5935	0.5935	0.5935	2.6044	2.6044	2.6044	0.5525	0.5525	0.5525	0.6724	0.6724	0.6724	1.9422	1.9422	1.9422	1.1563	1.1563	1.1563	0.6250	0.6250	0.6250	4.6581	4.6581	4.6581	0.5239	0.5239	0.5239	4.2878	4.2878	4.2878	0.8246	0.8246	0.8246	2.7844	2.7844	2.7844	0.8606	0.8606	0.8606	0.5855	0.5855	0.5855	0.1273
7 \lambda_{fq}	0.0387	0.0407	0.0397	0.0571	0.0571	0.0571	0.0354	0.0354	0.0354	0.0375	0.0375	0.0375	0.0391	0.0391	0.0391	0.0375	0.0375	0.0375	0.0455	0.0455	0.0455	0.0354	0.0354	0.0354	0.0420	0.0420	0.0420	0.0390	0.0390	0.0390	0.0355	0.0355	0.0355	0.0390	0.0390	0.0390	0.0403	0.0403	0.0403	0.0454	0.0454	0.0454	0.0356	0.0356	0.0356	0.0380	0.0380	0.0380	0.0403	0.0403	0.0403	0.0431	0.0431	0.0431	0.0430	0.0430	0.0430	0.0383	0.0383	0.0383	0.0412	0.0412	0.0412	0.0414	0.0414	0.0414	0.0413	0.0413	0.0413	0.0354	0.0354	0.0354	0.0380	0.0380	0.0380	0.0443	0.0443	0.0443	0.0359	0.0359	0.0359	0.0436	0.0436	0.0436	0.0443	0.0443	0.0443	0.0369	0.0369	0.0369	0.0388	0.0388	0.0388	0.0453	0.0453	0.0453	0.0364	0.0364	0.0364	0.0431	0.0431	0.0431	0.0365	0.0365	0.0365	0.0445	0.0445	0.0445	0.0360	0.0360	0.0360	0.0456	0.0456	0.0456	0.0428	0.0428	0.0428	0.1031
8 \vartheta_E	0.3042	0.4058	0.3550	0.8585	0.8585	0.8585	0.4506	0.4506	0.4506	0.4165	0.4165	0.4165	0.3347	0.3347	0.3347	0.7057	0.7057	0.7057	0.7427	0.7427	0.7427	0.5060	0.5060	0.5060	0.3929	0.3929	0.3929	0.6545	0.6545	0.6545	0.3075	0.3075	0.3075	0.1338	0.1338	0.1338	0.1383	0.1383	0.1383	0.6269	0.6269	0.6269	0.1874	0.1874	0.1874	0.5837	0.5837	0.5837	0.1072	0.1072	0.1072	0.5386	0.5386	0.5386	0.0717	0.0717	0.0717	0.0669	0.0669	0.0669	0.2543	0.2543	0.2543	0.0909	0.0909	0.0909	0.1953	0.1953	0.1953	0.4422	0.4422	0.4422	0.5703	0.5703	0.5703	0.1380	0.1380	0.1380	0.5359	0.5359	0.5359	0.4353	0.4353	0.4353	0.2565	0.2565	0.2565	0.2625	0.2625	0.2625	0.0911	0.0911	0.0911	0.1788	0.1788	0.1788	0.6089	0.6089	0.6089	0.3753	0.3753	0.3753	0.4217	0.4217	0.4217	0.0649	0.0649	0.0649	0.2088	0.2088	0.2088	0.5484	0.5484	0.5484	0.0456	0.0456	0.0456	0.2498
9 \gamma_L	0.0999	0.1014	0.1006	0.1005	0.1005	0.1005	0.0989	0.0989	0.0989	0.1022	0.1022	0.1022	0.1037	0.1037	0.1037	0.0987	0.0987	0.0987	0.0985	0.0985	0.0985	0.0971	0.0971	0.0971	0.1018	0.1018	0.1018	0.0968	0.0968	0.0968	0.1022	0.1022	0.1022	0.1042	0.1042	0.1042	0.1012	0.1012	0.1012	0.1004	0.1004	0.1004	0.1001	0.1001	0.1001	0.0986	0.0986	0.0986	0.1019	0.1019	0.1019	0.0972	0.0972	0.0972	0.1109	0.1109	0.1109	0.1059	0.1059	0.1059	0.0995	0.0995	0.0995	0.1013	0.1013	0.1013	0.0999	0.0999	0.0999	0.0979	0.0979	0.0979	0.1008	0.1008	0.1008	0.1104	0.1104	0.1104	0.0999	0.0999	0.0999	0.0994	0.0994	0.0994	0.1047	0.1047	0.1047	0.0999	0.0999	0.0999	0.1004	0.1004	0.1004	0.1037	0.1037	0.1037	0.0991	0.0991	0.0991	0.0994	0.0994	0.0994	0.1009	0.1009	0.1009	0.1097	0.1097	0.1097	0.1008	0.1008	0.1008	0.1017	0.1017	0.1017	0.1028	0.1028	0.1028	0.2202
10 k_L	6.1876	6.4075	6.2975	5.3073	5.3073	5.3073	6.2690	6.2690	6.2690	6.3796	6.3796	6.3796	6.1989	6.1989	6.1989	6.2451	6.2451	6.2451	6.1278	6.1278	6.1278	6.3359	6.3359	6.3359	6.7765	6.7765	6.7765	5.2615	5.2615	5.2615	6.2429	6.2429	6.2429	6.3655	6.3655	6.3655	6.7206	6.7206	6.7206	6.3999	6.3999	6.3999	7.5645	7.5645	7.5645	6.3001	6.3001	6.3001	6.2223	6.2223	6.2223	6.3253	6.3253	6.3253	6.1348	6.1348	6.1348	6.6005	6.6005	6.6005	6.2343	6.2343	6.2343	6.3855	6.3855	6.3855	6.3080	6.3080	6.3080	6.3469	6.3469	6.3469	5.8057	5.8057	5.8057	6.4774	6.4774	6.4774	6.0873	6.0873	6.0873	6.2348	6.2348	6.2348	6.2943	6.2943	6.2943	6.3867	6.3867	6.3867	6.2282	6.2282	6.2282	6.4131	6.4131	6.4131	6.3645	6.3645	6.3645	4.3096	4.3096	4.3096	6.4607	6.4607	6.4607	6.2822	6.2822	6.2822	6.2782	6.2782	6.2782	6.2949	6.2949	6.2949	6.5735	6.5735	6.5735	0.0898
11 \phi_T	4.4763	4.6144	4.5454	4.5165	4.5165	4.5165	4.5260	4.5260	4.5260	4.5342	4.5342	4.5342	4.5778	4.5778	4.5778	4.5467	4.5467	4.5467	4.5156	4.5156	4.5156	4.5469	4.5469	4.5469	4.4453	4.4453	4.4453	4.0447	4.0447	4.0447	4.5759	4.5759	4.5759	4.5247	4.5247	4.5247	4.5322	4.5322	4.5322	4.5156	4.5156	4.5156	4.5636	4.5636	4.5636	4.4603	4.4603	4.4603	4.5774	4.5774	4.5774	4.5040	4.5040	4.5040	4.6919	4.6919	4.6919	4.5456	4.5456	4.5456	4.6079	4.6079	4.6079	4.6095	4.6095	4.6095	4.6276	4.6276	4.6276	4.5406	4.5406	4.5406	4.4999	4.4999	4.4999	4.5527	4.5527	4.5527	4.4996	4.4996	4.4996	4.5248	4.5248	4.5248	4.5208	4.5208	4.5208	5.5762	5.5762	5.5762	4.5561	4.5561	4.5561	4.5793	4.5793	4.5793	4.4490	4.4490	4.4490	3.1294	3.1294	3.1294	4.5631	4.5631	4.5631	4.5463	4.5463	4.5463	4.5679	4.5679	4.5679	4.5451	4.5451	4.5451	4.6578	4.6578	4.6578	0.0746
12 \lambda_{qf}	0.0007	0.0008	0.0008	0.0004	0.0004	0.0004	0.0008	0.0008	0.0008	8.6625e-04	8.6625e-04	8.6625e-04	7.7896e-04	7.7896e-04	7.7896e-04	0.0008	0.0008	0.0008	0.0008	0.0008	0.0008	0.0007	0.0007	0.0007	7.1744e-04	7.1744e-04	7.1744e-04	0.0007	0.0007	0.0007	0.0008	0.0008	0.0008	7.0968e-04	7.0968e-04	7.0968e-04	7.7704e-04	7.7704e-04	7.7704e-04	0.0007	0.0007	0.0007	0.0007	0.0007	0.0007	0.0008	0.0008	0.0008	7.4564e-04	7.4564e-04	7.4564e-04	0.0006	0.0006	0.0006	7.8244e-04	7.8244e-04	7.8244e-04	0.0008	0.0008	0.0008	7.6673e-04	7.6673e-04	7.6673e-04	8.0438e-04	8.0438e-04	8.0438e-04	7.2553e-04	7.2553e-04	7.2553e-04	0.0009	0.0009	0.0009	0.0009	0.0009	0.0009	7.3578e-04	7.3578e-04	7.3578e-04	0.0011	0.0011	0.0011	7.6030e-04	7.6030e-04	7.6030e-04	7.6214e-04	7.6214e-04	7.6214e-04	0.0008	0.0008	0.0008	7.9562e-04	7.9562e-04	7.9562e-04	5.8146e-04	5.8146e-04	5.8146e-04	0.0012	0.0012	0.0012	6.9332e-04	6.9332e-04	6.9332e-04	0.0008	0.0008	0.0008	6.2453e-04	6.2453e-04	6.2453e-04	0.0009	0.0009	0.0009	0.0007	0.0007	0.0007	7.6600e-04	7.6600e-04	7.6600e-04	0.0908
13 \phi_{PH}	2.0090	2.0764	2.0427	2.0706	2.0706	2.0706	2.1173	2.1173	2.1173	2.1408	2.1408	2.1408	2.0916	2.0916	2.0916	2.1430	2.1430	2.1430	2.0730	2.0730	2.0730	2.1853	2.1853	2.1853	2.0911	2.0911	2.0911	2.0402	2.0402	2.0402	2.0731	2.0731	2.0731	1.4037	1.4037	1.4037	1.9517	1.9517	1.9517	2.2048	2.2048	2.2048	2.0772	2.0772	2.0772	2.1918	2.1918	2.1918	1.8919	1.8919	1.8919	1.9613	1.9613	1.9613	1.8723	1.8723	1.8723	1.9607	1.9607	1.9607	2.2937	2.2937	2.2937	1.9274	1.9274	1.9274	2.0262	2.0262	2.0262	2.1175	2.1175	2.1175	2.1324	2.1324	2.1324	2.0352	2.0352	2.0352	2.1908	2.1908	2.1908	2.1243	2.1243	2.1243	2.0453	2.0453	2.0453	1.9986	1.9986	1.9986	2.0286	2.0286	2.0286	1.9678	1.9678	1.9678	1.8308	1.8308	1.8308	1.9552	1.9552	1.9552	2.0522	2.0522	2.0522	1.9105	1.9105	1.9105	2.0017	2.0017	2.0017	1.9693	1.9693	1.9693	1.8763	1.8763	1.8763	0.0766
14 \delta	0.0561	0.0588	0.0574	0.0435	0.0435	0.0435	0.0528	0.0528	0.0528	0.0608	0.0608	0.0608	0.0616	0.0616	0.0616	0.0538	0.0538	0.0538	0.0521	0.0521	0.0521	0.0510	0.0510	0.0510	0.0581	0.0581	0.0581	0.0488	0.0488	0.0488	0.0589	0.0589	0.0589	0.0597	0.0597	0.0597	0.0547	0.0547	0.0547	0.0564	0.0564	0.0564	0.0568	0.0568	0.0568	0.0529	0.0529	0.0529	0.0604	0.0604	0.0604	0.0501	0.0501	0.0501	0.0759	0.0759	0.0759	0.0619	0.0619	0.0619	0.0560	0.0560	0.0560	0.0595	0.0595	0.0595	0.0556	0.0556	0.0556	0.0605	0.0605	0.0605	0.0563	0.0563	0.0563	0.0703	0.0703	0.0703	0.0577	0.0577	0.0577	0.0530	0.0530	0.0530	0.0621	0.0621	0.0621	0.0575	0.0575	0.0575	0.0620	0.0620	0.0620	0.0576	0.0576	0.0576	0.0531	0.0531	0.0531	0.0525	0.0525	0.0525	0.0553	0.0553	0.0553	0.0679	0.0679	0.0679	0.0604	0.0604	0.0604	0.0582	0.0582	0.0582	0.0573	0.0573	0.0573	0.2315
⋮

6. Uncertainty analisys

Finally, we validate the factor intervals by performing an uncertainty analysis (Montecarlo simulations)

Number of samples and the generation of the design matrix (combination of factor values)

N = 1000;

M = gsua_dmatrix(T,N,'Method','Sobol');

gsua_ua(M,T,'xdata',xdata,'ynom',ydata);

Starting parallel pool (parpool) using the 'local' profile ... Connected to the parallel pool (number of workers: 8). Progress: 100% Estimated processing time (h:m:s): 0:0:33 Remaining time (h:m:s): 0:0:0 Elapsed time (h:m:s): 0:0:33 Estimated stop time (h:m:s): 17:11:19 Number of simulations: 1000

Final table with estimated values and its respective confidence intervals

disp(T)
                               Range               Nominal           Est           index  
                      ________________________    __________    ______________    ________

    E^2_0                 769.17        955.18        862.17    [1×114 double]     0.20632
    P_0                   115.38         147.8        131.59    [1×114 double]     0.11307
    \beta_L              0.79725       0.80831       0.80278    [1×114 double]      0.1036
    \beta_T              0.12128       0.13959       0.13044    [1×114 double]     0.17691
    \beta_P              0.37379       0.39821         0.386    [1×114 double]    0.098067
    \phi_{EP}            0.68193        1.1527       0.91731    [1×114 double]     0.12731
    \lambda_{fq}        0.038696      0.040705      0.039701    [1×114 double]     0.10313
    \vartheta_E          0.30419       0.40582         0.355    [1×114 double]     0.24983
    \gamma_L             0.09987       0.10141       0.10064    [1×114 double]     0.22021
    k_L                   6.1876        6.4075        6.2975    [1×114 double]    0.089841
    \phi_T                4.4763        4.6144        4.5454    [1×114 double]    0.074593
    \lambda_{qf}      0.00073879    0.00079394    0.00076637    [1×114 double]    0.090782
    \phi_{PH}              2.009        2.0764        2.0427    [1×114 double]    0.076601
    \delta              0.056103      0.058779      0.057441    [1×114 double]     0.23153
    \eta_L               0.47726       0.53203       0.50465    [1×114 double]    0.074595
    \vartheta_P          0.50515       0.55588       0.53051    [1×114 double]     0.10979
    \gamma_H             0.12684        0.1479       0.13737    [1×114 double]     0.19741
    z                     6.2171        7.4512        6.8342    [1×114 double]     0.20624
    \phi_{PL}            0.24846       0.25472       0.25159    [1×114 double]      0.1034
    \eta_\vartheta       0.34482       0.41498        0.3799    [1×114 double]      0.1199
    a_L                   2.5927        2.7095        2.6511    [1×114 double]     0.21744
    b_L                   2.5315        2.6812        2.6063    [1×114 double]     0.21141
    a_\mu                 1.1701        1.1913        1.1807    [1×114 double]     0.13111
    b_\mu                 5.1995        5.4509        5.3252    [1×114 double]     0.13082
    \mu                  0.10983       0.11762       0.11373    [1×114 double]      0.2257
    a_H                   9.1968         10.28        9.7383    [1×114 double]      0.1527
    b_H                   2.0482        3.6998         2.874    [1×114 double]     0.14826
    \nu                   386.47        410.07        398.27    [1×114 double]     0.10128