update: different intervals of variation for the parameters

CRA_Updated_Version/CRA_Toolbox_bash/Examples/ode_covid19_v2.m

+%ODE model to describe the spread and clinical progression of COVID-19
+
+function dx=ode_covid19_v2(t,x,p,Tlock1,s01,s11)%,Tlock2,s02,s12)
+
+be = p(1);
+b0 = p(2);                
+b1 = p(3);
+b2 = p(4);
+b3 = p(5);
+a0 = p(6);
+a1 = p(7);
+f = p(8);
+g0 = p(9);
+g1 = p(10);
+p1 = p(11);
+g2 = p(12);
+p2 = p(13);
+g3 = p(14);
+u = p(15);
+%seas_amp = p(16);
+%seas_phase = p(17);
+%s0 = p(18);
+%s1=p(19);
+%se=p(20);
+%k_e0s=p(21);
+%g4=p(22);
+%u2 = p(23);
+
+%seas=(1 + seas_amp*cos(2*pi*(t-seas_phase)/365));
+
+% if t<Tlock
+%     s0 = 1;
+%     s1 = 1;
+% elseif t>=Tlock && t<Tlock2
+%     s0 = int_s0;
+%     s1 = int_s1;  
+% else
+%     s0 = v11
+%     s1 = v2;
+% end
+
+s0 = (t<Tlock1) + s01 * (t>=Tlock1); %* (t<Tlock2) + s01(2)*(t>=Tlock2);
+s1 = (t<Tlock1) + s11 * (t>=Tlock1); %* (t<Tlock2) + s11(2)*(t>=Tlock2);
+
+%susceptibles individuals (not infected)
+S = x(1);
+%exposed individuals, infected but no symptoms and no transmission
+E0 = x(2);
+%exposed individuals, infected but no symptoms and can transmit
+E1 = x(3);
+%infected individuals but asymptomatic infection
+I0 = x(4);
+%infected individuals, mild infection
+I1 = x(5);
+%infected individuals, severe infection
+I2 = x(6);
+%infected individuals, critical infection
+I3 = x(7);
+%recovered
+R = x(8);
+%dead
+D = x(9);
+
+
+dS = - (be *s0* E1 + b0*s0*I0 + b1*s1*I1 + b2*I2 + b3*I3)*S; % + k_e0s * E0;
+dE0 = (be *s0* E1 + b0*s0*I0 + b1*s1*I1 + b2*I2 + b3*I3)*S -a0*E0; % - k_e0s * E0; 
+dE1 = a0 * E0 - a1*E1; % -g4*E1;
+dI0 = f * a1 * E1 - g0*I0;
+dI1 = (1-f) * a1 * E1 - g1*I1 - p1*I1;
+dI2 = p1 * I1 - g2*I2 - p2*I2; % -u2*I2;
+dI3 = p2 * I2 - g3 *I3 - u * I3;
+dR = g0*I0 + g1*I1 + g2*I2 + g3*I3; % +g4 * E1;
+dD = u * I3; % + u2 * I2;
+
+
+
+dx=[dS;dE0;dE1;dI0;dI1;dI2;dI3;dR;dD];

CRA_Updated_Version/CRA_Toolbox_bash/main_covid19.m

+%SCRIPT FOR RUNNING THE CRA TOOLBOX WITHOUT USING THE GUI
+%The model is written as a Matlab function.
+
+%%%% THE FIRST THING TO DO FOR RUNNING THE SCRIPT IS TO ADD THE FOLDER
+%%%% CLASSES AND THE FOLDER Examples TO THE PATH %%%%
+
+%Parameters to be set by the user are:
+% 1-model_name: name of the model in .xml format. A struct is created to
+% store the following characteristics of an ODE model: nominal values and
+% names of the parameters, initial conditions and names of variables, input
+% values.
+% 2-stop_time: final time point for the model simulation
+% 3-step_size: time interval for the vector of time points
+% 4- ode_solver: type of ode_solver for simulating the model (possible
+% choices are: ode45, ode15s, ode23t, sundials, stochastic, explicit tau
+% and implicit tau
+% 5- Nr: number of independent realizations to perform 
+% 6- LBpi: lower boundary of the Latin Hypercube Sampling for perturbation
+% of the model parameter space
+% 7- UBpi: upper boundary of the Latin Hypercube Sampling
+% 8- Ns: number of samples of the Latin Hypercube (parameters n of the
+% lhsdesign function)
+% 9- variable_name: variable of the model to set as reference node to be
+% measured
+% 10- current_func: is the type of evaluation function. Currently, it is
+% possible to choose among
+% three evaluation functions: area under the curve, maximum value and time
+% of maximum for the time behavior of the selected variables. The user can
+% also define his evaluation function in a .m file by extending the
+% abstract class EvaluationFunction
+% 11- tail_size: number of samples to include in the upper and lower tail
+% when computing the probability density function of the evaluation
+% function
+% 12- current_tm: is the method for computing the tails of the pdf of the
+% evaluation function. Right now, it is possible to choose between two
+% methods: sorted() which sorts the values of the evaluation function and
+% selects the first and last samples according to tail_size; tmp_sum()
+% computes the tails by selecting the upper and lower quartile. When using
+% tmp_sum(), the parameter step_size needs also to be specified. Step_size
+% is the step for computing the lower and upper quartile of the pdf in an
+% iterative way, i.e. when the upper and lower tails do not have the number of
+% samples specified by the user, the threshold is increased of a quantity
+% equal to step_size and the calculation of the tails is repeated. 
+% The user can also define his own method for the tails computation by
+% extending the abstract class TailMethod().
+% 13- folder: name of the folder to create where the results are saved
+
+tic;
+
+model_name='ode_covid19_v2';
+
+stop_time=80;
+step_size=1;
+ode_solver='ode15s';
+
+%time axis for model simulation
+time_axis=[0:step_size:stop_time]';
+
+%parameters and initial conditions of the model
+
+N = 882000;         
+InitInf=1;
+
+E00 = InitInf;
+S0 = N-E00;
+x0 = [S0 E00 0 0 0 0 0 0 0];
+
+load('Examples/moda_parametri_inizio14step_LOG_100000_UMBRIA_I2I3DE.mat');
+nominal_input_parameters=moda_parametri_inizio14step_LOG;
+
+be = nominal_input_parameters(1);
+b0=nominal_input_parameters(2);
+b1=nominal_input_parameters(3);
+b2=nominal_input_parameters(4);
+b3=nominal_input_parameters(5);
+        
+be=be/(N);
+b0=b0/(N);
+b1=b1/(N);
+b2=b2/(N);
+b3=b3/(N);
+        
+FracSevere = nominal_input_parameters(6);
+FracCritical = nominal_input_parameters(7);
+FracMild = 1-FracSevere-FracCritical;
+FracAsym = nominal_input_parameters(8);
+               
+IncubPeriod = nominal_input_parameters(9);
+DurMildInf = nominal_input_parameters(10);
+
+DurAsym = nominal_input_parameters(11);
+   
+DurHosp = nominal_input_parameters(12);
+TimeICUDeath = nominal_input_parameters(13);
+ProbDeath=nominal_input_parameters(14);
+
+PresymPeriod=nominal_input_parameters(15);
+s01=nominal_input_parameters(16);
+s11=nominal_input_parameters(17);
+            
+CFR=ProbDeath*FracCritical/100; 
+            
+a1=1/PresymPeriod; %presymptomatic period of transmission
+a0=1/(IncubPeriod-PresymPeriod); %true latent period  
+
+f=FracAsym;
+    
+g0=1/DurAsym;
+
+g1=(1/DurMildInf)*FracMild;
+p1=(1/DurMildInf)-g1;
+    
+p2=(1/DurHosp)*(FracCritical/(FracSevere+FracCritical));
+g2=(1/DurHosp)-p2;
+    
+u=(1/TimeICUDeath)*(CFR/FracCritical);
+    
+g3=(1/TimeICUDeath)-u;
+
+nominal_parameters=[be b0 b1 b2 b3 a0 a1 f g0 g1 p1 g2 p2 g3 u]; 
+nominal_parameters=ones(1,15);
+parameters_name={'be', 'b0', 'b1', 'b2','b3', 'a0', 'a1', 'f', 'g0', 'g1', 'p1', 'g2', 'p2', 'g3','u'};
+
+Tlock1=23;
+
+num_observables=6;
+observables_name={'E1','I0','I1','I2','I3','D'};
+
+model=struct('name',model_name,'odesolver',ode_solver,'time',time_axis,'stop',stop_time,'step',step_size,'nominal_parameters',nominal_parameters,'parameters_name',{parameters_name},'num_observables',num_observables,'observables_name',{observables_name},'initial_conditions',x0,'Tlock1',Tlock1,'s01',s01,'s11',s11);
+
+Nr=3;
+
+%define specific boundaries for each parameter
+LBpi=[0.001 0.001 0.001 0.001 0.001 1e-5 1e-5 0.01 2 1 1 1 1 1 1];
+UBpi=[3 3 3 3 3 0.5 0.4 0.7 8 20 20 10 30 60 4];
+Ns=1000;
+
+variable_name='I0';
+current_func=Maximum(); %current evaluation function
+tail_size=10;  %number of samples for the lower and upper tail
+step_size=0.001;  %this parameter needs to be defined only when current_tm=tmp_sum(step_size)
+current_tm=sorted();
+
+folder='covid19_I0_results';
+
+%model simulation for each sample of the Latin Hypercube
+disp('Starting model simulation with perturbed parameters');
+[AllResults,AllPerturbations]=start_simulation_v2_apr2020(model,Nr,LBpi,UBpi,Ns);
+disp('All done! Model simulation completed!');
+
+%computation of the MIRI for the chosen evaluation function and model
+%variable
+disp('Starting computation of the MIRI for each parameter...');
+try
+  compute_MIRI(model,variable_name,current_func,tail_size,current_tm,Nr,Ns,AllResults,AllPerturbations,folder)
+catch ME
+    %break  
+    return
+end
+%plot and save probability density function of the evaluation function
+disp('Plot of the probability density function of the chosen evaluation function');
+plotpdf_evalfunc(folder,variable_name);
+
+%plot and save conditional probability density functions of the parameters 
+disp('Plot of the parameter probability density functions');
+plotpdf_param(folder,variable_name,model,Nr);
+
+toc;
\ No newline at end of file

CRA_Updated_Version/CRA_Toolbox_bash/start_simulation.m

@@ -34,7 +34,8 @@ for k=1:Nr
   psim_all=cell(Ns,1);  
   parfor i=1:Ns
      pi=perturbed_p(i,:);
-     modelsim=@(t,x)feval(model.name,t,x,pi,model.input,model.total_proteins);
+     %modelsim=@(t,x)feval(model.name,t,x,pi,model.input,model.total_proteins);
+     modelsim=@(t,x)feval(model.name,t,x,pi,model.Tlock1,model.s01,model.s11);
      [tsimi,psimi]=feval(model.odesolver,modelsim,model.time,model.initial_conditions);
     % for t=1:model.num_observables
      %  psim_all{i,t}=psimi(:,t);

CRA_Updated_Version/CRA_Toolbox_bash/start_simulation_v2_apr2020.m

+%Function for simulating the ODE model using the perturbed parameter
+%vector. It takes in input:
+%1. model which is the struct containing all the characheristics of the
+%model
+%2. Nr is the number of independent realizations to perform;
+%3. LBpi and UBpi are the lower and upper boundaries of the Latin
+%Hypercube. They are arrays since each parameter may have its own range of
+%variation;
+%4. Ns is the number of samples to generate for creating the Latin
+%Hypercube;
+%It returns in output the array AllResults containing all the output
+%variables simulated using the parameter vectors in AllPerturbations
+
+function [AllResults, AllPerturbations]=start_simulation_v2_apr2020(model,Nr,LBpi,UBpi,Ns)
+
+nominal_parameters=model.nominal_parameters;
+num_parameters=length(nominal_parameters);
+
+AllResults=cell(Nr,model.num_observables);
+
+AllPerturbations=cell(Nr,1);
+
+h = waitbar(0,'Please wait...');
+
+for k=1:Nr
+    
+  waitbar(k/Nr,h,['Realization number ',num2str(k)]);
+  
+  perturbation=zeros(Ns,num_parameters);
+  for i=1:length(LBpi)
+     hypercube=LatinHypercube(LBpi(i),UBpi(i),Ns);
+     perturbation_i=hypercube.generate_sample(1);
+     perturbation(:,i)=perturbation_i;
+  end  
+  perturbed_p=repmat(nominal_parameters,length(perturbation),1).*perturbation;
+  size(perturbed_p)
+  AllPerturbations{k,1}=perturbation;
+  disp(strcat('Generated a Latin Hypercube of size',{' '},sprintf('%.0f',size(perturbation,1)),'x',sprintf('%.0f',size(perturbation,2))));
+    %psim_all=cell(Ns,model.num_observables);
+  psim_all=cell(Ns,1);  
+  parfor i=1:Ns
+     pi=perturbed_p(i,:);
+     %modelsim=@(t,x)feval(model.name,t,x,pi,model.input,model.total_proteins);
+     modelsim=@(t,x)feval(model.name,t,x,pi,model.Tlock1,model.s01,model.s11);
+     [tsimi,psimi]=feval(model.odesolver,modelsim,model.time,model.initial_conditions);
+    % for t=1:model.num_observables
+     %  psim_all{i,t}=psimi(:,t);
+     %end
+     msgstr=lastwarn;
+     if isequal(msgstr,'')        
+        psim_all{i,1}=psimi;
+     else 
+        psim_all{i,1}=NaN(length(model.time),model.num_observables);
+     end
+      lastwarn('')
+   end 
+  
+ for j=1:model.num_observables
+    for w=1:Ns
+     AllResults{k,j}{w,1}=psim_all{w,1}(:,j);
+    end
+  end
+  %AllResults{k,1:end}=psim_all;
+  disp(strcat('Completed one realization. The array of results has size',{' '},sprintf('%.0f',size(psim_all,1)),'x',sprintf('%.0f',size(psim_all,2))));
+ end
+
+delete(gcp) %close parallel pool
+close (h)  %close the waitbar
+end
\ No newline at end of file