Instrumental variable method, noise-free system

SIRC.m

@@ -53,21 +53,21 @@ legend('S', 'I', 'R', 'C');
 %% Least square estimation of the noise-free modell
 
 % dSdt + dIdt = nu - mu*S - (gamma+mu)*I
-X_SI = [ones(size(y, 1)-1, 1), y(1:end-1, 1), y(1:end-1, 2)];
-Y_SI = y(2:end,1) - y(1:end-1,1) + y(2:end,2) - y(1:end-1,2);
+X_SI1 = [ones(size(y, 1)-1, 1), y(1:end-1, 1), y(1:end-1, 2)];
+Y_SI1 = y(2:end,1) - y(1:end-1,1) + y(2:end,2) - y(1:end-1,2);
 %Y_SI2 = y_noisy(2:end,1) + y_noisy(2:end,2);
-theta_SI = abs(lsq(X_SI, Y_SI));
+theta_SI = abs(lsq(X_SI1, Y_SI1));
 %theta_SI2 = lsq(X_SI, Y_SI2);
 
 % dIdt = beta*I*S + epsilon*beta*C*S - (gamma+mu)*I
-X_I = [y(1:end-1,1).*y(1:end-1,2), y(1:end-1,1).*y(1:end-1,4), y(1:end-1,2)];
-Y_I = y(2:end, 2) - y(1:end-1, 2);
-theta_I = abs(lsq(X_I, Y_I));
+X_I1 = [y(1:end-1,1).*y(1:end-1,2), y(1:end-1,1).*y(1:end-1,4), y(1:end-1,2)];
+Y_I1 = y(2:end, 2) - y(1:end-1, 2);
+theta_I = abs(lsq(X_I1, Y_I1));
 
 % dCdt = gamma*q*I - (Gamma-mu)*C
-X_C = [y(1:end-1, 2), y(1:end-1, 4)];
-Y_C = y(2:end, 4) - y(1:end-1, 4);
-theta_C = abs(lsq(X_C, Y_C));
+X_C1 = [y(1:end-1, 2), y(1:end-1, 4)];
+Y_C1 = y(2:end, 4) - y(1:end-1, 4);
+theta_C = abs(lsq(X_C1, Y_C1));
 
 [nu_lsq1, mu_lsq1, beta_lsq1, epsilon_lsq1, gamma_lsq1, q_lsq1, Gamma_lsq1] = deal(theta_SI(1), ...
     theta_SI(2), theta_I(1), theta_I(2)/theta_I(1), theta_SI(3)-theta_SI(2), ...
@@ -101,20 +101,60 @@ legend('S', 'I', 'R', 'C');
 %% Least square estimation after adding noise to the modell
 
 % dSdt + dIdt = nu - mu*S - (gamma+mu)*I
-X_SI = [ones(size(y_noisy, 1)-1, 1), y_noisy(1:end-1, 1), y_noisy(1:end-1, 2)];
-Y_SI = y_noisy(2:end,1) - y_noisy(1:end-1,1) + y_noisy(2:end,2) - y_noisy(1:end-1,2);
-theta_SI = abs(lsq(X_SI, Y_SI));
+X_SI2 = [ones(size(y_noisy, 1)-1, 1), y_noisy(1:end-1, 1), y_noisy(1:end-1, 2)];
+Y_SI2 = y_noisy(2:end,1) - y_noisy(1:end-1,1) + y_noisy(2:end,2) - y_noisy(1:end-1,2);
+theta_SI = abs(lsq(X_SI2, Y_SI2));
 
 % dIdt = beta*I*S + epsilon*beta*C*S - (gamma+mu)*I
-X_I = [y_noisy(1:end-1,1).*y_noisy(1:end-1,2), y_noisy(1:end-1,1).*y_noisy(1:end-1,4), y_noisy(1:end-1,2)];
-Y_I = y_noisy(2:end, 2) - y_noisy(1:end-1, 2);
-theta_I = abs(lsq(X_I, Y_I));
+X_I2 = [y_noisy(1:end-1,1).*y_noisy(1:end-1,2), y_noisy(1:end-1,1).*y_noisy(1:end-1,4), y_noisy(1:end-1,2)];
+Y_I2 = y_noisy(2:end, 2) - y_noisy(1:end-1, 2);
+theta_I = abs(lsq(X_I2, Y_I2));
 
 % dCdt = gamma*q*I - (Gamma-mu)*C
-X_C = [y_noisy(1:end-1, 2), y_noisy(1:end-1, 4)];
-Y_C = y_noisy(2:end, 4) - y_noisy(1:end-1, 4);
-theta_C = abs(lsq(X_C, Y_C));
+X_C2 = [y_noisy(1:end-1, 2), y_noisy(1:end-1, 4)];
+Y_C2 = y_noisy(2:end, 4) - y_noisy(1:end-1, 4);
+theta_C = abs(lsq(X_C2, Y_C2));
 
 [nu_lsq2, mu_lsq2, beta_lsq2, epsilon_lsq2, gamma_lsq2, q_lsq2, Gamma_lsq2] = deal(theta_SI(1), ...
     theta_SI(2), theta_I(1), theta_I(2)/theta_I(1), theta_SI(3)-theta_SI(2), ...
-    theta_C(1)/(theta_SI(3)-theta_SI(2)), theta_C(2)+theta_SI(2));
+    theta_C(1)/(theta_SI(3)-theta_SI(2)), theta_C(2)+theta_SI(2));
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+
+%% Instrumental variable method
+
+% Matrices composed from the instrumental variables
+KSZI_SI = [ones(size(y, 1)-1, 1), y_lsq1(1:end-1, 1), y_lsq1(1:end-1, 2)];
+KSZI_I = [y_lsq1(1:end-1,1).*y_lsq1(1:end-1,2), y_lsq1(1:end-1,1).*y_lsq1(1:end-1,4), y_lsq1(1:end-1,2)];
+KSZI_C = [y_lsq1(1:end-1, 2), y_lsq1(1:end-1, 4)];
+
+theta_SI = abs(iv4(X_SI1, Y_SI1, KSZI_SI));
+theta_I = abs(iv4(X_I1, Y_I1, KSZI_I));
+theta_C = abs(iv4(X_C1, Y_C1, KSZI_C));
+
+[nu_iv4, mu_iv4, beta_iv4, epsilon_iv4, gamma_iv4, q_iv4, Gamma_iv4] = deal(theta_SI(1), ...
+    theta_SI(2), theta_I(1), theta_I(2)/theta_I(1), theta_SI(3)-theta_SI(2), ...
+    theta_C(1)/(theta_SI(3)-theta_SI(2)), theta_C(2)+theta_SI(2));
+
+y_iv4 = zeros(size(y, 1), 4);
+y_iv4(1,:) = [S0, I0, R0, C0];
+for i = 2:length(y_iv4)
+    % dSdt = nu - (beta*I + epsilon*beta*C)*S - mu*S
+    y_iv4(i,1) = y_iv4(i-1,1) + nu_iv4 - (beta_iv4*y_iv4(i-1,2) + ...
+        epsilon_iv4*beta_iv4*y_iv4(i-1,4))*y_iv4(i-1,1) - mu_iv4*y_iv4(i-1,1);
+    % dIdt = (beta*I + epsilon*beta*C)*S - gamma*I - mu*I
+    y_iv4(i,2) = y_iv4(i-1,2) + (beta_iv4*y_iv4(i-1,2) + ...
+        epsilon_iv4*beta_iv4*y_iv4(i-1,4))*y_iv4(i-1,1) - ...
+        gamma_iv4*y_iv4(i-1,2) - mu_iv4*y_iv4(i-1,2);
+    % dRdt = gamma*(1-q)*I + Gamma*C - mu*R
+    y_iv4(i,3) = y_iv4(i-1,3) + gamma_iv4*(1-q_lsq1)*y_iv4(i-1,2) + ...
+        Gamma_iv4*y_iv4(i-1,4) - mu_iv4*y_iv4(i-1,3);
+    % dCdt = gamma*q*I - Gamma*C - mu*C
+    y_iv4(i,4) = y_iv4(i-1,4) + gamma_iv4*q_iv4*y_iv4(i-1,2) - ...
+        Gamma_iv4*y_iv4(i-1,4) - mu_iv4*y_iv4(i-1,3);
+end
+
+figure(5)
+plot(t, y_iv4);
+title('Instrumental variable method (noise-free modell)');
+xlabel('Time(days)');
+ylabel('Number of individuals');
+legend('S', 'I', 'R', 'C');
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