SIRC_noisyRLSQ.m

clc
%% System initalization
% Population
N = 1000;
% Initial number of infected individuals
I0 = 10;
% Initial number of recovered individuals
R0 = 0;
% Initial number of carrier individuals
C0 = 0;
% Everyone else is susceptible
S0 = N - I0 - R0 - C0;
% Initial parameteres
nu = 14;
mu = 0.013;
epsilon = 1.2;
beta = 0.0002;
gamma = 0.05;
Gamma = 0.03;
q = 0.8;

time = 200;
%t = linspace(0, time, time);

y0 = [S0, I0, R0, C0];
[t, y] = ode45(@(t, y) deriv(y, nu, mu, epsilon, beta, gamma, Gamma, q), [0 time], y0);


%% Add noise

[t_noisy, y_noisy] = ode45(@(t, y) noisy_deriv(y, nu, mu, epsilon, beta, gamma, Gamma, q), [0 time], y0);

figure(1)
plot(t_noisy, y_noisy);
xlabel('Time(days)');
ylabel('Number of individuals');
legend('S', 'I', 'R', 'C');

%% Recursive Least Square with ar additive noise

h = 1.0;
lambda0 = 0.99; % 0.95-0.99
alpha = 0.8;
N = length(y_noisy);

% dSdt = nu - beta*I*S - epsilon*beta*C*S - mu*S;
y_S = (y_noisy(2:end, 1) - y_noisy(1:end-1, 1)) / h;
phi_S = zeros(N-1, 4);
phi_S(:,1) = ones(N-1, 1);
phi_S(:,2) = y_noisy(1:end-1, 2) .* y_noisy(1:end-1, 1);
phi_S(:,3) = y_noisy(1:end-1, 4) .* y_noisy(1:end-1, 1);
phi_S(:,4) = y_noisy(1:end-1, 1);

[theta_S, p_S] = recursiveLSQ(phi_S, y_S, lambda0, alpha)
figure(2)
plot(theta_S);
legend('nu', '-beta', '-epsilon*beta', '-mu')

% dIdt = beta*I*S + epsilon*beta*C*S - (gamma + mu)*I;
y_I = (y_noisy(2:end, 2) - y_noisy(1:end-1, 2)) / h;
phi_I = zeros(N-1, 3);
phi_I(:,1) = y_noisy(1:end-1, 2) .* y_noisy(1:end-1, 1);
phi_I(:,2) = y_noisy(1:end-1, 4) .* y_noisy(1:end-1, 1);
phi_I(:,3) = y_noisy(1:end-1, 2);

[theta_I, p_I] = recursiveLSQ(phi_I, y_I, lambda0, alpha)
figure(3)
plot(theta_I);
legend('beta', 'epsilon*beta', '-(gamma+mu)')

% dRdt = gamma*(1-q)*I + Gamma*C - mu*R;
y_R = (y_noisy(2:end, 3) - y_noisy(1:end-1, 3)) / h;
phi_R = zeros(N-1, 3);
phi_R(:,1) = y_noisy(1:end-1, 2);
phi_R(:,2) = y_noisy(1:end-1, 4);
phi_R(:,3) = y_noisy(1:end-1, 3);

[theta_R, p_R] = recursiveLSQ(phi_R, y_R, lambda0, alpha)
figure(4)
plot(theta_R);
legend('gamma*(1-q)', 'Gamma', '-mu')

% dCdt = gamma*q*I - (Gamma + mu)*C;
y_C = (y_noisy(2:end, 4) - y_noisy(1:end-1, 4)) / h;
phi_C = zeros(N-1, 2);
phi_C(:,1) = y_noisy(1:end-1, 2);
phi_C(:,2) = y_noisy(1:end-1, 4);

[theta_C, p_C] = recursiveLSQ(phi_C, y_C, lambda0, alpha)
figure(5)
plot(theta_C);
legend('gamma*q', '-(Gamma+mu)')