ans =
Heelo neda,
To estimate the parameters ( s ), ( d ), and ( \gamma ) for your system of ODEs using lsqnonlin in MATLAB, you can follow these steps. This approach involves defining your system of ODEs, simulating it over time, and using lsqnonlin to minimize the difference between the simulated and observed data.
You can use the below steps as reference.
- Define the System of ODEs: Create a function that computes the derivatives based on your current estimates of the parameters.
- Simulate the ODEs: Use ode45 or another ODE solver to simulate the system over time.
- Objective Function: Define an objective function that computes the difference between the simulated results and your observed data.
- Use lsqnonlin: Call lsqnonlin to minimize the objective function and estimate the parameters.
function estimate_parameters
% Observed steady-state values
T0_ss = 2016;
T1_ss = 42;
% Initial guess for parameters [s, d, gamma]
initial_guess = [1000, 0.1, 0.1];
% Time span for simulation
tspan = [0, 100];
% Initial conditions for T0 and T1
T0_initial = 0;
T1_initial = 0;
initial_conditions = [T0_initial, T1_initial];
% Define the objective function
objective = @(params) model_error(params, tspan, initial_conditions, T0_ss, T1_ss);
% Set options for lsqnonlin
options = optimoptions('lsqnonlin', 'Display', 'iter', 'TolFun', 1e-6, 'TolX', 1e-6);
% Estimate parameters using lsqnonlin
[estimated_params, resnorm] = lsqnonlin(objective, initial_guess, [], [], options);
% Display the estimated parameters
fprintf('Estimated Parameters:\n');
fprintf('s = %.4f\n', estimated_params(1));
fprintf('d = %.4f\n', estimated_params(2));
fprintf('gamma = %.4f\n', estimated_params(3));
end
function error = model_error(params, tspan, initial_conditions, T0_ss, T1_ss)
% Unpack parameters
s = params(1);
d = params(2);
gamma = params(3);
% Solve the ODE system
[~, T] = ode45(@(t, y) odesystem(t, y, s, d, gamma), tspan, initial_conditions);
% Compute the error between the simulated steady-state values and the observed values
T0_error = T(end, 1) - T0_ss;
T1_error = T(end, 2) - T1_ss;
% Return the error as a vector
error = [T0_error; T1_error];
end
function dydt = odesystem(t, y, s, d, gamma)
% Unpack the variables
T0 = y(1);
T1 = y(2);
% Define the ODEs
dT0dt = s - d * T0 - gamma * T0;
dT1dt = 2 * d * T0 + d * T1 - gamma * T1;
% Return the derivatives
dydt = [dT0dt; dT1dt];
end
I hope it helps!
