- Define your SEIR model equations with time delay.
- Implement the model as a function that MATLAB can call.
- Set up parameter ranges for bifurcation analysis.
- Use dde23 to solve the delay differential equations for different parameter values.
- Extract and plot the steady-state solutions against the parameter values
Here's a basic outline of how you can achieve this:
% Define your SEIR model equations with time delay
function dydt = seir_delay(t, y, Z, beta, gamma, delta, tau)
% Extract variables
S = y(1);
E = y(2);
I = y(3);
R = y(4);
% Evaluate delayed variables
ZE = Z(:,1);
ZI = Z(:,2);
% Define model equations
dSdt = -beta * S * I - beta * delta * S * ZE(end);
dEdt = beta * S * I + beta * delta * S * ZE(end) - gamma * E;
dIdt = gamma * E - tau * I - gamma * I * ZI(end);
dRdt = tau * I;
% Return the derivatives
dydt = [dSdt; dEdt; dIdt; dRdt];
end
% Set up parameter ranges
beta_range = linspace(0, 1, 100); % Range for beta
gamma = 0.1; % Recovery rate
delta = 0.1; % Transmission rate with delay
tau = 0.1; % Recovery time
% Initial conditions
S0 = 0.8;
E0 = 0.1;
I0 = 0.1;
R0 = 0;
% Initialize arrays to store steady-state solutions
steady_states = zeros(length(beta_range), 4);
% Loop through each parameter value and solve the DDE
for i = 1:length(beta_range)
beta = beta_range(i);
% Define the delay function
delay_fun = @(t) [interp1(t_data, E_data, t-tau, 'linear', 'extrap');
interp1(t_data, I_data, t-tau, 'linear', 'extrap')];
% Solve the DDE
sol = dde23(@seir_delay, [tau, 20], @delay_fun, [S0; E0; I0; R0]);
% Store the steady-state solution
steady_states(i, :) = sol.y(:, end)';
end
% Plot the bifurcation diagram
figure;
plot(beta_range, steady_states(:, 3), 'b');
xlabel('Transmission rate (\beta)');
ylabel('Steady-state infected');
title('Bifurcation diagram of SEIR model with time delay');
thanks a lot but there is an error in this line
beta_range = linspace(0, 1, 10); % Range for beta
Kazi Redwan
Mohamed H Hassan The error likely occurs because the size of beta_range does not match the length of the steady_states array. The steady_states array is initialized with a length of 100 in the code, but you've changed the length of beta_range to 10. To resolve the error, you should ensure that the length of beta_range matches the length of steady_states.
You can do this by either changing the length of beta_range back to 100 or by adjusting the code accordingly. If you want to keep beta_range with a length of 10, you should also adjust the initialization of steady_states to match that length. Here's how you can do it:
beta_range = linspace(0, 1, 10); % Range for beta
% Initialize arrays to store steady-state solutions
steady_states = zeros(length(beta_range), 4);
Make sure you update the length of beta_range to 10 wherever it's used in the subsequent code, to ensure consistency.
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