How Are Eigenvalues and Eigenvectors Computed in Epidemiological Models?
The computation of eigenvalues and eigenvectors typically involves the following steps:
1. Constructing the Model First, an epidemiological model is formulated using a system of differential equations. For instance, in a SIR model, the equations describe how the number of susceptible, infected, and recovered individuals change over time.
2. Linearizing the System To analyze the stability of equilibrium points, the system of differential equations is linearized around these points. This involves finding the Jacobian matrix, which represents the partial derivatives of the system with respect to the state variables.
3. Solving for Eigenvalues and Eigenvectors The next step is to solve the characteristic equation of the Jacobian matrix to find its eigenvalues. Once the eigenvalues are known, the corresponding eigenvectors can be computed. Various numerical techniques and software tools, such as MATLAB or Python, can be used for these computations.
