How are Eigenvectors Used in Compartmental Models?

In compartmental models, the transition rates between different states (e.g., susceptible to infected) are often described by a matrix. By calculating the eigenvectors and eigenvalues of this matrix, epidemiologists can determine the stability of the disease-free equilibrium and the basic reproduction number (R0). For instance, if the largest eigenvalue (also known as the dominant eigenvalue) is less than one, the disease will eventually die out. Conversely, if it is greater than one, the disease will spread within the population.