An Analytical Approach to Determining the Coefficients in Lyapunov Direct Method: With Application to an Age-Structured Epidemiological Model

Lyapunov direct method has been employed as a powerful tool to show the global stability of an equilibrium in differential equations. One widely cited Lyapunov function for population dynamic models takes the form L(t)=∑_i=1ⁿc_iL_i(t) with L_i(t)=x_i(t)−x_i^∗−x_i^∗ln(x_i(t)/x_i^∗) (if x_i^∗>0) and L_i(t)=x_i(t) (if x_i^∗=0), a combination of functions involving different state variables x_i(t) of the model system with (x₁^∗,x₂^∗,…,x_n^∗) being an equilibrium. However, two challenges hinder the efficient applications of Lyapunov direct method: (a) determining the coefficients c_i; and (b) rearranging the time derivative L^′(t) along the trajectories of the system to show it is negative (semi-)definite. This study is to propose an easy-to-follow analytical approach to tackle these two challenges, which will be illustrated through an application to an epidemiological model with vaccination age. Furthermore, the Lyapunov functional for the endemic equilibrium can be reformulated to investigate the global stability for the disease-free equilibrium and a family of Lyapunov functionals can be proposed for the same purpose. It is expected that the approach can be further applied to other age-structured models and be extended to analyze more complicated models with other heterogeneous factors.