An analytical approach to applying the Lyapunov direct method to an epidemic model with age and stage structures.

Usually, it is very challenging to construct appropriate Lyapunov functionals for proving the global stability of age-structured models. In this paper, we propose an analytical approach to applying the Lyapunov direct method for such models. The novelty of this approach lies in successfully handling the two challenges when applying the method. On the one hand, according to the integral terms involved in the model, we propose an easy-to-follow way to determine the kernel functions in the Lyapunov functional candidate. On the other hand, we establish a new integral inequality, which is conducive to arranging the derivative of the functional so that it is easy to see whether the derivative is negative definite or negative semi-definite. As an application, we investigate the global stability of the endemic steady state of an age-structured epidemic model with two infectious stages. Moreover, the Lyapunov functional obtained for the endemic steady state is also helpful for proving the global stability of the disease-free steady state and the persistence of the disease. [ABSTRACT FROM AUTHOR]