Geometric approach to global asymptotic stability for the SEIRS models in epidemiology

In this paper, we present a more general criterion for the global asymptotic stability of equilibria for nonlinear autonomous differential equations based on the geometric criterion developed by Li and Muldowney. By applying this criterion, we obtain some results for the global asymptotic stability of SEIRS models with constant recruitment and varying total population size. Based on these results, we give a complete affirmative answer to Liu-Hethcote-Levin conjecture. Furthermore, an affirmative answer to Li-Graef-Wang-Karsai's problem for SEIR model with permanent immunity and varying total population size is given.