Turning points and relaxation oscillation cycles in simple epidemic models (Q2799039)

Assuming that the host population has a small intrinsic growth rate \(\varepsilon>0,\) the authors employ singular perturbation analysis to study a simple SIR epidemic model. Using the intrinsic growth rate \(\varepsilon\) as a perturbation parameter, they reformulate the standard SIR epidemic model as a singularly perturbed problem and apply to it techniques from geometric singular perturbation theory and global center manifold theory.NEWLINENEWLINEIt is shown that stable periodic oscillations can be observed in this simple SIR model for parameters from a biologically realistic range. Contrary to the traditional Hopf bifurcation analysis establishing the existence of bifurcating periodic solutions only of a small amplitude, the approach suggested in the paper allows to prove the existence of periodic solutions with large amplitudes of order \(O(1).\) The authors show that periodic solutions are of relaxation-oscillation type.NEWLINENEWLINEAn important characteristic of models satisfying the slow-growth assumption is the existence of a turning point on the slow manifold associated with the critical community size required to support an epidemic. In the model under study, the slow manifold is located in the disease-free region; the periods when solutions remain in the vicinity of the slow manifold correspond to the inter-epidemic period with a low disease incidence. The periods when solutions stay away from the slow manifold are associated with the fast dynamics; switches between slow motions towards and along the slow manifold and fast motions away from it characterize the global dynamics of the system and match empirical data.NEWLINENEWLINEThe authors point out two main difficulties in the analysis. The first is related to the existence of turning points at which two eigenvalues vanish, which leads to the loss of normal hyperbolicity of the one-dimensional slow manifold. The second is related to the necessity of dealing with the nonlinear dynamics in a large neighborhood of the slow manifold. The authors argue that the existence of turning points and the associated delay of stability loss due to the slow growth of the population offer a simple and robust mechanism for sustained oscillations of disease incidence. They plan to use singular perturbation analysis to investigate relaxation oscillations in a SEIR model.