Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases

Disease transmission can present significantly different cyclic patterns including small fluctuations, regular oscillations, and singular oscillations with short endemic period and long inter-epidemic period. In this paper we consider the slow-fast dynamics and nonlinear oscillations during the transmission of mosquito-borne diseases. Under the assumption that the host population has a small natural death rate, we prove the existence of relaxation oscillation cycles by geometric singular perturbation techniques and the delay of stability loss. Generation and annihilation of periodic orbits are investigated through local, semi-local bifurcations and bifurcation of slow-fast cycles. It turns out that relaxation oscillation cycles occur only if the basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document} is greater than 1, while small fluctuations and regular oscillations exist under much less restrictive conditions. Our results here provide a sound explanation for different cyclic patterns exhibited in the transmission of mosquito-borne diseases.