Open AccessSlow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases
Author(s) -
Chunhua Shan
Publication year - 2022
Publication title -
discrete and continuous dynamical systems. series b
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.864
H-Index - 53
eISSN - 1553-524X
pISSN - 1531-3492
DOI - 10.3934/dcdsb.2021097
Subject(s) - oscillation (cell signaling) , nonlinear system , relaxation (psychology) , mathematics , bifurcation , transmission (telecommunications) , annihilation , dynamics (music) , population , physics , biology , quantum mechanics , computer science , demography , telecommunications , neuroscience , genetics , sociology , acoustics
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.