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

This research was partially supported by Simons Foundation-Collaboration Grants for Mathematicians 523360

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  • 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 $ \mathcal{R}_0 $ 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.

    Mathematics Subject Classification: Primary: 34C26, 92D25; Secondary: 34C23.


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  • Figure 1.  (a) Bifurcation curves in $ (\beta_1, b) $ plane. (b) Pitchfork bifurcation occurs at $ E_0 $ when $ \beta_1 = \hat{\beta}_1 $ if $ b = \hat{b} $

    Figure 2.  Dynamics of layer problem (7) with $S_{h}^{0} <N^{0}$

    Figure 3.  Dynamics of system (7) on $ M(\mathcal{Z}_0) $, in which double arrow indicates the fast movement along the regular orbits, and single arrow indicates the slow moment on the slow manifold $ \mathcal{Z}_0 $. The blue line is the one of a family of slow-fast cycles

    Figure 4.  Two limit cycles are on the center manifold $ M^{\varepsilon} $, and the outer one is a relaxation oscillation cycle

    Figure 5.  (a) Hopf bifurcation curve in $ (\varepsilon, b) $-plane. (b) Hopf bifurcation curve in $ (\beta, b) $-plane, where $ \varepsilon = 4\times 10^{-5}, d_2 = 0.02, \mu_0 = 0.03, \mu_1 = 0.0305, N = 10000, M = 250000, \beta_2 = 0.025$

    Figure 6.  Limit cycles generated by Hopf bifurcation. For parameters, $ \beta_1 = 0.00115 $ and $ b = 2 $ in (a); $ \beta_1 = 0.0009760249 $ and $ b = 0.05 $ in (b)

    Figure 7.  Bifurcation diagram in $ (\beta_1, b) $-plane for $ \varepsilon>0 $ small

    Figure 8.  Generation and annihilation of limit cycles. Green curves signify stable limit cycles and pink curves signify unstable limit cycles

    Figure 9.  Relaxation oscillation cycle (blue curve) coexists with small fluctuation due to the unstable limit cycle (red curve). All parameters are chosen as those in Fig. 4 except $ \varepsilon = 10^{-5} $

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