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

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} $