Paradoxical phenomena and chaotic dynamics in epidemic models subject to vaccination

Figure 1.  Representation of $ \Upsilon(0.3,T) $ as a function of $ T $. Fixed parameters in both panels $ \beta = \gamma = \mu = 1 $. (Left) $ \lambda = 0.25 $. In this case, $ R_{0}<1 $ and $ \Upsilon(0.3,T)<1 $ for all $ T>0 $. (Right) $ \lambda = 3 $. In this case, $ R_{0}>1 $ and $ \Upsilon (0.3,T) $ exhibits a non-monotone behavior

Figure 2.  Illustration of the energy levels of system (28)

Figure 3.  Pictorial description of $ \Gamma_{x_{0}} $ (continuous curve) and $ t_{v}(\Gamma_{x_{0}}) $ (dashed curve)

Figure 4.  Pictorial description of $ \Gamma_{x_{0}} $, $ \Gamma_{x_{0}+\delta} $ (continuous curves), and $ t_{v}(\Gamma_{x_{0}}) $ $ t_{v}(\Gamma_{x_{0}+\delta}) $ (dashed lines). Geometrically, condition (31) means that the intersection of both annulus is below the line $ z = \widetilde{z}_{max} $, (red line)

Figure 5.  Pictorial description of the annulus $ \mathcal{A}_{1} $, $ \mathcal{A}_{2} $ and the topological rectangles $ \mathcal{R}_{1} $, $ \mathcal{R}_{2} $. Notice that $ \mathcal{Q}\subset \mathcal{R}_{2} $