Article Contents
Article Contents

# Paradoxical phenomena and chaotic dynamics in epidemic models subject to vaccination

The author is supported by the project MTM2017-87697P

• An alternative to the constant vaccination strategy could be the administration of a large number of doses on "immunization days" with the aim of maintaining the basic reproduction number to be below one. This strategy, known as pulse vaccination, has been successfully applied for the control of many diseases especially in low-income countries. In this paper, we analytically prove (without being computer-aided) the existence of chaotic dynamics in the classical SIR model with pulse vaccination. To the best of our knowledge, this is the first time in which a theoretical proof of chaotic dynamics is given for an epidemic model subject to pulse vaccination. In a realistic public health context, our analysis suggests that the combination of an insufficient vaccination coverage and high birth rates could produce chaotic dynamics and an increment of the number of infectious individuals.

Mathematics Subject Classification: Primary: 92B05, 34C28; Secondary: 34C60.

 Citation:

• 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 $\mathcal{Q}\subset \mathcal{R}_{2}$

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