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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.02306 |
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| _version_ | 1866916341984788480 |
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| author | de Carvalho, João P. S. Maurício Rodrigues, Alexandre A. |
| author_facet | de Carvalho, João P. S. Maurício Rodrigues, Alexandre A. |
| contents | We analyze a periodically-forced dynamical system inspired by the SIR model with impulsive vaccination. We fully characterize its dynamics according to the proportion $p$ of vaccinated individuals and the time $T$ between doses. If the basic reproduction number is less than 1 (i.e. $\mathcal{R}_p<1$), then we obtain precise conditions for the existence and global stability of a disease-free it $T$-periodic solution. Otherwise, if $\mathcal{R}_p>1$, then a globally stable $T$-periodic solution emerges with positive coordinates.
We draw a bifurcation diagram $(T,p)$ and we describe the associated bifurcations. We also find analytically and numerically chaotic dynamics by adding seasonality to the disease transmission rate. In a realistic context, low vaccination coverage and intense seasonality may result in unpredictable dynamics. Previous experiments have suggested chaos in periodically-forced biological impulsive models, but no analytic proof has been given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_02306 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pulse vaccination in a SIR model: global dynamics, bifurcations and seasonality de Carvalho, João P. S. Maurício Rodrigues, Alexandre A. Dynamical Systems We analyze a periodically-forced dynamical system inspired by the SIR model with impulsive vaccination. We fully characterize its dynamics according to the proportion $p$ of vaccinated individuals and the time $T$ between doses. If the basic reproduction number is less than 1 (i.e. $\mathcal{R}_p<1$), then we obtain precise conditions for the existence and global stability of a disease-free it $T$-periodic solution. Otherwise, if $\mathcal{R}_p>1$, then a globally stable $T$-periodic solution emerges with positive coordinates. We draw a bifurcation diagram $(T,p)$ and we describe the associated bifurcations. We also find analytically and numerically chaotic dynamics by adding seasonality to the disease transmission rate. In a realistic context, low vaccination coverage and intense seasonality may result in unpredictable dynamics. Previous experiments have suggested chaos in periodically-forced biological impulsive models, but no analytic proof has been given. |
| title | Pulse vaccination in a SIR model: global dynamics, bifurcations and seasonality |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2312.02306 |