On early extinction and the effect of travelling in the SIR model
Petra Berenbrink, Colin Cooper, Cristina Gava, David Kohan Marzagão, Frederik Mallmann-Trenn, Tomasz Radzik
Abstract
We consider a population protocol version of the SIR model. In every round, an individual is chosen uniformly at random. If the individual is susceptible, then it becomes infected w.p. $\beta I_t/N$, where $I_t$ is the number of infections at time $t$ and $N$ is the total number of individuals. If the individual is infected, then it recovers w.p. $\gamma$, whereas, if the individual is already recovered, nothing happens. We prove sharp bounds on the probability of the disease becoming pandemic vs extinguishing early (dying out quickly). The probability of extinguishing early, $\Pr{\mathcal{E}_{ext}}$, is typically neglected in prior work since most use (deterministic) differential equations. Leveraging on this, using $\Pr{\mathcal{E}_{ext}}$, we proceed by bounding the expected size of the population that contracts the disease $\mathbf{E}\left[R_\infty\right]$. Prior work only calculated $\mathbf{E}\left[R_\infty | \overline{\mathcal{E}_{ext}}\right]$, or obtained non-closed form solutions. We then study the two-country model also accounting for the role of $\Pr{\mathcal{E}_{ext}}$. We assume that both countries have different infection rates $\beta^{(i)}$, but share the same recovery rate $\gamma$. In this model, each round has two steps: First, an individual is chosen u.a.r. and travels w.p. $p_{travel}$ to the other country. Afterwards, the process continues as before with the respective infection rates. Finally, using simulations, we characterise the influence of $p_{travel}$ on the total number of infections. Our simulations show that, depending on the $\beta^{(i)}$, increasing $p_{travel}$ can decrease or increase the expected total number of infections $\mathbf{E}\left[R_\infty\right]$.
BibTeX
@InProceedings{pmlr-v180-berenbrink22a,
title = {On early extinction and the effect of travelling in the SIR model},
author = {Berenbrink, Petra and Cooper, Colin and Gava, Cristina and Kohan Marzag\~{a}o, David and Mallmann-Trenn, Frederik and Radzik, Tomasz},
booktitle = {Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence},
pages = {159--169},
year = {2022},
editor = {Cussens, James and Zhang, Kun},
volume = {180},
series = {Proceedings of Machine Learning Research},
month = {01--05 Aug},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v180/berenbrink22a/berenbrink22a.pdf},
url = {https://proceedings.mlr.press/v180/berenbrink22a.html},
abstract = {We consider a population protocol version of the SIR model. In every round, an individual is chosen uniformly at random. If the individual is susceptible, then it becomes infected w.p. $\beta I_t/N$, where $I_t$ is the number of infections at time $t$ and $N$ is the total number of individuals. If the individual is infected, then it recovers w.p. $\gamma$, whereas, if the individual is already recovered, nothing happens. We prove sharp bounds on the probability of the disease becoming pandemic vs extinguishing early (dying out quickly). The probability of extinguishing early, $\Pr{\mathcal{E}_{ext}}$, is typically neglected in prior work since most use (deterministic) differential equations. Leveraging on this, using $\Pr{\mathcal{E}_{ext}}$, we proceed by bounding the expected size of the population that contracts the disease $\mathbf{E}\left[R_\infty\right]$. Prior work only calculated $\mathbf{E}\left[R_\infty | \overline{\mathcal{E}_{ext}}\right]$, or obtained non-closed form solutions. We then study the two-country model also accounting for the role of $\Pr{\mathcal{E}_{ext}}$. We assume that both countries have different infection rates $\beta^{(i)}$, but share the same recovery rate $\gamma$. In this model, each round has two steps: First, an individual is chosen u.a.r. and travels w.p. $p_{travel}$ to the other country. Afterwards, the process continues as before with the respective infection rates. Finally, using simulations, we characterise the influence of $p_{travel}$ on the total number of infections. Our simulations show that, depending on the $\beta^{(i)}$, increasing $p_{travel}$ can decrease or increase the expected total number of infections $\mathbf{E}\left[R_\infty\right]$.}
}