# A Global Sharing Mechanism of Resources: Modeling a Crucial Step in the Fight against Pandemics

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Coupled SIR and DSP Model

#### 2.2. Sharing System

## 3. Results

#### 3.1. Equal Maximum Stocks

#### 3.2. Unequal Maximum Stocks

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Extended Methods

#### Appendix A.1. General Remark

#### Appendix A.2. Adapted SIR

#### Appendix A.3. DSP Model

#### Appendix A.4. Sharing Mechanism

Parameter | Symbol | Value | Comment |
---|---|---|---|

Population size | n | $1.7\times {10}^{7}$ | This parameter has no effect in the model but makes results more easily interpretable. It is roughly the population size of the Netherlands. |

Basic reproduction number | ${R}_{0}$ | $2.3$ | ${R}_{0}$ is the reproduction number of the virus in the absence of countermeasures and without immunity, i.e., at the start of the pandemic. In their June 2020 report [28] the WHO estimated this number in the range 2–4. In the Netherlands, the peak estimate [29] near the start of the pandemic is around $2.2$ but this varies significantly across countries and time. The most important thing about this parameter is that it should be larger than 1 in order to lead to exponential growth and thus a population-wide epidemic. |

Recovery rate | $\gamma $ | $1/6$ | This rate defines the average period ($1/\gamma $ days) during which an individual is infectious. In the same WHO report, the curve of virus-shedding appears to be high for about six days. More recent sources report a median of 8 days [30]. Together with the infection rate, this parameter induces the reproduction number, which in our simulation is the more important number. |

Initial production | $P\left(0\right)$ | $5\times {10}^{5}$ | This parameter has an arbitrary unit which has no effect on the model but makes results more easily interpretable. According to a news article from April 2020, the Netherlands were using about $4.5\times {10}^{6}/7$ face masks per day, which is the most well-known resource. Our value here is thus chosen to be on the order of magnitude of the number of face masks. |

Maximum production | ${P}^{\mathrm{max}}$ | $4\times P\left(0\right)$ | A community can increase the local production to this quantity. This includes all sources, such as increasing local production as well as importing more resources from global suppliers. |

Necessary PPE per infected case | w | 4 | Keeping to face masks for interpretability, numbers vary wildly: from over a hundred masks per day per corona patient in the intensive care unit to one or zero for infected persons who are either unaware of their infection or remain in quarantine at home. For interpretability, we turn to what would in principle be needed to prevent one infected person from infecting others as much as possible. According to various sources, a disposable mask should only be used for about three hours. This would mean that in principle, in order to cover one’s face for 12 h per day, about 4 masks would be needed, which is the (arbitrary) quantity we chose here. |

Reduction in ${R}_{e}$ for $S>0$ | r | $0.4$ | This factor multiplies with the infection rate and thus reduces the reproduction number. Its value is arbitrary with the most important property being that $r\xb7{R}_{0}<1$. In other words, this means that having sufficient PPE would stop the exponential growth of the epidemic. |

## Appendix B. Additional Figure

**Figure A1.**

**Influence of B maximum stock and sharing threshold on total infected ratio for different epidemic-onset time differences and fixed A maximum stock.**The epidemic is introduced in A at $t=0$, and in B at $t=\Delta t$. As a special case, the negative $\Delta t=-60$ thus indicates epidemic onset in B prior to its onset in A. The maximum stock of A is fixed at either Low stock (${S}^{\mathrm{max}}=1\times {10}^{7}$), Medium stock (${S}^{\mathrm{max}}=3\times {10}^{7}$), or High stock (${S}^{\mathrm{max}}=7\times {10}^{7}$). $n=1.7\times {10}^{7}$, ${R}_{0}=2.3$, $\gamma =1/6$, $r=0.4$, $w=4$, $P\left(0\right)=5\times {10}^{7}$, ${P}^{\mathrm{max}}=4\times {P}_{i}\left(0\right)$, $D\left(0\right)=P\left(0\right)$.

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**Figure 1.**

**Time evolution of epidemic development in two communities without and with sharing mechanism.**Shown are the proportions of infected (blue) relative to the total community population and the proportions of stock (green) relative to the maximum stock over time. The communities had an equal maximum stock of ${S}^{\mathrm{max}}=3\times {10}^{7}$. The only difference between the communities was epidemic onset, with epidemic introduction in A at $t=0$ and in B at $t=\Delta t=180$ days. Critical zero-stock periods are shaded in red in the no-sharing condition (left column). The lack of stock leads to an increase in epidemic spread and a high infected peak. The sharing threshold $\theta =0.6$ (yellow dotted line) is indicated in the sharing condition (right column). Here, sharing prevents the total depletion of stock and the associated high infected peak: both the local and total infected ratios are low. $n=1.7\times {10}^{7}$, ${R}_{0}=2.3$, $\gamma =1/6$, $r=0.4$, $w=4$, $P\left(0\right)=5\times {10}^{7}$, ${P}^{\mathrm{max}}=4\times P\left(0\right)$, $D\left(0\right)=P\left(0\right)$.

**Figure 2.**

**Influence of epidemic onset time difference and sharing threshold on total infected ratio for equal maximum stock communities.**The epidemic is introduced in A at $t=0$, and in B at $t=\Delta t$. The indicated infected ratio is the average of the two communities’ infected ratios. The red state indicates both communities reaching a high infected ratio and white means that one community (labeled with the red letter, here B) had a high infected ratio while the other remained low. In the blue state both communities had a relatively low amount of infected. For each phase plot, the two communities had the same maximum stock, being either ${S}^{\mathrm{max}}=1\times {10}^{7}$ (Low stock), ${S}^{\mathrm{max}}=3\times {10}^{7}$ (Medium stock) or ${S}^{\mathrm{max}}=7\times {10}^{7}$ (High stock). $n=1.7\times {10}^{7}$, ${R}_{0}=2.3$, $\gamma =1/6$, $r=0.4$, $w=4$, $P\left(0\right)=5\times {10}^{7}$, ${P}^{\mathrm{max}}=4\times P\left(0\right)$, $D\left(0\right)=P\left(0\right)$.

**Figure 3.**

**Influence of maximum stock size and sharing threshold on the total infected ratio for three distinct epidemic onset time differences between equal maximum stock communities.**In each simulation, the two communities had equal maximum stock sizes. The white state is labeled with the community that reached zero stock and a high infected ratio (B). The green dashed lines in the left panel indicate where cross-sections are taken for the three maximum stock categories used throughout this paper: Low stock with ${S}^{\mathrm{max}}=1\times {10}^{7}$, Medium stock with ${S}^{\mathrm{max}}=3\times {10}^{7}$ and High stock with ${S}^{\mathrm{max}}=7\times {10}^{7}$. The values at these lines correspond to those at the different maximum stock phase plots in Figure 2 for $\Delta t=0$. Within the $\Delta t=60$ and $\Delta t=180$ phase plots, cross-sections at the same stock values can be imagined for likewise comparison with the previous figure. $n=1.7\times {10}^{7}$, ${R}_{0}=2.3$, $\gamma =1/6$, $r=0.4$, $w=4$, $P\left(0\right)=5\times {10}^{7}$, ${P}^{\mathrm{max}}=4\times P\left(0\right)$, $D\left(0\right)=P\left(0\right)$.

**Figure 4.**

**Influence of time difference between epidemic onset and sharing threshold on total infected ratio for unequal maximum stock communities.**Respectively showing combinations of (

**a**) low- and medium-stock (

**b**) low- and high-stock and (

**c**) medium- and high-stock communities. The epidemic is introduced in A at $t=0$, and in B at $t=\Delta t$. In the white areas, a red letter indicates which of the two communities reached zero stock. The pink dashed line in (

**b**) indicates the settings taken for the time evolutions shown in Figure 5. $n=1.7\times {10}^{7}$, ${R}_{0}=2.3$, $\gamma =1/6$, $r=0.4$, $w=4$, $P\left(0\right)=5\times {10}^{7}$, ${P}^{\mathrm{max}}=4\times {P}_{i}\left(0\right)$, $D\left(0\right)=P\left(0\right)$.

**Figure 5.**

**Effect of increasing epidemic onset time difference on sharing success of unequal communities.**Indicated are the proportions of infected (blue) relative to the total community population and the proportions of stock (green) relative to the maximum stock over time. Critical zero-stock periods are shaded in red and the sharing threshold ($\theta =0.5$) is indicated by the yellow dotted line. The communities had unequal maximum stocks, with ${S}_{A}^{\mathrm{max}}=1\times {10}^{7}$ and ${S}_{B}^{\mathrm{max}}=7\times {10}^{7}$. By increasing the epidemic onset time difference, the system transitions from only high-stock community B weathering the epidemic with low infected ratios (for $\Delta t=0$ days), via both communities reaching high infected ratios (for $\Delta t=25$ days) and only low-stock community A reaching a low infected ratio (for $\Delta t=50$ days), to both having low infected ratios (for $\Delta t=75$ days). This behavior is observed for the specific settings, as indicated by the pink dashed line in Figure 4, but more generally illustrates the impact of epidemic onset time difference on the possible system outcomes. $n=1.7\times {10}^{7}$, ${R}_{0}=2.3$, $\gamma =1/6$, $r=0.4$, $w=4$, $P\left(0\right)=5\times {10}^{7}$, ${P}^{\mathrm{max}}=4\times P\left(0\right)$, $D\left(0\right)=P\left(0\right)$.

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**MDPI and ACS Style**

den Nijs, K.; Edivaldo, J.; Châtel, B.D.L.; Uleman, J.F.; Olde Rikkert, M.; Wertheim, H.; Quax, R.
A Global Sharing Mechanism of Resources: Modeling a Crucial Step in the Fight against Pandemics. *Int. J. Environ. Res. Public Health* **2022**, *19*, 5930.
https://doi.org/10.3390/ijerph19105930

**AMA Style**

den Nijs K, Edivaldo J, Châtel BDL, Uleman JF, Olde Rikkert M, Wertheim H, Quax R.
A Global Sharing Mechanism of Resources: Modeling a Crucial Step in the Fight against Pandemics. *International Journal of Environmental Research and Public Health*. 2022; 19(10):5930.
https://doi.org/10.3390/ijerph19105930

**Chicago/Turabian Style**

den Nijs, Katinka, Jose Edivaldo, Bas D. L. Châtel, Jeroen F. Uleman, Marcel Olde Rikkert, Heiman Wertheim, and Rick Quax.
2022. "A Global Sharing Mechanism of Resources: Modeling a Crucial Step in the Fight against Pandemics" *International Journal of Environmental Research and Public Health* 19, no. 10: 5930.
https://doi.org/10.3390/ijerph19105930