# Modeling the Propagation of Infectious Diseases across the Air Transport Network: A Bayesian Approach

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## Abstract

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## 1. Introduction

## 2. Characterizing the Air Transport Network

`R`[9]. This enables us to adeptly represent different epidemiological scenarios by adjusting the relevant parameters, eventually incorporating new input data as they arrive. Once our network model is trained and refined, it will empower us to forecast the progression and potential dissemination of epidemic outbreaks, akin to the global experiences of early 2020.

## 3. On-Board Transmission of Infectious Diseases

#### 3.1. Initial Number of Infected Passengers

#### 3.1.1. Estimation of ${p}_{\mathrm{infec}}$

- ${I}_{itd}$ is the incidence rate, defined as the cumulative number of confirmed cases of the infectious disease in the catchment area of airport i up to the current day t over the past d days. It is common to choose $d=7$ as the approximate length of the contagion period for an infected individual, as described in [15] for the case of COVID-19. Therefore, and unless otherwise stated, we use ${I}_{it7}$ in our calculations.
- ${N}_{i}$ is the population of the region of influence around the origin airport.
- $1<\upsilon \ll {N}_{i}/{I}_{it7}$ is the underestimating factor, which reflects the extent to which confirmed infected cases underestimate the actual number of cases.
- $\gamma <1$ is the asymptomatic factor, accounting for the proportion of infected individuals who are asymptomatic and their alleged lower infectiousness.
- $\eta <1$ is the healthy passenger factor, capturing the likelihood that the per capita rate of infected individuals among travelers boarding airplanes is lower than that in the overall population.

#### 3.1.2. Estimation of $\upsilon $

#### 3.1.3. Estimation of $\gamma $

#### 3.1.4. Estimation of $\eta $

#### 3.2. Final Number of Infected Passengers

- ${\rho}_{\mathrm{trans},\ell ,m}$ represents the unconditional probability that in the absence of masks and vaccines, a contact between infected passenger m and susceptible passenger ℓ will transmit the virus to the latter.
- ${\u03f5}_{m}\le 1$ is the effectiveness of masks.
- ${\u03f5}_{v}\le 1$ is the effectiveness of vaccines.

#### 3.2.1. Estimation of the Unconditional Transmission Probability, ${\rho}_{\mathrm{trans},\ell ,m}$

#### Estimation of ${\tau}_{0}$

#### Estimation of $\lambda $

#### Estimation of $\varphi $

#### 3.2.2. Estimation of the Effectiveness of Masks, ${\u03f5}_{m}$

#### 3.2.3. Estimation of the Effectiveness of Vaccines, ${\u03f5}_{v}$

#### 3.3. Relationship with the SIR Model and Its Variants

## 4. Estimating Imported Risks Using DES

- The initial daily imported risk ${r}_{i}$ at any airport i is 0.
- The first event must be a so-called initial departure, i.e., a flight k departing from airport $i\in \mathcal{I}$ and arriving at airport $j\in \mathcal{D}$ between 00:00 h and 23:59 h UTC of the incumbent day, with ${n}_{k}$ passengers onboard. Since this is the first event of the day at airport i, all passengers must come from its catchment area, out of which ${I}_{k}\left(0\right)$ are expected to be infected, as given by (1).
- The next event could be either another initial departure (which would be dealt with as in Step 2) or an arrival at airport j at time ${t}_{a}\in (t,t+\Delta t)$. For the latter, the initial number of infected passengers ${I}_{k}\left(0\right)$ might have increased during the flight time to ${I}_{k}\left({m}_{k}\right)={I}_{k}\left(0\right)+{\tilde{I}}_{k}\left({m}_{k}\right)$, with ${\tilde{I}}_{k}\left({m}_{k}\right)$ given by (9).
- (a)
- If the arrival is the last event of the day at airport j, we assume that all its passengers will leave the airport. Then, ${y}_{k}={I}_{k}\left({m}_{k}\right)$.
- (b)
- Else, we determine ${y}_{k}$ as given by (14).
- (c)
- In any case, we update the corresponding daily imported risk ${r}_{j}={r}_{j}+{y}_{k}$.
- (d)
- (e)
- Next, we need to distribute those connecting passengers among all the feasible flights that depart between one and three hours after the incumbent flight’s arrival. Let us assume that there are n such flights ${k}_{1},\dots ,{k}_{n}$ departing from airport j, connecting with new destinations ${j}_{1},\dots ,{j}_{n}$, and departing at times ${t}_{{d}_{1}},{t}_{{d}_{2}},\dots ,{t}_{{d}_{n}}$, such that $\lfloor {t}_{a}\rfloor +1<{t}_{{d}_{1}}<{t}_{{d}_{2}}\dots <{t}_{{d}_{n}}<\lfloor {t}_{a}\rfloor +3$ (times in hours). Therefore, we allocate the passengers in ${x}_{k}$ and ${z}_{k}$ by means of two multinomial distributions, with weights ${w}_{1},\dots ,{w}_{n}$ representing the relative nominal capacity of the corresponding aircraft compared to the other possible connecting flights, i.e.,$$\begin{array}{c}\left({\tilde{x}}_{{k}_{1}},{\tilde{x}}_{{k}_{2}},\dots ,{\tilde{x}}_{{k}_{n}}\right)\sim \mathcal{M}ult({x}_{k};{w}_{1},\dots ,{w}_{n}),\end{array}$$$$\begin{array}{c}\left({\tilde{z}}_{{k}_{1}},{\tilde{z}}_{{k}_{2}},\dots ,{\tilde{z}}_{{k}_{n}}\right)\sim \mathcal{M}ult({z}_{k};{w}_{1},\dots ,{w}_{n}),\end{array}$$
- (f)
- (g)

- The next event could be either an initial departure (Step 2), an arrival (Step 3) or a non-initial departure. For the latter, we must first accommodate infected and non-infected passengers connecting from other flights, given by ${\tilde{x}}_{k}$ and ${\tilde{z}}_{k}$ in (17) and (18), respectively. The rest of the passengers are drawn from the corresponding catchment area.
- We alternate steps 2–4 until there are no more flights in the database. The last event must be an arrival.

Algorithm 1: DES for the ATN performance | ||||

Data: Flights connecting airports in $\mathcal{I}$ and $\mathcal{D}$ plus internal connections in $\mathcal{D}$ | ||||

Result: Daily imported risk at the network nodes’ airsides ${r}_{j}$ | ||||

Initialization: ${r}_{j}=0$ | ||||

for all flights in the database do | ||||

if departure then | ||||

if flights arrived between 1 and 3 h before then | ||||

Fill partially aircraft with passengers connecting from other flights | ||||

Fill rest of aircraft from catchment area | ||||

else | ||||

Fill aircraft from catchment area | ||||

else | ||||

if arrival not final event then | ||||

Prop. ${\pi}_{t}$ of passengers leave airport, of which ${y}_{k}$ are infected | ||||

The rest (${x}_{k}$ and ${z}_{k}$) connect with other flights | ||||

else | ||||

100% passengers leave airport, of which ${y}_{k}={I}_{k}\left({m}_{k}\right)$ are infected | ||||

Update ${r}_{j}={r}_{j}+{y}_{k}$ |

## 5. The Impact of the Indian Outbreak in Europe

#### 5.1. Simulations of the Baseline Scenario

#### 5.2. Influence of High Occupancy

#### 5.3. Closing Airports/Routes

#### 5.4. Limitations of Our Model

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Composition scheme to sample the final epidemiological status ${I}_{k}\left({m}_{k}\right)$.

**Figure 5.**Flights arriving at MAD or BCN on 1 March 2021. Blue dots represent those airports connected by the suitable routes $\mathcal{R}$.

AMD | ATQ | BLR | BOM | CCU | COK | DEL | GOI | HYD | MAA |
---|---|---|---|---|---|---|---|---|---|

1 | 13 | 103 | 167 | 2 | 1 | 342 | 5 | 26 | 5 |

AMS | BHX | CDG | FCO | FRA | KBP | LGW | LHR | MAN | STN | SVO |
---|---|---|---|---|---|---|---|---|---|---|

57 | 1 | 112 | 10 | 174 | 3 | 1 | 274 | 2 | 3 | 28 |

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**MDPI and ACS Style**

Quirós Corte, P.; Cano, J.; Sánchez Ayra, E.; Joshi, C.; Gómez Comendador, V.F.
Modeling the Propagation of Infectious Diseases across the Air Transport Network: A Bayesian Approach. *Mathematics* **2024**, *12*, 1241.
https://doi.org/10.3390/math12081241

**AMA Style**

Quirós Corte P, Cano J, Sánchez Ayra E, Joshi C, Gómez Comendador VF.
Modeling the Propagation of Infectious Diseases across the Air Transport Network: A Bayesian Approach. *Mathematics*. 2024; 12(8):1241.
https://doi.org/10.3390/math12081241

**Chicago/Turabian Style**

Quirós Corte, Pablo, Javier Cano, Eduardo Sánchez Ayra, Chaitanya Joshi, and Víctor Fernando Gómez Comendador.
2024. "Modeling the Propagation of Infectious Diseases across the Air Transport Network: A Bayesian Approach" *Mathematics* 12, no. 8: 1241.
https://doi.org/10.3390/math12081241