key: cord-0808008-p2t2wd4t
authors: Cruz-Pacheco, Gustavo; Bustamante-Castaneda, Fernando J; Caputo, Jean Guy; Jimenez-Corona, Maria Eugenia; Ponce-de-Leon, Samuel
title: Dispersion of a new coronavirus SARS-CoV-2 by airlines in 2020: Temporal estimates of the outbreak in Mexico.
date: 2020-03-30
journal: nan
DOI: 10.1101/2020.03.24.20042168
sha: 53c53af4a88597f834ccf35dc5761d9cd962c351
doc_id: 808008
cord_uid: p2t2wd4t

On January 23, 2020, China imposed a quarantine on the city of Wuhan to contain the SARS-CoV-2 outbreak. Regardless of this measure the new infection has spread to several countries around the world. Here, we developed a method to study the dissemination of this infection by the airline routes and we give estimations of the time of arrival of the outbreaks to the different cities. In this work we show an analysis of the dispersion of this infection to other cities by airlines based on the classic model the Kermack and McKendrick complemented with diffusion on a graph composed of nodes which represent the cities and edges which represent the airline routes. We do several numerical simulations to estimate the date of arrival to different cities starting the infection at Wuhan, China and to show the robustness of the estimation respect to changes in the epidemiological parameters and to changes on the graph. We use Mexico City as an example. In this case, our estimate of the arrival time is between March 20 and March 30, 2020. This analysis is limited to the analysis of dispersion by airlines, so this estimate should be taken as an overestimate since the infection can arrive by other means. This model estimates the arrival of the infectious outbreak to Mexico between March 20 and March 30. This estimation gives a time period to implement and strengthen preventive measures aimed at the general population, as well as to strengthen hospital infrastructure and training of human resources in health.

On December 31, 2019, China reported to the country office of the World Health Organization (WHO) an outbreak of cases of pneumonia of unknown etiology; From December 31 to January 3, 2020, 44 cases of pneumonia of unknown origin in Wuhan city, Hubei Province, were reported to WHO. On Jan-. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03.24.20042168 doi: medRxiv preprint uary 7, the Chinese authorities announced that it is a new coronavirus (nCoV-2019), now called SARS-CoV-2. 2 7 This same model, supplemented with classical diffusion (which in the context of an epidemic is managed as a dispersion of the infection) has been quite useful in studies of the spread of some infections as, for example, rabies. 8 In this work we develop a new method to analyze the dissemination of a new virus as SARS-CoV-2 between different cities by airline routes with the purpose of estimating the arrival from one city to another. Our method uses the Kermack and Mckendrick model to simulate the way the outbreak grows and evolves when it arrives to the different cities and, also we use classical diffusion on a graph, to model the way the infection travels between cities by the airline routes. 9 The purpose of using a simple model for our analysis is to have the least number of parameters, but with sufficient precision to be able to give an estimate of the date of arrival of the outbreak for example, to Mexico City. By this date, it is well established that infection by this type of coronavirus is transmitted person to person through drops expelled by an infected person when coughing or sneezing, it is also possible to be transmitted by contact with surfaces or objects contaminated with the virus and then one puts his hands to his mouth, eyes or nose. 10 There is evidence that once the infectious outbreak has started there is some homogeneity in its . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03.24.20042168 doi: medRxiv preprint development in the affected city for these influenza-like infections. For these reasons, the Kermack and McKendrick model is applicable in the cities that we use as nodes of the graph in which the edges represent the flights between one city and another. Brockmann and Helbling, 1 used a similar model to study the dispersion of a general infection except that their transfer term is not a symmetric diffusion.

The basic reproductive number was estimated using the data reported for Hubei and Guangdong, 11 an adjustment was made to the data at the beginning of the epidemic outbreak to estimate the force of infection λ, which is related to the basic reproductive number as follows R 0 = 1 + λ / β. The setting of λ for Hubei and for Guangdong gives λ = 0.3. These estimates give an R 0 = 2.5, which is in very good agreement with the estimate of Joseph T Wu 12 of R0 = 2.53 and, is well into the interval of the estimations reported for JM Read. 13 The diffusion coefficient D was estimated using numerical simulations of the model and the fact that a new outbreak started in Singapore around February 15, 2020. This gives a diffusion coefficient D = 10 -6 . In Figure 1 the nodes of the graph represent cities that were chosen because their airports function as important distribution points to other airports. In the model, some of these nodes also function as a representation of other airports in the region, for example, the Paris node represents the main international airports of that European region: Paris, Amsterdam and, Frankfurt. Using this graph and the estimated epidemiological parameters, the model was solved numerically to study in several scenarios the time of arrival of the infection to Mexico City.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03.24.20042168 doi: medRxiv preprint Figure 1 Cities considered as nodes in this analysis, as well as the corresponding graph.

Using the parameters estimated the model was solved numerically to study in several scenarios the arrival of the outbreak to Mexico City. All our simulations have initial conditions in which only node 1 corresponding to the city of Wuhan has been infected and the other cities are not yet infected or represent a very small number with respect to its population. In the city of Wuhan, an infected number . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03.24.20042168 doi: medRxiv preprint of 20 is assumed, 14 the initial date for this analysis was taken on January 10, 2020, as the starting point of the numerical simulations.

We solved numerically this model, the result is shown in figure 2 is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03.24.20042168 doi: medRxiv preprint simulation in which R 0 has been changed as follows: R 0 = 2.1 (black in dashed line) and 1.9 (black). In both cases, the corresponding peaks at node 8 only gets slightly retarded.

To simulate the isolation measures imposed on Wuhan since the end of January, numerical simulations where performed with lowering the value of R 0 from 2.5 to 2.1 first and then to 1.9, This is shown in figure 2 by the black dashed curve for R 0 = 2.1 and by the black curve for R 0 = 1.9. As can be seen, this does not affect the high and width of the rest of the peaks and only delays the arrival of the rest of the nodes including node 8 of Mexico City by around of 4 days. This shows that changes not yet so small in R 0 do not substantially alter the result on the arrival of the outbreak at node 8. Simulations with R 0 = 3, when they era adjusted to match the outbreak in Wuhan also only anticipated the outbreak in Mexico City by 4 days. We performed other numerical simulations changing the R 0 in other nodes of the graph around 10% of its value, this changed the outbreaks at these nodes, but it did not change significantly, the date of arrival of the outbreak in Mexico City. Changes on the graph structure cutting some edges, meaning cancelling some airline routes, also did not modify significantly the date of arrival of the outbreak to the last node. Even cutting all the edges connecting to node 8 except one, only delayed the arrival of the outbreak to Mexico City around 6 days. Reducing the coefficient D has a larger impact on the delay of the outbreak, reducing D by one order of magnitude, from 10 -6 to 10 -7 , delayed the arrival of the outbreak to node 8 around 2 weeks.

The figure 2, also shows something else which is important, the time at which the first node reaches the maximum moves forward as the R0 decreases, this indicates that the isolation measures carried out in Wuhan only decreased a few tenths the R0, of the order of 5 tenths. This is because, according to the data presented by Johns Hopkins University, the report of cases begins to stabilize on approximately the in February 28, and therefore the maximum is expected to be reached close to this date. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 30, 2020. This model shows that the isolation measures that can be implemented in the cities where the outbreak first arrives, although they are very important to control it locally, do not affect significantly the time to arrival to other cities. The coefficient that best controls the spread of infection in the rest of the cities, when this dispersion is by airlines, is the parameter D. Therefore, surveillance at airports should be strengthened, with special emphasis on those connecting Mexico directly or indirectly with Asian countries. Finally, this model shows that all these measures can only delay the arrival of SARS-CoV-2, but if it can be delayed long enough, it would be very important to have as much time as possible to establish the appropriate prevention and control measures. it will allow better management of the outbreak.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03.24.20042168 doi: medRxiv preprint

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