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Results
Results
The final network has 18,361 nodes and approximately 460,250 edges representing approximately 91,725,790 individuals residing in and near these metropolitan areas. The outbreak simulation was initiated by the exposure of a single node within the Seattle metropolitan area and spread from there using a stochastic SEIR model based on social contact pattern estimates.
Arrival Times
Arrival Times
Arrival time was calculated as the time step during which the first node was exposed in a given metropolitan area. Seattle's average arrival time is 0 days because the outbreak is simulated to begin there. Los Angeles, Chicago, and New York will all have a case within 6 days of the first case despite random differences in network construction in 10 trials. Atlanta, on the other hand, has an average disease arrival time of 8.2 days, varied from 6 to 12 days across 10 different network constructions and patient zero locations in the Seattle community. These results illustrate that with the current model parameters, COVID-19 will have reached these other four metropolitan areas in approximately a week.
Table 1: Average, minimum, and maximum arrival times in days with no travel bans in place.
Travel Restrictions
Travel Restrictions
Below are the results of the baseline network simulation with no intervention of a travel restriction. Each bar shows the segmentation of number of nodes in each compartment of the SEIR model at weekly time steps, averaged over 10 trials.
Figure 1: Number of susceptible, exposed, infectious, and removed nodes at weekly intervals with no interventions. Here, we see all nodes reach the removed state by the end of six weeks.
The set of nine graphs below show compartment totals for different combinations of travel restrictions. The first row displays results for only a national travel restriction starting at three different time steps one week apart. The second row displays results for a national and between county travel restriction implemented one week after the other. The third row displays results for a national, between county, and within county travel restriction implemented one week after one another.
Figure 2: Number of susceptible, exposed, infectious, and removed nodes at weekly time steps depending on timing and scale of travel restrictions. A trend of earlier and more levels of travel restrictions show a greater reduction in overall cases. The national level travel restriction has the largest impact on overall case reduction, especially within 2 to 3 weeks of the first case. The between county restriction also has a significant impact when implemented within 2 to 3 weeks of the first case. The within county restriction still has an impact on overall cases, but it is less substantial when compared to the first two levels. It is notable that national and inter-county travel restrictions within two weeks of the first case lead to less than 500 infected nodes over the course of 100 days. Furthermore, national travel restrictions prevent the entire network of nodes from becoming infected.
Table 2: Total number of infected nodes after 100 days given different timing and scale of travel restrictions.
The total number of nodes infected in 100 days varied with national, county, or community-level travel restrictions. This number includes nodes who are exposed, infectious, or removed 100 days following the first case. In comparison to the values in Table 2, no travel restrictions leads to all 18,361 nodes infected in 100 days. Note that as the national restriction occurs longer after the first case, the total number of infected nodes increases. Furthermore, earlier implementation of inter-county and inter-community travel restrictions reduces the total number of infected nodes to an appreciable degree.
Table 3: Characteristic time periods in days related to travel restriction scale and timing. A range is given because total number of nodes infected was recorded at weekly, not daily intervals.
Characteristic time is defined as the time point at which 1/e (≈ 37%) fraction of the population has been exposed . In the context of our model, this is when approximately 6,755 nodes total are infectious, exposed, or removed. In comparison to the ranges in Table 3, no travel restrictions leads to a characteristic time between days 14 and 21. Note that delaying the national ban decreases the characteristic time, meaning that the time until reaching just over a third of the population being exposed is shorter. Also note that the timing of the inter-county and inter-community travel restrictions does not impact the characteristic time, as typically by that point in the outbreak the damage has either been done or the transmission has been substantially reduced already.
Discussion
Discussion
Implications
Implications
These simulations show that a disease can spread to several major metropolitan areas across the United States in just over a week, one consequence of the interconnected nature of our country and speed of current transportation. Our results also demonstrate the potential impact of travel restrictions on limiting the total number of infected individuals and the time it takes for approximately 37% of the population to be infected. If travel restrictions could be implemented within two weeks of the first case, the total number of cases would be substantially reduced. This quick turnaround is largely impractical as it requires quick identification of a new disease, abrupt shutdown of national travel, and likely has serious economic consequences. As such, we recognize that national travel restrictions are unlikely to be put in place by the federal government and won't be actively chosen by US citizens until the outbreak worsens, at which point reduction in national travel will have little effect on total cases.
Comparison to New York Times Reports
Comparison to New York Times Reports
The New York Times has been providing extensive free online coverage of the coronavirus outbreak. As part of their coverage, case related data is updated daily. From this, the Times creates graphs and maps of cases and deaths across the US and the world.
The graph below shows the daily report of new cases in the US. It starts on February 26, 2020 and carries through to April 23, 2020, so far covering 57 days. Although the scale of the Times' graph differs from our results (theirs is daily new cases over 57 days, our results show weekly total cases when combining yellow, red, and purple over 98 days), it is still interesting to compare the overall shapes of the graphs. We see that our graphs "National Ban Starting on Day 7" and "National Ban Starting on Day 14" most closely mirror the shape in the Times' graph below.
Figure 3: New York Times graph on US daily reported cases. https://www.nytimes.com/interactive/2020/us/coronavirus-us-cases.html
Future Adjustments
Future Adjustments
Scale Factor: Due to computational timing constraints, we could only scale our model down to 5,000 individuals per node. To improve the model, one node would represent one individual, thus serving a better approximation of the contact network and spread of the disease.
Blue and Red Edge Probabilities: Connections between counties were determined by trial and error. To improve the model, specific data regarding the flow of people between counties would lead to better approximations. Potential data could include commuting statistics like metro and bus rides or toll road records.
Beta: One aspect of our model that does not match the predictions of other modelers and the spread of COVID-19 observed of late, is that our disease spreads particularly quickly across the network. We might attribute this to the high beta value, complicated by the fact that each node represents 5,000 individuals, a sacrifice made due to limited computing power and time. In order to match the timeline of Imperial College's model [1] of no pharmaceutical intervention in the US, the outbreak would have to last approximately three months following the first case. They thus predict the number of deaths peaking in the middle of that three month period (around day 45 let's say), which could be interpreted as the number of infected individuals peaking then or slightly before that time given the delay in contracting the disease and succumbing to it. The beta value used in our model was 0.16 (see Methods and Implementation for how we calculated this value) and to observe outbreak peaks around day 45 (give or take a week), beta would have to be as low as 0.03 or 0.025 (data not shown). We could not justify this value with regard to our earlier calculations so we retained the beta value of 0.16 for the simulation and analysis. Given more time and computing power, we would further investigate the accuracy of our beta value in relation to our 5,000 individuals per node set up. Interplay between multiple parameters may also explain this difference in peak time and is something we would look into further given the chance.
Computation Time: As mentioned, computation time was one of the most limiting factors in the construction of this project. One run of the model, with the scale factor of 5,000 individuals per node, took about 3-4 minutes to complete. For all of our experiments, we ran 10 trials, so getting results for one set of travel restriction parameters took 30-40 minutes. Furthermore, we tried lowering the scale factor to 1,000 individuals per node but this led the system to crash. To improve, we'd run our code with software/hardware that can better accommodate these already high and potentially higher node counts or consider a different network construction approach.
Next: Broader Impacts
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