We formulate the transmission as a discrete Markov process during both day and night times. The daytime transmission lasts for ! day and the nighttime transmission " day ( ! + " = 1). Here, we assume daytime transmission lasts for 8 hours and nighttime transmission lasts for 16 hours, i.e., ! = 1/3 and " = 2/3. The transmission dynamics are depicted by the following equations.
Daytime transmission:
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Nighttime transmission:
Here, #$ , #$ , #$ & , #$ ( and #$ are the susceptible, exposed, reported infected, unreported infected and total population in the subpopulation commuting from county to county . CC-BY-NC 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.
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( ← ); is the transmission rate of reported infections; is the relative transmissibility of unreported infections; is the average latency period (from infection to contagiousness); is the average duration of contagiousness; is the fraction of documented infections; is a multiplicative factor adjusting random movement; 4 #$ = ( #$ + $# )/2 is the average number of commuters between counties and ; and # ' and # + are the daytime and nighttime populations of county . We integrate Eqs. 1-10 using a Poisson process to represent the stochasticity of the transmission process. A similar model has been used to generate forecasts for influenza in the United States 3 .
To account for reporting delay, we mapped simulated documented infections to confirmed cases using a separate observational delay model. In this delay model, we account for the time interval between a person transitioning from latent to contagious (i.e. E à # & ) and observational confirmation of that individual infection. To estimate this delay period, , , we examined line-list data from early-confirmed cases in China 4 . Prior to January 23, 2020, the time-to-event distribution of the interval (in days) from symptom onset to confirmation is well fit by a Gamma distribution 1 ( = 1.85, = 3.57, = −252.24). Consequently, we adopted a Gamma distribution to model , , but tested longer mean periods ( ) as symptom onset often lags the onset of contagiousness.
We calibrated the transmission model against county-level incidence data reported from February 21, 2020 through March 13, 2020 5 . Specifically, we estimated model parameters using an iterated filtering (IF) framework 6, 7 . The metapopulation model is high dimensional with 59,998 subpopulations. We therefore applied an efficient data assimilation algorithm -the Ensemble Adjustment Kalman Filter (EAKF) 8 , which is applicable to high dimensional models, in multiple iterations to infer parameters , , , , and . This iterated filtering (IF)-EAKF framework has been used to infer parameters in a large-scale agent-based model for antimicrobial-resistant pathogens 9 , as well as a metapopulation model depicting the spread of SARS-CoV-2 in China 1 . Details of its implementation can be found in Ref. In the inference, we fixed the shape parameter of the Gamma distribution for , as = 1.85, and vary the mean value of the distribution. We tested a range of mean , values from 6 days to 10 days.
To initialize the model, we seeded exposed individuals ( ) and unreported infections ( ( ) in counties with at least one confirmed case. Unlike the situation in China, where the outbreak originated from a single city, there was importation to multiple locations in the US that could have initiated community transmission. To reflect this potential ongoing community transmission before the reporting of the first local infection, for each county with confirmed cases, we randomly drew and & from uniform distributions [0, 12 ] and [0, 10 ] 8 days prior to the reporting date ( -) of the first case. Here is the total number of reported cases between dayand -+ 4.
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The rationale for this seeding strategy is as follows. If an average reporting delay of 8 days is assumed, we can estimate & on day -− 8 by
is the average number of daily cases during the first five days with reporting ( -to -+ 4). If we use the upper bound of the prior for D (i.e., 5 days), & is estimated as , which is also an upper bound. Using parameters obtained from China 1 , we assume the mean ( on day -− 8 is 5 , implying a reporting rate of 1/6=16.7%. Drawing ( from [0, 10 ] leads to a broader prior range of the reporting rate. As both & and ( were evolved from the exposed population , we draw from the range [0, 12 ]. This crude calculation gives us an estimate of seeding in US counties. During inference, this seeding can be adjusted up or down by the filter, and best-fitting models produce simulations that capture observed outcomes.
The inferred parameters for the best-fit model and the 95% CIs are reported in Table 1 . The mean reporting delay for the best-fit model is , = 6 days. We display the fitting of this model to national daily case data and cases for 5 other counties (King County WA, Westchester County NY, Santa Clara County CA, Middlesex County MA and Nassau County NY) in Fig. 1 .
To evaluate the impact of control measures, we examined how daily confirmed cases within 6 months of February 21 2020 were modulated by percent reductions of the contact rate (effected by isolation, quarantine, telecommuting, school closure, etc.) and travel restrictions (effected by reducing commuting and travel among counties). The reductions of contact rates disrupt normal mixing within metapopulation locations, whereas the travel restrictions impact mixing between locations. In particular, we ran model simulations with no intervention, a 25% reduction of contact rate, a 50% reduction of contact rate and a 95% reduction of cross-county mobility. The epidemic curve is more effectively flattened by reduction of the contact rate, as shown in Fig. 2 . In addition, we provide movies of each scenario (Movies S1-S4 10 ). Each shows a single stochastic simulation with best-fitting model parameters (excepting , which is adjusted in 2 of the scenarios). The projection outputs are available for download from GitHub 10 .
Disruption of mixing within locations, effected through a reduction of b, more substantially slows the spread and increase of confirmed cases than reductions of commuting and travel between locations. High reductions of commuting and travel between county locations (i.e. ³95%) are needed to reduce the spread and increase of infections.
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The copyright holder for this preprint this version posted March 23, 2020. . CC-BY-NC 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 23, 2020. . . CC-BY-NC 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 23, 2020. . https://doi.org/10.1101/2020.03.21.20040303 doi: medRxiv preprint Fig. 2 . National daily confirmed cases with no intervention, a 25% reduction of contact rate, a 50% reduction of contact rate and a 95% reduction of cross-county mobility.
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The copyright holder for this preprint this version posted March 23, 2020. . https://doi.org/10.1101/2020.03.21.20040303 doi: medRxiv preprint