key: cord-0856401-2bikuwwg authors: Wang, S.; Ramkrishna, D. title: On the Spread of Coronavirus Infection. A Mechanistic Model to Rate Strategies for Disease Management date: 2020-06-07 journal: nan DOI: 10.1101/2020.06.05.20123356 sha: 9c6f065bc297f23638ae68063f3f256e9bd08fa0 doc_id: 856401 cord_uid: 2bikuwwg Effective policy making based on ongoing COVID-19 pandemic is an urgent issue. We present a mathematical model describing the viral infection dynamics, which reveals two transmissibility parameters influenced by the management strategies in the area for control of the current pandemic. The parameters readily yield the peak infection rate and means for flattening the curve. Model parameters are shown to be correlated to different management strategies by employing machine learning enabling the comparison among different strategies. Treatment of population data with the model shows that restricted non-essential business closure, school closing and strictures on mass gathering influence the spread of infection. While a rational strategy for initiation of an economic reboot would call for a wider perspective of the local economics, the model can speculate on its timing based on the status of the infection as reflected by its potential for an unacceptably renewed viral onslaught. prospects of an economic breakdown of catastrophic proportions are a further complication that must also somehow influence the mode of confrontation of the pandemic. An essential prerequisite to facing the coronavirus pandemic is understanding of the various factors that have a potential contribution to limiting the spread. The spread of infection occurs in multifarious ways. Thus one that is cited the most frequently is spread of the virus 5 through droplets from coughing and sneezing (3) . Another is from unwitting contact with infected surfaces (4) such as glassware, boxes and so on. Intimate contact through handshakes and hugs are even more efficient ways to transmit infection. Each occurs through different scenarios that must be envisaged with their respective frequencies of occurrence for a model formulation. For symptomatic disease associated with a pathogen transmissibility (marked by a 10 basic reproduction number), different transmission routes are aligned to their implications for prevention; specifically, there may be four categories: symptomatic transmission, presymptomatic transmission, asymptomatic transmission, and environmental transmission. Given recent evidence of SARS-CoV-2 transmission by mildly symptomatic and asymptomatic persons (5) , its incubation period is about 5.1 days and about 12 days of infection from exposure to 15 symptom development (latent period). Therefore, unusually long term of latency period and presymptomatic transmission could have important implications for transmission dynamics (6). Analysis of data accumulated from numerous sources have provided the general features of the spread in terms of when to expect the peak infection rate and what it takes to flatten this curve. Yet this understanding must be said to be qualitative without notable predictive features. 20 A mathematical model is presented here of the spread of coronavirus in terms of three parameters that control the rate of its spreading and flattening the infection rate curve when intervention by a vaccine is not available. Our model is concerned with a specific geographic domain of the United States with a given population of specified density (number per unit area) of which a fraction is initially infected. The infected population contributes virus within the 25 domain which, for the present, is completely isolated from other domains. The spread of infection within the domain depends on the uninfected population and occurs at a rate governed by the extent of protective measures adopted to avoid infection from those infected. This spread obviously depends also on the viral population in the domain which grows by contribution from the infected (exhaled droplets, aerosol, contaminated surfaces, and possibly fecal-oral 30 contamination (7)) and disappears by death/isolation/herd immunity etc. We should note that All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 June 7, 2020. . https://doi.org/10.1101/2020.06.05.20123356 doi: medRxiv preprint while there are numerous reports on reinfection of COVID-19 (8) , majority of recovered patients retain certain immunity against the virus. Our goal here is to find a suitably simple framework to produce a mathematical model that contains a limited number of parameters which can be readily identified from gross observations. Furthermore, they should relate in some way to various strategies that may be 5 envisaged to control the spread of infection. To simulate both dynamics of viral and infected population, the modeling of the system in a considered geometric domain can be abstracted as its is the infected population density ( n ) normalized by the population density in Table 1 ) presented in above differential equations 15 compare the rates of different processes and have the capacity to control the spread of infection. Daily infection data must be fitted to the model by appropriate choice for the values of the dimensionless parameters (Figs. 1(B,C)). The socio-economic behavior has diversified the dynamics of the infection curve; Furthermore, major regulatory governmental strictures may enforce more discipline in public behavior thus seriously affecting the parameters. This effect, it 20 must be conceded, is buried in subtle empiricism of the model that we must seek to unearth. In doing so, we emulate the currently popular practice of machine learning towards estimating the parameters in each domain to assess the local government policy. In Fig. 1 (which was not certified by peer review) 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 June 7, 2020. . https://doi.org/10.1101/2020.06.05.20123356 doi: medRxiv preprint the spread of infection in the area under consideration. The different mechanisms of transmission of infection may operate to varying extents in different areas depending on how the infection is managed locally. Thus one must regard the model as only "broadly" mechanistic and the relationship of model parameters to different strategies would be somewhat diffuse. Therefore, in connecting the model to guide strategies we resort to a statistical methodology based on machine 5 learning tools, which could overcome the limitation just mentioned. We first examine role of model parameters in the spread of infection. The P1 duration reveals the period of pathogen transmission with limited prevention in the United States. The early state of virus transmissibility can be characterized by `R-naught' (R0), which is the basic reproduction number. Our estimate of R0 is about 2.8 (the median from data is 2.75; our model is 10 2.90) whose transmission is stronger than influenza (R0:1.4-1.6) (9) and weaker than Measles In general, counties with populous majority remain as small virulence during the entire period ( Fig. 1(D) ). Parameter γ represents the removal rate of infected patients (by recovery/death). Our model implies that γ is associated with / α β positively: despite the infection, its percentage in each county remains low (e.g. the percentage of infection at New York City is All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 June 7, 2020. p<0.05 considered as significant), which is consistent with the physical explanation of Table 1 infection rate has to be small (see Fig. 2 (C)). Our model recommends that this is accomplished 25 when / α β is low, suggesting the reduction of virus circulation (Fig. 2(D) ). We now proceed to model the effect of lockdown on COVID-19 transmissibility in New York City, as an example. Additionally, considering recently published policy of "Opening Up America Again" by the white house administration, we will study the effect of reopening All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 June 7, 2020. . https://doi.org/10.1101/2020.06.05.20123356 doi: medRxiv preprint economy on the dynamics of transmissibility in the New York metropolitan area. Here, we consider the influence of lockdown policy at New York City, where the isolation is determined by the geographic constraint of five boroughs (the Bronx, Brooklyn, Manhattan, Queens, and Staten Island) within the New York City (Fig. 3(A) ). From Fig. 3(B) , our model suggests that the mitigation brought about by lockdown is sensitive to the moment of implementation; an early 5 enforcement of lockdown could delay its occurrence. Our study also shows that if the implementation happens after the peak infection, the strategy of slowing works less efficiently. However, the lockdown likely results in a long-term dynamics and the associated economic damage has to be considered as well. During this pandemic, the financial center of the world--New York City area has been hit 10 by massive layoffs and anticipates looming recession (13) . This situation, spells some urgency for reopening the economy and resuming normal daily activity. However, we stress that opening the economy has to be cautious to the possible appearance of a second wave thus making the timing of the reopening very important. To simulate the impact of normal daily activity on the current dynamics of infection, we study the transmissibility in both New York City and Hudson 15 County within the metropolitan area. These two regions represent the most active interactions in the United States (leading out-computing in the metro area, NYC Planning 2018) and yet both have the leading coronavirus infections in their states. In the model, we relax the current government restraints and resume normal daily operations and activities, which allows the model to consider the worst scenario of the infection curve. Fig. 3(C) show that the economy reopening 20 (with the least precaution) inevitably brings the second wave and thereof more mortality. However, the extent of infection outbreak can be drastically reduced by delaying the opening date (35% increase at 5.5th week vs. 4% increase at 7.5th week). We note that an effective policy intervention may reduce the drastic increment of the infected population. In the next section, we discuss how to quantify the effectiveness of current implemented policy on coronavirus 25 transmissibility. With the U.S. administration declaring the social distancing guideline since March 16th, local governments have implemented more than 300 executive orders in fifty states, Puerto Rico, the District of Columbia, Guam, and the Virgin Islands. The executive actions and policies are related to declarations of states of emergency, school/business closure, prohibition of mass 30 gathering, stay at home order, etc. Central issues stand as the effectiveness of ongoing individual All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 June 7, 2020. . https://doi.org/10.1101/2020.06.05.20123356 doi: medRxiv preprint policy is unclear. In our study, the statistical inference suggests the relevance of several ongoing policies on the coronavirus transmission (Fig. 4(A) ). With our model empowered by machine learning tools, we perform the regression of both parameters / α β and x % based on all policy influences. By examining the weight associated with each policy measure and its significance (pvalue) in Fig. 4(B-1) , we should conclude that factors such as non-essential business closure, 5 gathering ban and school closure possess strong impact on x % (adjusted-R 2 =0.59, p=2e-6), which represents the total infected population. Both gathering ban and school closure emphasize the activity of population in young age, which is consistent with the recent finding that young people play a vital role in spread of COVID-19 (14, 15) . For virulence environment ( / α β ), while the severity of coronavirus spread is largely determined by the local population number, 10 nonessential business closure plays a role in its attenuation effort among other considered policies (adjusted-R 2 =0.30, p=1e-3; see Fig. 4(B-2) ). With the context of reopening the economy, the policies on certain non-essential business limitation, gathering ban and school closure may be continuously enforced. In this regard, the informative results delivered by combining both approaches (i.e. mechanistic model and machine learning) could promote 15 effective implementations against the transmission disease ( Fig. 4(C) ). In this report, we have proposed a new mechanistic model describing the transmission of COVID-19 in the United States. Our model is established in conjunction with administration policy, from which we propose two significant parameters. The parameter / α β quantifies the severity of the coronavirus circulation, and the parameter x % represents the projected total 20 infected fraction. To be consistent with CDC county-by-county guideline, we studied the infection dynamics of the leading county in each state. Our study shows that New York City in New York, Los Angeles county in California, and Wayne county in Michigan exhibit strong coronavirus circulation. By examining the peak infection rate, our suggested strategy of 'flattening the curve' has to deal with lowering / α β , meaning to drastically diminish the virus 25 population in the environment. Our further study of lockdown suggests that this policy has to be implemented before the peak infection arrives. We have quantified the impact of current social distancing policies with / α β and x % , suggesting that polices such as, restrictive non-essential business closure, a ban on gathering, and that of school closure are critical. This may strongly associate with the restricted activity of young people (young adults and teenagers). Although a 30 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 June 7, 2020. . https://doi.org/10.1101/2020.06.05.20123356 doi: medRxiv preprint rational analysis for an economic reboot should be based on a considerably expanded view of the local economics, it is possible to derive some useful guidelines from our model study. To this extent, we conclude, perhaps somewhat speculatively, that our suggestion for an economic reopening may be viable if non-essential business closure is conditional, mass gathering is New York Times github source, US coronavirus data (2020); https://raw.githubusercontent.com/nytimes/covid-19-data/master/us-counties.csv. draft: SW, DR; Writing, reviewing and editing: SW, DR. Competing interests: The authors declare no competing interests. Data and materials availability: US coronavirus data are publically available from the New York Times GitHub source: https://raw.githubusercontent.com/nytimes/covid-19-data/master/us-counties.csv; Weather data are publicly available from NOAA Global Surface Summary of the Day (GSOD): 5 https://data.nodc.noaa.gov/cgi-bin/iso?id=gov.noaa.ncdc:C00516; State and policy data to address coronavirus are publically available from Kaiser Family Foundation:https://www.kff.org/health-costs/issue-brief/state-data-and-policy-actions-to-addresscoronavirus/ 10 Supplementary Materials: Figures S1-S3 Tables S1-S3 15 References (6, 16-22) All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 June 7, 2020. . All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 June 7, 2020. 10 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 June 7, 2020. . Plots with shaded area are the modeling results where solid lines represent the median of the prediction and the shaded area indicates the uncertainty. The zeroth week is set at the moment when the total number of infections at the New York City is ten. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 June 7, 2020. (which was not certified by peer review) 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 June 7, 2020. Ratio of viral growth rate to its death rate All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 June 7, 2020. . https://doi.org/10.1101/2020.06.05.20123356 doi: medRxiv preprint Turbulent gas clouds and respiratory pathogen emissions: potential implications for reducing transmission of covid-19 Aerosol and surface distribution of severe acute respiratory syndrome coronavirus2 in hospital wards The incubation period of coronavirus disease 2019 (covid-19) from publicly reported confirmed cases: estimation and application Transmission dynamics and control of severe acute respiratory syndrome covid-19) outbreak: what the department of endoscopy should know Positive rt-pcr test results inpatients recovered from covid-19 The WHO Rapid Pandemic Assessment Collaboration, Pandemic potential of a strain of influenza a (h1n1): early findings The basic reproduction number (r0) of measles: a systematic review Compassionate use of remdesivir for patients with severecovid-19 Susceptible supply 15 limits the role of climate in the early sars-cov-2 pandemic The macroeconomics of epidemics Characteristics of covid-19 20 infection in Beijing Covid-19: identifying and isolating asymptomatic people helped eliminate virus in Italian village NOAA Global Surface Summary of the Day, Weather data for Kaiser Family Foundation, State Data and Policy Actions to Address Coronavirus Notes on R0 Revisiting the basic reproductive number for malaria and its implications for malaria control Extracting the time-dependent transmission rate from infection data via solution of an inverse ODE problem