key: cord-0908603-u7ao9dwx
authors: Goyal, Ashish; Reeves, Daniel B.; Cardozo-Ojeda, E. Fabian; Schiffer, Joshua T.; Mayer, Bryan T.
title: Wrong person, place and time: viral load and contact network structure predict SARS-CoV-2 transmission and super-spreading events
date: 2020-09-28
journal: medRxiv
DOI: 10.1101/2020.08.07.20169920
sha: 24ac1309cfb121d82a0a8fb8233c8983f169d34f
doc_id: 908603
cord_uid: u7ao9dwx

SARS-CoV-2 is difficult to contain because many transmissions occur during the pre-symptomatic phase of infection. Moreover, in contrast to influenza, while most SARS-CoV-2 infected people do not transmit the virus to anybody, a small percentage secondarily infect large numbers of people. We designed mathematical models of SARS-CoV-2 and influenza which link observed viral shedding patterns with key epidemiologic features of each virus, including distributions of the number of secondary cases attributed to each infected person (individual R(0)) and the duration between symptom onset in the transmitter and secondarily infected person (serial interval). We identify that people with SARS-CoV-2 or influenza infections are usually contagious for fewer than one day congruent with peak viral load several days after infection, and that transmission is unlikely below a certain viral load. SARS-CoV-2 super-spreader events with over 10 secondary infections occur when an infected person is briefly shedding at a very high viral load and has a high concurrent number of exposed contacts. The higher predisposition of SARS-CoV-2 towards super-spreading events is not due to its 1–2 additional weeks of viral shedding relative to influenza. Rather, a person infected with SARS-CoV-2 exposes more people within equivalent physical contact networks than a person infected with influenza, likely due to aerosolization of virus. Our results support policies that limit crowd size in indoor spaces and provide viral load benchmarks for infection control and therapeutic interventions intended to prevent secondary transmission.

The SARS-CoV-2 pandemic is an ongoing tragedy that has caused 700,000 deaths and 29 massively disrupted the global economy. The pandemic is rapidly expanding in the United States 30

and is re-emerging focally in many countries that had previous success in limiting its spread. transmissions numbers. 9,10 Differing shedding kinetics between the two viruses might explain 44 this distinction; SARS-CoV-2 is often present intermittently in the upper airways for many 45 weeks, 11,12 while influenza is rarely shed for more than a week. 13 Alternatively, SARS-CoV-2 46 aerosolization may predispose to wider exposure networks given the presence of an infected 47 person in a crowded indoor space. 48 variability, we generated a set of heterogeneous shedding curves in which the viral upslope, the 91 downslope of viral load after peak and the viral load during plateau phase were varied (Fig S2b) . 92 Overall, the model generated several distinct patterns of infection: rapid elimination after the initial 93 peak, a prolonged plateau phase with a low viral load, and a prolonged plateau phase with higher 94 viral load. We simulated the transmission model with and without imputed heterogeneity. transmitter has 2 exposure events at discrete timepoints resulting in 7 total exposure contacts and 101 3 secondary infections. Transmission is more likely at the first exposure event due to higher 102 exposure viral load. To model this process, the timing of exposure events and number of exposed 103 contacts is governed by a random draw from a gamma distribution which allows for heterogeneity 104 in number of exposed contacts per day (Fig S3) . Viral load is sampled at the precise time of each 105 exposure event. Probability of transmission is identified based on the product of two dose curves 106 (Fig S2C, D) which capture contagiousness (probability of viral passage to an exposure contact's 107 airway) and infectiousness (probability of transmission given viral presence in the airway). 108

Incubation period (Fig S4) of the transmitter and secondarily infected person is an input into each 109 simulation and is depicted graphically. Individual R0 is an output of each simulation and is defined 110 as the number of secondary infections generated by an infected individual. Serial interval is an 111 output of each simulated transmission and is depicted graphically. 112 113 114 Transmission dose response curves. We defined an exposure event in very specific biologic terms 115 as a discrete event consisting of sufficient contact in time and space between a transmitter and one 116 or more uninfected persons (exposure contacts) to allow for the possibility of a successful 117 transmission. We next designed hundreds of dose response curves which separately predict 118 contagiousness (CD curves) and infectiousness (ID curves) at a certain viral dose given an 119 exposure contact. Contagiousness is defined as the viral load dependent probability of passage of 120 virus-laden droplets or airborne particles from the airways of a potential transmitter to the airway 121 of an exposure contact. Infectiousness is defined as the viral load dependent probability of 122 transmission given direct airway exposure to virus in an exposure contact. Transmission risk is the 123 product of these two mechanistic probabilities derived from the ID and CD curves and results is a 124 transmission dose (TD) response curve. Each CD or ID curve is defined by its ID50 (l) or viral 125 load at which contagion or infection probability is 50% (Fig S2c) , as well as its slope (a) (Fig  126   S2d) . 29 The TD50 is defined as viral load at which there is 50% transmission probability. We 127 assumed equivalent curves for contagiousness and infectiousness for model fitting purposes. We 128 also considered a simpler model with only a single TD curve (for infectiousness) and obtained 129 qualitatively similar results (Supplement and Methods). Our model is inclusive of the hypothesis 130 Exposure contact rate simulations. We introduced heterogeneity of exposure contact rates among 133 possible transmitters by randomly selecting from a gamma distribution defined by mean number 134 of exposure contacts per day (q) and a scaling factor ( ) that controls daily variability (Fig S3) . 135 136 Transmission simulations. For each defined exposure contact, viral load in the transmitter was 137 sampled and transmission risk was then identified based on the product of the CD and ID curves, 138 or the TD curve (Fig S2e, f; Fig 1) . Based on these probabilities, we stochastically modeled 139 whether a transmission occurred for each exposure contact. This process was repeated when there 140 were multiple possible exposure events within a given discretized time interval and the total 141 number of exposures and transmissions within that interval was calculated. 142

For each successful transmission, we assumed that it takes days for the first infected cell 143 to produce virus. To inform simulated values of serial interval (SI or time between symptom onset 144 in the secondarily infected and transmitter), we randomly selected the incubation period (IP), for 145 both the transmitter and the newly infected person, from a gamma distribution based on existing 146 data ( Fig S4a) . 3,30 Incubation period was defined as time from infection to the time of the onset of 147 symptoms, where the mean incubation for SARS-CoV-2 is 5.2 days compared to 2 days for 148 influenza. 3,9,30 149 150 Model fitting. In order to identify the parameter set that best recapitulated the observed data, we 151 then simulated several hundred thousands of parameter sets with ~250 possible TD curves 152 defined by ID50 and CD50 (l) and slope (a), along with ~180 combinations of the mean 153 exposed contact rate per day ( ) and associated variance parameter ( ), and values of ∈ 154

[0.5, 1, 2, 3] days. We aimed to identify the parameter set that best recapitulated the following 155 features of the observed epidemiologic and individual-level data for SARS-CoV-2: mean R0 156 across individuals (R0 ∈ [1.4, 2.5]), 3,4,6,31,32 mean serial interval across individuals (SI ∈ 157 [4.0, 4.5]), 3,31,33 cumulative distribution functions of individual R0, 4,6,34-36 and cumulative 158 distribution functions of serial intervals derived from SARS-CoV-2 transmission pair studies that 159 were conducted early during the pandemic, 31 prior to any confounding influence of social 160 distancing measures. Here, we define individual R0 as the total number of secondary 161 transmissions from the transmitter in a fully susceptible population (Methods). Given that viral 162 RNA is composed mostly of non-infectious material, we further checked the closeness of the 163 solved ID curve with the observed relationship between viral RNA and probability of positive 164 viral culture from a longitudinal cohort of infected people. 37 165 166 Influenza modeling. Next, we performed equivalent analyses for influenza to explain the lower 167 frequency of observed super-spreader events with this infection. Influenza viral kinetics were 168 modelled using a previously data-validated model. 38 Incubation periods for influenza are lower 169 and less variable than for SARS-CoV-2 and were randomly selected for each simulation of the 170 model using a gamma distribution (Fig S4b) . 39 We again fit the model to: mean R0 across 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint observed individual R0 and serial interval histograms (Fig 2a, c) and cumulative distribution 179 functions (Fig 2b, d) . We re-ran the model to fit to a higher population R0 of 2.8 and arrived at a 180 similar set of parameter values but with a higher daily rate of exposure contacts ([ , , , , ] = 181 [0.8, 10 7.5 , 0.5, 20, 30]). Despite assuming that each infected person sheds at a high viral load for 182 a period of time (Fig 1, Fig S2b) , the model captured the fact that ~75% of 10,000 simulated 183 transmitters do not infect any other people and that each increase in the number of possible 184 transmissions is associated with a decreasing probability (Fig. 2a) Fig S5) : this adjustment was likely necessary because viral loads in the Dutch 198 study participants were notably higher than those in German, Singaporean, Korean and French 199 participants used in our intra-host model fitting. 25-28,37 200

The model also generated super-spreader events with 10,000 simulated transmissions 201 (Fig. 2b) . If super-spreaders are defined as those who produce at least 5 secondary infections, we 202 estimate that ~10% of all infected people and ~35% of all transmitters are super-spreaders. If 203

super-spreaders are defined as those who produce at least 10 secondary infections, we estimate 204 that ~6% of all infected people and ~25% of all transmitters are super-spreaders. If super-205 spreaders are defined as those who produce at least 20 secondary infections, we estimate that 206 ~2.5% of all infected people and ~10% of all transmitters are super-spreaders. If super-spreaders 207 are defined as those producing ≥5, ≥10, or ≥20 secondary infections, the contribution to all 208 secondary infections is estimated at ~85%, ~70%, or ~44%, respectively (Table 1) . 209

The model also recapitulated the high variance of the serial interval observed within 210 SARS-CoV-2 transmission pairs, including negative values observed in the data (Fig 2c, d) . We 211 next projected generation time, defined as the period between when an individual becomes 212 infected and when they transmit the virus, for all transmission pairs and identified that the mean 213 serial interval (4.4 days) provides an accurate approximation of mean generation time. However, 214 the variance of generation time was considerably lower and by definition does not include 215 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . negative values. A majority of generation times fell between 4 and 7 days, compared to -5 to 12 216 days for the serial interval (Fig 2e) 10 7 viral RNA copies and a moderately steep slope (Fig 3a) . The TD50 for SARS-CoV-2 was 226 slightly higher at 10 7.5 viral RNA copies (Fig 3a) . To assess the impact of these parameters on 227 transmission, we performed simulations with 10,000 transmitters and concluded that 228 transmission is very unlikely (~0.00005%) given an exposure to an infected person with an upper 229 airway viral load of <10 4 SARS-CoV-2 RNA copies, and unlikely (~0.002%) given an exposure 230

to an infected person with a viral load of <10 5 SARS-CoV-2 RNA copies. On the other hand, 231 transmission is much more likely (39%) given an exposure to an infected person who is shedding 232 >10 7 SARS-CoV-2 RNA copies, and 75% given an exposure to an infected person with a viral 233 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint load of >10 8 SARS-CoV-2 RNA copies. We obtain similar results (not shown) when we solve 234 our model using the assumption of homogeneous viral load trajectories as in Fig S2a . curve (transmission risk = Pt * Pt) curves for SARS-CoV-2. Transmission probability is a product 241 of two probabilities, contagiousness and infectiousness (Fig 1) . B-D. Three simulated viral 242

shedding curves. Heat maps represent risk of transmission at each shedding timepoint given an 243 exposed contact with an uninfected person at that time. the window of high probability transmission is limited to time points around peak viral load, and 249 some heterogeneity in regard to peak infectivity is noted between people (Fig 3b-d) . In general, 250 infected persons are likely to be most infectious (i.e., above TD50) for a ~0.5-1.0-day period 251 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint between days 2 and 6 after infection. We therefore conclude that the observed wide variance in 252 serial interval (Fig 2c) results primarily from the possibility of highly discrepant incubation 253 periods between the transmitter and infected person, rather than wide variability in shedding 254 patterns across transmitters. given an exposure viral load on log10 scale (x-axis) and number of exposed contacts per day (y-261 axis). The exposed contact network allows a maximum of 150 exposed contacts per day (black 262 dotted line) which is sufficient for multiple transmissions from a single person per day. B. 10,000 263 simulated transmitters followed for 30 days. The white space is a parameter space with no 264

transmissions. Each dot represents the number of secondary transmissions from a transmitter per 265

day. Input variables are log10 SARS-CoV-2 on the start of that day and number of contact 266 exposures per day for the transmitter. There are 1,154,001 total exposure contacts and 15,992 267

total infections. C. 10,000 simulated infections with percent of infections due to exposure viral 268 load binned in intervals of 0.5 intervals on log10 scale (x-axis) and number of exposed contacts 269 (y-axis).

Requirements for SARS CoV-2 super-spreader events. The solved value for exposed contact 273 network heterogeneity (r) is 40 indicating high variability in day-to-day exposure contact rates 274 ( Fig S3d) with a high average number of exposed contacts per day (q=4). We generated a heat 275 map from our TD curve to identify conditions required for super-spreader events which included 276 viral load exceeding 10 7 SARS CoV-2 RNA copies and a high number of exposure contacts on 277 that day. We observed an inflection point between 10 6 and 10 7 SARS CoV-2 RNA copies where 278 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint large increases in the number of daily exposure contacts had a more limited impact on increasing 279 the number of transmissions from a single person (Fig 4a) . The exposure contact network 280 occasionally resulted in days with ≥150 exposure contacts per day, which may allow an 281 extremely high number of secondary infections from a single person (Fig 4a) . 282

We next plotted transmission events simulated on a daily basis over 30 days since 283 infection, from 10,000 transmitters, according to viral load at exposure and number of exposure 284 contacts on that day (Fig 4b) . Secondary transmissions to only 1-3 people occurred almost 285 exclusively with daily numbers of exposure contacts below 10 with any exposure viral load 286 exceeding 10 6 RNA copies or with higher numbers of exposure contacts per day and viral loads 287 exceeding 10 5 RNA copies. Massive super-spreader events with over 50 infected people almost 288 always occurred at viral loads exceeding 10 7 RNA copies with high levels of concurrent 289 exposure contacts (Fig 4b) . 290 We next identified that over 50% of secondary infections were associated with a 291 transmitter who has a high number of exposed contacts (11-100 per day) and a viral load 292 exceeding 10 6 RNA copies (Fig 4c) , which is the mechanistic underpinning of why ~70% of all 293 secondary infections arose from transmitters who produced more than 10 secondary infections 294 (Table 1) . 295 296

parameter set most closely reproduced empirically observed histograms and cumulative 298 distribution functions for individual R0 and serial intervals for influenza: ( , , , , ) = (0.7, 299 10 5.5 , 0-0.5, 4, 1). ID50 values for influenza were lower than SARS CoV-2, but a direct 300 comparison cannot be made because tissue culture infectious dose (TCID) has been more 301 . 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) preprint

The copyright holder for this this version posted September 28, 2020. .

CoV-2. Nevertheless, TCID is a closer measure of infectious virus and it is thus reasonable that 303 ID50 based on TCID for influenza would be ~30-fold lower than ID50 based on total viral RNA 304 The other notable difference was a considerably lower value for influenza (Fig S3b) , 316 denoting much less heterogeneity in the number of exposure contacts per person while the 317 average daily exposure contact was the same for both viruses (4 per day). The model captures the 318 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint fact that 40% of influenza infected people do not transmit to anyone else and that each increase 319 in the number of individual transmissions is associated with a lower probability (Fig. 5a) . 320

Relative to SARS-CoV-2, super-spreader events involving 5 or more people were predicted to be 321 5-fold less common overall and 10-fold less common among transmitters (~2% of all infected 322 people and ~3% of transmitters) (Fig. 5b, Table 1 ). Super-spreaders defined as those infecting 323 ≥5 individuals contributed to only ~10% to all transmissions (Table 1) . 324

The model also recapitulated the lower variance of serial interval for influenza relative to 325 SARS-CoV-2 (Fig 5c, d) . We next identified that the mean and variance of the serial interval 326 provide good approximations of the mean and variance for generation time. A majority of 327 generation times fell between 2 and 6 days (Fig 5e) . (transmission risk = Pt * Pt) curves for influenza. Transmission probability is a product of two 333

probabilities, contagiousness and infectiousness (Fig 1) . B-D. Three simulated viral shedding 334

curves. Heat maps represent risk of transmission at each shedding timepoint given an exposed 335

contact with an uninfected person at that time. 336 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint 337 Viral load thresholds for influenza transmission. Based on the optimized TD curve for 338 influenza (Fig 6a) , we next plotted the probability of infection given an exposure to an infected 339 person. The TD50 for influenza was 10 6.1 TCID/mL. Under various shedding scenarios, the 340 window of high probability transmission was limited to time points around peak viral load ( Fig  341   6b-d) . In general, infected persons were likely to be most infectious (i.e., above TD50) for a 342 ~0.5-1.0 days period. The observed low variance in serial interval (Fig 5c) given an exposure viral load on log10 scale (x-axis) and number of exposed contacts per day (y-351 axis). The exposed contact network allows a maximum of 15 exposed contacts per day (black 352 dotted line) which is not sufficient for more than 15 transmissions from a single person per day. total infections. C. 10,000 simulated infections with percent of infections due to exposure viral 358 load binned in intervals of 0.5 intervals on log10 scale (x-axis) and number of exposed contacts 359 (y-axis).

Determinants of influenza individual R0. We generated a heat map from our TD curve to 362 identify conditions governing influenza transmission to multiple people including viral load 363 exceeding 10 6 influenza TCID and a high number of exposure contacts per day. The contact 364 network never resulted in days with more than 15 exposure contacts per day, which severely 365 limited the possible number of transmissions from a single person relative to SARS-CoV-2 ( Fig  366   7a, S3b) . 367

We plotted transmission events simulated on a daily basis over 30 days since infection 368 from 10,000 transmitters according to viral load at exposure and number of exposure contacts on 369 that day (Fig 7b) . Secondary transmissions to fewer than 5 people accounted for 90% of 370 infections (Table 1) and occurred with fewer than 10 daily exposure contacts and exposure viral 371 loads exceeding 10 4 TCID. Small scale super-spreader events with 5-10 infected people almost 372 always occurred at viral loads exceeding 10 5 TCID with 5-10 concurrent exposure contacts (Fig  373   7b) . 374 We next identified that over 50% of infections were associated with a transmitter who 375 had fewer than 10 exposure contacts per day and a viral load exceeding 10 4.5 TCID (Fig 7c) , 376 which is why no infected person ever transmitted to more than 10 other people (Table 1) . 377 378

spreader events. We sought to explain why SARS-CoV-2 has a more over-dispersed distribution 380 of individual R0 relative to influenza. To assess viral kinetics as a potential factor, we 381 comparatively plotted transmission risk per exposure contact as a function of time since infection 382 in 10,000 transmitters for each virus. The median per contact transmission risk was slightly 383 higher for influenza; however, 75% and 95% transmission risks were marginally higher for 384 SARS-CoV-2 compared to influenza with slightly higher peak transmission risk, and a longer tail 385 of low transmission risk beyond 7 days (Fig 8a) . The transmission risk was considerably higher 386 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint for the 25% of simulated SARS-CoV-2 infections with the highest viral loads, suggesting that a 387 substantial subset of infected people may be more pre-disposed to super-spreading. When plotted 388 as time since onset of symptoms, the variability in transmission potential was considerably larger 389 for persons with high SARS-CoV-2 viral load, owing to the variable incubation period of this 390 virus (Fig 8b) . 391 392 393 394 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . Fig 8c) . 423 . 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) preprint

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We next plotted the frequency of exposure contacts per day for both viruses and noted a 424 higher frequency of days with no exposed contacts (Fig 8d) , but also a higher frequency of days 425 with more than 10 exposure contacts (Fig 8e) for SARS-CoV-2 relative to influenza, despite an 426 equivalent mean number of daily exposure contacts. To confirm that this distribution drives the 427 different observed distributions of individual R0 values (Fig 8f) , we simulated SARS-CoV-2 428 infection with an assumed =1 and generated a distribution of individual R0 similar to that of 429 influenza (Fig S6a) . Similarly, we simulated influenza infection with an assumed =40 and 430 generated a distribution of individual R0 similar to that of SARS-CoV-2 (Fig S6b) . Under all 431 scenarios, predicted distributions of serial interval (Fig 8g, Fig S6) and generation time (Fig 8h,  432   Fig S6) were unchanged by shifts in the exposed contact network. 433 434 Projections of targeted physical distancing. Physical distancing is a strategy to decrease R0. We 435 simulated a decrease in the contact rate uniformly across the population and noted a decrease in 436 population R0 (Fig S7a) as well the percent of infected people who will transmit (Fig 7b) and 437 become super-spreaders (Fig S7c-d) . An approximately 40% decrease in the average exposed 438 contact rate decreased R0 below 1 (Fig S6a) . We further investigated whether lowering contact 439 rate among larger groups only, in particular by banning exposure events with a high number of 440 exposure contacts, could control the epidemic. We identified that limiting exposure contacts to 441 no more than 5 per day is nearly equivalent to limiting exposure contacts altogether and that only 442 a small decrease in mean exposure contact rate can achieve R0<1 if exposure events with less 443 than 20 contacts are eliminated (Fig S8) . 444 445 . 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) preprint

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for SARS-CoV-2 exists in the pre-symptomatic phase (Fig8b) which explains why 62% of 447 simulated transmissions occurred in the pre-symptomatic phase for SARS-CoV-2, compared to 448 10% for influenza. Similarly, 62% and 21% of SARS-CoV-2 and influenza super-spreader 449 events with secondary transmissions ≥5 and 39% of SARS-CoV-2 super-spreader events with 450 secondary transmissions R0≥10 fell in the pre-symptomatic period. 451 . 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) preprint

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Our results demonstrate that SARS-CoV-2 shedding kinetics are directly linked to the 453 virus' most fundamental epidemiologic properties. First, we identify a transmission dose 454 response curve which specifies that a nasal viral load below 10 5 RNA copies is unlikely to 455 commonly result in transmission. For SARS-CoV-2, this threshold is consistent with the overall 456 rarity of positive cultures at these levels. 37 We also predict a relatively steep TD curve such that While the duration of shedding for SARS-CoV-2 is often three weeks or longer, 11,12 we 463 predict that the duration of shedding above thresholds required for a moderate probability of 464 transmission per contact is much shorter, often less than half a day, and is comparable to that of 465 influenza. While transmission after the first week of infection is quite rare, our model is 466 consistent with the observation that transmissions commonly occur during the pre-symptomatic 467 phase of infection, 2 given the highly variable incubation period associated with SARS-CoV-2. 468

The observed high heterogeneity in serial interval is attributable almost entirely to the 469 variable nature of the incubation period, rather than transmission occurring extremely late after 470 infection. While our estimate for mean generation time is equivalent to that of mean serial 471 interval, it is notable that the range of SARS-CoV-2 serial intervals is much wider than the range 472 of generation times. This result is evident even though we built substantial heterogeneity into our 473 viral shedding curves beyond that observed in the somewhat limited existing shedding data. 474 . 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.

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The finding of limited duration of SARS-CoV-2 infectivity has practical implications. 475

First, considerable resources are being used in hospitals and skilled nursing facilities to isolate 476 patients with persistent SARS-CoV-2 shedding. We propose that a low nasal viral load, Another finding is that SARS-CoV-2 super-spreading events are dependent on a large 491 number of exposure contacts during the relatively narrow 1-2 days window during which a ~25% 492 subset of infected people is shedding at extremely high levels above the TD50. Because we 493 predict that super-spreader potential may be somewhat of a generalized property of infection, 494 rather than a characteristic of a tiny subset of infected people, this result also has practical 495 implications. A common experience during the pandemic has been early identification of a 496 cluster of infected people within a specific confined environment such as a senior living home, 497 . 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 this version posted September 28, 2020. . crowded work environment, athletic team, or restaurant. Our results demonstrate that newly 498 diagnosed people within small clusters may be past the peak of their super-spreading potential. 499

At this stage, many more infections have often been established and drastic quarantine 500 procedures should be considered. Other undiagnosed, pre-symptomatic infected people may have 501 super-spreader potential while the known infected person is no longer contagious, highlighting 502 the importance of effective contact tracing. 503

At the prevention level, school opening and work opening strategies should focus on 504 severely limiting the possible number of exposure contacts per day. Where large numbers of 505 exposure contacts are unavoidable, mandatory masking policies, perhaps with N95 masks that 506 may more significantly lower exposure viral loads should be considered. 23 507

Influenza infection is much less predisposed to super-spreader events than SARS-CoV-2. 508

Yet, influenza shedding at levels above those required for a high probability of transmission 509 occurs with only slightly lower frequency. Therefore, the markedly different probability of 510 super-spreader events between the two viruses is unlikely to relate to different viral host kinetics, 511 despite the fact that the overall duration of SARS-CoV-2 shedding exceeds duration of influenza 512 shedding often by more than two weeks. 513

Rather, our analysis suggests that the exposure contact networks of SARS-CoV-2 514 transmitters are highly variable relative to those of influenza. One possible explanation 515 underlying this finding is that SARS-CoV-2 is more predisposed to airborne transmission than 516 influenza. 44 Here our precise definition of an exposure contact (sufficient contact between a 517 transmitter and an uninfected person to potentially allow transmission) is of high relevance. Our 518 result suggests that a SARS-CoV-2 infected person in a crowded, poorly ventilated room, may 519 generate more exposure contacts than an influenza infected person in the same room, likely 520 . 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 this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint based on wider dispersal and / or longer airborne survival of the virus. Thus, our results suggest a 521 possible downstream quantitative effect of airborne transmission on SARS-CoV-2 epidemiology. 522

Another possibly important variable is that pre-symptomatic transmission, which is a common 523 feature of SARS-CoV-2 may predispose to multiple transmissions. This prediction reinforces 524 current public health recommendation to avoid crowded indoor spaces with poor air 525

On the other hand, a much higher proportion of SARS-CoV-2 infected people than 527 influenza infected people do not transmit at all. This result lacks a clear mechanistic explanation 528 but may imply that aerosolization occurs only in a subset of infected people. One theoretical 529 explanation is that high viral load shedding in the pre-symptomatic phase is defined by lack of 530 cough or sneeze leading to limited spatial diffusion of virus. Alternatively, it is also possible that 531 a proportion of infected people never shed virus at high enough viral loads to allow efficient 532 transmission. This possibility speaks to the need for more quantitative viral load data gathered 533 during the initial stages of infection. 534

Age cohort structure differs between the two infections, with a lower proportion of 535 observed pediatric infections for SARS-CoV-2. If adults have more high exposure events than 536 children, then this could also explain super-spreader events. We are less enthusiastic about this 537 hypothesis. First, SARS-CoV-2 super-spreader events have occurred in schools and camps and 538 would likely be more common in the absence of widespread global school closures in high 539 prevalence regions. Second, a sufficient proportion of influenza cases occur in adults to rule out 540 the presence of frequent large super-spreading events in this population. 541

Our analysis has important limitations. First, exposure contacts were assumed to be 542 homogeneous and we do not capture the volume of the exposing aerosol or droplet. For instance, 543 . 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 this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint if a large-volume droplet contains ten times more viral particles than an aerosol droplet, then the 544 exposure could be dictated by this volume as well as the viral load of the potential transmitter. It 545 is possible that under rare circumstances with extremely high-volume exposures, even persons 546 with extremely low viral loads may transmit. Second, based on the quality of available data, we 547 fit our models for SARS-CoV-2 and influenza to viral RNA and viral culture respectively. 548

Existing data suggest that kinetics of viral RNA and culture are similar during both infections, 549 with culture having lower sensitivity to detect virus. 37 Third, our intra-host model of SARS-550

CoV-2 was fit to heterogeneous data with different sampling techniques and PCR assays. 24 551

Moreover, R0 estimates have varied across the globe. Our estimates of TD50 are necessarily 552 imprecise based on available data and should serve only as a conservative benchmark. Most 553 importantly, we cannot rule out the possibility that a small minority of infected people shed at 554 sufficient levels for transmission for much longer than has been observed to date. Fourth, 555 contagiousness could have different dose response dynamics than viral load dependent 556 infectiousness and may require investigation in the future upon the availability of 557 epidemiologically relevant additional data. Finally, the model is intended to capture a general 558

property of SARS-CoV-2 infection but is not specific for local epidemics. The degree of R0 559 overdispersion in various countries and regions is likely to vary dramatically according to 560 numerous factors related to social contact networks that are not explicitly captured in our model. 561

In conclusion, fundamental epidemiologic features of SARS-CoV-2 and influenza 562 infections can be directly related to viral shedding patterns in the upper airway as well as the 563 nature of exposure contact networks. We contend that this information should be leveraged for 564 more nuanced public health practice in the next phase of the pandemic. 565 . 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. In the model, SARS-CoV-2-specific effector cells rise after 2 stages from precursors cells 578

( 1 and 2 ). The first precursor cell compartment ( 1 ) proliferates in the presence of infection 579 with rate ) and differentiates into the effector cell at a per capita rate during the next 580 intermediate stage. Finally, effector cells die at rate . The model is expressed as a system of 581 ordinary differential equations: 582 . 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. When we introduce simulated heterogeneity in cases of SARS-CoV-2 (by increasing the 587 standard deviation of the random effects of parameters β by 20, δ by 2, k by 2 and π by 5 in the 588 original distribution from 24 ), some of the viral shedding curves suggest that viral shedding could 589 continue for long period (over 6 weeks). Indeed, while median viral shedding duration has been 590 estimated at 12-20 days, shedding for many months is also observed commonly. 45 We assumed 591 that viral loads after day 20 drop to a exposure-level viral load level (i.e., (0)) as most viral 592 shedding observed after this point is transient and at an extremely low viral load. 46 The population 593 distribution of parameters to simulate artificial SARS-CoV-2 viral shedding dynamics is provided 594

in Table S1 . This model assumes = 0 and = 0 and can be expressed as a system of ordinary differential 599 equations: 600 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . contacts of an infected individual with viral load dependent infectiousness are physically exposed 615 to the virus (defined as exposure contacts), that only some exposure contacts have virus passaged 616 to their airways (contagiousness) and that only some exposed contacts with virus in their airways 617 become secondarily infected (successful secondary infection). Contagiousness and infectiousness 618 are then treated as viral load dependent multiplicative probabilities with transmission risk for a 619 single exposure contact being the product. Contagiousness is considered to be viral load dependent 620 based on the concept that a transmitter's dispersal cloud of virus is more likely to prove contagious 621 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . at higher viral load, which is entirely separate from viral infectivity within the airway once a virus 622 contacts the surface of susceptible cells. 623

We next assume that the total exposed contacts within a time step ( 9 # ) is gamma 624 distributed, i.e. 9 #~Q : ; , R Δ 5 , using the average daily contact rates ( ) and the dispersion 625 parameter ( ). To obtain the true number of exposure contacts with airway exposure to virus, we 626 simply multiply the contagiousness of the transmitter with the total exposed contacts within a time 627 step (i.e., 5 = 9 # 5 ). 628

Transmissions within a time step are simulated stochastically using time-dependent viral 629 load to determine infectiousness ( 5 ). Successful transmission is modelled stochastically by 630 drawing a random uniform variable ( (0,1)) and comparing it with infectiousness of the 631 transmitter. In the case of successful transmission, the number of secondary infections within that 632 time step ( 9 # ) is obtained by the product of the infectiousness ( 5 ) and the number of exposure 633 contacts drawn from the gamma distribution ( 5 ). In other words, the number of secondary 634 infections for a time step is 9 # = ( 5 ) 5 9 # . If we disregard contagiousness by assuming 5 = 635 1 in 5 , we identify that there are little to no differences on overall results other than the emergent 636 TD curve and optimal parameter set describing dose-response curve and exposed contact network, 637 which no longer agrees as closely with in vitro probability of positive virus culture (Fig S5) . 37 638

We obtain the number of secondary infections from a transmitter on a daily basis noting 639 that viral load, and subsequent risk, does not change substantially within a day. We then summed 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint Serial interval and generation time. We further assume that upon successful infection, it takes 644 days for the virus to move within-host, reach infection site and produce the first infected cell. 645

To calculate serial interval (time between the onset of symptoms of transmitter and secondarily 646 infected person), we sample the incubation period in the transmitter and in the secondarily infected 647 person from a gamma distribution with a shape described in the Fig S4. A modeling study that simulated observed outbreak sizes in ~40 affected countries during 662 the early phase of epidemics also confirmed that ~80% of secondary transmissions may have been 663

caused by a small fraction of infectious individuals (~10%). 4 The latter study provided the 664 distribution of individual R0 (Fig 2A) that we employed for fitting purposes. Using the data on is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint serial interval as well as the distribution of serial interval (Fig 2C) . 31 We employed this data for 667 fitting purposes. 668

The cumulative distribution function of individual R0 for influenza was obtained from a 669 modeling study that simulated the transmission dynamics of seasonal influenza in Switzerland . 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint

The parameter sets of ( , , , ) were simulated for 1000 infected individuals to determine how 688 well each set generates the summary statistics of mean R0, mean SI and the R0 histograms by 689 following a procedure explained in Fig S1 and below:  690 Step A: We visually checked whether our dose-response curve matched the observed probability 706 of positive virus culture. 37 We assumed that viral loads derived from positive culture 37 can be 707 considered equivalent to viral loads in the within-host model if divided by a positive integer. We 708

identified an integer of 25 to provide closest fit to the empirical data (Fig S5) . 709 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint

We performed a global sensitivity analysis to identify which parameter variability 710 accounted for fit to different components of the data. Only narrow ranges of l permitted close fit 711 to the mean of R0 and distribution functions of individual R0 (Fig S9) , while a specific value for 712 a was necessary to fit to mean serial interval and distribution functions of individual R0 (Fig  713   S9) . Only narrow ranges of q permitted close fit to the mean of R0 and distribution functions of 714 individual R0 (Fig S10) , while a specific value for r was necessary to fit to distribution functions 715 of individual R0 (Fig S10) . . 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint . 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint 757 758 Fig S3. Stochastic simulations of exposed contact frequency for varying dispersion (ρ). The 759

average number of exposed contacts is 4 per day in each example with imputed daily 760 heterogeneity based on an elevated value of ρ from a gamma distribution~Γ(4/ρ, ρ). 761 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint we divided these PCR values by 25 (light blue line), we identified high similarity between the 771 clinical data and our projected infectiousness dose response curve (red line). 772

. 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) preprint

The copyright holder for this this version posted September 28, 2020. interval, and generation time. A. SARS-CoV-2, and B. influenza. Lowering exposed contact 776 network heterogeneity to levels observed with influenza decreases SARS-CoV-2 individual R0 777 over-dispersion. Increasing exposed contact network heterogeneity to levels observed with 778 SARS-CoV-2 increases influenza R0 over-dispersion. Neither change impacts observed serial 779 interval or estimate generation time. 780 . 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) preprint

The copyright holder for this this version posted September 28, 2020. Transmitters are defined as infected people who generate at least one secondary infection. 786

. 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) preprint

The copyright holder for this this version posted September 28, 2020. networks on SARS-CoV-2 epidemiology. Simulations assume limitation of exposed contacts 791

only among daily exposures of more than 5, 10, 20 or 50 people. Mean reproductive number 792 decreases below one with only marginal decreases in overall rate of exposure contacts when 793

contacts are limited to fewer than 20 people. 794 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . 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) preprint

The copyright holder for this this version posted September 28, 2020. is the fixed effects (mean) and the bottom row is the standard deviation of the random effects. 811

We also fixed r=10, δE=1/day, q=2.4×10-5/day and c=15/day. 812 . 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) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.08.07.20169920 doi: medRxiv preprint

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