key: cord-0896404-09r4d3nu
authors: Kim, Kwang Su; Ejima, Keisuke; Ito, Yusuke; Iwanami, Shoya; Ohashi, Hirofumi; Koizumi, Yoshiki; Asai, Yusuke; Nakaoka, Shinji; Watashi, Koichi; Thompson, Robin N; Iwami, Shingo
title: Modelling SARS-CoV-2 Dynamics: Implications for Therapy
date: 2020-03-27
journal: nan
DOI: 10.1101/2020.03.23.20040493
sha: a64c4974c979487512a016eb09b1b4e0c8e5571e
doc_id: 896404
cord_uid: 09r4d3nu

The scientific community is focussed on developing antiviral therapies to mitigate the impacts of the ongoing novel coronavirus disease (COVID-19) outbreak. This will be facilitated by improved understanding of viral dynamics within infected hosts. Here, using a mathematical model in combination with published viral load data collected from the same specimen (throat / nasal swabs or nasopharyngeal / sputum / tracheal aspirate), we compare within-host dynamics for patients infected in the current outbreak with analogous dynamics for MERS-CoV and SARS-CoV infections. Our quantitative analyses revealed that SARS-CoV-2 infection dynamics are more severe than those for mild cases of MERS-CoV, but are similar to severe cases, and that the viral dynamics of SARS-CoV infection are similar to those of MERS-CoV in mild cases but not in severe case. Consequently, SARS-CoV-2 generates infection dynamics that are more severe than SARS-CoV. Furthermore, we used our viral dynamics model to predict the effectiveness of unlicensed drugs that have different methods of action. The effectiveness was measured by AUC of viral load. Our results indicated that therapies that block de novo infections or virus production are most likely to be effective if initiated before the peak viral load (which occurs around three days after symptom onset on average), but therapies that promote cytotoxicity are likely to have only limited effects. Our unique mathematical approach provides insights into the pathogenesis of SARS-CoV-2 in humans, which are useful for development of antiviral therapies.

due to MERS-CoV. In addition, as a median estimate, 65% inhibition of the initial 103 virus expansion is required to prevent the establishment of SARS-CoV-2 infection 104 (we provide a detailed analysis later). We also calculated the duration of infection in 105 which the viral load is above the detection limit ( ) in Table 1 , showing that SARS-106

CoV-2 is maintained in hosts for more than a week based on the median estimate. 107 108

To extend our analysis to include SARS-CoV, we analysed SARS-CoV viral 111 loads in nasopharyngeal aspirate reported by Peiris et al. (12) and MERS-CoV viral 112 loads reported by Oh et al. (8) in sputum or tracheal aspirate. The estimated 113 parameters, viral load at symptom onset, and the indices derived from the estimated 114 parameters are listed in Table 1 and from bootstrap t-test) (Fig. 2) . This demonstrates that in vivo viral dynamics of 121 SARS-CoV infection are similar to those for MERS-CoV in mild cases but not in 122 severe cases. Collectively, the findings from the viral load data analyses for the two 123 different specimens (throat/nasal swabs and nasopharyngeal/sputum/tracheal 124 aspirate) implied that SARS-CoV-2 also causes infection more effectively than 125 SARS-CoV. 126 127 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted March 27, 2020 . . https://doi.org/10.1101 

Our quantitative analyses provide insights into optimal usage of anti-SARS-129

CoV-2 therapies under development. In particular, it remains poorly understood how 130 a delay of treatment initiation after primary infection, or how incomplete blocking of 131 virus infection/replication, impacts the viral load dynamics. Based on our 132 mathematical model and estimated parameter values (Table 1) infections. This can be induced by drugs including human neutralising antibodies, 140 viral entry-inhibitors and/or antibodies raised by vaccination (13, 14) . For example, a 141 SARS-CoV-specific human monoclonal antibody has been reported to cross react 142 with SARS-CoV-2 (14). We conducted in silico experiments with varying drug 143 efficacy (considering inhibition rates from 10% to 100%, i.e. 0.1 ≤ ≤ 1) and with the 144 timing of initiation of therapy from 0 days (i.e., post-exposure prophylactic use of 145 antivirals) until 5 days after symptom onset (i.e., 0 ≤ * ≤ 5) (see Methods). Our 146 results show that early initiation of therapy (especially within two to three days) with 147 even a relatively weak drug (inhibition rates as low as 50%) might effectively reduce 148 the area under the curve of viral load (AUC) and prevent significant reductions in the 149 numbers of target cells because of cytopathic effects due to cell invasion. A therapy 150 of this type initiated four days after symptom onset, on the other hand, is not 151 predicted to induce a clear antiviral effect (Fig. 3AD) . This suggests that blocking de 152 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted March 27, 2020 . . https://doi.org/10.1101 novo infections is not likely to be effective unless the intervention is initiated before 153 the peak viral load. Hence, appropriate initiation timing (i.e., before or very soon after 154 symptom onset) is an important factor for suppressing viral load in addition to the 155 therapy having the potential for antiviral effects. 156

The majority of antiviral drugs inhibit intracellular virus replication. Although 159 their antiviral efficacies need to be confirmed, lopinavir/ritonavir (HIV protease 160 inhibitors), remdesivir (anti-Ebola virus disease candidate) and other nucleoside 161 analogues, and interferon have the potential to suppress SARS-CoV-2 by blocking 162 virus production (15, 16) . Interestingly, our results suggest that, even for relatively 163 small inhibition rates of around 30%, the AUC of viral load is partially reduced if 164 therapy is initiated early (within three days after symptom onset) (Fig. 3BE) . 165

However, if treatment is applied after the peak viral load, even drugs with 100% 166 inhibition rate are not able to reduce viral loads, which is similar to the predicted 167 outcomes of de novo blocking therapy. 168 169 (iii) Promoting cytotoxicity 170

Another possible antiviral mechanism is cytotoxic effects by adaptive 171 immunity including those mediated by cytotoxic T lymphocytes. Here, we assume 172 that promoting cytotoxicity increases the virus death rate by at most two times (i.e., 173 0.1 ≤ ≤ 1), that is, achieves up to 50% reduction of the mean length of virus 174 production. Compared with the other two therapies (blocking de novo infection and 175

virus production), the induction of cytotoxicity had relatively mild effects on the AUC 176 reduction if initiated before peak viral load. However, cytotoxicity induction initiated 177 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted March 27, 2020 . . https://doi.org/10.1101 after peak viral load could effectively reduce the viral load AUC (Fig. 3CF) . This 178 implies that there is an optimal time to apply this therapy, and that significant antiviral 179 effects are expected unless the promoting rate is too low or therapy is initiated either 180 too early or too late. However, large reductions of target cells due to ongoing de 181 novo infection cannot be avoided even with very early initiation (i.e., immediately 182 after symptom onset) of the therapy. 183 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted March 27, 2020 . . https://doi.org/10.1101 

There are a number of potential transmission routes of SARS-CoV-2, 185 including direct person-to-person transmission due to viral particle inhalation and 186 contact transmission due to contact with nasal/oral/eye mucuous membranes. The virus production effectively reduces AUC of SARS-CoV-2 load; for example, if the 193 therapy can reduce more than 90% of de novo infections and is initiated 3 days after 194 symptoms onset, the viral load AUC is expected to be reduced by 81.4% (Fig. 3DE) . 195

However, if therapy is initiated after peak viral load (more than 2-3 days following 196 symptom onset), the effect on viral load AUC is limited. Compared with either 197 blocking de novo infection or virus production, promoting cytotoxicity showed 198 relatively mild effects on AUC reduction, however initiation of that therapy after the 199 peak viral load has the potential to still reduce viral load AUC (Fig. 3F) . 200

The effectiveness of the hypothetical drugs can be evaluated in detail using a 201 reported for SARS-CoV-2, a number of animal models exist for SARS-CoV including 206 mice, hamsters, ferrets, and macaques (17). Animals could be used to verify the 207 conclusions from our models, by monitoring the viral loads in animals treated with 208 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in load data would allow further investigation of the effectiveness of drugs with different 210 action mechanisms, which would be informative for development of appropriate 211 treatment strategies (i.e., the optimal dose/timing of antivirals) for SARS-CoV-2 212

In conclusion, effective treatment of SARS-CoV-2 infections requires an 214 appropriate choice of class-specific drugs; otherwise, the antivirals do not alter the 215 viral load significantly and are wasted. Identification of SARS-CoV-2-specific virus 216 characteristics is needed to design optimal treatments and to ensure that limited 217 resources are deployed effectively. Additionally, effective combinations of anti-218 SARS-CoV-2 drugs and vaccines will maximise the impacts of control, reduce the 219 required drug dose and potentially limit side effects, all of which are highly desirable. 220

Our theoretical approach could complement ongoing experimental investigations into 221

19 treatment. To our knowledge, previous studies have neither characterised SARS-223

CoV-2 dynamics in humans using viral dynamics models, nor compared the resulting 224 dynamics against those of other coronaviruses (i.e., SARS-CoV and MERS-CoV). 225

Our mathematical modelling approach has led to an improved understanding of the 226 characteristics of SARS-CoV-2 in vivo, and can be used to test possible treatments 227 for COVID-19 further going forwards. 228 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted March 27, 2020. . https://doi.org/10.1101/2020.03.23.20040493 doi: medRxiv preprint

The data examined in our manuscript came from studies of SARS-CoV-2, 231 MERS-CoV and SARS-CoV by Zou et al. (1) , Oh et al. (8) and Peiris et al. (12) , 232 respectively. To extract the data from images in those publications, we used the 233 program datathief III (version 1.5, Bas Tummers, www.datathief.org). We excluded 234 patients for whom data were measured on only one day, and assumed that viral load 235 values under the detection limit were set as the detection limit for the purposes of 236 fitting the model. We converted cycle threshold (Ct) values reported in Zou et al. (1) , 237

Oh et al. (8) and Peiris et al. (12) to viral RNA copies number values, where these 238 quantities are inversely proportional to each other (18). 239 240

To parameterise coronavirus infection dynamics from patient viral load data, 242

we derived a simplified mathematical model from the following virus dynamics 243 model: 244 , , , and represent the rate constant for virus infection, the death rate of infected 250 cells, the viral production rate, and the clearance rate of the virus, respectively. Since 251 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted March 27, 2020. . https://doi.org/10.1101/2020.03.23.20040493 doi: medRxiv preprint the clearance rate of virus is typically much larger than the death rate of the infected 252 cells in vivo (10, 19, 20) , we made a quasi-steady state (QSS) assumption, 253 ( )⁄ = 0 , and replaced Eq.(3) with ( ) = ( )⁄ . Because data on the 254 numbers of coronavirus RNA copies, ( ), rather than the number of infected cells, 255 ( ), were available, then ( ) = ( )⁄ was substituted into Eq.(2) to obtain 256

Furthermore, we defined the ratio of the number of uninfected target cells at time to 258 the initial number of uninfected target cells (0) , that is,

Accordingly, we obtained the following simplified mathematical model, which we 260 employed to analyse the data in this study: 261 CoV were fitted using a nonlinear mixed-effect modelling approach (described 271 below), which uses the whole sample to estimate population parameters but also 272 account for inter-individual variation. 273 274 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in CoV-2 therapy, the median parameter sets were used to predict the expected 286 outcome of each therapy. In other words, even though no drug administration trials 287 have been conducted yet, we were able to infer the efficacy of each drug treatment 288 based on our in silico experiments. We implemented the different mechanisms of 289 action in the model as follows: 290 (i) Blocking de novo infection. The antiviral effect of blocking de novo infection 291 therapy (0 < ≤ 1. = 1 implies de novo infection is 100% inhibited) initiated at * 292 days after symptom onset was modelled by assuming: 293

where ( ) is a Heaviside step function defined as ( ) = 0 if < * : otherwise 296 ( ) = 1. We evaluated the expected antiviral effect of the therapy under different 297 inhibition rates ( ) and initiation timings ( * ) using our estimated parameter values. 298 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted March 27, 2020. . https://doi.org/10. 1101 The mean reduction of cumulative virus production, i.e., the area under the curve of 299 viral load (AUC: ∫ ( ) 28 0

: because the observed durations COVID-19 infection are 300 longer than previous coronavirus infection (i.e., SARS-CoV and MERS-CoV), we 301 used the maximum length of observations 28 days as the upper bound for 302 integration), induced by blocking de novo infection for SARS-CoV-2 was calculated. 303

Note that the expected values at day 0 after symptom onset ( * = 0) corresponds to 304 the antiviral effect of therapy initiated immediately after symptom onset. 305

(ii) Blocking virus production. Alternatively, we assumed an inhibition rate of 306 virus production of 0 < ≤ 1. The antiviral effect by blocking virus production (0 < 307 ≤ 1. = 1 indicates that the virus reproduction from the infected cells are perfectly 308 inhibited) is modelled as follows: 309

Note that the difference between blocking de novo infection and virus production is 311 that the former reduces , whereas the latter reduces in the full model (1) likelihood estimation procedure for parameters in a nonlinear mixed-effects model, 320

was employed to fit to the viral load data. Nonlinear mixed-effects modelling 321 approaches allow a fixed effect as well as a random effect describing the inter-322 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted March 27, 2020. . https://doi.org/10.1101/2020.03.23.20040493 doi: medRxiv preprint patient variability. Including a random effect amounts to a partial pooling of the data 323 between individuals to improve estimates of the parameters applicable across the 324 population of patients. By using this approach, the differences between viral 325 dynamics in different patients were not estimated explicitly, nor did we fully pool the 326 data which would bias estimates towards highly sampled patients. In this method of 327 estimation, each parameter estimate (= × ) depends on the individual where 328 is fixed effect, and is random effect with an assumed Gaussian distribution with 329 mean 0 and standard deviation Ω. Population parameters and individual parameters 330 were estimated using the stochastic approximation expectation-maximisation 331 algorithm and empirical Bayes' method, respectively. Individual estimated 332 parameters and initial values for patients are summarized in Table S1 . Using parameters were compared and tested using the bootstrap t-test. Due to the small 340 sample size for viral loads in sputum/tracheal aspirate for MERS-CoV, we assumed 341 the fixed effect was the same for mild and severe MERS-CoV cases. Otherwise, the 342 process was exactly the same as that described for the throat swab data. 343

The computation of , , * and 345

Based on the estimated parameter distributions, we calculated several 346 quantities: the duration of virus production ( ), the basic reproduction number ( 0 ), 347 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted March 27, 2020. . https://doi.org/10.1101/2020.03.23.20040493 doi: medRxiv preprint and the critical inhibition rate ( * ). We also calculated the period during which the 348 viral load was above the detection limit ( ) from the in silico simulations with 349 individual estimated parameters and an initial viral load equal the detection limit (i.e. 350 numerical experiments began at the point at which the virus became detectable). 351

The distributional estimates of 0 , , * and were calculated separately for 352 SARS-CoV-2 and SARS-CoV, as well as for severe and mild cases of MERS-CoV. 353 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted March 27, 2020 . . https://doi.org/10.1101 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted March 27, 2020 . . https://doi.org/10.1101 perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted March 27, 2020. . 

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The severe acute respiratory 429 syndrome

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) Virus dynamics

SARS-CoV-2 MERS-CoV(severe) MERS-CoV(mild)

SARS-CoV-2 MERS-CoV(severe) MERS-CoV(mild)

SARS-CoV-2 MERS-CoV(severe) MERS-CoV(mild)

MERS-CoV(severe) MERS-CoV(mild) SARS-CoV MERS-CoV(severe) MERS-CoV(mild)

severe) MERS-CoV(mild) Supplementary Appendix Modelling SARS-CoV-2 Dynamics: Implications for Therapy