Study data

The longitudinal viral load data examined in our study were extracted from the published studies of SARS-CoV-2: Young and colleagues [27], Zou and colleagues [28], Kim and colleagues [29], and Wölfel and colleagues [30]. For consistency, the viral load data measured from upper respiratory specimens were used. We excluded patients who received antiviral treatment and for whom data were measured on only 1 or 2 days (because 1 or 2 data points are not enough to estimate parameters). We converted cycle threshold (Ct) values to viral RNA copy number values, where these quantities are inversely proportional to each other [28]. In total, we use the data from 30 patients. To extract the data from the images in those papers, we used the software DataThief III (version 1.5, Bas Tummers, www.datathief.org).

Mathematical model for virus dynamics without and with antiviral treatment

SARS-CoV-2 virus dynamics without antiviral treatment is described by a mathematical model previously proposed in [24,31,32]: (1) (2) where f(t) is the relative fraction of uninfected target cells at time t to those at time 0, and V(t) is the amount of virus at time t, respectively. Both f(t) and V(t) are in linear scale. The parameters β, γ, and δ represent the rate constant for virus infection, the maximum rate constant for viral replication, and the per capita death rate of virus-producing cells, respectively. Note that δ implicitly includes the effects of the immune response in killing infected cells, e.g., by cytotoxic T lymphocytes. All viral load data were fit using a nonlinear mixed-effect modeling approach, which estimates population parameters while accounting for interindividual variation in virus dynamics (see the next section for details). The day from symptom onset was used as a timescale (i.e., t = 0 at symptom onset).

The virus dynamic model under antiviral treatment (which we assume blocks virus replication) initiated at t* days after symptom onset can be described based on the above model as follows: (3) (4) where H(t) is a Heaviside function indicating off and on treatment, defined as H(t) = 0 if t < t* (i.e., before treatment initiation); otherwise, H(t) = 1. ε is the fraction of virus production inhibited by the therapy (0 < ε ≤ 1). ε = 1 when the virus replication from the infected cells is totally inhibited (i.e., the antiviral effect is 100%). We evaluated the expected antiviral effect of the treatment on the outcomes (duration of virus shedding and cumulative viral load measured on a log scale) under different inhibition rates (ε) and initiation times (t*). The effect of drugs that blocking de novo infection can be modeled by inhibiting both the βf(t)V(t) and γf(t)V(t) terms, and a drug promoting cytotoxicity can be modeled by increasing δV(t), as we discussed in [31]. Unfortunately, because sufficient viral load data under antiviral drug therapy are not available yet, the antiviral effect (ε) of drugs in preclinical development and in clinical trials are still unknown. Therefore, we chose to examine hypothetical examples of drugs with 50%, 95%, or 99% efficacy. We used 50% and 99% efficacy to illustrate the difference in viral dynamics between patients with and without treatment (see section “SARS-CoV-2 virus dynamics and antiviral effect”) and 95% and 99% efficacy in “Simulation mimicking a randomized controlled trial for antiviral drugs.” Since clinical trials are performed only for drugs with sufficient efficacy (i.e., there is no reason to test drugs with weak efficacy), we believe this value range is reasonable.

Parameter estimation with the nonlinear mixed-effects model

A nonlinear mixed-effects model was used to fit the viral dynamic model given by Eqs 1 and 2 to the longitudinal viral load data. The model included both a fixed effect (constant across patients) and a random effect (different between patients) in each parameter. Specifically, the parameter for patient k, is represented as a product of ϑ (a fixed effect) and (a random effect). π_k follows the normal distribution with mean 0 and standard deviation Ω. Fixed effects and random effects were estimated using the stochastic approximation expectation–maximization algorithm and empirical Bayes method, respectively. The conditional distribution of the vector of individual parameters was estimated for each patient using the Metropolis–Hastings algorithm and was used to calculate the 95% predictive interval of the viral load curve in Fig 1. The mixed-model approach is becoming more common in longitudinal viral load data analysis [26,33], because it can capture the heterogeneity in virus dynamics, and parameter estimation is feasible even for those with limited data. Fitting was performed using MONOLIX 2019R2 (www.lixoft.com) [34]. To account for data points under the detection limit (see the red dots in S1 Fig), the likelihood function reflected the likelihood that the data are in the censoring interval (0 to the detection limit) given parameter values with a right-truncated Gaussian distribution [35].

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Fig 1. Observed and fitted viral load data for individual patients.

Viral loads were measured using nasal swabs (China), pharyngeal swabs (Germany), nasopharyngeal swabs (Singapore and Korea), and oropharyngeal swab (Korea) for hospitalized SARS-CoV-2–infected patients. Note that the detection limits of the PCR assay for SARS-CoV-2 were 68.0 copies/ml (Singapore and Korea), 15.3 copies/ml (China), and 33.3 copies/ml (Germany), respectively, and are shown as dotted horizontal lines. The closed dots and curves correspond to the observed and the estimated viral load for each patient using their individual parameters given in S2 Table, respectively. Shaded regions correspond to 95% predictive intervals. Different colors of the dots and the lines (light blue, black, and pink) correspond to the 3 different types of patients characterized by rapid, medium, and slow viral load decay, respectively. The red dots represent the data at or under the detection limit regardless of the group. Patient IDs are the same as in the original papers if available. The underlying data for this figure can be found in S1 Data. SARS-CoV-2, Severe Acute Respiratory Syndrome Coronavirus 2.

https://doi.org/10.1371/journal.pmed.1003660.g001

Clustering of individual viral load dynamics

As observed in Fig 1, the virus dynamics has huge heterogeneity between patients. For some patients, viral load declines rapidly, but for others, it persists for almost 1 month. It is ideal if the longitudinal viral load data can be directly compared between patients; however, the data collection intervals are not the same between patients, and the data under the detection limit are not quantifiable. Therefore, we used the fitted viral load every day since symptom onset, which is available from the best fit curve, for comparison. The fitted daily viral load values of each patient were rescaled by their maximum values and log-transformed. Then, hierarchical clustering was performed on the rescaled-transformed fitted daily viral load using the linkage function with Ward method [36] in SciPy [37]. Once multiple clusters are identified, estimated parameter distributions among the clusters were compared by ANOVA to assess the source of the difference in virus dynamics. Pairwise comparison was subsequently performed using Student t test. The p-values of the pairwise Student t test were adjusted by the Bonferroni correction.

Simulation mimicking a randomized controlled trial for antiviral drugs

We mimicked randomized controlled trials using the model including the effects of an antiviral drug. The allocation ratio is assumed as 1:1 (control:treatment). A total of 20,000 parameter sets were randomly sampled from the estimated distributions of individual parameters. A longitudinal viral load time series for each individual was created based on their chosen parameter set. Note that for those in the treatment group, the antiviral effect (ε) was assumed to be constant. For sensitivity analysis, we used 2 different values of ε, 95% and 99%. To obtain a realistic simulation, treatment was initiated following the distribution of time from symptom onset to hospitalization obtained from Bi and colleagues: lognorm (1.23, 0.79) (the mean is 4.64 days) [38], where the treatment was assumed to be initiated immediately after hospitalization. We also used a truncated distribution to mimic the randomized controlled trials including only patients recruited and treated early (within 0.5, 1, 2, 3, and 4 days after symptom onset).

We used 2 quantities as outcome measures: the duration of virus shedding from the onset of symptoms until the time the virus becomes undetectable (T_D) and the log10-transformed cumulative viral load, i.e., the area under the curve (AUC) of viral load (). Many clinical studies have used the duration of viral shedding as a primary outcome (see S1 Table), and previous theoretical studies have quantified the AUC. Both of the outcomes we use here are expected to be reduced under effective antiviral treatment.

From our simulations, we obtained 10,000 outcomes for each group (duration of virus shedding and cumulative viral load). The sample size was computed for different values of ε using the 2-tailed Welch t test with significance level and power as 0.05 and 80%, respectively.

Heterogeneity in SARS-CoV-2 virus dynamics

SARS-CoV-2 viral load data were analyzed using a mathematical model to quantify the heterogeneity in virus dynamics among patients and to examine the source of the heterogeneity. Longitudinal viral load data from 30 patients from different countries were fitted simultaneously using a nonlinear mixed-effects modeling approach. With the estimated parameters for each patient (listed in S2 Table), viral loads since the time of symptom onset were fully reconstructed even when the viral load was missed or under the detection limit (Figs 1 and 2A). This reconstruction allowed us to quantitatively compare viral load dynamics between patients. The viral loads over time, which were reconstructed based on the mathematical model with the estimated parameters, were analyzed with a clustering approach (Fig 2B) and placed into 3 groups. Patient “China O” was detected as an outlier and was excluded from further analysis.

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Fig 2. Characterizing and clustering COVID-19 patients using viral load data.

(A) Schematic illustration for data fitting with a virus dynamics model. Longitudinal SARS-CoV-2 RNA load data (i.e., clinical data) were extracted from published papers. The data were analyzed by the mathematical model, and then virus dynamics parameters were estimated for each patient (i.e., phenotyping). Daily viral load since symptom onset for each patient was simulated by running the model with the estimated parameters. (B) Clustering patients using daily viral load. Daily viral load obtained through simulation was used for clustering of the 30 patients. In the dendrogram, the height from the bottom to the point where 2 or more patients are joined indicates the distance (i.e., dissimilarity) between patients. For example, “Singapore 11” and “Germany 2” are very close, and those are far from “Singapore 6.” As a result, 3 different patient groups were identified, and “China O” was detected as an outlier. The heatmap next to the dendrogram (“Virus dynamics parameters”) shows the estimated parameters and initial condition (γ, β, V(0), δ) for each patient. Light blue and pink correspond to high and low values, respectively. Statistically significant between-group differences were found in the maximum rate constant for viral replication, γ (ANOVA p-value: 3.61 × 10⁻³), and the death rate of virus-producing cells, δ (ANOVA p-value: 3.23 × 10⁻¹³); moreover, there were statistical differences between all pair groups for δ. The death rate is highlighted by the dotted square. The right heatmap shows the daily viral load for each patient. Green and purple correspond to high and low values, respectively. “Group 1” maintained a high viral load for a longer period compared with the other groups. The underlying data for this figure can be found in S2 Data. COVID-19, Coronavirus Disease 2019; SARS-CoV-2, Severe Acute Respiratory Syndrome Coronavirus 2.

https://doi.org/10.1371/journal.pmed.1003660.g002

To understand the source of the difference in virus dynamics between groups, we tested the differences in the estimated parameters (i.e., β, γ, δ, and V(0)) among the groups. Statistically significant between-group differences were found in the maximum rate constant for viral replication, γ, and in the death rate of virus-producing cells per day, δ (S1 Fig). The differences in γ are related to the growth of viral load (a larger γ indicates more rapid growth); however, the difference in γ between groups was sufficiently small that its influence on virus dynamics especially after the viral load peak (or symptom onset) is negligible. The difference in δ manifests in the speed of viral load decay, i.e., a small value of δ corresponds to a slow decay in viral load (Fig 2B). Thus, we named the 3 groups as rapid, medium, and slow viral load decay groups (Fig 3A). The mean decay rates were 1.17 d⁻¹ (95% CI: 1.06 to 1.27 d⁻¹), 0.777 d⁻¹ (0.716 to 0.838 d⁻¹), and 0.450 d⁻¹ (0.378 to 0.522 d⁻¹) for the 3 groups, respectively. The minimum and maximum of the decay rates for the 3 identified groups were 0.270 d⁻¹ to 0.616 d⁻¹ (slow), 0.700 d⁻¹ to 0.914 d⁻¹ (medium), and 0.993 d⁻¹ and 1.30 d⁻¹ (rapid), respectively. The border value of the decay rate between groups is defined as the mean of the highest value in the lower group and the lowest value in the higher group. Thus, the border value of the slow and medium groups was 0.658 d⁻¹ [(0.616 + 0.700)/2] and that of the medium and rapid groups was 0.953 d⁻¹ [= (0.914 + 0.993)/2].

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Fig 3. Patient variability and difference in therapeutic response.

(A) Viral load trajectories since symptom onset for the 3 groups (black: medium decay group, light blue: rapid decay group, and pink: slow decay group) obtained through simulation (without antiviral treatment). (B) Viral load trajectories since symptom onset for the 3 groups under antiviral treatment with different inhibition rates and different timing of treatment initiation. The left 3 panels are viral load trajectories when the treatment is initiated at 0.5 days (“Early initiation”) since symptom onset. The right 3 panels are vial load trajectories when treatment is initiated at 5 days (“Late initiation”) since symptom onset. Blue and red dotted lines correspond to the trajectories with 50% and 99% inhibition rate, respectively. The bolded lines are the trajectory without treatment shown for comparison. The dotted horizontal lines are the D.L. D.L., detection limit.

https://doi.org/10.1371/journal.pmed.1003660.g003

SARS-CoV-2 virus dynamics and antiviral effect

Using our mathematical model and the estimated parameter distribution for each patient (S3 Table), we conducted in silico experiments to determine the possible therapeutic response, measured in terms of virus dynamics, of drug treatments blocking virus replication. Clinical outcomes are known to be related to the timing of initiation of antiviral treatment in general, and, especially, for influenza [31,39–42], and the antiviral effects of a treatment are dependent on dose and the patients’ immune system [43,44]. Thus, we studied several different scenarios in which we varied the time of treatment initiation (0.5 or 5 days from symptom onset, which were before and generally after the estimated peak viral load in our dataset) and the inhibition rate (99% or 50%). We resampled a total of 1,000 parameter sets from the estimated parameter distributions for this simulation and separated the individuals according to the value of the viral load decay rate (i.e., rapid, δ > 0.953 d⁻¹; medium, 0.658 d⁻¹ ≤ δ ≤ 0.953 d⁻¹; or slow, δ < 0.658 d⁻¹). We found that early initiation of antiviral treatment with a high inhibition rate (i.e., 99%) immediately reduced the viral load after initiation (Fig 2B, S3 Fig). However, if the inhibition rate was low (i.e., 50%), the viral load kept increasing, and the viral load decay rate after the peak was slower or equivalent to that without treatment. This was because viral replication was not efficiently inhibited and thus it continued albeit with a lower rate even after treatment initiation and continued long after the peak. In contrast, virus dynamics was not much influenced if treatment was initiated after the peak regardless of the inhibition rate or the patient type (Fig 3B, S2 Fig), because the number of uninfected targeted cells remaining at this stage of infection is limited. It is intriguing that a weak antiviral effect was observed for patients with rapid decay even when the treatment was initiated after the peak. Because the virus is removed rapidly during the course of infection for patients with rapid decay, more uninfected cells remain compared with the other groups. Therefore, antiviral drugs can mitigate replication of the virus even when treatment is initiated after the peak to some extent. Note that these findings are not unique to SARS-CoV-2; similar findings for virus dynamics and antiviral effects have been suggested in other infectious diseases [25,45].

Observational studies for antiviral drugs cannot yield significant results owing to heterogeneity in virus dynamics

We explored why compassionate use programs do not yield significant findings when using the duration of virus shedding as an outcome. Duration of virus shedding is one of the most frequently used outcomes for assessing antiviral treatment for SARS-CoV-2 infection (S1 Table) [12,13,15,45]. The distribution of the duration of virus shedding without treatment in the different virus decay groups is shown in Fig 4A and S3A Fig. As can be expected from the difference in viral load dynamics, the duration of virus shedding without treatment is longer in the group with slow viral load decay: The averages in the groups with medium, rapid, and slow decay were 12.3 (SD: 1.06), 8.86 (SD: 1.40), and 22.5 (SD: 8.15) days, respectively. As a sensitivity analysis, we also computed and compared the cumulative viral load (AUC). We confirmed the same trend in the cumulative viral load in log scale (S3B and S4A Figs). We further compared the outcomes under antiviral treatment (inhibition rate was set as 50% and 99%). Regardless of viral decay rate group, we consistently observed that both outcomes were improved by early treatment initiation (day 0.5) but not by late treatment initiation (day 5), as illustrated in Fig 4B and S4B Fig.

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Fig 4. Duration of virus shedding in the 3 different groups.

(A) The relative density distributions of duration of virus shedding since symptom onset for the 3 groups (light pink: medium decay group, light blue: rapid decay group, and pink: slow decay group) without treatment obtained through simulation. (B) The relative density distributions of duration of virus shedding since symptom onset for the 3 groups under antiviral treatment with different inhibition rates and different timing of treatment initiation. The left 3 panels are when antiviral treatment is initiated at 0.5 days (“Early initiation”) since symptom onset. The red dotted line corresponds to the distribution with a 99% inhibition rate. The distribution without treatment is shown in the back for comparison. The right 3 panels are when antiviral treatment is initiated at 5 days (“Late initiation”) since symptom onset. The distributions are represented as “relative density” to reflect different proportions of the 3 groups. The underlying data for this figure can be found in S3 Data.

https://doi.org/10.1371/journal.pmed.1003660.g004

If a patient possesses strong viral defenses, including immune-mediated defenses, the virus-producing cells are removed quickly, which corresponds to a shorter duration of virus production and rapid viral load decay. Indeed, the duration of virus shedding in respiratory samples has been associated with disease severity [46] and differs between symptomatic and asymptomatic cases [47]. Taken together, these findings suggest that both treatment allocation and clinical outcomes in compassionate use programs are associated with severity; thus, severity is a potential confounding variable. Further, there may be other confounding variables in the assessment of treatment efficacy in compassionate use programs; however, controlling all of them in the analysis is not possible. For example, heterogeneous immune responses, which are partially represented by the death rate of infected cells in the model, can confound the inference. However, quantifying the immune response is difficult. We need to be careful when interpreting the results from compassionate use programs.

Randomized controlled trials need to enroll patients early after symptom onset to observe significant antiviral effects

In contrast to observational studies, randomized controlled trials may not be influenced by confounding variables and could provide valid inference. However, clinical trials for COVID-19 should consider the timing of treatment initiation in the design (i.e., inclusion–exclusion criteria), because differences in outcomes are unlikely to be observed under late treatment initiation as we demonstrated in the previous section (Fig 4B, S4B Fig).

We computed the sample size needed to observe a statistically significant difference in outcomes with 80% power and a significance level of 0.05 assuming patients are randomly assigned and treated (with antiviral or placebo) immediately after hospitalization with different antiviral effect (95% and 99%) (Fig 5A) and with different inclusion and exclusion criteria for the timing of enrollment. We primarily used the duration of virus shedding as an outcome, but the results are qualitatively consistent for cumulative viral load as an outcome. The sample size is strongly dependent on the criterion of the timing of enrollment, i.e., the sample size can be reduced if patients are enrolled early after symptom onset (S4 Table). The distribution of duration of virus shedding under the different criteria is shown in Fig 5B and 5C. If patients are enrolled regardless of the time of treatment initiation, the sample sizes are 13,603 and 11,670 per group when the inhibition rate is 95% and 99%, respectively, which is much larger than the empirical sample size of the randomized controlled trials of antivirals for SARS-CoV-2 (S1 Table). This large sample size is needed given that the treatment is initiated 4.6 days after onset of symptoms on average, which is after the viral load peak. If we enroll only the patients treated within 1 day of the onset of symptoms, the sample size is reduced to 584 and 458 per group when inhibition rate is 95% and 99%, respectively. Note that antiviral drugs with a weaker inhibition rate will require larger sample sizes. The trend was similar when cumulative viral load in log scale was used as an outcome (S4 Table, S4C and S4D Fig).

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Fig 5. Simulation mimicking randomized controlled trial for anti-SARS-CoV-2.

(A) Schematic illustration for the simulation mimicking randomized controlled trials for antiviral drugs. A total of 20,000 parameter sets were sampled from the estimated parameter distributions. The parameter sets were randomized into control (“No treatment,” 10,000 individuals in total) or treatment (“Anti-SARS-CoV-2 treatment,” 10,000 individuals in total) groups. For the treatment group, the treatment is initiated randomly to reflect the delay of treatment initiation since symptom onset. We have also used a truncated distribution (red area) to mimic the randomized controlled trials including only patients recruited and treated early (“Early initiation”). Then the outcomes (duration of virus shedding and the cumulative viral load in log scale) were calculated for each patient. (B) The distributions of duration of virus shedding for patients without and with treatment are shown in black and red curves, respectively. For antiviral treatment, a 99% inhibition rate of virus replication was assumed. The red dotted curve is when all patients are included regardless of the timing of recruitment and treatment initiation. The red dashed curve is the case when only patients recruited and treated within 0.5 days since symptom onset are included. (C) Means and standard deviation of the duration of virus shedding under different inclusion criteria (not treated, included regardless of the timing of recruitment and treatment initiation, and early treatment initiation [within 0.5, 1, 2, 3, and 4 days since symptom onset]) are shown. The underlying data for this figure can be found in S4 Data. COVID-19, Coronavirus Disease 2019; SARS-CoV-2, Severe Acute Respiratory Syndrome Coronavirus 2.

https://doi.org/10.1371/journal.pmed.1003660.g005

Timing of randomization of clinical studies of antiviral drugs for SARS-CoV-2

To validate our findings from a practical perspective, we checked the clinical trials investigating antiviral efficacy registered in ClinicalTrials.gov. As of May 22, 2020, we identified 176 clinical trials with the search terms “antiviral” and “COVID.” Among them, 46 studies did not investigate the efficacy of antiviral drugs (the effect of anti-inflammatory drugs were investigated, for example), and 20 studies did not directly investigate the efficacy of antivirals (such as vaccine studies and safety studies). Among the remaining 110 studies investigating antiviral effect, including remdesivir, chloroquine, and lopinavir/ritonavir, only 17 studies (15%) explicitly stated the time from symptom onset in the inclusion or exclusion criteria. The average time from symptom onset to randomization was 7.2 days, which our findings suggest is too late to observe a statistically significant antiviral effect with a reasonable sample size.