Relationship between the inclusion/exclusion criteria and sample size in randomized controlled trials for SARS-CoV-2 entry inhibitors

Abstract

The coronavirus disease 2019 (COVID-19) pandemic that has been ongoing since 2019 is still ongoing and how to control it is one of the international issues to be addressed. Antiviral drugs that reduce the viral load in terms of reducing the risk of secondary infection are important. For the general control of emerging infectious diseases, establishing an efficient method to evaluate candidate therapeutic agents will lead to a rapid response. We evaluated clinical trial designs for viral entry inhibitors that have the potential to be effective pre-exposure prophylactic drugs in addition to reducing viral load after infection. We used a previously developed simulation of clinical trials based on a mathematical model of within-host viral infection dynamics to evaluate sample sizes in clinical trials of viral entry inhibitors against COVID-19. We assumed four measures as outcomes, namely change in log10-transformed viral load from symptom onset, PCR positive ratio, log10-transformed viral load, and cumulative viral load, and then sample sizes were calculated for drugs with 99 % and 95 % antiviral efficacy. Consistent with previous results, we found that sample sizes could be dramatically reduced for all outcomes used in an analysis by adopting inclusion/exclusion criteria such that only patients in the early post-infection period would be included in a clinical trial. A comparison of sample sizes across outcomes demonstrated an optimal measurement schedule associated with the nature of the outcome measured for the evaluation of drug efficacy. In particular, the sample sizes calculated from the change in viral load and from viral load tended to be small when measurements were taken at earlier time points after treatment initiation. For the cumulative viral load, the sample size was lower than that from the other outcomes when the stricter inclusion/exclusion criteria to include patients whose time since onset is earlier than 2 days was used. We concluded that the design of efficient clinical trials should consider the inclusion/exclusion criteria and measurement schedules, as well as outcome selection based on sample size, personnel and budget needed to conduct the trial, and the importance of the outcome regarding the medical and societal requirements. This study provides insights into clinical trial design for a variety of situations, especially addressing infectious disease prevalence and feasible trial sizes. This manuscript was submitted as part of a theme issue on “Modelling COVID-19 and Preparedness for Future Pandemics”.

1. Introduction

The coronavirus disease 2019 (COVID-19) pandemic, of which severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the pathogen, that began in 2019 is still ongoing and how to control it is one of the international issues to be addressed. Although vaccination and its booster shots have spread rapidly, primarily to prevent severe disease, vaccine efficacy and vaccination coverage have not been sufficient to prevent an epidemic. Each country has undertaken different policies against the spread of the disease. In a situation where the spread of infection continues, antiviral drugs that reduce the viral load in terms of reducing the risk of secondary infection are important. It is also important to establish a process for the evaluation of potential therapeutic agents to counteract emerging infectious diseases, including those that may emerge in the future, which will allow the rapid treatment of infected individuals.

This study evaluated the efficacy of drugs that inhibit viral entry into target cells. Currently, the only drugs approved for oral administration to patients with mild COVID-19 are those that inhibit viral replication or particle formation within cells (Jayk Bernal, 2022, Hammond, 2022). The combined use of drugs with different mechanisms of action is expected to have synergistic effects. In addition, drugs such as entry inhibitors are expected to be effective at preventing infection when administered to those who are likely to encounter an infected person, such as family members of infected persons and health care workers. Anti-SARS-CoV-2 monoclonal antibodies recognize viral spike proteins and inhibit their entry into host cells (Hansen, 2020). Several of these drugs have been approved for limited therapeutic purposes and their preventive efficacy has been demonstrated (Gupta, 2021, Weinreich, 2021, O'Brien, 2022).

Changing the method used for the treatment of mild or asymptomatic patients to ensure economic activity and reduce the burden on medical institutions, as well as the approval of drugs that can reduce the severity rate, will make it more difficult to conduct large-scale clinical trials. In our previous mathematical modeling studies, we performed computer simulations that mimicked a randomized controlled trial of a viral replication inhibitor against COVID-19 and showed that inclusion/exclusion criteria that limited the trial population to only patients in the early post-infection period dramatically reduced the sample size required to detect drug efficacy (Iwanami, 2021, Akao, 2021). We also showed that sample size varied in a measurement schedule-dependent manner depending on the characteristics of the outcome used in the trial (Akao, 2021). This method is expected to be useful for efficient trial planning prior to clinical trials by assuming that the antiviral efficacy of candidate drugs is based on viral infection dynamics quantified from the viral load of COVID-19 patients.

We adapted this mathematical model-based method of evaluating drug efficacy by simulation to investigate the relationship among inclusion/exclusion criteria, outcome measurement schedules, and sample size for clinical trials of drugs that inhibit viral entry into target cells. By calculating sample sizes for multiple outcomes, we performed an exhaustive analysis of the evaluation of antiviral drug efficacy based on viral load. This study provides insights into clinical trial design for a variety of situations, especially addressing infectious disease prevalence and feasible trial sizes.

2. Materials and methods

2.1. A mathematical model for the treatment of SARS-CoV-2 infection with entry inhibitors.

We used a simple mathematical model derived from a basic virus dynamics model to describe SARS-CoV-2 infection dynamics in COVID-19 patients. The following model was used for the simulation of randomized controlled trials of virus entry inhibitors (Iwanami, 2021, Akao, 2021, Kim, 2021):

$$\frac{df\left(t\right)}{\mathit{dt}}=-\left(1-\epsilon \times H\left(t\right)\right)\beta f\left(t\right)V\left(t\right),$$ (1)

$$\frac{dV\left(t\right)}{\mathit{dt}}=\left(1-\epsilon \times H\left(t\right)\right)\gamma f\left(t\right)-\delta V\left(t\right).$$ (2)

The variables, $f\left(t\right)$ and $V\left(t\right)$, are the fraction of uninfected cells compared with number of cells at symptom onset and the amount of virus, respectively, at time $t$ after symptom onset. Parameters, $\beta ,\gamma$, and $\delta$, correspond to the rate constant for infection, maximum replication rate, and death rate of infected cells, respectively. The virus entry inhibition rate of drugs is described as parameter $\epsilon ,$ which is equal to 0 without treatment of drugs or treatment with placebo, and it is equal to 1 if the drug completely inhibits virus entry (i.e., $0\le \epsilon \le 1$). The function $H\left(t\right)$, is a Heaviside step function, defined as $H\left(t\right)=0$ if $t<t^{\ast }$ otherwise $H\left(t\right)=1$, which indicates the presence or absence of treatment. Here, $t^{\ast }$ is the time of treatment initiation.

2.2. Simulation of randomized controlled trials of entry inhibitors for the treatment of SARS-CoV-2 infection

For simulations of randomized controlled trials to evaluate the treatment efficacy of entry inhibitors against SARS-CoV-2 infection, we used a previously developed approach based on a mathematical model of virus dynamics (Iwanami, 2021, Akao, 2021, Kim, 2021, Kim, 2021). The entry inhibition rates of candidate drugs were set as 99 % and 95 % (i.e., $\epsilon =0.99$ or $0.95$) assuming sufficient antiviral efficacy. To calculate the SARS-CoV-2 viral load in hypothetical COVID-19 patients, parameter values for each patient, $\beta ,\gamma ,\delta$ and $V\left(0\right)$, were randomly sampled from the distributions of model parameters summarized in Table S1 which was estimated in our previous study (Iwanami, 2021). The estimation of the distribution was based on the time-course data of SARS-CoV-2 viral load obtained from the upper respiratory tract swab on COVID-19 patient without antiviral treatment by using PCR test which were published in early stages of pandemic (Young, 2020, Zou, 2020, Kim, 2020, Wölfel, 2020). The distribution of time from symptom onset to viral load peak and time from symptom onset to clearance without antiviral treatment calculated by our model and parameters randomly sampled from estimated distributions were showed in Figs. S1 and S2, respectively, and their median were summarized in Table S1. Overall, 20,000 parameter sets were generated, and they were divided into two groups: drug treatment or placebo. We assumed that treatment with the drug or placebo started immediately after trial participation at time $t^{\ast }$ after symptom onset which was also assumed to be concurrent with the hospital visit and enrollment in the study. The times of treatment initiation for each patient, $t^{\ast }$, were assumed to follow a log-normal distribution, $\mathrm{\Lambda }\left(1.23,0.79\right)$, estimated as the distribution of the time from symptom onset to hospitalization in the previous study (Bi, 2020). From the generated model parameters and times of treatment initiation, the individual dynamics of viral load from symptom onset, $V\left(t\right)$, were computed by Eqs. (1), (2) which assumed to be measured by PCR test with a detection limit of 100 copies/mL. Then, the outcomes in simulated randomized controlled trials were calculated to evaluate the sample sizes for candidate drugs.

2.3. Modeling measurement error

Viral load may have random noises due to the errors occurred on measurement process, e.g., sampling and quantification. We added the following measurement error to the viral load calculated from mathematical model, $V\left(t\right)$.

$${\log }_{10}\widehat{V}\left(t\right)={\log }_{10}V\left(t\right)+\zeta ,$$ (3)

where $\widehat{V}\left(t\right)$ is the observed viral load. $\zeta$ is the measurement error which is assumed to follow a normal distribution with mean 0 and variance ${\sigma }^2$. The variance of the error, ${\sigma }^2$, was estimated as the variance of residual errors between the raw viral load data and the simulated data with individual parameters (${\sigma }^2=1.19$ Table S1 and Fig. S3).

2.4. Calculation of the difference in viral load from baseline and sample sizes.

We used the difference in viral load from baseline, defined as the viral load at the time of treatment initiation, calculated using the simulated longitudinal viral load, to evaluate the statistical differences between the placebo and treatment groups. We determined the difference in log scale viral load a few days after treatment initiation, which is given by

$$\begin{array}{cc}{\log }_{10}\left(\frac{\widehat{V}(t^{\ast }+i)}{\widehat{V}\left(t^{\ast }\right)}\right)& \left(i=1,2,\cdots ,7\right)\end{array},$$ (4)

where $t^{\ast }$ is the time of treatment initiation and $i$ is the elapsed days post treatment initiation. Of note, we considered the difference in viral load 1–7 days after treatment initiation. Here, we assumed the detection limit of the viral load to be 100 copies/ml. Then, we calculated the minimum sample size that would cause a significant difference between each pair of placebo and treatment groups using the two-sided Welch’s $t$-test with a significance level of 0.05 and a power of 80 %. Welch’s $t$-test was implemented by the AB_t2n_prop() function in the R package pwrAB.

3. Results

3.1. Planning inclusion/exclusion criteria and measurement schedules to obtain small sample sizes

In previous studies, we found that sample sizes could be reduced in clinical trials by including only patients with a short time from symptom onset and by starting treatment early in randomized controlled trials of antiviral drugs with replication-inhibiting effects against COVID-19 (Iwanami, 2021, Akao, 2021). Using a similar approach of simulating randomized controlled trials, we investigated how inclusion/exclusion criteria affected the sample size of clinical trials for entry inhibitors. In this simulation, we used the change in the log10-transformed viral load compared with viral load at treatment initiation (change in viral load from baseline) as a primary outcome because this value was used as an endpoint in the evaluation of antiviral drugs and is based on viral load (Jayk Bernal, 2022, Hammond, 2022, Yotsuyanagi, et al., 2022). The time from symptom onset to peak viral load and time to PCR negative calculated by our model, Eqs. (1–2) in the absence of treatment are shown and summarized in Fig. S1, S2 and Table S1. In addition to the inclusion/exclusion criteria, the effect of the timing of data measurement on sample size, which may affect longitudinal outcomes, was examined.

We calculated the time-change of the mean ± SD of the change in viral load from baseline, Eq. (4), on each day after treatment initiation in the randomized controlled trials for the entry inhibitor drugs (99 % or 95 % entry inhibition) with different inclusion/exclusion criteria (all patients or patients within 0.5, 1, 2, 3, or 4 days after symptom onset) (Fig. 1 A, Materials and Methods). The changes in viral load from baseline monotonically decreased for the placebo and drug groups in all trials. When simulated trials included only patients soon after symptom onset (within 0.5 or 1 day), the mean viral load in the placebo group at 1 day after treatment initiation was increased compared with baseline because the infection had not reached its peak. Then, the sample sizes required to detect a significant effect of entry inhibitor drugs were calculated and compared (Fig. 1B, Materials and Methods). Because differences in the change in viral load between the placebo and drug groups were larger using the criteria of earlier treatment initiation, the sample sizes were smaller for trials with tight inclusion/exclusion criteria. Interestingly, we found that measuring viral load 2–4 or 3–5 days after treatment initiation reduced the required sample size for drugs with 95 % and 99 % entry inhibition, respectively, because the average of the change in viral load were larger than that measured at earlier and later times and because the viral load increased from baseline in patients enrolled in the placebo group before the viral load reached its peak. The smallest sample sizes were 148 at 2 days and 96 at 3 days after treatment initiation for 95 % and 99 % entry inhibitor drugs, respectively, in the trial including patients within 0.5 days after symptom onset (Fig. 1B and Table S2). Compared with a trial that included all patients, the sample size would be 189–11052 and 1017–30639 patients smaller for 95 % and 99 % entry inhibitor drugs, respectively, with the inclusion/exclusion criteria considered in this study even if the patients included in the trial were included within 4 days after symptom onset.

Fig. 1.

The time-change of viral load and sample sizes in simulated randomized controlled trials for entry inhibitor drugs. (A) The time-change of the mean ± SD of the change in the log10-transformed viral load at each day after treatment initiation in randomized controlled trials for entry inhibitor drugs (95 % or 99 % entry inhibition) with different inclusion/exclusion criteria (all patients or patients within 0.5, 1, 2, 3, or 4 days after symptom onset). (B) The number of samples required to detect significant differences in the change of the log10-transformed viral load between groups treated with placebo or entry inhibitor drugs.

3.2. Sample sizes calculated from different outcomes

In studies that assume the use of a drug that inhibits viral replication, we found that sample size and optimal measurement schedules vary depending on the outcome measures (Iwanami, 2021, Akao, 2021). In addition to the change in viral load from baseline, we calculated the required sample sizes using four other indicators as endpoints: 1) the ratio of patients who tested positive on PCR (PCR positive ratio, Fig. S4); 2) log10-viral load (Fig. S5); 3) time from treatment initiation in the trial to viral load reaching the detection limit for the first time (duration of virus shedding, Fig. S6); and 4) cumulative viral load during the phase of virus shedding (area under the curve [AUC] Fig. S7) (see Supplementary Information for the calculation process). The PCR positive ratio and the log10-viral load were obtained daily from the time of treatment initiation, whereas the duration of virus shedding and the AUC were defined as one value per patient. Sample sizes were calculated for these outcomes but not the duration of virus shedding because the duration might be longer in the entry inhibitor groups compared with the placebo groups (see Discussion).

For all outcomes, the sample sizes were smaller when the inclusion/exclusion criteria included patients soon after symptom onset as observed for the change in viral load from baseline (Fig. 2 and Tables S3–S5). This is explained by the reduced viral load in those receiving early treatment before the infection reached its peak. When the PCR positive ratio was used as an outcome, the sample size was smaller with the measurements at later time points from treatment initiation (i.e., day 4–6) (Fig. 2 and Table S3). Sample sizes immediately after treatment initiation were larger because most patients were PCR negative and there was little difference between the drug and placebo groups (Fig. S4). Sample sizes calculated from the log10-viral load were the smallest earlier after the start of treatment (i.e., day 2–4). Consistently across all outcomes, the sample size was largest when no inclusion/exclusion criteria were established. The sample size was reduced to 0.417 %, 0.303 %, and 2.31 % (95 % entry inhibitor drugs) and 0.138 %, 0.587 %, and 0.418 % (99 % entry inhibitor drugs) at its smallest for PCR positive ratio, log10-viral load, and AUC, respectively, compared with the largest sample size when no inclusion/exclusion criteria were used. These results emphasized that inclusion/exclusion criteria based on days after symptom onset dramatically reduced the sample size calculated from outcomes based on viral load, and that there were optimal measurement schedules depending on the characteristics of the endpoints.

Fig. 2.

Sample sizes in simulated randomized controlled trials for entry inhibitor drugs. The number of samples required to detect significant differences between placebo and entry inhibitor drug groups related to the PCR-positive ratio (left), log10-transformed viral load (center), and AUC from symptom onset to test negative (right) for each entry inhibition drug efficacy (99 % and 95 %) and inclusion/exclusion criteria (all patients or within 0.5, 1, 2, 3, or 4 days after symptom onset).

3.3. Strategies to reduce sample size by considering several outcomes

We investigated the relationship between study design factors, namely the inclusion/exclusion criteria and measurement schedule, and sample size in a randomized controlled trial of an entry inhibitor against COVID-19. Additionally, we asked what outcomes would lead to more efficient clinical trials. If no inclusion/exclusion criteria were used, the trial could be conducted with the smallest sample size of 3051 patients for a drug with 99 % entry inhibition using the changes in viral load from baseline with comparing each day until day 7 after treatment initiation (Table S2). In contrast, when strict inclusion/exclusion criteria (only patients within 0.5 or 1 day after symptom onset) were used, the change in viral load from baseline, log10-transformed viral load, and AUC returned reasonable sample sizes, which were 96–535, 59–298, and 60–155, respectively (Tables S2, S4, and S5). The PCR positive ratio is a useful option if measurements are taken long after the start of treatment (5, 6, or 7 days), because it returned relatively smaller sample sizes than the other outcomes (Table S3). A similar trend was observed for a drug with 95 % entry inhibition. These results suggest that comparisons between outcomes using these sample sizes allow for the efficient design of clinical trials for drug evaluation.

4. Discussion

Attempts to quantify the inter-individual variability in SARS-CoV-2 infection dynamics have been made in studies using various models and data (Goyal et al., 2020, Chatterjee et al., 2022, Néant, et al., 2021, Gonçalves, 2020). These quantitative models have been used to predict the efficacy of antiviral treatment and to investigate the factors affecting individual differences in outcomes in clinical trials as we did in this study. We previously quantified SARS-CoV-2 infection dynamics as four parameters of a minimal mathematical model: rate constant for infection, $\beta$, maximum viral replication rate, $\gamma$, decay rate of infected cells, $\delta$, and initial concentration of viruses, $V\left(0\right)$ (Iwanami, 2021). A good handle on the types of data used to quantify the model and the elements included in the model can yield important clinical implications. Exploring the entire range of parameters in a simple model with inflammation and immune response that return possible clinical values of biomarkers could also explain symptom severity and treatment effects (Sanche, 2022). A recent study explained the COVID-19 vaccine efficacies from phase 3 trials by constructing the shape space of in vitro response of neutralizing antibody against SARS-CoV-2 (Padmanabhan et al., 2022). Interpretation of clinical data using mathematical analysis is expected to provide new insights into treatment and prevention methods to overcome rapidly spreading infectious diseases.

We focused on the relationship between each inclusion/exclusion criterion and sample size to detect significant differences in the efficacy of entry inhibitors as candidate drugs against COVID-19. Using a computational simulation of randomized controlled trials with a mathematical model of virus infection dynamics, we investigated better measurement processes for clinical trials. For sample sizes calculated using the daily change in log10-transformed viral load from treatment initiation, the sample sizes of trials for drugs with 99 % entry inhibition were reduced by 96.7 %–99.5 % and 33.3 %–85.8 % with the most strict or lax inclusion/exclusion criteria (including patients within 0.5 days or 4 days from symptom onset), respectively, compared with trials without inclusion/exclusion criteria. Moreover, we found a 1.61–3.67 times difference in the sample size if the measurement date was changed in a trial with certain inclusion/exclusion criteria. Calculations of the sample sizes were also performed using the PCR-positive ratio, log10-transformed viral load, and AUC during the phase of virus shedding. A comparison of sample sizes indicated optimal outcomes depending on the conditions of the inclusion/exclusion criteria and measurement schedules.

Sample sizes for the change in viral load from baseline, the PCR positive ratio, and the log10-transformed viral load were calculated for the first 7 days after treatment initiation. Because the viral load approaches the detection limit over time, it is expected that differences in the change in viral load from baseline and log10-transformed viral load between drug and placebo groups will become smaller with time, and that the sample size using these outcomes will continue to increase after day 7 of trials. The sample size calculated using the PCR positive ratio was smallest at 4–6 days after treatment initiation. When the PCR positive rate was used to evaluate replication inhibitors, the sample size was smallest around day 7 from the start of treatment (Akao, 2021).

For the actual measurement of the AUC, which is related to the duration of virus shedding, multiple measurements are required to determine the accumulation of the consecutive viral load until a patient has a negative result by PCR. In actual clinical trials, the duration is substituted by a defined period under appropriate rules, such as by performing interval-based tests (e.g., a trial of a Neutralizing Antibody Cocktail (Weinreich, 2021) or by using improvement in clinical symptoms as a criterion. A mathematical modeling study have also shown that variations in the frequency and number of measurements affect the power of detection (Watson, 2022). This problem on the measurement schedule is strongly related to the study design of clinical trials and will require exhaustive simulation taking into account all possible measurement schedules and its clinical validation. Because the sample sizes calculated from the change in viral load from baseline, log10-transformed viral load, and AUC are similar, it is necessary to choose an appropriate outcome based on the cost of conducting a clinical trial, including required personnel and expenses. We used four different measurements calculated directly from viral load to evaluate the drug efficacy in this simulation, but better outcomes may exist for reducing the sample size. Alternatively, it may be possible to create new clinical indexes including features that can efficiently detect significant differences, from commonly used outcomes. For example, by weighting the calculated features and converting them to new features, it might be possible to create features that detect significant differences with even smaller sample sizes. A modeling study showed that clearance rate of viruses estimated by assuming mathematical model is more robust index to detect the difference from antiviral treatment with small sample sizes rather than the time to clearance of viruses (Watson, 2022). This model-based estimate may be a candidate of outcome for clinical trials with reasonable size. However, the choice of outcome may largely depend on its clinical importance and the requirement of society based on the status of the pandemic and the development of other drugs.

Our study design has some mathematical and biological limitations. In our computational simulation, the duration of virus shedding was sometimes greater in the drug treatment groups compared with the placebo groups, and drug efficacy could not be evaluated under the assumption that drugs shortened the duration. One reason for this is that the effect of entry inhibition was assumed to act on two terms in our mathematical model: de novo infection and viral replication, which prolongs infection by maintaining a number of uninfected target cells. This is one limitation of the mathematical model and therefore it needs to be validated with clinical data. In addition, the viral load during treatment was calculated on the assumption that high drug efficacy was maintained, but whether a sufficiently high concentration of drug was achieved in the target tissue immediately after its administration depends on the nature of the drug. Furthermore, depending on the pharmacokinetics of the candidate drug, the concentration may be temporarily reduced to a value that has an insufficient antiviral effect. From the biological aspects, a single-entry inhibitor may not achieve the entry inhibition rate, 95 % and 99 %, which was assumed in our model, because SARS-CoV-2 can enter cells via independent pathways related to two proteases, TMPRSS2 and Cathepsin B/L (Hoffmann et al., 2018, Hoffmann, 2020). A modeling study suggested that combination use of entry inhibitors targeting two different entry pathways may have synergistic effect for the SARS-CoV-2 infection to cells (Padmanabhan et al., 2020).

Our results might depend on the measurement of SARS-CoV-2 viral load. The values of the parameters of the virtual patient population used to evaluate sample sizes were based on the dynamics of the viral load obtained from the upper respiratory tract. Previous studies of SARS-CoV-2 human challenge and infection experiment in non-human primate showed that the time from infection to peak of viral load obtained from throat was shorter than that from nasal (Kim, 2021, Killingley, 2022, Williamson, 2020, Munster, 2020). Whether the viral load peaks after symptom onset in the site where the drug is effective is open to argument. Moreover, our estimation was based on hospitalized symptomatic patients so that our framework for the simulation of randomized controlled trials fails to consider the existence of asymptomatic patients. Some papers showed a comparison of asymptomatic and symptomatic patients may have a difference in the kinetics of SARS-CoV-2 infection (Kissler, 2021). If the symptom onset cannot be defined, as in asymptomatic patients, the viral load at enrollment may be used as an indicator of inclusion/exclusion criteria as an alternative to time since onset. However, it may be easier to use time since symptom onset only for symptomatic patients because it is expected that even when viral load is high, the peak of viral load may have passed in many cases (Fig. S8).

In our model, we considered only that administration of an entry inhibitor inhibits de novo viral infection of uninfected cells. A mathematical modeling study suggested that there is a large difference in the sensitivity of immune responses related to CD8T cell exhaustion due to viral infection between patients with mild and severe cases (Chatterjee et al., 2022). If suppression of viral load with antiviral treatment restores such immune cell activity and leads to a greater rate of elimination of infected cells, viral load may be greatly reduced as our previous mathematical modeling study have shown (Kim, 2021). The importance of this activation of the immune response by antiviral drug treatment has been discussed for treatment of infections by human immunodeficiency virus, hepatitis C virus and others (Conway and Perelson, 2015, Desikan et al., 2020, Baral et al., 2018, Baral et al., 2019). Identifying conditions for efficient drug evaluation based on viral infection dynamics will help to establish new treatments in a pandemic. Rapid drug discovery is required for emerging infectious diseases and mutant strains with different properties over time. Establishing a platform where such mathematical methods can be used in conjunction with infectious disease outbreaks is a key element necessary to achieve a society resilient to infections.

Funding source

This study was supported in part by Grants-in-Aid from JSPS Scientific Research (KAKENHI) B 18KT0018 (to S. Iwami), 18H01139 (to S. Iwami), 16H04845 (to S. Iwami), Scientific Research in Innovative Areas 20H05042 (to S. Iwami); AMED CREST 19gm1310002 (to S. Iwami); AMED Japan Program for Infectious Diseases Research and Infrastructure, 20wm0325007h0001 (to S. Iwami), 20wm0325004s0201 (to S. Iwami), 20wm0325012s0301 (to S. Iwami), 20wm0325015s0301 (to S. Iwami); AMED Research Program on HIV/AIDS 19fk0410023s0101 (to S. Iwami); AMED Research Program on Emerging and re-emerging Infectious Diseases 19fk0108050h0003 (to S. Iwami), 19fk0108156h0001 (to S. Iwami), 20fk0108140s0801 (to S. Iwami) and 20fk0108413s0301 (to S. Iwami); AMED Program for Basic and Clinical Research on Hepatitis 19fk0210036h0502 (to S. Iwami); AMED Program on the Innovative Development and the Application of New Drugs for Hepatitis B 19fk0310114h0103 (to S. Iwami); JST MIRAI (to S. Iwami); Moonshot R&D Grant No JPMJMS2021 (to S. Iwami) and JPMJMS2025 (to S. Iwami); PRESTO Grant Number JPMJPR21R3 (to S. Iwanami); Mitsui Life Social Welfare Foundation (to S. Iwami); Shin-Nihon of Advanced Medical Research (to S. Iwami); Suzuken Memorial Foundation (to S. Iwami); Life Science Foundation of Japan (to S. Iwami); SECOM Science and Technology Foundation (to S. Iwami); The Japan Prize Foundation (to S. Iwami); Daiwa Securities Health Foundation (to S. Iwami). The funders had no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.

CRediT authorship contribution statement

Daiki Tatematsu: Formal analysis, Investigation, Software, Validation, Visualization, Writing – original draft. Marwa Akao: Formal analysis, Investigation, Software, Validation, Visualization. Hyeongki Park: Data curation, Formal analysis, Investigation, Project administration, Software, Validation, Visualization, Writing – original draft. Shingo Iwami: Conceptualization, Funding acquisition, Methodology, Resources, Writing – review & editing. Keisuke Ejima: Conceptualization, Funding acquisition, Methodology, Writing – review & editing. Shoya Iwanami: Conceptualization, Funding acquisition, Investigation, Methodology, Project administration, Software, Supervision, Writing – original draft.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data

The following are the Supplementary data to this article: