Quantifying Efficiency Gains of Innovative Designs of Two-Arm Vaccine Trials for COVID-19 Using an Epidemic Simulation Model

Clinical trials of a vaccine during an epidemic face particular challenges, such as the pressure to identify an effective vaccine quickly to control the epidemic, and the effect that time-space-varying infection incidence has on the power of a trial. We illustrate how the operating characteristics of different trial design elements maybe evaluated using a network epidemic and trial simulation model, based on COVID-19 and individually randomized two-arm trials with a binary outcome. We show that “ring” recruitment strategies, prioritizing participants at an imminent risk of infection, can result in substantial improvement in terms of power in the model we present. In addition, we introduce a novel method to make more efficient use of the data from the earliest cases of infection observed in the trial, whose infection may have been too early to be vaccine-preventable. Finally, we compare several methods of response-adaptive randomization (RAR), discussing their advantages and disadvantages in the context of our model and identifying particular adaptation strategies that preserve power and estimation properties, while slightly reducing the number of infections, given an effective vaccine.

The network operates at the level of the individuals. Each individual belongs to a household H h and every individual in a household is connected to every other individual in the household, such that the induced subgraph G 1 [H h ] is completely connected: Let H = {H h , h = 1, ..., N H } be the set of N H households, which forms a partition of the vertices V (all subsets are mutually disjoint and their union is equal to the set). Let a 1 (i) be the household of individual i, i.e. a 1 (i) = h ⇔ i ∈ V[H h ]. Then the size of a household is We start with N H = 500 households with O(H h ) people in each. The age and number distribution follows (Office for National Statistics, 2011). That is, each person has an age attribute, a 2 (i) ∈ {1, 2, 3} where a 2 (i) = 1 if person i is under 19, a 2 (i) = 2 if person i is 19 to 65, and a 2 (i) = 3 if person i is over 65.
There are A 3 = i 1 a 3 (i)=1 ≈ 731 people in the workforce. We define A 3 /15 ≈ 49 workplaces to which people with a 3 (i) = 1 (that is, all those aged 19 to 65, and 1/5 of people over 65) are assigned following a multinomial distribution. We choose to place on average 15 people in a "workplace", W w , which represents not their employment structure but a close shared use of the infrastructure. A useful guide might be how many toilets per person a place of work should have.
The workplaces are completely connected: Finally, 1000 totally random edges are added, which amounts to around ten per person: .
These edges correspond to potential transmission encounters that would not be recalled or anticipated through contact tracing. The result is an average of 20 connections per person, of which ten have a weight of 1 and ten have a weight of 0.1.

A.1.2. Parametrisation
The parameters we have used and their provenance are listed in Table 1. A susceptible individual i belonging to arm x becomes "exposed" (i.e. transitions to state E) with rate where h refers to household, w to workplace, and n to random. χ h > 0 is the edge weight for household contacts, χ w > 0 is the edge weight for workplace contacts, and χ n > 0 is the edge weight for random contacts. One could specify a different χ h for pre-and post-symptomatic infectiousness, or even a changing profile over time, as described in He et al. (2020). x i = 1 if person i is vaccinated and 0 otherwise. That is, we assume that the vaccine prevents infection, and does not change infectiousness or the likelihood of asymptomatic disease among those who become infected after vaccination or a prophylactic effect among those who were infected before. η ≤ 1 is the vaccine efficacy, and β > 0 is the per-contact rate at which infectious people infect their susceptible contacts is similarly defined as the number of contacts who are infectious but not symptomatic.
Note that there is no infection other than from a contact (with the exception of the simulated "time trends", Figure 2). Thus any infection within a contact network came (directly or indirectly) from the contact network's index case.
Our model and parameter choices yield new infection events as occurring from pre-symptomatic people 58% of the time and from symptomatic people 42% of the time, omitting transmissions from people who never become symptomatic. In comparison to reported estimates (44% from 77 recorded transmission pairs (He et al., 2020), and 48 and 62% in Singapore and Tianjin, respectively (Tapiwa et al., 2020), we confirm that our simulation scenario is consistent with one with quick quarantine of close contacts (He et al., 2020).

A.2.2. Trial rules
In our trial we define contact tracing as identifying only existing relationships ("acquaintances", or "those met before", in the terminology of Kucharski et al. (2020)) that might be, or might have been, a means for transmission. That is, a newly diagnosed person is asked to recall all of their contacts, and these individuals are contacted and asked likewise to list all their contacts. This makes our simulation and trial design most like the "self-isolation and manual contact tracing of acquaintances" of Kucharski et al. (2020). The relationships that are of interest will depend on the society and any concurrent actions, guidance or instruction from the state, which in our simulation include home and workplace contacts, while random contacts remain unknown. We assume unknown relationships are not recalled in contact tracing and, similarly, cannot be anticipated for recruitment of trial participants. We assume work and home relationships are recalled perfectly. By recruiting "contacts" and "contacts of contacts" into the trial, we are recruiting the people the index case lives with and their colleagues, and those the case works with and their housemates.
For our purposes, an index case for a contact network is a person identified as being in state I S,· after the initiation of the trial. Eligible people are traced as described in European Centre for Disease Prevention and Control (2020) and those who consent are enrolled as soon as they are identified and give their consent.
It takes on average ten days from symptom onset to enrol a contact network, which includes the time to report, the time to contact trace, and the time to enrol (Henao-Restrepo et al., 2017). Susceptible and exposed people are eligible for enrolment if they are a "known" contact of the index case and they are not already enrolled in the trial. Symptomatic I S,U people are excluded on the basis of their symptoms. R U are excluded on the basis of their history, which we assume a perfect knowledge of in our simulation. This could result either from people being able to identify having had COVID-19, or from there being an accurate and reliable antibody test. Inclusion of R U people in the simulated trial would result in a dilution of infections and would therefore require enrolment of more participants to maintain power. Their inclusion is advocated for reasons of safety testing.
The enrolment rate is 0 ≤ ≤ 1, which is the probability for each eligible person to enrol, where we use only contact structure and lack of symptoms to define eligibility. We use = 0.7. 2 Enrolled participants are randomised to the experimental arm with probability 0 ≤ π 1 ≤ 1, and π 1 + π 0 = 1. Enrolment following symp-tom onset for the index case takes time α, which we describe following observed enrollment times (Henao-Restrepo et al., 2017). Alternatives include a Poisson distribution with parameter ∼ Uniform(1.5, 2.5) (Fyles et al., 2021). Ideally, this parameter would be determined according to observed contact-tracing efforts and the expected capacity of the trial team.
Those vaccinated have an additional wait time before reaching state S V , which is development of immunity (or time to seroconversion or detectable antibodies), and which takes time τ . Transition from S U to E U is possible in this wait time. We assume that the vaccine effect before seroconversion is zero and that the vaccine effect after is the full effect of the vaccine.
Individuals who are in state E and are unenrolled are enrolled with the same probability ( ) as the susceptibles S U , as they are asymptomatic. The same wait time for enrolment applies, but time to seroconversion does not, as the individual is already infected. If the participant transitions to I S,U before the recruitment time elapses, they will be excluded from the trial, as they will be showing symptoms and can be tested for confirmation.
Result accrual relies on surveillance and self reporting. For our simulations we assume that a fraction 1 − δ = 0.2 of infectious individuals are asymptomatic. These infection events go unreported in the trial results (but still contribute to onward transmission). Our simulation mirrors the rules given to the participants: that upon the onset of symptoms, they should stay at home, and notify the trial team, who will organise the PCR test. In this way, the day of symptom onset is recorded (to be confirmed by PCR). We assume that this information will be available at the end of the follow-up period (25 days). This means that we assume that a person who becomes symptomatic on day 25 is tested and confirmed on the same day. In our simulation we assume that the rules are followed without error.
We simulate one network at a time, beginning when the index case is identified. Transmission in each network is independent of all other networks in the trial. The networks are related only through the time reference, in that one contact network is initiated on each day.
Finally, we assume that randomisation, enrolment and vaccination all happen on a single day for each individual, although the day will differ between individuals.

B. Analyses, and exclusion criterion implemented at analysis points
Here we detail all equations to accompany Section 2.4, which describes the different ways we explore to analyse the outcome. We present the methods in the same order and use a single framework that describes all the methods in the same way.

B.1. Analysis of raw data
We have j = 1, ..., N I individuals. Each has a vaccination status, x j , and a disease status y j , where the vaccination status is dictated by the trial design and the disease status from the underlying epidemic model: x j = 0 person j not vaccinated 1 person j vaccinated and y j = 0 person j not diagnosed 1 person j diagnosed at the end of the trial.
The test statistic is a standard normal variable Z, where p v is the true probability of not being a confirmed case if in arm v, andp v is our estimate of it, defined as the proportion of people in arm v not confirmed, where v = 0 is the control arm and v = 1 is the experimental arm; N v is the total number in arm v; and σ v is the variance of the estimatorp v . Power is defined as the proportion of Z values that exceed 1.64, which is the 95th quantile of a standard normal distribution. ω j is the weight of person j which, for the unweighted method, is 1 for all participants. In the descriptions that follow, we see that the weights ω define the retrospective exclusion criterion so that all methods use the same calculation and each is defined only by the definition of the weights.

B.2. Analysis using binary weighting
The binary-weighting method proceeds as above but considers also the day of commencement of symptoms, s j . Individuals are excluded if s j < 9 relative to a randomisation day of 0. We write this as the weight, ω j , for each individual j, so that a weight of 0 equates to exclusion: ω j = 0 s j < 9 1 s j ≥ 9 or y j = 0 These are used together with Equations 1, 2, 3, and 4 as before.

B.3. Analysis using continuous weighting
Given person j's symptoms began on day s j relative to their randomisation day of 0, τ and ξ are their unknown time to seroconversion 3 and incubation time, respectively. The probability they were infected after the trial began is P (τ + ξ < s j ). We assume τ and ξ are distributed between individuals as Γ(shape = 3, rate = 1) and 2+Γ(shape = 13.3, rate = 4.16). Hence the distribution of τ +ξ is estimated by matching moments using the Welch-Satterthwaite equation, as described in Box (1954).
We estimate the vaccine efficacy 0 ≤η ≤ 1 aŝ Suppose from the cumulative distribution function of the gamma distribution we have a nominal probability, i.e. neglecting the effect of the vaccine, q j that the day person j was infected, d j , was after seroconversion on day D c . We write the complement, the probability that person j was infected before day D c , as r j = 1 − q j . We re-estimate their probability given that person j was vaccinated (x j = 1): because, if there is some efficacy, then they are more likely to have been infected before being vaccinated than after (relative to a vaccine that has no effect: Pr(d j > D c |x j = 0) = q j ). We solve this iteratively forη with reference to all observations j. 4 Then See Figure 1 for an illustration of the relationship between weight and vaccine effect. The resulting weights are used in Equations 1, 2, 3, and 4 as before.

Determining the inclusion weight for a vaccinated person
The inclusion weight for a vaccinated person j (q j = 1), whose symptoms began after D c (randomisation date plus an assumed time from vaccination to seroconversion) is the probability that they were infected after D c . To determine this, first we make explicit the condition that their infection date d j is less than their symptom date D s . The conditional probability we want can then be decomposed into probabilities unconditional on the symptom date, as follows, Then splitting the denominator into the probabilities of being infected in two different periods (before versus after D c ) gives Since the probability of being infected before D c doesn't depend on whether person j was vaccinated, P (d j < D c |q j = 1) = P (d j < D c |q j = 0). We additionally  Figure 1: The probability that a person became infected on day x, given the day they showed symptoms, whether or not they were vaccinated, and the effect of the vaccine (where we omit vaccine-induced antibody response for simplicity of presentation). The inclusion weight given the day of symptom onset is the area under curve to the left of the randomisation day. The lines illustrate two participants who show symptoms on the same day. One (turquoise) is vaccinated; the other (navy) is not. Given the distribution of the incubation period, the navy person has a probability of 0.83 of having been infected after randomisation day. Their weight is therefore 0.83. Given that the vaccine efficacy estimate exceeds zero, the turquoise person is less than 83% likely to have been infected after randomisation day -67%, for an estimated vaccine efficacy of 0.6. In the binary-weighting case, both these participants have weight 0 (if day of symptom onset is less than the threshold of nine days) or weight 1 (otherwise).
assume that where ψ is the relative risk of infection between a vaccinated and unvaccinated person, assumed to be constant through time, giving Expressing the right hand side in terms of probabilities conditional on symptoms, by dividing the numerator and denominator by P (d j < D s |q j = 0), then gives the weight for a vaccinated person j as

C.1. Dependence of ring recruitment on contact tracing
As ring recruitment relies on contact tracing, here we present a sensitivity analysis that demonstrates how the success of the method depends on the "known fraction". We define the "known fraction" of a network as r s = e s w s /(e s w s + e u w u ), the weighted sum of known edges over the total weighted sum of edges, where there are e s known and e u unknown edges, with relationship weights w s and w u respectively. The net result is that the known fraction is the proportion of new cases that are in the contact networks of recent cases. Therefore the known fraction is entirely determined by how "contact tracing" is implemented: any contact that is not traced is by definition transient. The known fraction is 0.91 for our standard network (see Section A).
In Table 2 we consider three possible values for the known fraction r s by holding the edges fixed and adjusting the relationship weights. To show power as a function of the known fraction, we fix the trial size to be 100 contact networks of participants. The results show the extent to which power diminishes as the fraction of known transmissions decreases. As contacts become more "known" (thus more contacts are recruited into the trial), the number of confirmed cases within the trial increases, especially in the "null case" where there is no vaccine effect (VE=0). This is a result of the recruitment method better targeting those at risk.
Where it is possible to predict who is at imminent risk of infection, ring recruitment designs are advantageous compared to random recruitment. We expect to see a high known fraction r s for diseases whose transmission depends on exchange of or exposure to bodily fluids, such as EVD, as well as for COVID-19 in societies under "lockdown", where public spaces are closed and people stay at home, and/or there is extensive quarantining. We expect that the more restrictions to movement and activity there are, the greater the fraction of known transmission events.  Figure 2 shows the robustness of one adaptive trial design to a time trend, demonstrating the effect of the correction of Simon and Simon (2011). We choose to illustrate with linear trends rather than something more realistic, undulating, or stochastic in order to stress test the method. In addition, we choose a trend that we would expect to most favour the Thompson sampling methods. Indeed, we find that uncorrected Thompson sampling sees an increase in type 1 error, which is corrected by the resampling method of Simon and Simon (2011). As TS is a more "aggressive" response-adaptive method than TST, we expect it to be more susceptible to time trends, and therefore the loss in power following correction to be exaggerated.   Simon and Simon (2011). Type 1 error rate and power as a function of the trend in the background rate for a trial with a response-adaptive randomisation rate. "Background rate" can be interpreted as the rate of infection by individuals unknown in the context of the trial (e.g. source population or unknown contact). We illustrate with linear trends rather than something more realistic, undulating, or stochastic in order to stress test the method. In addition, we choose a trend that we would expect to most favour the Thompson sampling methods. As TS is a more "aggressive" responseadaptive method than TST, we expect it to be more susceptible to time trends, and therefore the loss in power following correction to be exaggerated. Left: Five different time trends for the background rate. The trend is that background rates diminish over time to zero. Middle: The gradient of the trend is shown on the x axis. On the y axis is the type 1 error rate. Right: The gradient of the trend is shown on the x axis. On the y axis is the power.  Table 3.

C.4. Thompson sampling methods without early stopping
In Table 3, we show the Thompson sampling methods again, where we do not conclude the trial early for efficacy. This results in much longer trials: on average 6 and 17 days longer (for TST and TS, respectively) than the trials that do stop early, and much harsher power corrections due to the extended periods spent at unequal randomisation, in comparison to Main Text Table 3.

C.5. Fixed-duration trials
We additionally compare the response-adaptive trials when there is a strict limit on their duration and there exists an efficacious vaccine, for example for a trial that lasts at most 85 days, and we evaluate the trial impact in terms of total people vaccinated up to 65 days after the trial's end. First we consider that the same number of people (N P = 32) can be vaccinated per day, and then that ten times as many (N P = 320) can be. Table 4 compares the designs after the total duration of 150 days, where the trial has a fixed duration of at most 85 days, and the number of people vaccinated in the remaining time depends on the probability that the trial successfully identified that the vaccine was efficacious, and the number of people that can be vaccinated per day post-trial. With limited time, the trials that allocate most participants to the efficacious experimental arm -TS and TST -would be preferable in terms of vaccinating the most people during the trial, and up to day 150, if the rate of vaccination after the trial is the same as the rate of enrolment into the trial. These methods also see fewer exported infection events in the course of the trial -both in total and per day. However, if, after the trial, ten times as many people can be vaccinated per day as were enrolled in the trial, the designs with higher powers are preferable, as they are better able to identify an efficacious vaccine, and therefore will vaccinate more people in the period after the trial.  Table 4: Comparison of response-adaptive trials that last at most 85 days. We compare their profiles in terms of the totals vaccinated up to day 150, which includes a maximum of 85 days of vaccinations as part of the trial and a minimum of 65 days of vaccination roll-out given a successful trial. "Vaccinated in trial" is the expected number vaccinated during the trial, and "Vaccinated up to day 150: assuming 32 (320) per day" is the expected number vaccinated after the trial, and up to day 150, estimated as 32 (320) people vaccinated per day, for the days remaining after the end of the trial, multiplied by the power (the probability to have concluded efficacy and rolled out the vaccine -bold text indicates whether uncorrected or drift-corrected power is used). Standard deviations for 10,000 simulations in brackets. Correction for time trend uses the resampling method of Simon and Simon (2011