Optimising Vaccine Dose in Inoculation against SARS-CoV-2, a Multi-Factor Optimisation Modelling Study to Maximise Vaccine Safety and Efficacy

Developing a vaccine against the global pandemic SARS-CoV-2 is a critical area of active research. Modelling can be used to identify optimal vaccine dosing; maximising vaccine efficacy and safety and minimising cost. We calibrated statistical models to published dose-dependent seroconversion and adverse event data of a recombinant adenovirus type-5 (Ad5) SARS-CoV-2 vaccine given at doses 5.0 × 1010, 1.0 × 1011 and 1.5 × 1011 viral particles. We estimated the optimal dose for three objectives, finding: (A) the minimum dose that may induce herd immunity, (B) the dose that maximises immunogenicity and safety and (C) the dose that maximises immunogenicity and safety whilst minimising cost. Results suggest optimal dose [95% confidence interval] in viral particles per person was (A) 1.3 × 1011 [0.8–7.9 × 1011], (B) 1.5 × 1011 [0.3–5.0 × 1011] and (C) 1.1 × 1011 [0.2–1.5 × 1011]. Optimal dose exceeded 5.0 × 1010 viral particles only if the cost of delivery exceeded £0.65 or cost per 1011 viral particles was less than £6.23. Optimal dose may differ depending on the objectives of developers and policy-makers, but further research is required to improve the accuracy of optimal-dose estimates.


S2. Sensitivity
We attempted to account for uncertainty in the data and models.

S2.1. Distribution of parameters
Available seroconversion data gave the number of individuals per dosing group and the number that was seroconverted. Following a bayesian perspective of the data and a parametric bootstrapping approach, we consider each dosing group as being sampled from a binomial distribution with n = 36 and the unknown true probability of seroconversion p. The likelihood distribution of p was calculated for each of the dosing groups, and we can consider that the true probability of seroconversion follows these likelihood distributions for each group. This was repeated for the adverse event data ( Figure S2).
We sampled from each of these likelihood distributions 5000 times to create 5000 bootstrap dose-response data sets. For each of these data sets, we calibrated a sigmoid curve and recorded MaxResponse, Scale, and Dose 50 for each. This gave a distribution of the values of MaxResponse, Scale, and Dose 50 for the seroconversion that were reasonable giving the observed data. This was repeated for the adverse event data to give a distribution of the values of Scale and Dose 50 for the safety curve ( Figure S3).

Figure S3.
Distribution of model parameters following bootstrapping process with 5000 samples.

The two
Scale parameters appeared to be the least well identified, and MaxResponse appeared well identified. We calculated a non-parametric 95% confidence interval for parameters by finding the 2.5th and 97.5th percentile of the parameter distributions. This is given in the varying range column of

S2.2. Parameter Sensitivity
These parameters define the utility function. To determine the sensitivity of the optimal dose prediction to a parameter, θ , we fix all other parameters at the calibrated/literature derived value and allow θ to vary in the region around it that we just defined [ Table S1].
The optimal dose for each of these varying θ values were calculated and plotted. We did this analysis for both the costless and cost utility functions.

S2.2.1. Costless
The utility function was most sensitive to variance in the Dose 50 (Seroconversion) and Scale (Safety) parameters, but some uncertainty in optimal dose may also be caused by variance in the estimated Scale (Seroconversion) parameter.

S2.3. Optimal dose Confidence Interval
We resampled with replacement from the bootstrap dose-response data calibrated parameter sets. We did this for a combined 10,000 seroconversion/safety parameter sets. For these we calculated the optimal dose as defined by all utility functions. This was used to calculate an approximate 95% confidence interval for the optimal dose of both utility functions.

Figure S5.
Distribution of optimal dose based on herd immunity from parametric bootstrapping of the data.

Figure S6.
Distribution of optimal dose without including cost from parametric bootstrapping of the data.

Figure S7.
Distribution of optimal dose (log10 scale) including cost from parametric bootstrapping of the data.

S3. Population demographics
Population demographics of age, gender, and pre-existing adenovirus neutralising antibody titre as described in the body of work the data were extracted from [15]. S4. Variability in the data.
We note that in plot 2b the data shows that for the three dosing groups (5.0 x10 10 , 1.0 x 10 11 and 1.5 x 10 11 ), 86%, 83%, and 75% of individuals experienced any grade adverse events respectively.
This represents respectively that for each of the three dosing groups of size N=36 (31,30, and 27) individuals of individuals experienced any grade adverse events. There is a qualitative downwards trend, which our strictly-increasing sigmoid model would be unable to model.
We considered this data using whilst taking the interpretation that individuals are independent samples of an underlying Bernoulli process we can calculate the 95% confidence interval on the true probability of experiencing any grade adverse events, using a similar approach to that c) For dose 1.5 x 10 11 ; 75% (58%,88%) As these confidence intervals do overlap, we did not believe that there was sufficient justification to consider the possibility that an increased dose could reduce the number of adverse events experienced, even given the downward trend observed. We believe it more likely that all three data points have approximately similar probabilities of any grade adverse events.
To illustrate this point please consider the below plot, which shows the data described overlaid with the 95% confidence intervals for Bernoulli trials assuming that our underlying model is correct. As all of the points are within these bounds, again this model seems reasonable with the available data. Figure S8. A plot of the expected any-grade adverse event data compared to observed data The black line plots the calibrated curve. The red area plots the 95% confidence interval for the percentage of individuals that would experience any-grade adverse events in a 36-per-group trial assuming that this is the true model (for example, if the true probability of an adverse event for a given dose is 0.5, then approximately 95% of trials of that dose with size 36 would have between 13(=36%) and 23(=63%) individuals experiencing adverse events) However, further investigation into the relationship between dose and proportion of individuals experiencing adverse events would be useful if there was sufficient data.

S5. Threshold analysis, Bivariable
We considered varying both of the Cost Delivery and Cost per 10 11 viral particles parameters in the +/-3 orders of magnitude range at the same time, and found that for high values of Cost Delivery the optimal dose was independent of Cost per 10 11 viral particles ( Figure S6). If these plots are censored to include only points where the predicted optimal dose is less than 10 11 , 5 x 10 11 and 10 10 VP, we find the behaviour observed in Figures S7, S8  Hence, we can suggest that, assuming no uncertainty in the safety and seroconversion related model parameters; If the cost per 10 11 VP is greater than 0.2 times the cost per vaccination that is independent of dose, optimal dose is less than 10 11 VP.
If the cost per 10 11 VP is greater than 1.3 times the cost per vaccination that is independent of dose, the optimal dose is less than 5 x 10 10 VP.
If the cost per 10 11 VP is greater than 17.9 times the cost per vaccination that is independent of dose, the optimal dose is less than 10 10 VP.
In all other cases optimal dose is greater than 10 11 , with the largest recommended dose across all costing parameters was 1.5 x 10 11 VP, which is the dose recommended by the results in objective 2.

Figure S9.
Optimal predicted dose for +/-3 orders of magnitude (log10 scale) around Cost per 10 11 viral particles and Cost Delivery .The left has Cost per 10 11 viral particles at a log10 scale and the right scaled normally. Figure S10. Pairs of Cost per 10 11 viral particles and Cost Delivery for which the optimal predicted dose was less than 10 11 VP. Black line represents the estimated decision boundary.

Figure S11. Pairs of Cost per 10 11 viral particles and
Cost Delivery for which the optimal predicted dose was less than 5 x 10 10 VP. Black line represents the estimated decision boundary. Figure S12. Pairs of Cost per 10 11 viral particles and Cost Delivery for which the optimal predicted dose was less than 10 10 VP. Black line represents the estimated decision boundary.

S6. Weighted Utility Functions
We suggested that there alternate approach to the utility function, where we weight the expected discomfort of a SARS-CoV-2 infection relative to the expected discomfort of receiving a vaccination. This approach requires defining such a weighting, which would require making additional assumptions and introducing complexity that we did not believe added to the main body of this work. Whilst establishing reasonable weightings are beyond the scope of this work, we suggest potential utility functions with pseudo-arbitrary values for the weighting. Hence, whilst these utility functions would not be useful presently for decision-making, if weights could accurately be determined they may be informative. Hence potential weighted utility functions are proposed. We note that the likely method of determining weighting is through a questionnaire of experts and decision makers or through group discussion, as is typical for determining weightings in multi-criteria decision analysis [9].

S6.1. 2:1 Ratio
The utility functions recommended in 3.4 and 3.5 assume that the only desirable outcome of vaccination is seroconversion without experiencing grade 3+ adverse events. This, implicitly, assumes that both seroconverting and avoiding grade 3+ adverse events are equally as desirable. Alternatively, we could consider all outcomes of vaccination with relative weightings of utility. We (pseudo-arbitrarily) choose a 2:1 weighting, where seroconversion is twice as desirable as avoiding a grade 3+ adverse event. The possible outcomes are namely; • Not seroconverting or experiencing grade 3+ adverse events.
The below table indicates the relative 'scores' of each of these outcomes. So defining as the probability of seroconversion, as the probability of no P S P S′ = 1 P − S seroconversion, as the probability of experiencing grade 3+ adverse events, as P A P A′ = 1 P − A the probability of not experiencing grade 3+ adverse events, we have the following utility function.
Below is the dose-utility function for this utility function. We see that under this function and weighting the utility increases with dose, before decreasing. For sufficiently large doses the utility tends to , as the model predicts 100% of individuals experience Score(SA) 1 = seroconversion and grade 3+ adverse events.

S6.2. Expected discomfort
Alternatively, we can look to expressly minimise expected discomfort. We can consider that an individual has two sources of potential discomfort, namely discomfort arising from the vaccination and discomfort arising from the disease that the vaccine aims to prevent or minimise symptoms of.
We consider that for these two sources of potential discomfort, the discomfort could be rated as adverse event' models discussed in the main body of this work. As we have no data to estimate the relationship between dose and the other two outcomes, we assume that the vaccine cannot cause either of these outcomes.
These outcomes are each assigned weights for discomfort, which are not based on literature but represent the idea that critical sickness or death are significantly worse outcomes than mild sickness.
We can define the expected discomfort of contracting SARS-CoV-2 as We can also estimate that an individual has a 65.5% (=0.655) (the herd immunity threshold) probability of contracting SARS-CoV-2 if they are not protected. However, this may be reduced depending on the percentage of individuals in the population that have previously contracted or received a vaccine for SARS-CoV-2 (which could be investigated by considering epidemiological models).
Hence a vaccinated individual is predicted to experience expected discomfort as a function of dose: Where is the probability of not seroconverting and hence not being protected as a P (Dose) S′ function of dose. The plot below shows this relationship.
For these weights, the following behaviour is observed. As the dose increases from 0, the increasing discomfort of vaccination, whilst small, is not justified by the possible reduction in SARS-CoV-2 risk, as there is no meaningful level of seroconversion. For doses at approximately 10 11 , we see a reduction in expected discomfort. At higher doses, whilst seroconversion is probable, the probability of grade 3+ adverse events is large enough that vaccination at this dose may be considered to be less comfortable than the average SARS-CoV-2 infection. Figure S14. Dose-Utility for the expected discomfort utility function. Black dots represent the empirically tested doses.
We can also consider the expected reduction in discomfort from baseline by subtracting the dose-dependent expected discomfort from the zero-dose expected discomfort. Further, we can consider only the doses where the discomfort reduction is predicted to be greater than 0. Hence, by dividing by the 'dose-cost' model found in the main body of this work we can also estimate the expected reduction in discomfort per GBP spent on the vaccine at each dosing level. Figure S16. Baseline subtracted Dose-Utility for the expected discomfort utility function, divided by cost and censored if discomfort reduction is less than 0. Black dots represent the empirically tested doses.