Fig 1.
Simple example calculation of overall protection conferred by a transmissible vs. a traditional vaccine.
First, clusters are vaccinated with either a transmissible or traditional vaccine. In the second stage, the transmissible is allowed to circulate for all animals. Finally, an epidemic occurs, and the number of infected animals of the cluster, among both initially vaccinated and initially unvaccinated animals, is counted. We assume there is no interference between clusters, i.e., The condition of vaccination or infection of an animal of one cluster affects either the condition of vaccination or probability of infection in another cluster. In this simple example 10% of each cluster is initially vaccinated for illustrative purposes, whereas in our Results 5% of each cluster is initially vaccinated.
Fig 2.
Differences in cumulative infection burden depend on whether the trial is anticipatory or reactionary, and if the contact structure is overdispersed.
Left panel: anticipatory trial with no overdispersion in contact structure. Middle panel: reactionary trial, where vaccination starts once 0.5% of the population has been infected, with no overdispersion in contact structure. Right panel: anticipatory trial with overdispersion in contact stricture (k=1). Observation period is 150 days. Parameters set at: R0,v = 1.1, R0,w = 2, proportion initially recovered = 0, vaccine efficacy = 80%, initial vaccinated proportion, α = 5%.
Table 1.
Estimated protection or risk difference from and required sample sizes, NT*, for trials comparing a transmissible vaccine to a traditional vaccine. Prior to the outbreak, α%=5% of each cluster is vaccinated with either a transmissible or traditional, non-transmissible vaccine. The contact structure is Poisson distributed and trials are anticipatory. The estimand of interest is noted in the first column: Δ1(5) refers to the overall protection, Δ3(5) refers to the spillover protection, θ1(5) refers to the difference in risk of infection for indirectly vaccinated animals in the transmissible case compared to those unvaccinated in the traditional case, and θ2(5) refers to the difference in risk of infection for never vaccinated animals in the transmissible case. Vax. Eff. is the true direct efficacy of the vaccine to decrease susceptibility of vaccinated animals. R0,v refers to the R0 of the vaccine. Estimate protection or risk difference is the mean estimate of the noted estimand comparing transmissible vaccines to traditional vaccines. <.> indicates the mean 95% confidence interval width. Req. sample size from simulation, NT*, is the number of clusters and number of animals required to estimate the effect in order to achieve at least 80% power. Power is the percentage of simulations with a p-value ≤ 5% across 1000 trial simulations. Type I error is the percentage of simulations with a p-value ≤ 5% across 3000 trial simulations for this sample size with a traditional vaccine.
Table 2.
Estimated overall protection or risk difference from and required sample sizes, NT*, for trials comparing a transmissible vaccine to a traditional vaccine when clusters have overdispersed contact structures. Prior to the outbreak, α%=5% of each cluster is vaccinated with either a transmissible or traditional, non-transmissible vaccine. The contact structure is moderately overdispersed (k=1) and trials are anticipatory. The estimand of interest is noted in the first column: Δ1(5) refers to the overall protection, Δ3(5) refers to the spillover protection, θ1(5) refers to the difference in risk of infection for indirectly vaccinated animals in the transmissible case compared to those unvaccinated in the traditional case, and θ2(5) refers to the difference in risk of infection for never vaccinated animals in the transmissible case. Vax. Eff. is the true direct efficacy of the vaccine to decrease susceptibility of vaccinated animals. R0,v refers to the R0 of the vaccine. Estimate Protection or risk difference is the mean estimate of the noted estimand comparing transmissible vaccines to traditional vaccines. <.> indicates the mean 95% confidence interval width. Req. sample size from simulation, NT* is the number of clusters and number of animals required to estimate the effect in order to achieve at least 80% power. Power is the percentage of simulations with a p-value ≤ 5% across 1000 trial simulations. Type I error is the percentage of simulations with a p-value ≤ 5% across 3000 trial simulations for this sample size with a traditional vaccine.
Table 3.
Estimated overall protection from and required sample sizes, NT* for reactionary trials comparing a transmissible vaccine to a traditional vaccine. When 0.5% of the cluster is infected, α%=5% of each cluster is vaccinated with either a transmissible or traditional, non-transmissible vaccine. The contact structure is Poisson distributed. Δ1(5) is the overall protection. Vax. Eff. is the true direct efficacy of the vaccine to decrease susceptibility of vaccinated animals. R0,v refers to the R0 of the vaccine. Estimate protection is the mean estimate of the overall protection conferred by transmissible vaccines compared to traditional vaccines. <.> indicates the mean 95% confidence interval width. Req. sample size from simulation, NT*, is the number of clusters and number of animals required to estimate the effect in order to achieve at least 80% power. Power is the percentage of simulations with a p-value ≤ 5% across 1000 trial simulations. Type I error is the percentage of simulations with a p-value ≤ 5% across 3000 trial simulations for this sample size with a traditional vaccine.
Table 4.
Estimated overall protection from and required sample sizes, NT*, for anticipatory trials comparing a transmissible vaccine to a pure control group with no vaccination. Clusters are either vaccinated at a rate of α%=5% prior to the outbreak with a transmissible vaccine or do not receive any vaccination. The contact structure is Poisson distributed and trials are anticipatory. Δ1(5) is the overall protection. Vax. Eff. is the true direct efficacy of the vaccine to decrease susceptibility of vaccinated animals. R0,v refers to the R0 of the vaccine. Estimate protection is the mean estimate of the overall protection conferred by transmissible vaccines compared to traditional vaccines. <.> indicates the mean 95% confidence interval width. Req. sample size from simulation, NT*, is the number of clusters and number of animals required to estimate the effect in order to achieve at least 80% power. Power is the percentage of simulations with a p-value ≤ 5% across 1000 trial simulations. Type I error is the percentage of simulations with a p-value ≤ 5% across 3000 trial simulations for this sample size with a traditional vaccine.
Fig 3.
Required sample sizes, NT*, for overall protection estimand (Δ1(5)) will depend on the transmissible vaccine’s protective strength, as well as on the trial design.
Our approximate formula reveals that as the R0 of the vaccine increases (left panel) or number sampled in each cluster, s, increases (middle panel), NT* for trials that measure the overall protection that transmissible vaccines confer relative to traditional vaccines will decrease. NT* will increase as the between-cluster variance in the final size proportion increases. All other parameters are set at R0,v = 1.1, number sampled from each cluster, s = 100, between cluster variance = 0.01, R0,w = 2, proportion initially recovered = 0, vaccine efficacy = 80%, initial vaccinated proportion, α = 5%.