Suppose that the recipient acquires virions from the donor in the transmission event, a mixture of strain and strain . For completeness (and we will return to this in the Discussion), these may be clumped in multiple infectious entities or separate. If clumped, then each infectious entity may itself be a mixture of strain and strain virions. In any case, a straightforward calculation shows that the expected (mean) composition of this load immediately upon infection is as follows:

Lodgement-site specific factors within the host (independent of strain ( or )) will prevent all but a proportion of the virions from invading a susceptible cell. Of those that do initiate replication, the expected proportion of strain that survive stochastic extinction is , and similarly for strain [15].

The surviving progeny are then assumed to undergo exponential growth. We neglect mutation from one strain to another. Some subtleties in the process just described are explored in the Discussion. Thus after units of generation-time the number of strain virus particles is:(1)and likewise for the number of strain particles. Thus, after some algebra, the proportion of particles being strain after units of generation-time is:(2)where(3)We will usually write for (and similarly ) unless we are explicitly considering the generation-time in our analysis. We identify (the ordinate in Figure 1) with .

If then . Further, is an increasing function of and a decreasing function of , so that if then , and vice versa. Moreover, if strain is less secreted than strain then we have . Overall then, having means that strain would outgrow strain in the recipient, given an even mixture in the donor.

We write as a shape parameter and introduce the functional form:(4)where .

The shape parameter, , may be interpreted as an overall relative viral fitness of strain compared to strain , accounting for three factors: (1) secretion from the donor, (2) the initial extinction probability in the recipient and (3) subsequent growth in the recipient.

For example, consider the case in which a drug-resistant mutant (strain ) is seeded within a population where the wild-type (strain ) is currently circulating. The strains are antigenically similar (and assumed identical for the sake of this argument) removing any possibility of immunologic selection. In the absense of any drug-selective pressure, if is positive then the wild-type has an advantage over the drug-resistant mutant and so we expect it to continue to dominate the epidemiology. Conversely, if is negative, then the mutant has a fitness advantage and over time would be expected to replace the wild-type strain in the population.

We have constructed from two components, and , for mathematical convenience and the form has allowed us to interpret the meaning of and . We also find it convenient to decompose the overall measure of relative fitness, , another way. From the three biological factors identified above (labelled (1), (2) and (3)), we consider two components: a within-host “replication fitness” (3) captured by the exponential term in and a relative “transmission fitness” of strain compared to strain , accounting for secretion (1) and extinction (2) given by:(5)For completeness, although not used herein, we write . Figure 2 shows four representative curves for the mean behaviour expected from the model, for the simplifying case , in which drops out of the model and .

If we are willing to assume that there is no measurement error for the donor and recipient strain proportions – that is, that the only factor leading to the observed deviation of the recipients' proportions from the model predictions is the inherent stochastic nature of the transmission process – then we can make an estimate for .

The usual estimation quantity for a binomial process is the probability of success, given the number of draws. Here, we are interested in the converse: the number of transmitted virions (equivalently, “draws”), given information on the probability of infection per virion (equivalently, “success”).

We consider transmission events, each defined by a tuple , , plotted as in Figure 1. First, we determine the best fit model (equation 4), with corresponding estimate, , for the shape parameter, and calculate the residual sum of squares, , for the data from the model:(6)

Having determined the RSS of the data given the model, , we now ask: What inoculum size, , when used to generate synthetic data from the best-fit model with shape parameter , gives an RSS closest to ?

As a function of (with up to , with some sufficiently large integer), we simulate the random draw of particles of strain from , where is identified as the probability of selecting a strain virion from the transmitted virions. We obtain a simulated data set , with .

We calculate the residual sum of squares, , for the simulated data (compared to the model) as a function of :(7)We then choose the minimum difference in RSS as our estimate for :(8)

Finally, we repeat, scanning over for many generated synthetic data sets , on each occasion selecting that with minimum difference in RSS as our estimate . We obtain an empirical distribution of best fit _s, from which we can calculate an average, , and variance.

The strain (mutant) proportion as measured using the rRT-PCR assay is accurate to within 1–3 percentage points across experimental repeats [8]. Furthermore, per assay, the result is extremely accurate as the following analysis reveals. Per cycle in the rRT-PCR assay, the amount of product doubles (prior to saturation). The cycle-threshold (Ct) is defined as the number of cycles required for the product (marked by a fluorescent signal) to exceed a specified threshold. A lower Ct value indicates a larger quantity of product in the sample as fewer doubling cycles are required to reach the threshold. The strain proportion is determined from the differences in Ct values for the strain and strain strains as follows:(9)The Ct values are measured to two decimal places in the experiments reported by Hurt et. al. [8]. A simple numerical calculation probing the worst-case scenario (attempting to maximise (or minimise) the difference ) shows that with this accuracy on the recording of the Ct values, the mutant proportion as calculated using this formula and reported by Hurt et. al. [8] would be unchanged (at two significant figures for the proportion) in almost all cases. A few of Hurt et. al. 's reported values could, in the worse case, fluctuate by just 1 percentage point.

In summary, based on repeatability of rRT-PCR assay, we may expect a small () percentage point error to be introduced.

The second issue proves more interesting. With a daily sampling window for swabbing the animals, we potentially introduce a systematic bias into the reported paired transmission results. Figure 3A represents a donor (“Donor”) infected at day and infecting a recipient ferret during the time window . We show two different – that is non-concurrent and non-interacting – hypothetical transmission events to a recipient, termed either “Recipient T1” or “Recipient T2” as appropriate. All swabs for all animals are taken at days . A transmission event occurring in the time window will be recorded as or for Recipient T1 or T2 respectively, and by construction, both of these pair-values will be the same. However, as shown in Figure 3, Recipient T1 was infected just after time point while Recipient T2 was infected just prior to time point . Figure 3B plots a square at location as recorded in the experiment. Assuming that the within-host transmission fitness of the mutant is less than that of the wild-type (), and that stochastic processes do not overwhelm this systematic effect of a reduced replication rate for the mutant, the arrows (RT1 for Recipient T1 and RT2 for Recipient T2) show the direction in which the true transmission pair would lie. We argue as follows:

The diagonal arrow shows the “correction-direction” for an infection occurring at an arbitrary infection time during the time window for the case when . Note that if the mutant were more fit than the wild-type (), the bias (arrows) would be in the opposite direction. For equally fit viruses, there is no bias introduced due to the measurement period.

The preceding argument assumed that the within-host dynamics proceeded in a systematic noise-free way, with the measurement for or over time being well behaved and well predicted by the difference between and . If, however, stochastic processes overwhelm the systematic within-host differences between strains, then the potential bias introduced due to the sampling may not be evident in the data. The revised mutant proportion (in either the donor or recipient) may be either larger or smaller than the reported value. In this case examination of the experimental data on a case-by-case data is necessary and then only “Recipient T2” type events may be reliably assessed as without data at time point for a “Recipient T1” event no revised estimate for is available.

The third issue arising from the experimental design – and one which is easily avoidable in future experiments – is that we do not know which donor ferret was responsible for the transmission event to the first recipient. The mutant proportion for the donor shown in Figure 1 is the average of the mutant proportions in the two donor ferrets at the sample time directly proceeding confirmation of infection in the recipient. As it may be that the mutant proportion as measured each day in any given ferret is subject to substantial fluctuations, and that each donor ferret has a unique experience, there is significant potential for this source of error to be large.

For the five relevant transmission events in the H274Y experiment we have the following donor pairs (not pairs) at the time of transmission: (0.00,0.00), (0.09,0.09), (0.38,0.31), (0.56,0.64) and (0.96,0.94). In this small sample, we have up to a 4 percentage point difference between the average across donors and individual recorded values.

In general, there is scope for significant uncertainty due to multiple potential sources of infection and the statistical inference should take this into account.

We now account for the three sources of error discussed in Section Effects due to data ascertainment limitations. Firstly, the rRT-PCR assay introduces a small (1–3 percentage point) uncertainty into our data (both and ). Secondly, as reported on in detail by Hurt et. al., data were collected at daily intervals from the ferrets [8]. An examination of that data indicates that stochastic, rather than systematic, effects dominate. The mutant (strain ) proportion measured at adjoining time points shows significant variation over and above the systematic reduction expected from the marginally reduced within-host fitness of the mutant. In consequence, while we cannot make an assessment of the “Recipient T1” error (see below), we can still make a direct estimation of the “Recipient T2” error from the daily mutant proportion data. We estimate (in a semi-quantitative way) that we may introduce up to a 10–15 percentage point error in due to this stochastic variation, but it may be far less for any individual data point. Conservatively, in the absence of information to inform the “Recipient T1” error we allow for an additional 10 percentage point error on . Finally, we can also assess the possible modifications to model based inferences due to averaging across the two donor ferrets for the first transmission event in each experiment. We estimate that the variation due to this is of the order of 5–10 percentage points, again an influence on and variable from data point to data point. The data in [8] allows us to make a direct estimate of this on a transmission-event by transmission-event basis.

Listing the transmission events shown in Figure 4A from left to right on the -axis, Table 1 shows our estimate for the plausible range for that tuple's and values. For clarity, we list all entries as percentage points, rather than proportions.

To make an assessment of the influence that these uncertainties may have in estimating the relative transmission fitness of strain , we take a simulation approach. We generate 1,000 equivalent synthetic data sets in which, for each of the 10 observed transmission events, and for the tuple are drawn independently and randomly from the uniform distribution over the range identified in Table 1. For each of the 1,000 synthetic data sets we fit the model, recording the best fit shape parameter . Reporting the relative transmission fitness for convenience, we recover (0.51, 1.73), where the 95% confidence interval is empirically determined from the distribution of recovered best fit values for over the 1,000 simulations. Similarly, to determine the number of transmitted virions, , we apply equations 6–8 to each synthetic realisation, and combine the empirical distributions, obtaining a mean of 4.3 and variance of 9.8. Figure 5 shows the results.

Comparing these results to those in Figure 4, the key difference is an extension in the tail of the distribution for the number of virions transmitted. Over the 1,000 samples of the synthetic data, some will fall closer to the model curve, and so the residual variation, attributed to the stochastic nature of the transmission process, will be smaller. The consequence is a larger estimate for . A minority of synthetic data sets based on the distributions for and as shown in Table 1 have this reduced residual variance, while the mean is almost unchanged, suggesting that the method first presented in which the three sources of experimental error are ignored is valid, at least for this data set.

For reporting convenience we here assume that and present the relative transmission fitness, rather than the shape parameter, . If one needs to make an estimate of the reproduction numbers by other means to separate out the within-host and transmission comfponents of the overall relative fitness of strain compared to strain .

To assess the possible variation in observed data and implications for inference, we generate 1,000 simulated data sets for each specified combination of parameters shown in Table 2. We then extract the best fit shape parameter, and number of transmitted virions, , for each simulated data set. Determination of the average best fit shape parameter and average best fit number of virions is done numerically in order to avoid unnecessary assumptions of normality when estimating confidence intervals.

Scenario , based on the H274Y data, provides the reference point for all other scenarios. Scenarios , and explore a reduction in the relative transmission fitness to 0.9, 0.8 and 0.5 respectively, while keeping the number of transmission events, , fixed at 10. In scenarios the number of observed transmission events is increased by a factor of three to 30, while all other biological and experimental parameters are kept unchanged relative to the -scenarios. Scenarios examine a further increase in the number of observed transmission events to 90. Scenarios and explore if more limited strain proportion ranges in the donors (the set ) may change the precision of the estimates for and/or . Scenarios and examine hypothetical mutants which transmit a larger number of virions () in a typical transmission event.

Figure 6 shows a summary of the recovered estimates for the relative transmission fitness, and the transmitted inoculum size, , for the 12 simulations through listed in Table 2. Results for scenarios and are presented in Text S1. Each boxplot shows the median, 25th and 75th centiles with tails extending to the upper and lower adjacent values and outliers shown as crosses. The asterisk marks the true value used in the simulation. Note that a log scale has been used for due to the extended right tail. Text S1 contains detailed results for all simulation runs.

Figure 6A shows that we have an unbiased estimator for the relative transmission fitness, . As expected, the uncertainty in the estimate for reduces with increasing number of observations (e.g. compare simulations to ). To exclude from the estimate (of key relevance to public health) a larger number of transmission observations need to made when the true value of is closer to 1.

We have an accurate, although slightly biased estimator for the transmitted inoculum size, . The mean and median estimates for are always larger than the true value, while the mode is unbiased (see Text S1 for more details). Figure 6B indicates that the bias reduces as the number of observed transmission events, , increases. If the true value for the transmitted inoculum size is greater, the estimator is far less able to exclude very large values for (see Text S1, simulations and )