article: We present a method to measure the relative transmissibility (“transmission fitness”) of one strain of a pathogen compared to another. The model is applied to data from “competitive mixtures” experiments in which animals are co-infected with a mixture of two strains. We observe the mixture in each animal over time and over multiple generations of transmission. We use data from influenza experiments in ferrets to demonstrate the approach. Assessment of the relative transmissibility between two strains of influenza is important in at least three contexts: 1) Within the human population antigenically novel strains of influenza arise and compete for susceptible hosts. 2) During a pandemic event, a novel sub-type of influenza competes with the existing seasonal strain (s). The unfolding epidemiological dynamics are dependent upon both the population' s susceptibility profile and the inherent transmissibility of the novel strain compared to the existing strain (s). 3) Neuraminidase inhibitors (NAIs), while providing significant potential to reduce transmission of influenza, exert selective pressure on the virus and so promote the emergence of drug-resistant strains. Any adverse outcome due to selection and subsequent spread of an NAI-resistant strain is exquisitely dependent upon the transmission fitness of that strain. Measurement of the transmission fitness of two competing strains of influenza is thus of critical importance in determining the likely time-course and epidemiology of an influenza outbreak, or the potential impact of an intervention measure such as NAI distribution. The mathematical framework introduced here also provides an estimate for the size of the transmitted inoculum. We demonstrate the framework' s behaviour using data from ferret transmission studies, and through simulation suggest how to optimise experimental design for assessment of transmissibility. The method introduced here for assessment of mixed transmission events has applicability beyond influenza, to other viral and bacterial pathogens.
  Under the selective pressure from the host immune system on the influenza haemagglutinin (HA) and the ecological environment, antigenically novel HA ‘drift variants’ of influenza A (IAV) generated by random mutation during replication emerge and circulate in the human population. Intermittent cross-species transmission and/or re-assortment events have the potential to generate antigenically novel (at least for the HA and NA genes) mutant strains of IAV. If transmissible such strains have pandemic potential [1]. As seen in 1918/19,1957 and 1968, following a brief period of co-circulation with the existing seasonal strain, the pandemic strain typically drives to extinction the previously circulating seasonal variant, to which there is greater prior immunity. However, replacement is not a necessity. In 1977 the existing H3N2 strain continued to circulate after re-emergence of H1N1. In 2010, indications are that the 2009 pandemic H1N1 has replaced the seasonal H1N1 but not the seasonal H3N2 [1], [2]. The use of antiviral drugs has the potential to provide an additional selective pressure on the influenza virus. Between 2007 and 2009 widespread resistance to oseltamivir, the most common neuraminidase inhibitor (NAI), emerged for the H1N1 seasonal IAV [3]–[6], although this replacement event occurred despite the absence of widespread use of oseltamivir in the human population. The mutation in these oseltamivir-resistant H1N1 viruses is a histidine-to-tyrosine mutation at residue 274 (H274Y) in the neuraminidase (NA) gene. The epidemiological observation of replacement indicates that it has a transmission fitness similar to, or greater than, that of the sensitive strain. Other NAI-resistant viruses can have significantly reduced fitness, such that they are unlikely to spread through the population. For example, an arginine-to-lysine mutation at residue 292 (R292K) in the NA gene of seasonal H3N2 viruses confers oseltamivir resistance, but this mutant has not been observed to readily transmit from one host to another [7], [8]. At present, the 2009 pandemic H1N1 strain remains largely sensitive to oseltamivir and has almost entirely replaced the previously circulating NAI-resistant (H274Y) H1N1 strain [2], [3], [9]. Accordingly, during times of co-circulating seasonal strains, emergence of new antigenic seasonal variants, seasonal to pandemic transitions or emergence of drug-resistant strains, assessment of whether or not one strain will out-compete the other and come to dominate the human epidemiology is of relevance to public health planning and response. Key to making predictions on the dynamics of such events is the relative transmission fitness of one strain compared to another. Our group has previously published, in the context of NAI-resistant and NAI-sensitive strains, observations of both growth of mixtures within an animal model and sustained transmission of mixtures over multiple host generations of infection [8]. The observation of transmission of mixtures allow us to measure the relative transmissibility (“transmission fitness”) of one strain compared to the other. Govorkova et. al. employed a similar experimental design to compare the within-host fitness of an oseltamivir-sensitive and oseltamivir-resistant H5N1 influenza virus pair, but as H5N1 does not transmit readily in the ferret, they were unable to make an assessment of transmission fitness [10]. Duan et. al. have also employed the technique to compare drug-sensitive and -resistant H1N1 pandemic strains [11], probing both contact and respiratory droplet transmission routes. We introduce a mathematical and statistical framework to capture the key characteristics of transmission of mixtures. Our framework allows us to derive a causative-model based estimate of the transmission fitness-cost (or fitness-gain as the case may be) for one strain compared to the other and suggest a method for designing optimised animal experiments. Furthermore, we derive an estimate for the number of successfully infecting units transferred from the donor to recipient, an important measurement for helping parameterize epidemiological models of influenza transmission that consider, for example, the emergence and subsequent spread of NAI-resistant strains [12]–[14].
  Figure 1 (adapted from the author' s results and reproduced with permission [8]) presents data from a series of contact transmission studies in naïve ferrets. The abscissa shows the proportion of the infecting ferret' s viral-load at the time of transmission that is oseltamivir-resistant (the “mutant” strain). The remainder is oseltamivir-sensitive (the “wild-type” strain). The ordinate shows the proportion of the infected ferret' s viral-load that is the mutant. Briefly (see [8] for experimental details, and Section Effects due to data ascertainment limitations for a detailed discussion of the consequences), the ferrets were swabbed daily, and infection identified by positive real-time RT-PCR. We plot the mutant-proportion in the infected ferret' s first positive swab against the mutant-proportion in the infecting ferret' s swab from the day prior. We see that the R292K mutant (circles) does not transmit: no mutant virus is detectable in any of the infected ferrets, the data covering a wide range for the infector' s mutant proportion. For the H274Y mutation we have two generations of transmission. A donor ferret experimentally inoculated with a mixture transmits to a recipient ferret (squares). This ferret is then the infector for the second generation of transmission to another ferret (triangles). The data are consistent with the hypothesis that the infectee' s mutant proportion is given by the infector' s mutant proportion at the time of transmission. For convenience, we will always refer to the infector as the donor and the infectee as the recipient. We will use to represent the mutant proportion in the donor and the mutant proportion in the recipient. A data set will be said to have transmission events, . Three key observations can be made for the H274Y data (squares and triangles in Figure 1): We now introduce a model framework to help understand these observations. We consider a model with two viral strains, and. For example, could be the oseltamivir-resistant mutant (either R292K or H274Y) and the oseltamivir-sensitive wild-type, as in Figure 1. We introduce the following notation: Equation 4 provides the mean estimate for the recipient' s strain proportion given the donor' s strain proportion. We wish to derive an estimate for the number of virions transmitted (in a successfully infecting way) and to this end make use of the variation about the mean predicted by equation 4. We present an approach for the estimation of, the number of virions transmitted and that initiate infection per exposure event. Note that and likely, where is the number of virions secreted from the donor and that enter into the recipient' s airways. The method relies on repeated observations of successful transmission events. Initially we assume that there is no experimental error on the data points and that the number of virions transmitted is fixed across each exposure event. Before proceeding, we introduce some useful nomenclature. Scripted variables (e. g.) will represent simulated data, non-scripted variables (e. g.), calculations from model equations. Sets will be denoted as, with the element of a set given by. Until now, we have implicitly assumed that an observation is without error. However, experimental limitations, including animal ethics principles and practicalities within the laboratory environment, introduce uncertainty into any measurement of a transmission event. Three sources of experimental error are of potential significance: Each needs to be assessed and if necessary accounted for when performing the proposed analysis on a new data set. We address each issue in turn, in a semi-quantitative way, using the H274Y data to illustrate the issues. With only a limited amount of data available from ferret experiments, we use results from simulation-estimation studies to optimize future experimental design and make predictions on the likely precision of estimates for shape parameters from future experimental observations. Our simulation model has four free parameters: Synthetic data, with, for transmission events, each from a donor transmitting virions with strain proportion at the time of transmission, is generated as follows. First, we specify the shape parameter for equation 4, defining. If is a random integer drawn from and is a random integer drawn from then our synthetic data, , are given by (): (10) If then. We estimate the shape parameter, , by fitting the model (equation 4) to either real or synthetic data, by application of non-linear least squares regression (routine nlinfit in MATLAB R2010a). We estimate the number of virions, , using equations 6 through 8.
  Applying our model equation 4 to the data for the H274Y mutation shown in Figure 1 (and assuming the data are recorded without error – see below for a generalised analysis) provides an estimate for the shape parameter of with a 95% confidence interval of (0. 259,2. 30), calculated under the assumption of i. i. d Gaussian residuals, which is not strictly true. Assuming – which is a valid approximation given the within-host analysis of this data set [8] – the corresponding estimate for the relative transmission fitness for the H274Y mutant, in a direct-contact scenario, is (0. 44,3. 9). We predict that an average 3. 8 virions were responsible for each transmission event, with a variance for our estimate of 5. 9, but we caution that this estimate is based on an assumption of data recorded without error. Figure 4A shows the model fit to the H274Y data and Figure 4B shows a histogram of the best' s.
  The relative transmissibility of one strain of influenza compared to another is important in a number of different contexts: seasonal/seasonal interactions, seasonal/pandemic interactions and NAI-resistant/NAI-sensitive interactions. Given the competitive mixtures animal model paradigm is applicable in all three contexts, the mathematical framework presented here has wide applicability in influenza research. Our results indicate that application of equation 4 and equations 6–8 to transmission data from mixed infection studies allows estimation of both the transmission fitness of a new strain of interest (strain) relative to that of the existing wild-type (strain) and the transmitted inoculum size. Our consideration of the uncertainties inherent in the data source due to experimental procedures indicate that the method is robust to these issues, although we caution that any application of the method should consider each source of uncertainty in the data on its merits. The simulation studies show that the method accurately recovers the relative transmission fitness, of strain compared to strain. As expected, the closer the true value of is to unity, the greater the number of transmission events that need to be observed to exclude 1 from the 95% confidence interval for. For recovering the number of virions transmitted and successfully initiating infection, the mode of the distribution is unbiased, while the mean and median are consistently slightly larger, due to the extended right-tail of the distribution. It follows that this method can be used in experiment design to ensure that studies are sufficiently powered to provide estimates of a specified precision for the transmission fitness. In our proof-of-principle analysis of the H274Y data set [8], for both simplicity and because the data indicate that it is a good approximation, we have assumed that and so in equation 4. If the within-host replication fitnesses of the two strains were not equal, then an estimate of the effective reproduction numbers (and) and the generation time, , for the within-host dynamics would be necessary to correctly infer the relative transmission fitness, , from the estimate of the shape parameters, . Furthermore, a simulation analysis allowing for uncertainty in infection time (along the lines of the argument laid out in Section Effects due to data ascertainment limitations) would also be necessary. Our method does not explicitly require knowledge of the immune status of the ferrets, the shape parameter providing information on the effective fitness of strain compared to strain given the natural history of infection and/or vaccination of the animals. Differences in immunity to strains and manifest as changes in the effective reproduction numbers (and). Consequently (as just discussed above), to determine the relative transmission fitness, , one must obtain independent knowledge of, and. This may be achieved by conducting an analysis of the within-host dynamics of competitive-mixtures infection in naïve and non-naïve animals. Estimation of the within-host reproduction number is a key focus of the rapidly developing within-host influenza dynamics literature [16]–[20]. Our method indicates a number of potential improvements in competitive-mixtures experimental design. Firstly, the use of multiple donor ferrets – introduced by Hurt et. al. as a “reliability” factor in the face of uncertainty of infection following inoculation – introduces unnecessary complications and should be avoided given that experimental inoculation has been shown to reliably result in infection. Secondly, regular sampling is essential so as to tie down as closely as possible the strain proportion in both donor and recipient at the time of transmission. This is of particular importance if as just discussed. An alternative way to more accurately ascertain may be to introduce recipient ferrets to the infectious donor for controlled and limited times. Such a design has the additional appeal of allowing determination of the donor' s time-dependent infectivity, of particular relevance to public health planning in influenza (see, e. g. [21]). We see scope for improvement to the mathematical framework in a number of ways. Firstly, the number of virions transmitted and successful at initialising infection in the recipient was assumed constant (in equation 10). Clearly this is not strictly true and it would be possible to probe the effects of the Poisson nature of the process, from transmission event to transmission event, in more detailed simulation studies. We have also assumed that virions are deposited and die or replicate independently of one another. However, it is plausible that virions and indeed mixtures of strain and strain virions, clump together in the process leading to their secretion from the donor and transmission to the recipient, for example, in droplet or aerosol particles. Our analysis of the H274Y contact transmission data indicates that, the number of virions that successfully establish infection is small, although the result has not been established for aerosol or droplet transmission. While we are not aware of any direct experimental evidence from influenza studies allowing comparison, Abrahams et. al. report that for HIV infection, 78% of infections involved a single variant (using their terminiolgy, a single “infectious unit” and so we tentatively suggest a single virion), while the remaining infections were established by between 2 and 5 (median 3) “infectious units” [22]. While we estimate that is small, it is entirely plausible that the total number of potentially infectious virions transmitted, , may be somewhat or perhaps even much larger (). Each transmitted entity (or “unit”), itself containing perhaps multiple virions, may either lodge at a site compatible with replication or not. Of those that do (essentially the proportion in equation 1), local within-host competitive processes between strain and virions will then come into play. If multiple entities are successful in establishing local sustained growth within the host, the eventual establishment of a mixed-infection may be due to a hybrid of the heterogeneity in mixtures across entities, and the local within-host processes at play as each entity establishes infection. How to probe such possibilities, both experimentally and theoretically, presents as an interesting future research opportunity. The third avenue for improvement concerns accounting for re-assortment of strains and during the within-host dynamics, in both the donor (s) and recipient (s). For the H274Y data, this is not of concern, due to the similarities between the two strains [8], but more general experiments, say comparing a new pandemic strain to an existing seasonal strain of a different sub-type, would need to be analysed accounting for the additional complexity. With the limited data available from the H274Y experiment, we were unable to meaningfully constrain the relative transmission fitness of the NAI-resistant strain ( (0. 44,3. 9) ). We also caution that the result is for contact transmission, and there are indications that respiratory droplet and/or aerosol transmission experiments may yield different results, as demonstrated recently by Duan et. al. [11]. However, our simulation analysis, which can be considered as a computational power calculation, indicates that we should be able to identify a relative transmission fitness – in either contact or droplet transmission studies – of 0. 9 or less with a large but not prohibitive number of ferrets (). It should be kept in mind that our analysis can be performed on pooled data from multiple experiments, each with other primary end-points of interest, as long as those experiments do not probe interventions that may be expected to modify the relative transmissibility of one strain compared to the other (such as immune changes, vaccination etc.). Finally, the method presented here is applicable to pathogens other than influenza as any series of observations of transmission of mixtures may be represented as in Figure 1. The most critical factor requiring thought will be the within-host dynamics of the two strains, and in particular, appropriate consideration of any competitive or synergistic effects due to co-infection.
summary: Determining which of two related viruses will spread from human to human more efficiently – e. g. an influenza virus that is treatable with drugs and one that is resistant to them – is important when forecasting the potential impact of an emergent novel virus or developing public health intervention strategies. However, making such measurements of relative transmissibility directly through observation, even using an animal model, is difficult. We have recently developed and published an experimental technique in which an animal is infected with both viruses of interest at once, and then allowed to mix with other animals and so transmit the infection. These experiments provide the necessary data for analysis using the novel mathematical framework that we introduce here. Our mathematical and computational results exploit the power of the experimental system, and allow us to make a quantitative estimate of the relative transmissibility of a drug-resistant influenza virus compared to its drug-sensitive counterpart. Through computer simulation, we demonstrate the wider application of our mathematical technique, and suggest design criteria for future experiments designed to measure the transmissibility of one virus (or other type of pathogen) compared to another.
keywords: animal models of infection
  theoretical biology
  viral transmission and infection
  virology
  co-infections
  biology
  microbiology
title: A Mathematical Framework for Estimating Pathogen Transmission Fitness and Inoculum Size Using Data from a Competitive Mixtures Animal Model