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All authors conceived and designed the experiments and wrote the paper. AH performed the experiments and analyzed the data.

The authors have declared that no competing interests exist.

Neuraminidase Inhibitors (NI) are currently the most effective drugs against influenza. Recent cases of NI resistance are a cause for concern. To assess the danger of NI resistance, a number of studies have reported the fraction of treated patients from which resistant strains could be isolated. Unfortunately, those results strongly depend on the details of the experimental protocol. Additionally, knowing the fraction of patients harboring resistance is not too useful by itself. Instead, we want to know how likely it is that an infected patient can generate a resistant infection in a secondary host, and how likely it is that the resistant strain subsequently spreads. While estimates for these parameters can often be obtained from epidemiological data, such data is lacking for NI resistance in influenza. Here, we use an approach that does not rely on epidemiological data. Instead, we combine data from influenza infections of human volunteers with a mathematical framework that allows estimation of the parameters that govern the initial generation and subsequent spread of resistance. We show how these parameters are influenced by changes in drug efficacy, timing of treatment, fitness of the resistant strain, and details of virus and immune system dynamics. Our study provides estimates for parameters that can be directly used in mathematical and computational models to study how NI usage might lead to the emergence and spread of resistance in the population. We find that the initial generation of resistant cases is most likely lower than the fraction of resistant cases reported. However, we also show that the results depend strongly on the details of the within-host dynamics of influenza infections, and most importantly, the role the immune system plays. Better knowledge of the quantitative dynamics of the immune response during influenza infections will be crucial to further improve the results.

Neuraminidase Inhibitors (NI) are currently the most effective drugs against influenza [

Two important parameters, for which there are currently no good estimates, are: (i) the initial generation of resistance, defined here as the number of resistant infections caused by a patient receiving NI treatment who was initially infected with a sensitive strain; (ii) the subsequent spread of resistance, defined as the number of resistant infections caused by a patient initially infected with the resistant strain.

The best data we currently have for the generation of NI resistance come from clinical studies that report the fraction of treated patients from which resistant strains could be isolated [

For the spread of NI resistant strains, some insights have been obtained from studies with ferrets, where it was shown that certain resistant strains are transmissible, while others are not [

The dilemma is obvious: We need good parameter estimates to understand and model the potential spread of NI resistant influenza, but we do not want to wait until such spread has occurred and epidemiological data are available that would allow us to obtain good parameter estimates. Therefore, it is important to find alternative ways to estimate these parameters.

Here, we use a conceptual framework that links within-host infection dynamics to between-host epidemiological parameters [

The simplest way of modeling the dynamics of viral infections is based on ordinary differential equations, an approach that has a long and successful history [

This TD model does not include an immune response. Since the immune response likely contributes to viral clearance [

Equations for the Two Models Describing the within-Host Dynamics

Parameters for the Two Models Describing the within-Host Dynamics

Most parameter values for the two models are obtained by fitting viral load data from human volunteers [

Symbols in (A) and (B) are data from humans infected with influenza A/Texas/91 (H1N1). Early drug treatment starts at around 29 h, late treatment at around 50 h post-infection. The dotted horizontal line shows the limit of assay sensitivity. The solid lines show total viral load as obtained from (A) the target-cell depletion (TD) model and (B) the immune response (IR) model. The dashed lines show the resistant subpopulation. Further details on the experimental data are given in [

While we tried to obtain all parameters through fitting, the available data are not sufficient to obtain reliable estimates. We therefore decided to use estimates obtained from the literature for those parameters where such information was available. One parameter that we estimate from independent experimental studies is the mutation rate, μ, at which NI resistant mutants are produced. The mutation rate per base pair per replication for influenza A has been estimated to be about 7 × 10^{−5} [^{−6} [^{−6} to 10^{−4}. For most of our simulations, we choose ^{−5}. We also investigate how variation in μ affects the results.

Another independently estimated parameter is the cost in fitness,

The initial number of uninfected epithelial cells has been estimated previously to be _{0} = 4 × 10^{8} [^{2} [^{−7} cm^{2} [_{0} = 4 × 10^{8}. While this estimate comes with a certain amount of uncertainty, for our purposes, knowing the exact value for _{0} is not critical since a different value would simply lead to a rescaling of some of the parameter values.

Lastly, the constant

Since resistant virions are initially not present and, upon initial generation, are at low numbers, stochastic effects can become important. It is therefore useful to also use stochastic versions of the deterministic models described in the previous section. One problem with using stochastic simulations is the issue of units. With the deterministic models introduced above, we are able to work in the experimentally reported units of 50% tissue culture infectious doses per milliliter (TCID_{50}/ml) of nasal wash. However, if we want to study the impact of stochastic effects, we need to convert to numbers of infectious virions at the site of infection. Both TCID_{50} measurements as well as our models only deal with viable, infectious virions. Non-infectious viral particles, which are known to be created in rather large quantities due to the segmented nature of the influenza virus, can therefore be ignored. Still, it is unclear how TCID_{50}/ml of nasal wash convert to numbers of infectious virions. At the minimum, one TCID corresponds to a single infectious virion, but it is more likely that on average more than one virion is needed to establish an infected cell culture. We estimate that 1−100 virions correspond to one TCID. Next, virions/ml of nasal wash need to be converted to virions/ml at the site of infection, which for uncomplicated influenza infections is mainly the upper respiratory tract. Not much information is available; based on circumstantial data ([_{50}/ml of nasal wash corresponds to about 10^{2}–10^{5} virions at the site of infection. Calling this conversion factor γ (with units of (TCID_{50}/ml)^{−1}), the variables of the deterministic model are rescaled for the stochastic model according to V →

An important issue is the fact that even for the largest estimate of γ, the best fit value for the initial viral load in the TD model (_{0} ^{−7} TCID_{50}/ml) corresponds to an inoculum size of less than one virion. While some studies suggest that a few virions are enough to start an infection, clearly a value below one makes no sense. It is likely that the initial estimate for _{0} obtained from the model is wrong. Indeed, experimental data from mice suggests that instead of having minimum viral load at the start of infection, the viral load first drops over the course of a few hours, before it starts to increase again [_{0}. Another reason suggesting that the value for _{0} obtained from the TD model is too low comes from the fact that for the IR model, the value of _{0} is orders of magnitude larger. While this indicates problems with the TD model, the data used here do not allow us to conclusively reject it. We return to the problem of model discrimination and lack of data in the Discussion. To allow comparison between the deterministic and stochastic versions of the TD model, we bound _{0} by 10^{−5} TCID_{50}/ml from below. This leads to a fit that is only a few percent worse than the unbounded fit, and by setting ^{5}, we can then compare deterministic and stochastic results in the TD model. Further, to investigate the impact of a lower γ, we set ^{3} for the IR model.

The deterministic models are fitted to the viral load data using several fitting routines (lsqnonlin, fmincon, and nlinfit) provided by Matlab R2006b (The Mathworks). To obtain the results shown below, we perform both stochastic and deterministic simulations. The deterministic ODEs are implemented in Matlab R2006b, the stochastic simulations are written in Fortran 90. A purely stochastic simulation (Gillespie algorithm) would be prohibitively slow, due to the large numbers of cells/virions. Therefore, the simulation is implemented using a partitioned leaping algorithm [

The models described in the

Here, _{tot}_{1} = 6.1, _{2} = 4.8, _{3} = 2.6, and _{4} = 1.5. The data and best fit are shown in ^{2} is 0.75 for the Hill function and 0.73 for a linear model). We also prefer Equation 1 on biological grounds, due to its threshold and saturation effects for low and high viral load, respectively.

Data are from [

We then obtain the total amount of viral shedding by multiplying virus concentration with the amount of discharge at every time point and integrating over the duration of infection. The equation for the total amount of shedding is given by
_{s}_{r}

The results obtained for viral shedding allow us to estimate the two quantities of interest, the initial generation of resistance and its subsequent spread through the population. Both quantities can be expressed in terms of the average number of new infections caused by an infected host, the reproductive number, _{s}_{r}_{s}_{s}S_{s}_{r}_{r}_{r}S_{r}

Estimates for the reproductive number of influenza (_{s}_{s}_{s}_{r}_{r}_{s}_{r}

We first consider the generation of resistance during treatment as described by _{s}_{s}

Results are shown for both deterministic and stochastic simulations. Unless varied, treatment starts 24 h post-infection, and the other parameters are as shown in

Several observations are notable. First, the results show that more effective treatment, which better removes the sensitive strain and thereby allows the resistant strain to grow, increases the danger of resistance generation (

Second, a change in fitness cost or mutation rate has little impact on the TD model for a wide range of parameter values but does affect the results for the IR model (

Third, the TD model consistently predicts values for resistance generation above those for the IR model. This can be understood as follows: in the TD model, the resistant strain competes with the sensitive strain for resources (target cells). In the presence of the sensitive strain, the resistant strain is outcompeted. If the sensitive strain is removed, the resistant strain will infect most target cells and reaches high levels. In contrast, the immune response in the IR model acts against both the sensitive and resistant strains. If sensitive virus is suppressed by NI, the mounting immune response will still act against the resistant strain, preventing it from reaching high levels [

Fourth, stochastic effects become important either if treatment occurs early and wipes out the sensitive population before resistance has been created, or if mutation rates are low. As mentioned in the

While the parameter governing the generation of resistance is important, the parameter describing the subsequent spread of resistance is arguably more important. Even if generation of resistance is infrequent, only a few resistant infecteds could be enough to start a resistant outbreak. The possibility for such an outbreak is determined by _{r}_{r}_{r}_{s}_{r}_{r}_{r}_{s}_{r}_{r}

Note that only variation in the fitness cost has an impact on _{r}

We have demonstrated that it is possible to combine data from infected individuals with mathematical models to obtain estimates for important between-host parameters, without the need for epidemiological data.

The results we obtained suggest that to minimize the danger of resistance generation, treatment at the very beginning of the infection (i.e., prophylaxis) is best (

Unfortunately, several shortcomings currently do not allow us to obtain precise results. The main problem is our lack of understanding of the dynamics governing within-host influenza infections. Both the TD model without immunity and the IR model with immunity are able to fit the data; however, the estimated parameters differ. Additionally, parameters such as the conversion rate between TCID_{50} and number of virions, or the rate of mutation, are based on estimates that come with a significant amount of uncertainty. The problem of unrealistic parameter estimates, such as the very low initial viral load obtained for the unbounded TD model, further reinforce the fact that more data is needed to better discriminate between models.

The inability to discriminate between models would not be too problematic if the two different models produced similar results. While the results are somewhat similar for the spread of resistance, as well as the impact of treatment on the sensitive strain (see Appendix B: The Impact of Treatment on the Spread of the Sensitive Strain), they differ significantly with regard to the initial generation of resistance (

Also needed are further studies that investigate the ability of the resistant strains to transmit. Specifically, it is necessary to understand if reduced transmission is due to reduced shedding, reduced survival of the resistant strain during transmission, or other factors such as changes in contact rates. To that end, further studies in ferrets seem to be the most promising approach.

While the lack of better data currently prevents us from obtaining quantitative results, these limitations can be overcome. Provided enough experimental data on within-host dynamics and some transmission data between individuals are available, the approach discussed here can produce parameter estimates that can then be used to simulate and study potential spread of novel emerging pathogens. Crucially, this can be done before the pathogen has produced outbreaks large enough to reliably obtain parameter estimates from epidemiological data. Such an approach will be important if we want to be one step ahead of NI resistant influenza, a potential H5N1 outbreak, and other newly emerging diseases for which epidemiological data are lacking. In the best case, we will be able to prevent this data from ever existing.

In the previous text, we connected viral shedding and the number of new infections by a simple proportionality,

We can express the symptom score _{1}_{2}_{10}(_{1} = 0.15 and _{2} = 0.77 (

Data are system symptom score values and viral load from [

We then obtain

The integral expression now represents shedding, adjusted for behavioral changes. The constant _{i}_{s}_{r}_{r} = D_{s}^{t}

Shown are results for the model without (^{−5}.

While the main focus of this study is on the generation and spread of NI resistant influenza, the framework can also be used to study how treatment affects shedding and therefore transmission of the sensitive strain, providing an approach that is complementary to existing ones [

Unless varied, treatment starts 24 h post-infection, and antiviral efficacy is

immune response

neuraminidase inhibitor

tissue culture infectious dose

target cell depletion