Using simple models for the kinetics of influenza A virus infection, we have been able to fit experimentally derived viral titer data and estimate parameters characterizing viral production and clearance in experimentally infected individuals. Tables and present the patient-specific and geometric mean parameter estimates that correspond to the best fits of the models with and without delay to the data. These estimates were sensitive to choice of initial parameter guesses, which suggests that there may be other parameter estimates that can also fit the data reasonably well. Examining the surface generated by plotting the sum of squared residuals versus various pairs of parameters, we find that rather than having a deep, well-defined minimum, the surface tends to be flat at the minimum, thus explaining how different initial parameter guesses can give rise to different parameter estimates. Nonetheless, the estimates we found generally were consistent among patients, had reasonably tight 95% confidence intervals, and generally fell within experimentally determined ranges. Studies with more-frequent sampling as well as studies that independently estimated some parameters, such as infected cell life span, would be useful in confirming the parameter estimates we report here.
We studied both a version of the standard target cell-limited model of viral infection developed for the characterization of HIV infection and a more accurate variant that incorporated a delay from time of cell infection to beginning of viral production. From a statistical standpoint, the improvement of the fit with the delay model (which had one more parameter) was not great enough to justify adding an additional parameter. Nevertheless, the target cell-limited model with a delay in viral production is more realistic than the model without the delay, and it yielded more-reasonable parameter estimates. The estimate of the life span of infected cells in the model without delay was 6 h, which is not realistic. Incorporating the eclipse phase of the viral life cycle lengthened the total life span to close to 12 h. Thus, we prefer the model with delay and below discuss only the results of this model.
The delay model suggests that influenza virus is produced by an infected cell for a period of about 5 h, that it takes about 6 h between infection of target cells and virion release, and that infected cells have an average life span of about 12 h (Table ). If cells stop producing virus without dying, their life spans would be longer, as our model tracks the states of cells only from time of infection to loss of viral production. Such a noncytolytic loss of viral production might be caused by IFN or other antiviral cytokines.
Estimates of the average lifetime of infected cells derived from our model differed from experimental values. While one report suggests that cells infected with influenza virus have an average lifetime of ~24 h (52
), our target cell-limited model with delay yielded an estimated average lifetime of infected cells of ~12 h. Whether this discrepancy is real needs to be examined with more-refined experiments in which the half-life of infected cells is more precisely defined, especially since the experimental estimates (52
) were derived over 30 years ago.
Our data and models allowed us to estimate the basic reproductive number, R0, for influenza A virus infection of susceptible nasal epithelial cells within an individual. The estimated R0 was about 22 in the target cell-limited model with delay. This high value of R0 suggests that initial infection spreads rapidly and would be difficult to extinguish.
As we illustrated with our models (Fig. ), treatment of influenza infection with NIs reduces viral load and hence should reduce period of symptomatic disease. Further, prophylactic use with a highly effective NI is predicted to prevent infection. Hayden et al. (15
) observed that zanamivir (GG167) administered prophylactically as intranasal drops is highly protective against infection, as only 3 of 44 volunteers receiving zanamivir became infected, whereas 24 of 33 volunteers receiving placebo became infected. A similar study with oseltamivir showed that prophylactic therapy protected 8 of 21 volunteers from infection, whereas 8 of 12 volunteers receiving placebo became infected (16
In light of the fact that adaptive immune responses in primary influenza infection are not detected until 6 to 8 DPI (10
), rapid innate immune response may play a major role in resolving influenza virus infection. However, the first line of defense to virus is probably physical clearance due to mucociliary action. The half-life of free virus in the target cell-limited model with delay averaged 3.2 h (Table ). If mucociliary clearance is the major explanation for viral clearance, then clearance rate of free virus, c
, would be expected to be lower in individuals such as smokers and people with emphysema, asthma, and cystic fibrosis, all of whom have decreased capacity for mucociliary clearance (33
). In our model, a decrease in the parameter c
, which reflects decreased physical clearance, results in a faster increase of virus titer and a higher peak titer during influenza infection and hence may be correlated with the increased seriousness of influenza infections noted for people with decreased capacity for mucociliary clearance.
The target cell-limited models we described can easily fit virus titer data when there is only one peak. However, in approximately 50% of individuals studied here and in another study, we observed bimodal virus titer profiles, which could be explained by the incorporation of an IFN response. Our model with IFN generally produced a second virus titer peak 36 to 48 h after the initial peak. When the second peak appears later than 48 h after the initial peak, more-complex models that incorporate explicit immune responses along with the possible generation of escape mutants may be needed. Other possible explanations could also account for the second virus titer peak. The extension of active viral replication to a previously uninvolved site in the upper respiratory tract, e.g., sinus or nasal passages, could result in a second peak. If the free virus migrates into a new area, then the number of susceptible target cells increases and allows the virus to undergo another surge in viral titer. Also, the types of cells infected in the nasal mucosa differ by virus type (avian versus human influenza viruses) and likely over time with human influenza viruses (25
). The sequential infection of different cell types could also produce viral titer peaks at different times.
To increase our understanding of the control of influenza infection in humans, additional data in which antigen-specific T cells, antibody titers, NK cells, IFN levels, and viral titers are measured frequently (maybe more often then daily) need to be collected. Also, it is important to keep in mind that the experimental data used in our analysis are derived from upper respiratory viral infection, and hence, parameters characterizing the kinetics of natural infection in the lower respiratory tract may be different from those derived here.
In summary, we have shown that simple target cell-limited models of virus infection, as previously developed to study HIV, can be applied to improve our understanding of influenza virus infection. We estimated, using our model with delay from time of infection to viral production, that during upper respiratory tract infection, influenza virus initially spreads rapidly with 1 cell, on average, infecting ~22 others. The infection slows as target cells are consumed, and by the time of the virus peak at days 2 to 3, the vast majority of the initial target cells have been destroyed. Thus, influenza A infection could be self-limiting.
In the case of therapy for hepatitis C virus infection, simple target cell-limited models have been used to estimate the antiviral efficacy of interferon (30
) and the effects of ribavirin (8
). We believe that models of the type introduced here, once more fully validated, can be used to estimate the efficacies of established antivirals, such as oseltamivir and zanamivir, as well as new agents for the treatment of influenza. We illustrated this by comparing model predictions with data in which the neuraminidase inhibitor zanamivir was used to treat experimentally infected adult volunteers.