In this work, we have developed a differential equation model of influenza A virus infection and fitted this model to a comprehensive data set from primary IAV infection in the mouse. The data set is unique in that it represents the results from simultaneous high frequency (every 12 to 24 h) sampling of lymphocytes, antibody, and influenza virus in a large number of animals (n = 340), with multiple replicates at each data point. The large number of frequent measurements gives much greater power in fitting the model to data and led to several key findings, which we summarize here.
Our model divides the immune response to IAV infection into the following two phases: the preadaptive (innate) and adaptive immune response phases. During the innate response phase (days 0 to 5), adaptive immune response elements (IAV-specific Ig and lung-infiltrating CTLs) are negligible. Although clonal expansion of IAV-specific CTLs and B cells may be starting in the lymph nodes and spleen prior to day 5, measurements of the effectors in the lung are below detection limits during this period. It is during the innate phase that the viral infection is spread among susceptible respiratory epithelial target cells, and each infected cell becomes a viral production source, thus contributing to peak viral load. The total number of epithelial cells includes those that are susceptible to viral infection (targets), those that are infected, those that are immune to infection, and those that are regenerating cells (either target or immune cells). A priori, key parameters that determine the viral load during the innate phase are virus infectivity and replication rates, host parameters such as the number of infected, target, and regenerating epithelial cells, and the rate of epithelial cell death. The adaptive phase of the immune response limits viral load by active viral clearance and by the killing of IAV-infected cells. We use our model to investigate the sensitivity of the mathematical models to changes in specific parameters and to estimate kinetic parameters that are difficult to measure directly.
A key set of model predictions, regarding the factors that affect the peak viral load, are as follows: (i) virus infectivity changes the timing but not the magnitude of the peak viral load, (ii) peak viral load is target cell limited, and (iii) peak viral load is very sensitive to the viral replication rate but not to infectivity. The magnitude of peak viral load is relatively insensitive to infectivity, which may seem counterintuitive. However, within physiologic parameter ranges, the viral replication rate is much more significant, and moderate to high rates of replication during this period of low viral clearance are quantitatively the primary determinant of peak viral load. These results imply that viruses with higher production would be more pathogenic than those with high infectivity. Although not investigated here, infectivity may be more relevant to spread of the virus from person to person rather than a primary determinant of virus levels within an individual.
Consistent with these findings regarding the significance of virus production versus spread in the host is the prediction that peak viral load is limited by the available target cells. Thus, once established in an epithelial cell, the pace of virus production and the amount of virus produced have a greater influence than cell-to-cell spread, particularly as the infection progresses and the number of available targets decrease. During the adaptive phase, the actions of antibody and CTLs outpace virus replication and spread, causing titers to decline. Thus, we predict that therapies that impede or slow virus replication would have the most significant impact during the innate phase of infection. Similarly, robust therapies that limit target cell availability may also have a strong effect. In fact, the type I interferon system is an example of how biology has evolved a mechanism to achieve exactly this effect.
Using several techniques, our modeling predicts that epithelial cell proliferation has a negligible impact on determining the kinetics of IAV infection. Parameter ρE
, the net growth rate of uninfected epithelial cells per day, is included in the absolute number of target cells available for the virus to infect.
The reliability of the estimate of ρE
is greatly affected by the magnitude of the entire term, reflecting the net growth of uninfected epithelial cells ρEEp
. Once the infection is established, estimating the term for epithelial proliferation becomes unreliable, primarily because the number of infectible target cells, Ep
, is nearly zero. Even if the proliferation rate of the epithelial cells increases, the value of the entire term (ρEEp
) is still low once most of the target cells are infected (e.g., Ep
≈ 0) and has no effect on the adaptive model predictions. Similar analysis holds for estimation of the number of viral infections per unit time (βα
). Essentially, a very small number of target cells (Ep
) and a large amount of virus (V
) cannot be used to reliably estimate the epithelial proliferation (ρE
) or infection rates (βα
) because most of the target cells have become infected (
In our models, infected epithelial cell death occurs through the following two modes: killing by CTLs and death by other factors that are a composite of immune (complement, ADCC, and NK cells, etc.) and natural (necrosis and apoptosis) factors. Considering the overall infection (days 0 to 14), death by factors other than CTL killing contributed the most to loss of infected cells. However, during the adaptive phase (days 5 to 14), model selection analysis suggests that the contribution of CTL killing is more significant than the contribution of other factors. Specifically, based on the adaptive phase-fitting results, CTLs are the main contributors to the clearance of infected epithelial cells, with a killing rate (kE) of 6.4 × 10−5 per CTL per day. The antigen-specific CD8 T-cell number peaks at >5 × 104 cells, yielding an effective peak killing rate (kE·TE) of >3.2 day−1. Comparing the innate to the adaptive phase of the immune response, we found that the half-life of infected epithelial cells decreases from an average of 1.16 to 0.52 days, primarily due to the effects of CTLs. The death rate (δE*) of infected epithelial cells due to factors other than CTLs is 0.16 per day during the adaptive phase, while δE* equals 0.6 per day during the first 5 days. Whether this is a real effect or an artifact caused by the much stronger effect of CTL killing is uncertain. Nonetheless, since most of the infected epithelial cells are killed by CTLs more efficiently, the other causes of infected cell death do not alter the infected epithelial cell half-life significantly. To further evaluate the impact of the CTL killing on viral clearance, we also performed computer simulations to compare the outputs of model 2 with and without CTL killing (equivalently, kE equaling 6.4 × 10−5 per cell per day and kE equaling 0 per cell per day). We find that the virus can be completely cleared about 10 days after infection when CTL killing is included in the model. When we omit the CTL killing effect, however, our model predicts that the virus cannot be cleared (instead, the virus titer reaches a nadir of about 3.5 log10 EID50 on day 9 and then starts to rebound). This observation suggests that the CTL effect has a big impact on influenza virus clearance.
Our estimates of killing rates for murine CTLs are consistent with estimates of the killing rates for CD8 T cells for lymphocytic choriomeningitis virus (LCMV)-infected cells (4.5 × 10−5
per CTL per day), even with differences in the target cells and virus (13
). Others have also estimated CD8 CTL killing rates in murine LCMV infection by fitting a closed-form model to experimental data, with the estimated half-life of infected cells being about 10 min (6.9 × 10−3
). There are several factors that could explain the difference in target cell half-life between the two systems. LCMV infection was measured in the spleen, where infected target cells and effector CTLs are in close contact with each other, with greater freedom to migrate. In the IAV-infected epithelium, target cells are arranged in an essentially two-dimensional surface, and CTL migration to infected target cells in the airways may be more limited. Therefore, the effective half-life of the target cells is longer in the influenza model because of limited target accessibility, even though the per cell killing rates are similar.
Our results suggest that IgM is crucial for IAV clearance in a primary infection, whereas the contribution of IgG is less important, despite the eventual presence of higher and more sustained levels of IAV-specific IgG. During the adaptive phase of infection (>5 days), parameter estimates show IgM-mediated clearance of free virus (kVM) at a rate of 4.4 ml/(pg·day). In contrast, IgG antibody had a relatively low contribution to viral particle clearance, with a rate (kVG) of 1.0 × 10−5 ml/(pg·day). During the adaptive phase, our estimate of the clearance rate of virus particles due to factors other than antibody neutralization (cV) is <0.01 per day, which is much lower than that during the first 5 days of infection. This translates into a shortened half-life of infectious viral particles from 4 h (days 0 to 5) to 1.8 min (days >5), on average, predominantly due to IAV-specific IgM. Even if we fix the value of cV at the innate level when estimating the rates of virus inactivation by IgG and IgM (kVG and kVM, respectively), the estimates nearly do not change, indicating that the overall model is insensitive to changes in cV. This is a particularly interesting finding in that early IgM responses would likely be without significant affinity maturation and thus have lower affinity for the viral antigens than later IgG. In considering these observations, keep in mind that virus titers drop 3 or more orders of magnitude from days 5 to 8. Experimental observations show that class-switched IgG antibody, which is generally dependent on virus-specific CD4+ T helper cells, does not become detectable in significant quantities in our data until day 7, a time when virus titers are relatively low. Therefore, the majority of free virus elimination occurs when IgM, plus the uptake of virus by target cells, phagocytosis, and mucociliary processes, numerically dominate. Nevertheless, we realize that the measurement of total IgG and IgM antibodies may overestimate the amount (and type) of antibodies ultimately involved in clearance of viral particles. Future experiments focused on measuring virus-antibody complexes may yield better insights.
It is worth discussing some of the unexpected ways in which these predictions and fitting results work together. If the viral load is dependent primarily on the rate of virus production, but not spread, then it becomes intuitive that the elimination of the virus “factories” by CTLs has the largest impact on reducing virus titers. This prediction is borne out by both the biological observations (2
) and the model fitting of the adaptive phase to the experimental data. In the same way, the effect of antibody-mediated inhibition of virus spread by IgG would have relatively little influence since it effectively diminishes infectivity. Infectivity of the virus was not found to significantly predict influenza viral titers.
Our results are also consistent with the other previously published modeling studies of the IAV immune response in human subjects (3
). Despite many biological and physical differences between mice and human subjects, many of the parameter estimates are remarkably close. In our study, the estimated overall infection rate of epithelial cells is 5.1 × 10−6
, while the overall death rate of infected epithelial cells is 1.2 per day. Baccam et al. (3
) investigated the kinetics of influenza A virus infection in human subjects using a target cell-limited model fitted to data on nasal IAV shedding obtained from six human subjects. The infection rates were estimated to be from 2.7 × 10−6
to 1.1 × 10−3
, which are in the same range of our estimates. The death rates of infected target cells were estimated to be between 2.1 and 13.5 day−1
, higher than our estimate of 1.2 day−1
. The difference in the epithelial cell death rates could be due to the relatively high virus production value assigned in our study, which was chosen based on direct measurements from in vitro
systems using cultured airway epithelial cells (6
). Based on these estimates, we fixed virus production at 100 EID50
and estimated innate virus clearance (cV
) as 4.2 day−1
. Baccam et al. (3
) estimated the viral production rates per cell to be 3.0 × 10−3
to 5.9 × 10−1
and the viral clearance rate to be 2.1 to 13.5 day−1
. In addition, Handel et al. (16
) investigated the development and spread of neuraminidase inhibitor resistance in humans infected by influenza virus. The estimated death rate of infected cells shown in the work of Handel et al. (16
) is 0.5 to 1.3 day−1
, which is very close to our estimates listed in Table (i.e., 0.60 day−1
for the early phase, 0.15 day−1
for the adaptive phase, and 1.2 day−1
for the combined two phases). The estimated virus death rate shown in the work of Handel et al. (16
) is 0.081 to 3.0 day−1
, which is lower than our estimated innate virus clearance rate of 4.2 day−1
In conclusion, we have constructed and validated a robust and quantitative model of primary influenza infection. Our data suggest that an effective adaptive immune response appears to begin around day 5 after infection, when viral titers are high. Our model indicates that the viral replication rate is more important than viral infectivity, with respect to viral titers, for a primary infection. In addition, virus-specific IgM and CD8 cytotoxic T lymphocytes are the most important factors in the adaptive immune response to clear virus in a primary infection. These findings suggest that future work might focus on increasing the early IgM response and vaccines which boost virus-specific CD8 T-cell production and memory.