A review of mathematical models of influenza A infections within a host or cell culture: lessons learned and challenges ahead

The most basic models considered to capture the dynamics of influenza infections, both in vivo and in vitro, consist of sets of ordinary differential equations (ODEs), namely

These models describe the dynamics of susceptible target cells, T, which become infected at rate β by the free virions, V . In model (1), the newly infected cells, I, immediately begin to produce virus, whereas in model (2), newly infected cells first undergo a latent or eclipse phase, E, before they become infectious, I, after an average time 1/k has elapsed. Infectious cells, I, are assumed to produce virions at a constant rate p, until they undergo apoptosis after an average time 1/δ. Finally, virus produced by infectious cells is eventually lost after an average time 1/c due to clearance mechanisms that include loss of infectivity (if the viral titer is measured in units of infectious virus, e.g., pfu, TCID₅₀), and binding with antibodies or mucus when analyzing in vivo experiments. Note that these models make the assumption of exponentially distributed latent and infectious periods, which were shown to be incorrect as they cannot reproduce the kinetics of certain experimental influenza infection assays (see Applications to in vitro systems). The use of more appropriate distributions in implementing these delays can alter the model behavior and estimates obtained from data fitting [28‐30].

The typical kinetics of these models is illustrated in Figure 2 for the model including a latent phase using the averaged parameters presented in Table 3 of [15]. These parameters correspond to the geometric average of a nonlinear fit of model (2) to viral titer from 6 human volunteers infected with influenza A/Hong Kong/123/77 (H1N1). Viral titer grows exponentially, peaks around 2–3 dpi, before decaying exponentially. Target cells are consumed rapidly, with the population of infected cells peaking around the same time as viral titer.

Because this type of model does not explicitly incorporate an immune response (IR), it is said to be target-cell limited, i.e. the virus load reaches its peak and subsequently declines once most cells have been infected and few susceptible cells remain. More accurately, the peak is reached when βTV ≈ δI. The target-cell limited nature of the models is clearly illustrated in Figure 2 where most target cells have been depleted by 54 hpi, around the time of viral titer peak. This almost complete depletion of target cells needs to be understood in the context of susceptibility: Cells susceptible to the virus and able to produce progeny virions as described by the model do not necessarily directly correspond to all epithelial cells in the respiratory tract. Indeed, it is not well known which cells contribute to the infection dynamics in the way described by the model. This likely varies between different influenza strains and hosts. Some cells might not be susceptible or be productively producing virus for instance due to reduced affinity of the virus for the types of receptors the cells express on their surface [6, 9‐11], or due to the protection provided by the presence or emergence of an IR not explicitly represented in the model.

The absence of an explicit IR in target-cell limited models is equivalent to assuming that either the effect of the IR on viral titer levels is negligible, or that its effect is somewhat constant through the course of the infection. In the latter case, the immune system can then implicitly be taken into account through parameters δ and c, which control the rate of loss of infectious cells (I) and virus (V ), respectively. And as mentioned above, the target-cell limitation itself can also act as an implicit IR by limiting the number of cells available for infection owing to the protective effect of the IR. Since viral titer in an influenza infection peaks around 48 hpi, whereas the primary adaptive IR is not detectable before approximately 5 dpi [31, 32], it might be a reasonable assumption to ignore the IR, though its role on infection clearance is still not fully resolved. We will return to this point later when we discuss models which incorporate an IR.

To our knowledge, the first mathematical model proposed to describe the within-host dynamics of an influenza infection was introduced by Larson et al. in 1976 [33]. The 7-compartment, 5-parameter model was fitted against the viral titer for mice infected with A/Aichi/2/68 (H3N2). The model could successfully reproduce the viral titer curves for virus sampled from the lung, the trachea, or the nasopharynx. While most of the five parameters could not directly be related to specific infection mechanisms (e.g., viral production rate, infected cell lifespan), two of them were of particular interest. Parameter P₁, which corresponds to the initial viral titer (V at time t = 0) is of interest because the compartment (lung, trachea, or nasopharynx) with the largest initial viral inoculum likely corresponds to the primary area of deposition of the infectious dose administered. Parameter P₂, which corresponds to the exponential viral titer growth rate is of interest because the compartment with the largest viral titer growth rate is that in which the virus reproduces most effectively. The fit of the model to the various viral titer curves indicated that virus replicated most effectively in the trachea, then in the lungs, with the poorest replicative efficiency found in the nasopharynx. Unfortunately, the viral titer sampling was sparse (every 24 h) often providing only one or two viral titer points from which to characterize the initial viral inoculum (P₁) and the exponential viral titer growth rate (P₂). Yet this work shows the early interest in mathematical modeling, and the promise it holds to characterize infection kinetics in a more quantitative way.

Thirty years after the Larson et al. model, Baccam et al. performed a study where they fitted a set of simple differential equation models to experimental viral titer for the course of an influenza infection within a host [15]. They first applied the target-cell limited models (1) and (2), which had previously been applied successfully to HIV [34, 35] and HCV [36‐38], to fitting viral titer of primary infection of human volunteers with influenza A/Hong Kong/123/77 (H1N1) [15]. Because the variables (target cells, latently infected cells, infectious cells, and viral titer) and parameters (e.g. viral clearance rate, cell lifespan) of the models correspond directly to biological processes, the parameter values obtained from the fits of the models to viral titers provided novel, quantitative information about the kinetics of the infection. While the reported best fit parameter values in this study largely agree with known biology (e.g., Table 1), the values should be used with caution due to the problem of overparametrization, which we will discuss below. It is also important to note that the authors did not perform a sensitivity analysis on the parameters of their models. A year later, Handel et al. also used a simple target-cell depletion model to fit human influenza data in the context of a study of neuraminidase inhibitor resistance emergence [39]. However, the study suffers from the same overparametrization issue as [15]. In addition, since the main focus of that study was not parameter estimation, the authors did not perform as careful an analysis as was done in [15]. For instance, no confidence intervals for parameter values were provided. Since then, additional modeling studies have been performed which incorporated components of the IR with varying levels of details. We discuss these models below.

More recently, Dobrovolny et al. [40, 41] have considered a simple extension of model (2) in which two cell populations are represented: a default and a secondary population. The two target cell model can capture the kinetics of uncomplicated infections as well as that of sustained and/or severe infections by incorporating the IR in an implicit way via the secondary cell populations. The default cell population is used to represent the readily accessible cells typically consumed by an influenza infection. The secondary population represents cells that are protected from infection in the case of seasonal infection, but are consumed to varying degrees in severe or chronic infections, perhaps due to differing cell tropism, lack of pre-existing immunity, or an aberrant IR. The two target cell model can also be applied to the study of the effect of cell tropism (different virus strains having different preferences for different cell populations) in cell cultures such as human tracheobronchiolar epithelium (HTBE) cells.

While models that ignore host factors such as the adaptive IR constitute an approximation of in vivo systems, they more accurately describe in vitro infections. In vitro experiments have long been used to carefully characterize specific aspects of the infection process, which could not be studied easily in vivo. The application of mathematical models to the analysis of in vitro infection systems allows for simple models, which can focus on the kinetics of cell-virus interactions alone, without the need to additionally consider a wide array of host factors, such as the IR.

Several studies of mathematical modeling of in vitro influenza infections, combined with experiments, have been undertaken by the group of Reichl and colleagues [42‐45]. They have focused their attention on studying the growth of influenza virus within microcarrier cell cultures, with the goal of characterizing and maximizing viral titer yield in these systems meant to produce virus for use in influenza vaccines. In 2005, Möhler et al. proposed a simple model for the kinetics of infection in a microcarrier of MDCK cells infected with an equine influenza A (H3N8) virus [42]. The model is similar to model (1), but includes the death and regeneration of target (uninfected) cells, the loss of virus due to adsorption onto target cells, and incorporates a fixed delay of 4.5 h between cell infection and the start of viral production, instead of an explicit eclipse phase as in model (2). The model in [42] provides a good fit to HA titer data. From their study, the authors conclude that viral yield in these systems can be most effectively maximized by increasing the total number of susceptible cells, upregulating viral production rate, and delaying the apoptosis of infected cells. In a follow-up study, Sidorenko et al. developed a Monte Carlo model for viral titer growth in microcarrier MDCK cultures infected with an equine influenza A (H3N8) virus that incorporates both intra- and inter-cellular infection kinetics. Rather than explicitly representing all aspects of intra-cellular viral replication (a topic the group addressed in a separate study [46]), they characterize a cell’s intracellular infection state as different classes representing cells that contain different numbers of intra-cellular virus equivalents. This allowed the authors to fit not just the viral HA titer, but also the fluorescence distribution for the population of infected cells measured through flow cytometry of cells stained using antibodies against the virus M1 and NP. These and further studies by this group [42‐45] offer a unique look at the process of influenza viral infection in microcarrier cell cultures and a rare opportunity to develop and refine intra-cellular models for influenza viral replication by providing high quality experimental data.

The analysis of in vitro data using mathematical models can also reveal infection parameters buried within experimental data. For example, in 2008, Beauchemin et al. used models (1) and (2) to analyze the viral titer over the course of experimental infections of MDCK cells with influenza A/Albany/1/98 (H3N2) in a hollow-fiber reactor under different concentrations of the antiviral drug amantadine [47]. The aim of the work was to characterize the effect of amantadine treatment on the course of the infection. Using different variants of the target-cell limited mathematical models, Beauchemin et al. were able to determine the IC₅₀ (0.3–0.4 µM) and maximum efficacy (56–74%) of amantadine at blocking the infection of susceptible cells. Research by Beauchemin and colleagues has also focused heavily on the application of mathematical models to the analysis of in vitro infections, with special attention to the properties of in vitro assays. In Holder et al., two mathematical models were constructed to reproduce the course of an influenza infection in two different viral assays: a plaque and a viral yield assay [48]. The aim of the project was to determine if and how the fitness of the oseltamivir-sensitive wild-type (WT) A/Brisbane/59/2007 (H1N1) differs from that of its H275Y (N1 notation) oseltamivir-resistant counterpart. Interestingly, while the plaque assay suggested that the WT strain was fitter (exhibited a more rapid plaque growth), the viral yield assay suggested that instead the H275Y mutant was fitter (exhibited a larger exponential viral growth rate). Using mathematical equivalents of the assays to run a large number of simulated experiments, Holder et al. uncovered that plaque assays, due to the spatial restriction of the overlay on infection spread, were most sensitive to the length of the cell’s eclipse phase, whereas the viral yield assay was equally sensitive to virus infectivity, viral production rate, and the length of the cell’s eclipse phase. This difference in the sensitivity of the assays to different aspects of the viral replication cycle explains why the two assays appear to provide contradictory conclusions about the fitness of the two strains. Thus, mathematical models can help shed light on the limitations or caveats of in vitro assays. In return, in vitro assays can teach us about fallacies in the formulation of our mathematical models. Holder et al. investigated how different in vitro assays can inform model development [30, 49]. Notably, they showed that use of either exponential or Dirac-delta distributions for the times spent by cells in the latent or productively-infected state are not consistent with experimental results from single-cycle viral yield experiments, whereas normal and lognormal time distributions are. From these studies, it is clear that a greater synergy between experiments and mathematical models is highly desirable.

As discussed earlier, the kinetics of the viral titer over the course of an influenza infection is well captured with a simple model that does not include an IR. Instead, one can account for the decline in viral load by attributing it solely to the complete depletion of target cells. While complete cell depletion — even if restricted to a localized patch of cells — may appear excessive, at least one histological study of influenza infection of ferrets supports this idea. In [50], Francis and Stuart-Harris examined the lungs of ferrets infected intranasally with a sub-lethal inoculum of influenza virus and noted desquamation of the tracheal area by 2 dpi, which progressed to a complete destruction of the epithelium. Despite this severe insult, the animals survived and their epithelial tissue fully regenerated within a few weeks. While ferrets are generally considered a good animal model for human influenza infections, it is not clear how applicable this result is to influenza infections in humans [51]. Reports of immunocompromised patients shedding influenza virus for prolonged periods suggest that the IR plays an important role in clearing infection, or at least in preventing chronic and potentially fatal outcomes [52‐55]. The IR is likely to be especially important in more severe influenza cases where infection also involves the lower respiratory tract [56, 57], as observed in infections with avian-origin H5N1 [58, 59], and 1918 pandemic influenza strains [60, 61]. In fact, it is quite likely that some of the virulence of influenza strains such as H5N1 or the 1918 H1N1 is due to an over- or mis-responding IR causing immunopathology, though the details are yet to be resolved [62‐66]. Further evidence of the crucial role of the IR comes from animal studies in which components of the IR are depleted. Those studies taken together suggest that both innate and adaptive IR are important [32, 67, 68]. Thus, to gain a comprehensive understanding of the progression of an influenza infection within a host, i.e. to understand specifically how host factors shape the course and outcome of an influenza infection, models incorporating an IR are essential. Several recent modeling studies have taken the IR into account, we will discuss those in the next section.

To our knowledge, the first influenza modeling study that included components of the IR was a very detailed ODE model developed by Bocharov et al. in 1994 [69]. The model tracked macrophages, CD4⁺ T-cells and CD8⁺ Cytotoxic T Lymphocytes (CTL), B-cells, antibodies and interferon (IFN) in a very detailed manner. The authors used the model to analyze how different components of the IR affect infection kinetics. In particular, they found that a 50-fold increase in specific antibodies and CTLs could prevent an infection from occurring. Another model that is similar in detail to the Bocharov et al. model has recently been developed and studied by Hanciglou et al. [70]. The authors analyzed the effect of initial viral load on the infection kinetics and found that for small initial viral load the disease progresses through an asymptomatic course, for intermediate value it takes a typical course with constant duration and severity of infection but variable onset, and for large initial viral load the disease becomes severe. Two other models, by Chang et al. [71] and Tridane at al. [72], are simpler models based on a variant of the target-cell limited model (2), with an additional component to describe the dynamics and effect of CTLs. The Chang et al. study also included IFN. Chang et al.’s model suggested that the time and level of virus peak is influenced by the innate (IFN) response, while the duration of the infection and clearance phase is determined by the CTL response. Tridane et al. focused in their study on investigating the impact of different model choices for the CTL response on the infection dynamics and found that slight changes in how the CTL dynamics is implemented can influence the resulting dynamics.

Some of the results obtained from these models could have important implications for treatment or vaccine strategies. However, large uncertainty with regards to parameter values and overall model structure make it difficult to evaluate the validity of the models and their predictions. While all these models [69‐72] were constructed and parametrized based on the known biology of influenza infections, until such models are brought into direct contact with data for model validation or falsification, the results should mainly be considered as providing conceptual insights.

Several other modeling studies have been performed that were based on a direct connection between models and data. In the study by Baccam et al. [15] already mentioned above, the authors employed one model that included an IR component, namely IFN. Fitting such a model to data, they found that while the fits obtained from this model were not statistically significantly better, they could reproduce a double peak in virus load, something that was observed in a few of the patients they studied. In another study dealing with the issue of drug resistance emergence during infection, Handel et al. [39] fitted both a model with and without an IR. The latter included a very simplistic version of an antibody response. The authors found that the available data did not permit discrimination between the two models.

More detailed and comprehensive models that compared various IR models to data have since been published. Lee et al. [73] developed a model that included all the major effectors of the adaptive IR (CD4⁺ and CD8⁺ T-cells, B-cells/antibodies) as well as explicit descriptions of the IR activation process mediated by dendritic cells. They were able to compare their model to (sparse) data and thereby partially validate it. A follow-up study by the same group made use of extensive experimental data specifically collected for the purpose of fitting the model [31]. In this study, Miao et al. used several ingenious ideas for fitting their models to the data. They divided the course of the infection into an early and late phase, corresponding to infection kinetics in the absence and presence of an adaptive IR, respectively. They also did not attempt to construct ODEs to capture the dynamics of IgA and IgG antibodies, and CTLs. Instead, they fitted smooth curves to the data sets, which were then used as given quantities in their ODE model. The separation of infection dynamics into an early and late phase and the large amount of data the authors had available allowed for a more detailed analysis of the importance of different aspects of the IR (innate, adaptive) during the course of the infection, and the authors were able to estimate important kinetic parameters and how they varied during the infection (see Table 1).

Another recent study by Saenz et al. used a unique dataset of virus load and infected cell levels in ponies to study a model that included an IFN response [74]. The study suggested that inclusion of an IFN response was sufficient to describe the dynamics, while a model without it did not fit the data well. It is still possible, and quite likely, that other model alternatives with adaptive IR components might fit the data equally well, therefore the relative importance of IFN is not yet fully established.

Yet another modeling study using data from two different experimental studies in mice with intact and compromised IRs was conducted by Handel et al. [75]. The authors compared different models with and without both innate and adaptive IR components. They found that both the innate and adaptive IRs are required to provide adequate explanation of the data. For one of the datasets investigated, the authors found that they could not describe the IFN data using a simple equation such as those used in other studies [15, 74]. Instead, Handel et al. used the IFN level as given and fitted the remaining data to the dynamical model using an approach similar to that used in [31]. The study also showed that a discrimination between different types of adaptive IRs (i.e. T-cell versus B-cell/antibody) was not possible based on the available data [75].

The studies described thus far are based on differential equations. Another class of models, spatial agent-based models, were also proposed [76, 77]. In 2005, Beauchemin et al. introduced an agent-based model for the spread of influenza within a host. In 2006, Beauchemin used this model to explore how spatially localized effects can come to shape and dominate the course and outcome of the infection. While the model is more akin to a toy model, its analysis did yield some insight into the effect of spatial versus well-mixed implementation of cell regeneration and CTL expansion on chronic infection outcomes.