Department of Physics, Ryerson University, Toronto, ON, M5B 2K3, Canada

Department of Epidemiology and Biostatistics, College of Public Health, University of Georgia, Athens, GA 30602, USA

Abstract

Most mathematical models used to study the dynamics of influenza A have thus far focused on the between-host population level, with the aim to inform public health decisions regarding issues such as drug and social distancing intervention strategies, antiviral stockpiling or vaccine distribution. Here, we investigate mathematical modeling of influenza infection spread at a different scale; namely that occurring within an individual host or a cell culture. We review the models that have been developed in the last decades and discuss their contributions to our understanding of the dynamics of influenza infections. We review kinetic parameters (e.g., viral clearance rate, lifespan of infected cells) and values obtained through fitting mathematical models, and contrast them with values obtained directly from experiments. We explore the symbiotic role of mathematical models and experimental assays in improving our quantitative understanding of influenza infection dynamics. We also discuss the challenges in developing better, more comprehensive models for the course of influenza infections within a host or cell culture. Finally, we explain the contributions of such modeling efforts to important public health issues, and suggest future modeling studies that can help to address additional questions relevant to public health.

Introduction

The influenza A virus causes annually recurring epidemic outbreaks, most people become infected multiple times over their lifetime

An influenza A infection is typically initiated following the inhalation of respiratory droplets from infected persons. These droplets containing influenza virions (virus particles) first land on the mucus blanket lining the respiratory tract ^{+} cytotoxic T lymphocytes (CTL) are first observed around 5 dpi, peaking around 7 dpi, whereas in a secondary infection Abs and CTLs can respond as early as 3 dpi

Kinetic parameters for influenza obtained from both fitting mathematical models to data and by direct estimation from experimental data.

**Parameter**

**Values [References]**

Mathematical models to fit experimental data

Average lifespan of an infected cell

39h

Average infectious lifespan of a virion

111h

Length of the latent (eclipse) phase

6h

Rate of epithelial cell (re)growth per day

0.72 ^{-8} and 0.34

Drug efficacy

0.97 and 0.99

Lifespan of interferon

3.5h

Direct experimental measures

Average lifespan of an infected cell

12–48h

Average infectious lifespan of a virion

0.5–3h

Length of the latent (eclipse) phase

3–12h

Lifespan is defined as the inverse of the rate parameters (the sometimes alternatively used half-life contains an extra factor of log(2)). Note that some of the studies are

Course of an influenza infection within a host.

**Course of an influenza infection within a host.** The timings of the adaptive immune response, namely Antibodies (Abs) and cytotoxic T lymphocytes (CTL), for both a primary (PR) and secondary (SR) response to an influenza infection are indicated.

Several aspects of influenza infections are still unresolved. For instance, the contributions of strain-specific cell tropism, pre-existing immunity, and host genetic factors in shaping the virulence and transmissibility of a particular influenza strain are not well understood

Much work has been done on attempting to capture the dynamics of influenza A using mathematical models; almost all of these models are on the host population level and are concerned with transmission between infected hosts. These models can be used as tools to inform public health decisions with respect to pandemic planning: whom, how and when to quarantine, vaccinate, treat with antivirals, and how much and what to stockpile

Here, we focus on a lesser known application of mathematical modeling to the study of influenza kinetics, that aimed at understanding and quantifying the processes involved in determining the severity, duration, and outcome of the progression of the infection within a host or a cell culture. These types of models provide information of a different nature, but, as we will outline below, the information they provide can be equally critical for better treatment and management of the disease. Furthermore, the development of reliable within-host models is critical to improving epidemiological models since the latter relies on the former to more accurately capture the diversity of infection severity, latency, and symptoms.

We first present a survey of the published literature on within-host and in vitro modeling of influenza infections (see also

Mathematical models of within-host influenza dynamics and their contributions

Simple models without an immune response

Overview of the models

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, _{50}), 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

The typical kinetics of these models is illustrated in Figure

Typical kinetics exhibited by the target-cell limited model with a latent phase predicting the course of an influenza infection within a host.

**Typical kinetics exhibited by the target-cell limited model with a latent phase predicting the course of an influenza infection within a host**. We can see that the target cells (^{-}^{5} ([V] ^{-}^{1}^{-}^{1}^{-}^{1}^{-}^{2} [V]^{-}^{1}), and initial conditions, (_{t}_{=0} = (4 ^{8}^{2} [V]), where [V] is TCID_{50}/mL of nasal wash are from Table 3 of

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

Applications to in vivo systems

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

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 _{1}, which corresponds to the initial viral titer (_{2}, 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 (_{1}) and the exponential viral titer growth rate (_{2}). 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

More recently, Dobrovolny et al.

Applications to in vitro systems

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

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 _{50} (0.3–0

More extensive models which incorporate an immune response

The importance of the immune response to influenza infections

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

Influenza models incorporating an immune response

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 ^{+} 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.

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

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.

More detailed and comprehensive models that compared various IR models to data have since been published. Lee et al. ^{+} 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

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

Yet another modeling study using data from two different experimental studies in mice with intact and compromised IRs was conducted by Handel et al.

The studies described thus far are based on differential equations. Another class of models, spatial agent-based models, were also proposed

Lessons learned and challenges ahead

Conceptual insights and parameter estimates obtained from the models

At the most fundamental level, models can be used to explore a complicated dynamical system, and to gain basic insights into the relative importance of host and viral factors. Such models can either be simple and try to capture only the most fundamental interactions making up the kinetics of the infection, or they can be detailed and try to integrate most of the known biological processes. Unfortunately, the quantity and diversity of available data is usually limited and the results from many modeling efforts therefore remain predominantly conceptual and qualitative. These models are still excellent tools that can help shape our understanding of infection kinetics. The power of modeling has been well documented for HIV

The spatial influenza models described at the end of the previous section

Once sufficient understanding of a system has been obtained and data are available, one can formulate mathematical models that encapsulate specific mechanistic hypotheses. By comparing the models with data, one can discriminate between those hypotheses. For instance, in the work by Saenz et al.

Once a model has been found that can be trusted to reasonably approximate the biology (i.e. the model is well-supported by a fair amount of data), one can fit the model to experimental data to obtain estimates for the kinetic parameters of a system. This is especially helpful if such parameters cannot easily be obtained through direct experimental measurements. The parameters one can estimate depend on the specific model and the data available. Almost all models published so far include parameters for the average lifespan of an infected cell and the half-life of virions. Other parameters that have been estimated from some models include the average length of the eclipse (latent) phase, the growth or replacement rate of new susceptible cells and the efficacy of drug treatment. In Table

Data diversity and quantity and its effect on parameter identifiability

Quantitative knowledge of infection parameters could provide much needed answers to many important questions. For example, say one could determine quantitatively from experiments the rate at which virions of a given influenza strain are produced in a given cell line. Using this information one could compare strains with respect to their replication efficiency, could map how specific mutations within a given strain affect specifically viral production, or what concentration of an antiviral is required to block viral production by a specific amount. What, then, are the roadblocks in making this goal a reality? One is insufficient data, both in terms of diversity, quality, and quantity. While more complete and complex mathematical models can be developed readily, use of these models to predict infection kinetics or to estimate unknown parameters is questionable if critical aspects of the model or key parameters are unknown or too poorly supported by experimental data. Thus, if one is to use mathematical models to extract parameter values, one can only add as many components as can be determined from data, and this, ultimately, is what limits the complexity of the models

Consider, for example, the target cell limited models (1) and (2). These models are well-suited to the analysis of viral titer curves from in vitro or in vivo uncomplicated infections because, like the models themselves, these curves typically follow a simple shape: a period of exponential viral growth (_{g}_{p}_{p}_{d}_{0}_{g}_{d}_{g}

These expressions illustrate how the determination of

Ultimately, since viral titer courses can be well described using just four parameters

Reconciling the disconnect between experimental measures and model variables

To allow easy comparison with experimental data, mathematical models are typically described in terms of variables, which correspond to, or can easily be related back to, experimentally measurable quantities. Unfortunately, these quantities (e.g., plaque forming units, fluorescence level) are at best relative measures of quantities of interest (e.g., infectious viral titer levels, proportion of cells infected) and are often related to quantities of interest in nontrivial ways. For example, the number of plaque forming units (pfu) in a viral sample is thought to correspond to the number of virions that are infectious to the cells in the culture used to measure the viral pfu. If the virus solution is collected from a ferret but the viral pfu is measured in MDCK cells, one has not determined the number of virions infectious to ferrets, but rather those infectious to MDCK cells. When using a count of the total number of virions rather than the number of infectious virions (using RT-PCR, for example), models typically assume that the fraction of infectious to non-infectious virions over time remains constant. However, since influenza virions loose infectivity faster than they lose RNA integrity

The units used to measure the experimental quantities will often “contaminate” the model’s parameters, greatly limiting the usefulness of their values. One way to emphasize the effect of relative measures on a model’s parameters is to perform a rescaling of each variable to see how each parameter will be affected. For example, using model (2), let us consider the rescaling of the cell population _{m}_{m}_{m}_{m}

where only the virus infectivity,

While knowing the relative value of

It is also important to understand that parameters extracted by applying a mathematical analysis to experimental data can sometime also be extracted from experiments. However, the parameters extracted experimentally may not be equivalent to those extracted through mathematical modeling of experimental data. For example, several in vitro assays are traditionally used to estimate the IC_{50} of a drug against a particular virus strain. In such assays, the IC_{50} represents the drug concentration required to half the viral titer or fraction of dead cells observed experimentally compared to that seen for an untreated infection at a given time post-infection. Since in these experimental assays the IC_{50} is defined as the concentration required to half a certain experimental observable, the IC_{50} estimated in this manner varies between different techniques and assays, and cannot readily be compared. In contrast, mathematical models define the IC50 as the concentration of drug required to half a specific viral replication parameter (e.g. virus production rate by an infected cell) _{50} estimates thus obtained are more robust and should be readily comparable for a given cell-virus strain pair, irrespective of the details of the experimental procedure followed. As such, mathematical models may present a preferable approach to extracting the IC_{50} for a given drug-strain pair.

Discussion

Public health contributions of mathematical models

The usefulness of mathematical modeling for public health has long been recognized at the between-host population level. Starting as early as the work of Bernoulli

Future directions

It is obvious that a general direction for the future is to perform more and more detailed modeling studies, preferably in close contact with appropriate data (i.e. data of high quality, quantity, and diversity). The recent combined experimental and modeling work by Miao et al.

While understanding the infection dynamics per se is a useful and necessary first step, in the end we are interested in outcomes that are important from a medical or public health perspective. For instance, can we develop within-host models that can produce and predict quantities such as “virulence” or “transmissibility” as readouts? Several tentative steps in this direction have recently been made

To further improve treatment and intervention strategies, we need to better understand their impact and consequences. With the increasing level of antiviral resistance in circulating influenza strains, much activity is currently ongoing to investigate the usefulness of drug combination therapy for the treatment of influenza, a strategy similar to that already employed with HIV

Another area of interest is to better understand vaccines and vaccine efficacy

One active area of research in biology in general is the development of multi-scale models

While the development of new and more detailed models and data will be important, it is equally important to improve the rigor with which models are analyzed. For instance, extensive sensitivity analysis, as has been used in infectious disease modeling

While our focus in this review has been on

Summary

In our opinion, mathematical and computational models are powerful tools to study the infection dynamics of infections. The last few years have seen increased interest in such modeling studies, and we are likely going to see further increases in such studies in the future. We believe such studies can do for influenza what similar studies have already done for infections such as HIV or HCV. To achieve this, it will be crucial that the models be connected to data as tightly as possible, and that the model type and complexity is appropriate for the question one wants to address. As long as these simple rules are followed, we have no doubt that modeling will continue to provide important insights into the infection dynamics and will eventually help us address several of the questions mentioned in the previous section, as well as many others. In addition, much of the progress will not only benefit our understanding of influenza, but will also help to study other acute infections on the within-host level, an area that is still much less developed compared to similar studies at the between-host level.

List of abbreviations

IR: Immune response; HA: Hemagglutinin; NA: Neuraminidase; dpi: Days post-infection; hpi: Hours post-infection; CTL: Cytotoxic T lymphocyte; Abs: Antibodies; ODE: Ordinary differential equation; TCID_{50}: 50% Tissue culture infectious dose; IC_{50}: 50% Inhibitory Concentration; pfu: Plaque forming units; MDCK: Madin Darby canine kidney; IFN: Interferon

Competing interests

AH declares that he has no competing interests. In the preparation of

Authors' contributions

CAAB and AH were both responsible for drafting the manuscript and contributed equally. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (CAAB).

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