Volume 11 Supplement 1
Exploring the effect of biological delays in kinetic models of influenza within a host or cell culture
© Holder and Beauchemin; licensee BioMed Central Ltd. 2011
Published: 25 February 2011
For a typical influenza infection in vivo, viral titers over time are characterized by 1–2 days of exponential growth followed by an exponential decay. This simple dynamic can be reproduced by a broad range of mathematical models which makes model selection and the extraction of biologically-relevant infection parameters from experimental data difficult.
We analyze in vitro experimental data from the literature, specifically that of single-cycle viral yield experiments, to narrow the range of realistic models of infection. In particular, we demonstrate the viability of using a normal or lognormal distribution for the time a cell spends in a given infection state (e.g., the time spent by a newly infected cell in the latent state before it begins to produce virus), while exposing the shortcomings of ordinary differential equation models which implicitly utilize exponential distributions and delay-differential equation models with fixed-length delays.
By fitting published viral titer data from challenge experiments in human volunteers, we show that alternative models can lead to different estimates of the key infection parameters.
In the past decade, mathematical models of viral infection have been successfully applied to a number of problems on the periphery of the annual public health problem that is influenza . In the laboratory, mathematical models have aided the development of efficient vaccine production techniques  and improved the quantitative characterization of antiviral drug action . Mathematical models have also improved our understanding of the course of the disease within human  and animal hosts . Because these models serve as a bridge between the microscopic scale (where virus interacts with cell) and the macroscopic scale (where the infection is manifested as a disease) they will inevitably be applied in the future to pressing public health questions such as the estimation of virulence and fitness for emerging strains, the spread of drug resistance and, more generally, the connections between viral genotypic information and clinical data.
The success of a within-host virus infection model depends on an accurate representation of biological reality. This allows a model not only to describe the phenomenon under consideration, but also to make reliable predictions about unobserved consequences. For example, in 1995 a simple model of HIV dynamics was applied to describe the observed exponential clearance of virus under the administration of a drug suppressing viral production . The primary result of this work, however, was not the description of viral clearance itself, but the prediction of dynamics in the absence of drug, i.e., that high viral clearance must be balanced by high viral production, which in turn allows for extremely rapid mutation of the virus strain. This conclusion had important implications for the development of therapy, specifically the necessity of a “drug cocktail”. For influenza infections, the primary clinical data available to a mathematical modeler is the viral titer over the course of an infection, usually obtained by a daily nasal wash collected from an infected patient. This data generally follows a simple functional form in time which can be reproduced by a variety of dynamical models. Thus, if meaningful information is to be extracted from such data, the model applied must already be a trusted simulator of the underlying infection kinetics. In this paper, we consider evidence from laboratory infection experiments which must inform the construction of a mathematical model, focusing specifically on the implementation of the time spent by a cell in each of the various stages of infection.
The implementation of a particular dynamical structure on this basic model requires a more detailed specification of the biological processes. The infection of cells (the transition of target cells to latently-infected cells) has been observed to be a Poisson process where the rate of infection is proportional to the local virus concentration  and it is implemented in the model as a continuous representation of that stochastic process. Virus production by infectious cells can be assumed to proceed at a constant rate and the infectivity of free virus is known to decrease exponentially in time [3, 9], leading to a simple equation for virus dynamics. To complete the dynamical description, one must specify how a latently infected cell becomes infectious and for how long infectious cells produce virus. In other words, one must specify the distribution of the delays, t L and t I , between the states of infection.
In an epidemiological context, the problem of implementing generic delays between infected classes was first considered by Kermack and McKendrick in their seminal 1927 work on infectious disease dynamics . Hethcote and Tudor  introduced a general approach to the problem, using a probability density function for the time spent in a given state, which has been applied frequently in the field of mathematical epidemiology (see, e.g., [12–14] and references therein). Here, we will apply the same approach to within-host influenza viral infections, resulting in a model with differential equations to describe target cell and virus dynamics, and integral equations to describe the latent and infectious cell populations (a similar approach was considered for HIV in ).
Despite questions about their biological appropriateness, ODE models have had success in describing in vivo infection data (for influenza see, for example, [4, 5]). Models with non-exponential delays have been similarly successful, including those with Dirac delta function transition distributions, leading to a delay-differential model [3, 4, 22]; and multi-compartmental ODE models (with n sequential phases of infection) yielding delays with a gamma-function distribution [23–25] Here, we consider a set of in vitro experiments which allows for some discrimination between models, namely the single-cycle viral yield assay. By fitting models with different transition distributions (Figure 2) to single-cycle assay data, we show that the correct implementation of delays is crucial to the success of a model in describing these assays. Using these results, we consider in vivo data from challenge experiments in humans to explore how the choice of delays affects the parameter values extracted when fitting the model to experimental data.
where T, L, I are the populations of cells in the target, latently-infected and infectious (virus-producing) states, respectively, and N the total number of cells in the system. V is the virus concentration, β and p are the rates of infection and virus production, respectively, and c is the viral clearance rate. The function P L (t) is the probability that a cell remains in the latent state for at least a time t before transitioning to the infectious state, and P I (t) is the probability that a cell remains in the infectious state for at least a time t before transitioning to the dead state (i.e., before it ceases to release virus). The transition profiles for different choices for the expression of P(t) are illustrated in Figure 2. f L is the probability density function for the time a cell will spend in the latent state before transitioning to the infectious state (f L = –dP L /dt). Note that f I (t) does not explicitly appear in the model.
When a Dirac delta function is used for both f L and f I (fixed-delay), such that the times spent by cells in the latent state and the infectious state are exactly τ L and τ I , respectively, Equations (1a–1d) reduce to a set of delay differential equations (DDE) .
where erfc(x) is the complementary error function (i.e., erfc(x) = 1 – erf(x)) and the standard deviations σ L and σ I are dimensionless quantities.
and for the truncated normal, the median is found by setting its cumulative distribution function to one-half (there is no simple analytical expression).
Numerical evaluation of the model in Equation (1) was performed using a modified Euler technique. At every time step of length ∆t, newly infected cells, Lnew = βTV ∆t, were removed from the target population. The passage of these Lnew cells through the latent and infectious states was then calculated for all future times using f L , P L and P I and added to that of previously transitioned cells. Virus dynamics at each time step were calculated according to the Euler approximation of Equation (1d). Simulations of single-cycle in vitro experiments were initialized with L(0) = N; simulations of in vivo infections were initialized with T(0) = N and V (0) = V0.
Model fitting and parameter extraction
In any model-fitting exercise, a number of considerations must be made to ensure the reliable extraction of parameter values from experimental data. First, one must consider the question of parameter identifiability [36, 57, 58]: If the experimental system were to exactly reproduce the dynamics of the model equations, could the parameters be uniquely identified from the available observations? A number of techniques have been introduced to address identifiability for ODE models [37, 59], but these are not directly applicable to the more general system considered here (Equation (1)). Nevertheless, for each experiment considered here we attempted to reduce consideration to an identifiable set of parameters. For the early phase single-cycle viral yield experiments, we fixed parameter values involving viral clearance and infectious cell death, and fitted only those parameters related to viral production and the transition of cells from the latent to infectious state. In the single-cycle, single-history experiment, independent information on viral clearance allowed for that parameter to be fixed in fitting. For the in vivo volunteer patient infection data, we have arbitrarily fixed the product of the viral production and infection rates in order to obtain a unique solution in the fitting procedure.
A second consideration when fitting experimental data is the question of model error: Is the mathematical model an appropriate representation of experimental data? To address this question, we performed least squares fitting of the model equations to the data sets using the Octave 3.2.4  implementation of the Levenberg-Marquardt algorithm, leasqr. For all data presented here, fits were performed to the viral titer data in log-space and the sum of squared residuals (SSR) was calculated as
where k is the number of model parameters, n is the number of data points and AICc is the “corrected” form of the AIC for small sample sizes .
Finally, to account for measurement error, we calculated 95% confidence intervals for each reliably extracted parameter value by fitting 1000 bootstrap replicates . Confidence intervals were not calculated for models fits with large error (high SSR and AICc) or for fits to volunteer patient infection data, where arbitrary assumptions were made to constrain the fitted parameter values.
Calculation of the basic reproductive number
It can also be shown using the ODE that this quantity is equal to the commonly-quoted definition of the reproductive number: the number of secondary infections caused by one infectious cell, in a completely susceptible cell population. For other delay models, where an analytical form is not readily available, we calculated R0 numerically according to that statement, i.e., for a given set of parameters, we disallowed latent to infectious transitions, initialized the simulation with one infectious cell, and determined the number of cells in the latent state as t → ∞.
Results and discussion
General features of single-cycle viral growth
Single-cycle growth (SCG) viral yield experiments provide a unique view of viral replication. By initiating infection with a viral inoculum of high concentration (a multiplicity of infection (MOI) much larger than one), all cells are infected simultaneously, and the experimentalist effectively synchronizes the cells’ passage through the phases of latency, viral production and death. The resulting viral production curve can then be viewed as that of the average cell. This is in sharp contrast to “multiple-cycle” yield experiments (MOI ≪ 1), where only a few cells are initially infected, leading to successive cycles of infection; the resulting exponential growth of both infected cells and virus over time effectively masks the dynamics of a single cell. Virus infections of humans and animals are similar to this latter experiment in that they are likely initiated by the infection of a only a few cells [26, 27], leading to the exponential consumption of a large target cell population. Thus, SCG experiments demonstrate an artificial infection dynamic which would never occur in nature. However, their depiction of the average virus production of a cell makes them an invaluable tool for model building and for isolating specific components or parameters of the viral replication cycle.
Information on the individual single-cycle viral yield experiments plotted in Figure 3
(A) A/Udorn/307/72 (H3N2)
(8.2 ± 0.3) × 105
10.9 ± 0.8
(B) A/Udorn/307/72 (H3N2)c
(2.7 ± 0.6) × 105
12 ± 5
(C) A/Udorn/307/72 (H3N2)
(3.1 ± 0.9) × 106
(6 ± 3)
(D) A/Udorn/307/72 (H3N2)d
(2.6 ± 0.1) × 106
5.8 ± 0.4
(E) A/Udorn/307/72 (H3N2)
(5.6 ± 0.3) × 106
(6.1 ± 0.4)
(F) A/PR/8/34 (H1N1)
(4 ± 1) × 103
6 ± 3
(G) A/PR/8/34 (H1N1)
(7.5 ± 0.4) × 105
(9.8 ± 1)
(H) A/X-31 (H3N2)
(1.34 ± 0.01) × 106
(3.75 ± 0.01)
(I) A/NWS/33 (H1N1)
(1.26 ± 0.07) × 105
10.8 ± 1
We will consider below the characterization of the SCG experiment using various dynamical models, but some simple analysis can be done using only the empirical relation above. For example, two experiments (Figure 3A-B and Figure 3C-D) considered the growth of influenza A/Udorn/307/72 (H3N2) and a counterpart strain possessing a single mutation in the NS1 gene (T215A and R83A, respectively) [28, 29]. In each case, the experiment reveals a significant reduction of the approximate viral production rate p* for the mutant, without a significant change in the approximate latency period. This shows that the single-cycle experiment can highlight important biological characteristics of a virus strain, with very little mathematical analysis.
Characterizing the latent infection period from single-cycle growth assays
Fits to two single-cycle viral yield experiments (Figure 4)
A/Udorn/307/72 (H3N2) 
τ L (h)
2.4 × 106
8.1 × 103
7.7 × 105
3.3 × 103
8.3 × 105
3.4 × 103
8.2 × 105
1.8 × 106
4.8 × 103
2.6 × 106
5.3 × 103
5.2 × 106
A/NWS/33 (H1N1) 
τ L (h)
1.4 × 1026
4.8 × 1023
3.3 × 1023
1.2 × 105
4.7 × 102
1.5 × 105
5.3 × 102
2.3 × 105
4.5 × 1031
9.1 × 1028
6.3 × 1028
1.8 × 103
9.6 × 104
9.3 × 102
3.8 × 105
1.1 × 103
1.2 × 106
Both the normal- and lognormal-delay models provide an adequate description of the data over the entire range of values and for both values of viral clearance, although the SSR and AICc values are smaller for fits using a normal distribution. When the fits to the log-valued virus are viewed in linear-space (inset graphs), the normal fits appear to be a more reasonable approximation of the data in that the linear SSRs of these fits, which depend most sensitively on the larger virus values, are also smaller. When viral clearance is neglected, the fitted values of the viral production rate, p, are close to the approximate values of p* (Table 1). Non-zero viral clearance leads to a larger fitted production rate, as expected. The fitted values for the median latent infection period, , vary depending on the distribution type, but are always as long as the approximate values of for both experiments (10.9 h for  and 10.8 h for ). The introduction of a non-zero viral clearance leads to even longer latent infection periods, ranging from 8 h to 24 h. The fitted standard deviations, σ L , are between 1.5 and 3.0 h for the normal distribution and between 0.15 and 0.44 for the lognormal distribution.
Characterizing the infection cycle from a single-cycle, single-history yield assay
In 1968, an in vitro experiment was performed which, to our knowledge, is unique in the literature . Like the experiments presented in the previous sections, a SCG experiment was prepared: ~ 107 cells were incubated with a high titer (MOI = 10) of influenza A/PR/8 (H1N1) virus such that almost all cells were infected and then the infection medium was removed. Unlike typical SCG experiments, however, viral titer was not measured by sacrificing independent wells at each sampling time to titrate their overlay. Instead, the liquid overlay from the same well was removed in its entirety and replaced with fresh, virus-free medium and the infection was allowed to continue. Thus, titrations of the collected overlay medium provided a measure of the amount of virus being produced by a single cell culture at the time of collection. Application of this sampling protocol to SCG experiments — which we refer to as a single-cycle, single-history yield experiment (SCSH) — mitigates complications tied to the accumulation of virus, and brings into focus the viral production of the cell culture as a series of snapshots over time, all sharing a common kinetic history.
τ L (h)
τ I (h)
1.8 × 106
7.5 × 105
5.4 × 106
2.0 × 106
The resulting transformed data is shown in Figure 5. Viewed in this way, one can explicitly see the growth of the infectious cell population, including a steep rise from 4 h to 12 h as cells transition from latent to infectious, and a long decline between 24 h and 48 h as infectious cells cease viral production. The model dynamics for pI/N, generated using parameters from the fit to the raw data, agree well with the transformed data, validating the use of Equation (1d) and the assumptions made in the above transformation.
While the normal-delay and lognormal-delay models both lead to an adequate description of the experimental data, a consideration of the fitted parameter values for the median infectious cell lifespan, , reveals vastly different underlying dynamics. In the normal-delay case, the median infectious lifespan is predicted to be short but the associated standard deviation is large. The long decay of infectious cells at late times is therefore explained by a broad distribution in the times spent by cells in the infectious state. Using the lognormal-delay model, however, the best fit is nearly that of a fixed-delay (σ I ≪ 1), with a long median infectious lifespan Under this assumption, the decay of infectious cells is thus completely determined by the long tail in the distribution of latently infected cell lifespans. With only a single data set, it is not possible to discriminate between these two extreme cases: short infectious lifespans on average with a broad distribution, versus a long average infectious lifespan with a narrow distribution of transitions. The results of SCG experiments, where viral titer is observed to grow linearly over 10 to 20 h (Figure 3), suggest that the former is unlikely since infectious cell death would lead to a turnover in the viral titer curve. It would be useful to duplicate this unique experiment in parallel with a typical SCG experiment such that these biologically distinct possibilities can be distinguished.
Effect of delay assumptions in fitting clinical data
Constrained model fits to the human influenza infection data in Figure 6.
A/Texas/36/91 (H1N1) 
τ L (h)
τ I (h)
A/Bethesda/1/85 (H3N2) 
τ L (h)
τ I (h)
1.3 × 105
7.7 × 103
1.2 × 104
8.0 × 103
A comparison of the fitted parameter values shows a clear delineation between the results obtained under the assumption of exponentially-distributed delays and those of the other three models, which enforce longer delays for all cells. The median lifespan of a latently infected cell, , is shorter under the exponential assumption (3.7 h vs. an average of 7.6 h for the A/Texas/36/91 (H1N1) data set  and 6.4 h vs. an average of 9.4 h for A/Bethesda/1/85 (H3N2) ). The same is true for the median infectious lifetime, . Extracted values for the viral production rate are larger for the exponential-delay model, by a factor of ~ 2 in one data set and by an order of magnitude in the other. The estimated basic reproductive number is smaller for the exponential-delay model, by a factor of 2–10, than the other three delay types. The fitted value of the viral clearance, c, was independent of delay choice for one data set (and equal to the viral decay rate, λ d ), but differed for the exponential-delay model in the other.
Mathematical models of viral infections within a host or cell culture have helped shed light on several aspects of cell-virus interactions [6, 40, 41]. Most frequently, models have been used to extract values for the parameters controlling small-scale infection kinetics from experimental data [4, 42, 43]. This has allowed mathematical modeling to play an ever-growing role in virological and immunological studies, with an increasing number of publications in these fields incorporating some amount of modeling [44–46]. At present, however, modeling work often lags experimentation and primarily serves an explanatory role. It is desirable that models take a more predictive role in the future, where the simulation of viral infections may aid, for example, in the prediction of virulence for new strains or the kinetic mechanisms of untested antiviral therapies. Before models can take on a predictive role, however, the mathematical implementation of viral infection dynamics must be tested against a diverse set of experimental conditions to ensure that biological reality is faithfully represented.
Here, we have investigated the implementation of the progress of a cell through the states of infection: from latently infected to infectious to dead. We have focused on the characterization of two times: the time a cell is latently infected but not yet releasing virus, and the time a cell is infectious (releasing virus) before infection-induced death. We explored four different distributions for these state lifetimes: exponential, Dirac delta (fixed-delay), normal, and lognormal. The validity of each distribution was assessed by fitting the associated model to data from single-cycle growth (SCG) viral yield experiments. These experiments provide a unique view of the average dynamics of a single cell due to the synchronous infection of all cells. We have shown that ODE models which implement exponential delays and DDE models with fixed delays are unable to describe this experimental data, whereas normal-delay and lognormal-delay models both provide a good fit to the data. In addition to the classic SCG experiment, we have also considered delay dynamics for a “single-cycle, single-history” (SCSH) experiment , which reveals average cellular virus production as a function of time. We have shown that, like the SCG experiment, both normal- and lognormal-delay models provide an adequate description of the dynamics while exponential- and fixed-delay models do not. The origin of the inability of ODEs and DDEs to exhibit single-cycle dynamics can be seen quite clearly in the experimental data (Figures 3, 4, and 5): virus production begins only after a long delay following the infection of a cell, a feature which ODE models cannot replicate, and the transition of cells into and out of the infectious phase follows a smooth distribution which DDE models cannot reproduce.
The median values of the latent infection period, determined by fitting normal- and lognormal-delay models to the data from these SCG experiments, ranged from 8 h to 24 h, which is significantly longer than the 4 to 6 h values typically quoted in the literature (e.g., ). There are a number of possible reasons for this discrepancy. First, the latent infection period in the model includes both the typical eclipse period prior to virus production and any additional time required for viral release. Second, viral production in a culture is likely detectable much earlier. For example, under the assumption of normal delays the median length of the latent phase was found in the SCSH experiment to be with σ L = 4.6 h. Thus, 6% of cells (~ 600, 000) had already begun releasing virus after only 6 h; using an alternative definition, the phase of latency could be declared over much earlier. Finally, the model describes a system where each cell releases no virus until the start of the infectious phase at which point virus is produced at a constant rate common to all cells. This is obviously a simplified version of reality. In fact, there is significant evidence from flow cytometry fluorescence experiments that different cells produce virus at different rates, perhaps over several orders of magnitude ; it is also likely that the rate of virus production varies over the course of its infectious lifespan.
The median infectious cell lifespan, determined in fits of the normal- and lognormal-delay models to the SCSH data, ranged from 6 to 18 h, but the application of these two distributions types implied disparate dynamical scenarios. Under the assumption of normal delays, the infectious lifespan was small but the distribution was broad. In contrast, the lognormal assumption predicted nearly a fixed infectious lifetime for all cells and the observed slow decline of infections cells was completely ascribed to a long tail in the transition from the latent to infectious phase. Biologically, the infectious cell lifespan is variously characterized in the literature, as is influenza-induced cell death in general. When cell death is due to apoptosis, for example in MDCK cell cultures , the median time of cell death ranges from 12 to 48 h after infection, depending on the influenza strain subtype. When virus-induced cell death is caused by necrosis, as in cultures of some lung and intestinal epithelial cells, this range increases significantly with cells living 2–3 d post-infection [20, 49, 50]. The death of cells during an infection in vivo is obviously much more difficult to measure. There is some evidence that influenza-induced cell death is caused by apoptosis [51–53] but details of the timing and strain dependence are unknown. Future elaboration of SCSH experiments, in concert with the type of analysis performed here, could aid in the quantitative characterization of virus-induced cell death.
Using the results from our analysis of in vitro experiments, we considered the effect of delay distribution choice when fitting viral titer data from in vivo infections (experiments performed on human volunteers). To allow for the identification of some kinetic parameters, we restricted the analysis by assuming a fixed value for the viral infectivity (defined as the product of the infection and production rates, pβ) and used the values for the parameters σ L and σ I determined when fitting the in vitro data. Under these assumptions, we found a clear difference between the extracted parameter values for models enforcing a delay in transitions (fixed-, normal- and lognormal-delay) and the ODE model with exponential transitions. Specifically the ODE predicted larger virus production rates, shorter latent and infectious phase lifespans, and a lower value of the basic reproductive number, R0 (Table 4). Although this result depends on rather arbitrary assumptions — infectivity may vary by orders of magnitude and there is little reason to assume that parameter values determined in vitro should be the same in vivo — it demonstrates that the choice of delay distribution has a significant effect on the conclusions drawn from model-fitting of in vivo data. This analysis also suggests that while a fixed-delay model cannot reproduce the continuous dynamics of cell transitions observed in single-cycle experiments, it may be a reasonable substitute for the more complicated normal and lognormal models in some contexts.
Our analysis of the in vivo experiments demonstrates the difficulty in extracting reliable information from viral titer data alone. Consideration of the SSR values for the model fits to in vivo data (Table 4) shows that each model adequately describes the data, despite the arbitrary constraints imposed. This highlights both the weakness of using viral titer data for model selection and the difficulty in uniquely identifying parameters from such data, given a particular model. Precise statements about the parameters controlling an in vivo infection can only be made by either imposing constraints to reduce the considered parameter space (as we have done here), or by obtaining complementary data (for example information about infectious cell population dynamics). Collection of infected cell data over the course of an influenza infection has only recently been considered in animal models (e.g., ) and such data will likely be unobtainable for human infections. Full characterization of models using in vitro experiments will therefore remain an important direction of future research. The introduction of infection models with an explicit immune response and the parallel measurements of immune system quantities (which are easier to obtain than infected cell populations) is a promising direction not only for a more complete understanding of the influenza infection but also the complete parametrization of these models [5, 54, 55] This will, of course, depend on the development of simple models for which the added complexity is warranted by the available data.
In the past ten years, tools for dynamical measurements of in vitro viral infections have improved quickly. The measurement of infection within individual cells by fluorescence microscopy has become routine . The spatial spread of virus infections on cell culture can be viewed in real time, both at the level of cellular deformation and at the level of individual virus particles . Individual virus particles have been tracked as they enter a cell, and repeated observations have allowed for a statistical characterization of the timing of events in the early stages of infection . These detailed views of the influenza infection will be invaluable for the construction of the next generation of viral infection models. Models, in turn, will provide a cohesive picture of the overall infection process and, crucially, make connections between the known small-scale details of the virus-cell interaction and the infection at the level of the organism, where it is manifested as a disease.
List of abbreviations
Ordinary differential equation
Delay differential equation
Sum of squared residuals
Akaike information criterion
Akaike information criterion, corrected for small samples
Multiplicity of infection
Single-cycle, single-history growth
Madin Darby canine kidney
This study was supported by the Canadian Institutes for Health Research [funding reference number 86937] and by the Natural Sciences and Engineering Research Council of Canada (CAAB). The authors wish to thank Dr. Shingo Iwami for his careful reading of the manuscript and helpful comments.
This article has been published as part of BMC Public Health Volume 11 Supplement 1, 2011: Mathematical Modelling of Influenza. The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2458/11?issue=S1.
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