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 [9], 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 the experimental data (Figures 3, 4, and 6): 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., [2]). 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 [47]; 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 [48], 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, *R*
_{0} (Table 1). 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 1) 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., [54]) 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 [47]. 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 [56]. 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 [16]. 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.