Respiratory syncytial virus (RSV) is the most common cause of bronchiolitis and pneumonia in children younger than 1 year of age in the United States. Moreover, RSV is being recognized more often as a significant cause of respiratory illness in older adults. Although RSV has been studied both clinically and in vitro, a quantitative understanding of the infection dynamics is still lacking. In this paper, we study the effect of uncertainty in the main parameters of a viral kinetics model of RSV. We first characterize the RSV replication cycle and extract parameter values by fitting the mathematical model to in vivo data from eight human subjects. We then use Monte Carlo numerical simulations to determine how uncertainty in the parameter values will affect model predictions. We find that uncertainty in the infection rate, eclipse phase duration, and infectious lifespan most affect the predicted dynamics of RSV. This study provides the first estimate of in vivo RSV infection parameters, helping to quantify RSV dynamics. Our assessment of the effect of uncertainty will help guide future experimental design to obtain more precise parameter values.
Respiratory syncytial virus (RSV) is a major cause of lower respiratory tract disease among infants, a frequent pathogen in elderly and immunosuppressed patients, and a major public health concern worldwide [
Historically, mathematical and computational methods have not played a large role in immunology and virology. This is now changing, and impressive advances have come from the use of simple models applied to the interpretation of quantitative data [
Initial modeling studies for any virus often use a system of nonlinear differential equations and the models are typically fit to viral time course data to generate estimates of viral kinetic parameters. However, given the large amount of experimental error in viral titer measurements [
Uncertainty in differential equations has been considered in recent decades in a wide variety of applied areas, such as physics, chemistry, biology, economics, sociology, and medicine [
This paper is organized as follows. Section
The experimental data used in this paper was first published in Lee et al. [
In this paper we will use a model based on an autonomous system of nonlinear ordinary differential equations to characterize the in vivo infection dynamics of the A2 strain of RSV. The model is an extension of the basic viral infection model for influenza described in [
Viral kinetic model. The virus,
In order to characterize the RSV replication cycle and extract kinetic parameter values, we need to fit the mathematical model to the patient data [
It is sometimes difficult to compare parameter estimates from different experiments since the units of viral titer depend on the details of measurement. There is no universal standard viral titer unit, making comparison of parameters such as
The reliability of the parameter estimates depends mainly on two factors. The first one is the accuracy of the experimental data, which is out of our hands, but it is known that these types of viral data are subject to uncertainty [
We have chosen to study the effect of parameter uncertainty on viral dynamics using a Monte Carlo method. The Monte Carlo method allows us to study random effects with different distributions in ordinary differential equation models. We employ a conceptually simple Monte Carlo approach by running numerical simulations separately for each parameter which results in a set of corresponding plausible simulation predictions. These predictions can be used to characterize the uncertainty in viral titer time course due to inaccuracy in either single parameters or combinations of parameters [
One of our aims is to estimate the parameters of the mathematical model (
Estimated parameters for RSV in vivo infection.
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SSR |
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(h) | (/h) | (h) | (h) | (h) | (h) | (TCID50/mL/mL) | ||
1 | 2.4 | 0.28 | 7.2 | 7.2 | 5.1 | 5.1 | 0.17 | 5.6 |
2 | 4.3 | 0.78 | 5.3 | 19 | 2.2 | 11 | 0.010 | 5.3 |
3 | 1.9 | 0.13 | 4.8 | 9.6 | 2.8 | 4.8 | 0.013 | 9.5 |
4 | 3.8 | 0.20 | 7.9 | 11 | 5.6 | 7.8 | 0.011 | 8.8 |
5 | 0.72 | 0.063 | 18 | 24 | 10 | 9.1 | 0.010 | 1.1 |
6 | 2.4 | 0.20 | 10 | 17 | 4.5 | 8.5 | 0.012 | 4.3 |
7 | 6.2 | 0.096 | 14 | 17 | 9.9 | 9.8 | 0.030 | 0.6 |
12 | 1.7 | 0.44 | 13 | 10 | 3.4 | 5.0 | 0.16 | 2.3 |
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Median | 2.4 | 0.20 | 9.0 | 14 | 4.8 | 8.2 | 0.13 | 4.8 |
Numerical simulations of the virus kinetic model (
Despite the overfitting, the extracted parameter values seem to be biologically reasonable. While we have no previous RSV data for comparison, we can examine our parameter estimates in the context of what has been found for influenza. Influenza is also an upper respiratory tract viral infection that typically causes mild disease in healthy adults, so the two infections have been compared before [
Several studies have determined estimates of the infecting time for influenza ranging from 0.02 h to 2.5 h [
One parameter for which we have some RSV data for comparison is the viral clearance rate. Viral decay rate can be determined from mock infection experiments. In these experiments, the virus is placed in a dish without cells and infectious virus is measured every few hours. Mock infection experiments for RSV indicate clearance rates in the range of 0.016–0.034 /h [
Finally, we determined estimates for the standard deviation in the eclipse duration and the infectious lifespan. There are not many estimates of these parameters, even for influenza, since most models assume exponential transitions between the phases of the cell life cycle. Our estimates of the standard deviation in infectious lifespan of RSV infected cells range from 2.2 h to 10 h and our estimates of the standard deviation of the eclipse duration during RSV infection range from 4.8 h to 11 h. For influenza, infectious lifespan standard deviation estimates range between 1.4 h and 9.7 h [
Given the range of estimated parameter values and the known experimental error in viral titer measurements [
Fit of the viral kinetics model (
To begin the Monte Carlo process, we need to assume probability density functions for each of the parameters. In this case, we assumed that the parameters follow a Gaussian distribution. The mean of the Gaussian distribution is set to be the parameter value determined through fitting of the median data set. The variance is assumed to be proportional to the mean value of the distribution,
Monte Carlo numerical simulations of the virus kinetic model using the parameter values extracted from the fit to the median virus data. The graphs present the mean (dashed blue line) and
Figure
This paper presented the first fits of a viral kinetics model to in vivo RSV infections. This allowed us to extract viral kinetic parameters for an in vivo RSV infection. While it is difficult to judge the accuracy of our parameter estimates since there are no similar studies for RSV, we can compare RSV parameter estimates to parameter estimates for influenza. Both diseases are viral infections of the respiratory tract that most often cause mild illness but have the potential to cause serious illness and death. A previous study comparing the two infections in an in vivo challenge study noted that influenza viral load peaked about 3 dpi before RSV and that RSV appeared to have a longer incubation period. These findings agree with our quantitative findings of a longer infecting time and longer eclipse phase duration for RSV than for influenza. Our study also suggests, however, that the infectious cell lifespan is shorter for RSV than for influenza, showing that not all processes take longer for RSV. Some of this decreased lifespan in RSV could potentially be accounted for by the actions of cytotoxic T lymphocytes (CTLs). CTLs kill infected cells, but it takes several days after infection (~5–8 dpi) for them to appear in substantial numbers [
This paper also investigated the effect of parameter uncertainty on RSV dynamics. It is important to understand the limits of the predictive capabilities of mathematical models since they are more often being used to assess the effect of vaccines and drug treatment regimens [
While we have used individual fits to patient data to estimate RSV parameters, other methods may be used to estimate the parameter values. Mixed effect models are increasingly used to estimate parameters [
While this paper focused on fitting a particular data set for RSV, the assessment tools presented here will be widely applicable to other systems of differential equations. The Monte Carlo assessment of the effect of parameter uncertainty can guide experimentalists in developing experiments that will produce more accurate predictive models.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the anonymous reviewers for their helpful suggestions and remarks.