The authors have declared that no competing interests exist.
Current address: Exploratory Science Center, Merck & Co., Inc., Cambridge, Massachusetts, United States of America
Experimental Zika virus infection in non-human primates results in acute viral load dynamics that can be well-described by mathematical models. The inoculum dose that would be received in a natural infection setting is likely lower than the experimental infections and how this difference affects the viral dynamics and immune response is unclear. Here we study a dataset of experimental infection of non-human primates with a range of doses of Zika virus. We develop new models of infection incorporating both an innate immune response and viral interference with that response. We find that such a model explains the data better than models with no interaction between virus and the immune response. We also find that larger inoculum doses lead to faster dynamics of infection, but approximately the same total amount of viral production.
The relationship between the infecting dose of a pathogen and the subsequent viral dynamics is unclear in many disease settings, and this relationship has implications for both the timing and the required efficacy of antiviral therapy. Since experimental challenge studies often employ higher doses of virus than would generally be present in natural infection assessment of this relationship is particularly important for translation of findings. In this study we used mathematical modelling of viral load data from a multi-dose study of Zika virus infection in a macaque model to describe the impact of varying the dose of Zika virus on model parameters, and developed a novel mathematical model incorporating viral interference with the innate immune response.
Zika virus (ZIKV), which was the cause of an outbreak in South America that was classified by the World Health Organization as a Global Public Health Emergency in 2016 [
A primary experimental readout of the severity of infection with ZIKV and many other viruses is the plasma viral load (VL): the amount of viral RNA present in one ml of plasma. It is thought that the majority of ZIKV infections do not result in clinical symptoms, and when symptoms do occur onset is between 3 and 11 days post exposure [
The plasma VL dynamics after infection with ZIKV in the NHP model (e.g. [
It is known that ZIKV infection elicits a robust innate immune response, with in vitro studies demonstrating IFN production from a variety of infected cell types [
The choice of inoculum dose in animal infection models is an important aspect of study design. In an experimental setting, higher doses are generally expected to provide more reliable infections, more rapid development of clinically relevant signs and less variability between animals. However, in the case of infections primarily transmitted by a mosquito vector, the natural challenge dose is likely relatively small [
The relationship between infection dose and viral dynamics has been explored both experimentally and in mathematical modeling studies. In a mouse model of norovirus infection, higher inoculum doses result in higher and earlier peak viral loads in intestine, mesenteric lymph nodes and spleen [
A thorough mathematical modeling study of dose effects [
In the data set we analyze, 28 rhesus macaques were infected subcutaneously with 103, 104, 105 or 106 PFU of Brazilian (BR, Brazil-ZIKV2015, Genbank KU497555) or Puerto Rican (PR, PRVABC59, Genbank KU501215) strains of ZIKV, as reported by Aid et al. [
The limit of detection of the experimental assay is 100 RNA cp /ml, and when ZIKV was undetectable in a sample it is shown here at the limit of detection for illustrative purposes.
Correlations between inoculum dose and VL characteristics are tested by Pearson correlation for each strain separately, and where this is found to be significant at the α = 0.05 level after Bonferroni correction for multiple testing (
The area-under-the-curve (AUC) of the log10 plasma VL kinetics can be used as a proxy for the total viral shedding. Interestingly, we found no relationship between AUC and inoculum dose or viral strain (
Mathematical models based on target cell limitation have been used to describe acute infection plasma viral loads in HIV [
Target cell limited models, either with or without the eclipse phase as modelled here, have the property that the VL AUC is determined primarily by the number of target cells available, independently of the initial dose [
We used non-linear mixed effects modeling, as described in Methods, to fit the plasma VLs in all 28 monkeys simultaneously and obtain estimates of the population distributions for model parameters. We found that models incorporating an average eclipse phase length of four hours or shorter (
Despite not explicitly including any effect of inoculum dose in this model fit, we found that the estimated initial plasma viral load,
As a more thorough analysis of the effect of inoculum dose or viral strain on the viral dynamics parameters, we allowed population parameters to depend on dose or strain as covariates in the model fitting. We then used an iterative approach to determine which covariate relationships should be retained, as described in Methods. This approach accepts covariate structures that describe statistically significant relationships between dose and parameter value while maintaining at least as good a model fit. We found statistical support only for a relationship between estimated initial plasma viral load,
An explicit covariate relationship between inoculum dose and initial viral load
Target cell limited model ( |
Innate immune model ( |
Viral interference model (Eqs |
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Parameter | Population estimate | Variability estimate | Population estimate | Variability estimate | Population estimate | Variability estimate | ||||||
2.59 | (16%) | 0.0229 | (572%) | 3.44 | (12%) | 0.0284 | (888%) | 4.16 | (9%) | 0.033 | (405%) | |
δ | 8.83 d-1 | (57%) | 0.211 | (60%) | 3.31 d-1 | (21%) | 0.116 | (103%) | 2.25 d-1 | (10%) | 0.064 | (85%) |
1724 d-1 | (65%) | 0.596 | (33%) | 637 d-1 | (24%) | 0.536 | (24%) | 450 d-1 | (16%) | 0.514 | (26%) | |
γ | 3.36 | (49%) | 0.859 | (52%) | 0.010 | (59%) | 2.12 | (24%) | ||||
τ | 5.37 d | (5%) | 0.239 | (15%) | 2.68 d | (3%) | 0.107 | (29%) | ||||
log10 |
-1.51 ml-1 | (23%) | 0.329 | (28%) | -1.00 ml-1 | (8%) | 0.341 | (26%) | -0.96 ml-1 | (34%) | 0.359 | (21%) |
We noted that the parameter estimates for the death rate of productively infected cells, δ, gave a population median of 8.8 d-1 (
The target cell limited model (
We extended the viral dynamic model given by
There are a number of ways that the action of the innate immune response can be included in the mathematical model. We initially tested models where the innate response affects the viral dynamics in one of a number of ways: (i) reducing viral infectivity (Eq. S2), which also models making target cells less susceptible to infection, (ii) increasing the death rate of infected cells (Eq. S3), a possible action of natural killer cells or (iii) reducing the rate of viral production from infected cells (Eq. S4), an activity of IFNα when used to treat hepatitis C virus infection [
The model where the innate immune response reduces the rate of viral production from infected cells is described by the following system of ODEs:
Without loss of generality and for the sake of identifiability we set the coefficient of production of the innate response factor,
Fitting the innate immune model (
When considering the individual estimated parameters under this model, we didn’t observe any statistically significant relationships between parameter values and inoculum dose or viral strain (
Fitting the model including a dose dependency in τ (
To mechanistically describe the ability of ZIKV to subvert the innate immune response in the context of our viral dynamics model we modified the form of the immune restriction of viral production, rewriting
This form of the reduced viral production rate
We again tested each possible additional covariate relationship in this model and under our selection criteria (see
This viral interference model describes the individual viral dynamics in each animal well (
For each animal the observed plasma viral loads (markers), the predicted viral load from the immune response model with viral interference (
In
Each inoculum dose is indicated by color, with the mean predicted dynamics within an inoculum group shown. The solid line shows the model dynamics with the estimated parameters. The dotted line shows the model dynamics with estimated parameters but with
The fraction
The estimated parameter distributions from each of the models considered in this study (
Estimated parameter distributions for the three model fits: immune response model with viral interference (
The only dose dependency that is supported in the viral interference model is in the initial viral load
Experimental challenge models are used to study many viral infections, and in order to obtain robust infections it is often the case that the inoculum doses used are many times higher than exposure in a natural setting. In order to translate findings from experimental to natural infection settings the impact of the inoculum dose needs to be understood.
Here we studied the effect of inoculum dose on plasma viral loads (VLs) after Zika infection of non-human primates. We found dose-dependent behavior in the viral load dynamics that is not readily apparent from summary statistics of the VL measurements, demonstrating the value of careful mathematical modelling analyses.
The subcutaneous viral inoculum dose given (measured in PFU) was found to be well correlated with the estimated effective initial viral load in circulation (measured in RNA cp/ mL), and across the dose range used in this study we saw no evidence of any saturation in the transport process from tissue to circulation. With an appropriate transformation, the log10 inoculum dose can be considered to be equivalent to the initial log10 viral load in the model (e.g
The data presented in this analysis, in contrast to our previous study [
The statistical approach of estimating population parameter distributions that was used in this study allows for an analysis of whether the inoculum dose size has an influence on any of the model’s parameter values. We saw that introducing a relationship between the inoculum dose and the time delay in the immune response improved the model fit and was statistically significant. The time delay in the model can be thought of as the time it takes for the appropriate cytokine signaling, cytokine production and the cytokine’s effect to occur after a cell becomes infected. The predicted relationship between this time delay and the inoculum dose was such that at higher doses there was a longer delay. Rather than reflecting a true biological dose-dependence, we took this relationship as providing clues for additional mechanisms to include in the model. Zika virus is able to interfere with the host immune system, and modeling this interference with a Hill function-like dependency on viral concentration allowed us to describe the observed data better.
Our novel viral interference model is able to capture more of the biology of the dynamics seen in these acute infections. It removes the surprising relationship between inoculum dose and the timing of the immune response seen with the innate immune response model, as well as providing a framework for a quantitative description of the effect of the experimentally observed degradation of cytokine signaling by ZIKV.
However, we were not able to accurately identify the shape of the relationship between viral load and suppression of the immune response. The half-maximal concentration was seen to be able to range over 3 orders of magnitude while providing equivalently good fits, and Hill coefficients of 0.25 and 0.5, meaning shallow slopes in the Hill function
In our mathematical model, the viral concentration exerts effects on itself, beyond the simple viral replication through cell infection, via two competing mechanisms. First, free virus infects target cells in a concentration dependent manner and the infected cells initiate an immune response that restricts production of new free virus. Second, free virus is able to inhibit the immune response through the viral interference mechanism, increasing viral production back towards the baseline rate. The combined impact of these two mechanisms on the course of infection will be complex and depends on both the timing and magnitude of the immune response and viral interference.
These two mechanisms do not depend on viral concentration in the same manner. As discussed above, the per-virion effect of interference decreases with higher viral concentrations, while the per-virion effect of immune response does not vary with viral concentration although there is a complicating factor of the time delays in the immune response. These differing concentration dependencies mean that the ratio of immune response effect to viral interference is more heavily weighted towards the viral interference mechanism at low viral loads than at high viral loads. In natural infection settings, the initial viral concentration from a mosquito bite is likely to be substantially lower than in experimental infection models. Dudley et al [
This framework for considering the effect of the immune response on viral dynamics might have an impact on assessments of antiviral efficacy from high dose experimental challenge models. A treatment that reduces viral load will also presumably reduce both the innate response and viral interference with the innate response. The variable balance of the effect of these mechanisms at a lower challenge dose might change the effectiveness of the therapy. The dynamics of the host immune response, and how the virus interferes with it are important to uncover in order to further our understanding of how ZIKV infection is usually effectively controlled.
Plasma viral load (VL) measurements, assessed using an RT-PCR assay, were collected after subcutaneous infection with ZIKV as described in [
Statistical significance was assessed with a threshold of α = 0.05, accounting for multiple testing via the Bonferroni correction where appropriate.
Ordinary differential equation (ODE) based compartmental models were used to describe the plasma viral dynamics, as detailed in the text.
As in our previous Zika modeling work [
We typically have seven positive data points per animal, thus it is likely that not all parameters are practically identifiable [
For dengue virus, a similarly structured flavivirus, a mathematical modeling study estimated the clearance rate to be around 5 d-1 [
Similarly, the length of the eclipse phase is unlikely to be identifiable from these viral load data, since the samples are taken at most daily and the average eclipse phase length is likely to be less than 1 day given the high VLs observed on day 1 in many monkeys. As such, VL measurements within the first day post infection would be required to fully estimate the eclipse phase length. Hamel et al. [
To select values of
Additionally, it is known that in this model only the product
Following Snoeck et al. [
Model selection was based on log likelihoods (natural logarithms) to compare different model structures and Wald tests for inclusion of covariates, as provided by Monolix. Comparisons between nested models were performed using the log likelihood ratio test (LLRT).
To assess the effect of inoculum dose and viral strain we allowed for model parameters to depend on those two covariates (dose and strain) using the following standard procedure. Each possible covariate structure was added to the model one at a time and the parameter estimation algorithm was run for each. The possible covariate structures include relationships between each fitted parameter and either inoculum dose or viral strain. Inoculum dose was included as a continuous covariate. For a lognormally distributed parameter
We selected covariate relationships to incorporate into our model structure based on both statistical significance and model fit. A covariate relationship is included in the model structure if it both has a significant p-value (at a threshold of 0.05 after correction for multiple testing) by the Wald test as provided by Monolix and provides at least as good a log likelihood as the model without the relationship. If more than one covariate relationship fulfilled these criteria, the one with the lowest
We visually inspected model fits to data at the individual level by obtaining predicted viral kinetics from estimated individual parameters and plotting these with the observed data measurements. At a population level, we used visual predictive checks (VPCs), whereby parameter sets are repeatedly selected at random (and independently) from the estimated distributions and predicted viral load dynamics are recorded for comparison with observed data. The range of these predicted viral kinetics are then plotted alongside the aggregated observed data measurements.
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Where viral RNA was undetectable in a sample it is indicated at the limit of detection of the assay, 102 RNA copies/ml.
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Only a covariate relationship been inoculum dose and initial plasma viral load
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The value of τ shown is the one that was found to provide the maximum likelihood. The
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Relative standard errors are shown in parentheses. An explicit covariate relationship between inoculum dose and initial viral load
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Note that the
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Relative standard errors are shown in parentheses. Explicit covariate relationships between inoculum dose and initial viral load
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No covariate relationships fulfil both criteria (log likelihood and
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At each pair of values for
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Results from each implementation of the fitting algorithm (as described in SF1) are shown by markers, medians are shown by horizontal bars and interquartile ranges are shown by vertical lines (often not visible due to tightly distributed estimates). The 100 implementations of the fitting algorithm shown here are each from different initial parameter guesses and random seeds.
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Individual estimated parameters are derived from the population fit of the target cell limited model (
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Individual estimated parameters are derived from the population fit of the target cell limited model (
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100,000 repeated random parameter sets were selected from the estimated parameter distributions for each inoculum dose, and predicted viral loads given these parameter values were recorded. Black lines show median predicted viral load, grey shaded region shows 2.5th– 97.5th percentiles of predicted viral loads and black points indicate experimentally observed viral loads. The limit of detection of the experimental assay is shown with a horizontal dashed line and where experimental measurements failed to detect ZIKV in a sample it is shown with an open marker at this limit of detection.
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Inoculum dose indicated by color where relevant (light blue = 103 PFU, dark blue = 104 PFU, orange = 105 PFU, red = 106 PFU).
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Correlations between parameters and inoculum dose are assessed via the Pearson correlation, and where this is significant after Bonferroni correction the linear regression line is shown (dashed) and the
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Differences between parameters by viral strain are assessed by the Mann Whitney U test, and no significant relationships after Bonferroni correction are observed. Markers for individual animals are colored by the viral strain (BR: green triangles, PR: purple circles).
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Color and marker shape indicate inoculum strain (BR: green triangles, PR: purple circles) and inoculum dose is indicated top left of each panel. Observed VLs are shown by markers and model prediction is shown by the solid line. The limit of detection of the experimental assay is shown by the horizontal dashed line and where ZIKV is not detectable in a sample it is shown with an open marker at this value.
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Model fits are of the target cell limited model (
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100,000 repeated random parameter sets were selected from the estimated parameter distributions for each inoculum dose, and predicted viral loads given these parameter values were recorded. Black lines show median predicted viral load, grey shaded region shows 2.5th– 97.5th percentiles of predicted viral loads and black points indicate experimentally observed viral loads. The limit of detection of the experimental assay is shown with a horizontal dashed line and where experimental measurements failed to detect ZIKV in a sample it is shown with an open marker at this limit of detection.
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Inoculum dose indicated by color where relevant (light blue = 103 PFU, dark blue = 104 PFU, orange = 105 PFU, red = 106 PFU).
(PDF)
Correlations between parameters and inoculum dose are assessed via the Pearson correlation, and where this is significant after Bonferroni correction the linear regression line is shown (dashed) and the
(PDF)
Differences between parameters by viral strain are assessed by the Mann Whitney U test, and no significant relationships after Bonferroni correction are observed. Markers for individual animals are colored by the viral strain (BR: green triangles, PR: purple circles).
(PDF)
Color and marker shape indicate inoculum strain (BR: green triangles, PR: purple circles) and inoculum dose is indicated top left of each panel. Observed VLs are shown by markers and model prediction is shown by the solid line. The limit of detection of the experimental assay is shown by the horizontal dashed line and where ZIKV is not detectable in a sample it is shown with an open marker at this value.
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Each circle represents the model fit from one implementation of the fitting algorithm with randomly selected initial guesses and random seed. In the likelihood panel, the horizontal line shows the maximum likelihood with fixed α = 2 d-1 (
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100,000 repeated random parameter sets were selected from the estimated parameter distributions for each inoculum dose, and predicted viral loads given these parameter values were recorded. Black lines show median predicted viral load, grey shaded region shows 2.5th– 97.5th percentiles of predicted viral loads and black points indicate experimentally observed viral loads. The limit of detection of the experimental assay is shown with a horizontal dashed line and where experimental measurements failed to detect ZIKV in a sample it is shown with an open marker at this limit of detection.
(PDF)
Inoculum dose indicated by color where relevant (light blue = 103 PFU, dark blue = 104 PFU, orange = 105 PFU, red = 106 PFU).
(PDF)
Correlations between parameters and inoculum dose are assessed via the Pearson correlation, and where this is significant after Bonferroni correction the linear regression line is shown (dashed) and the
(PDF)
Differences between parameters by viral strain are assessed by the Mann Whitney U test, and no significant relationships after Bonferroni correction are observed. Markers for individual animals are colored by the viral strain (BR: green triangles, PR: purple circles).
(PDF)
Color and marker shape indicate inoculum strain (BR: green triangles, PR: purple circles) and inoculum dose is indicated top left of each panel. Observed VLs are shown by markers and model prediction is shown by the solid line. The limit of detection of the experimental assay is shown by the horizontal dashed line and where ZIKV is not detectable in a sample it is shown with an open marker at this value.
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Those fits which are indistinguishable by log likelihood, within 2 points of the maximum, are highlighted in white.
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100,000 repeated random parameter sets were selected from the estimated parameter distributions for each inoculum dose, and predicted viral loads given these parameter values were recorded. Black lines show median predicted viral load, grey shaded region shows 2.5th– 97.5th percentiles of predicted viral loads and black points indicate experimentally observed viral loads. The limit of detection of the experimental assay is shown with a horizontal dashed line and where experimental measurements failed to detect ZIKV in a sample it is shown with an open marker at this limit of detection.
(PDF)
Inoculum dose indicated by color where relevant (light blue = 103 PFU, dark blue = 104 PFU, orange = 105 PFU, red = 106 PFU).
(PDF)
Correlations between parameters and inoculum dose are assessed via the Pearson correlation, and where this is significant after Bonferroni correction the linear regression line is shown (dashed) and the
(PDF)
Differences between parameters by viral strain are assessed by the Mann Whitney U test, and no significant relationships after Bonferroni correction are observed. Markers for individual animals are colored by the viral strain (BR: green triangles, PR: purple circles).
(PDF)
Color and marker shape indicate inoculum strain (BR: green triangles, PR: purple circles) and inoculum dose is indicated top left of each panel. Observed VLs are shown by markers and model prediction is shown by the solid line. The limit of detection of the experimental assay is shown by the horizontal dashed line and where ZIKV is not detectable in a sample it is shown with an open marker at this value.
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Top: the area under the curve (AUC) of the immune-response-restricted viral production rate (
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Correlations between inoculum dose and viral characteristic are assessed via a Pearson correlation, with
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