The author has declared that no competing interests exist.
Conceived and designed the experiments: RRR. Performed the experiments: RRR. Analyzed the data: RRR. Wrote the paper: RRR.
To assess the efficacy of HIV vaccine candidates or preventive treatment, many research groups have started to challenge monkeys repeatedly with low doses of the virus. Such challenge data provide a unique opportunity to assess the importance of exposure history for the acquisition of the infection. I developed stochastic models to analyze previously published challenge data. In the mathematical models, I allowed for variation of the animals' susceptibility to infection across challenge repeats, or across animals. In none of the studies I analyzed, I found evidence for an immunizing effect of non-infecting challenges, and in most studies, there is no evidence for variation in the susceptibilities to the challenges across animals. A notable exception was a challenge experiment by Letvin et al. Sci Translat Med (2011) conducted with the strain SIVsmE660. The challenge data of this experiment showed significant susceptibility variation from animal-to-animal, which is consistent with previously established genetic differences between the involved animals. For the studies which did not show significant immunizing effects and susceptibility differences, I conducted a power analysis and could thus exclude a very strong immunization effect for some of the studies. These findings validate the assumption that non-infecting challenges do not immunize an animal — an assumption that is central in the argument that repeated low-dose challenge experiments increase the statistical power of preclinical HIV vaccine trials. They are also relevant for our understanding of the role of exposure history for HIV acquisition and forecasting the epidemiological spread of HIV.
Individuals are exposed to Human Immunodeficiency Virus (HIV) many times before they contract the virus. It is not known what an instance of exposure, which does not result in infection, does to the host. Frequent exposures to the virus are hypothesized to immunize an individual, and result in resistance to infection with HIV. This hypothesis may explain the resistance observed in some individuals despite frequent exposure to the virus. Since it is very difficult to monitor the HIV exposure and infection status of humans, this question is easier to address in animal models. I took data from previously published infection experiments of monkeys with Simian Immunodeficiency Virus (SIV) and analyzed them with newly developed mathematical models. I found that there is no evidence that challenging monkeys with the virus reduces their susceptibility to infection. These findings have important repercussions for the testing of HIV vaccines in monkeys, and also for our understanding of the role of exposure history for the acquisition of HIV.
Before being tested clinically, vaccines or preventive treatment strategies against human-immunodeficiency virus (HIV) are assessed in non-human primates. The paradigm of how vaccine candidates are assessed preclinically has been shifting in recent years. A decade ago, vaccine protection was not quantified directly by determining how much the vaccine candidate reduced the susceptibility of the animal hosts to infection. Instead, indirect measures were used, such as the level of virus-specific immune responses, or as the reduction of the viral set-point induced by vaccination.
Unfortunately, there are many uncertainties about how these immunological and virological measures correlate with protection. As a consequence, vaccines have recently been tested using repeated low-dose challenge experiments
The challenge data generated in such trials are usually analyzed to infer the efficacy of a vaccine candidate, or post-exposure prophylaxis. In addition to information on treatment efficacy, however, these data contain information on other, very relevant aspects of the transmission of HIV. In the present study, I analyzed challenge data that had been published previously. I focused on the challenge data from control animals that did not receive any vaccine or treatment.
My first main question was if hosts are immunized by repeated challenges. In SIV challenge experiments, potential immunization is usually studied by measuring systemic or localized immune responses in an animal after a non-infecting challenge. This approach relied on the strong assumption that these immune measures are causatively linked to protection. To my knowledge, however, no such link has been systematically ascertained for SIV infection to date.
In this paper, I adopted an alternative approach to assess if immunization has occurred after challenge: I essentially compared the susceptibility of animals before and after challenge. Therefore, throughout this paper,
The second main question I posed was if there are differences in susceptibilities between animals. Again my approach focused on differences of animals in terms of their susceptibility to infection, and did not involve the quantification of target cells or their susceptibility in relevant anatomical sites.
As is common in mathematical epidemiology, I conceptualized infection as a stochastic event that occurs with a given probability. More precisely, I described infection as a Bernoulli process. While the infection probability can be thought of as a trait of an individual animal, its estimation requires data of more than one animal. Repeated low-dose challenge data allow us to estimate the infection probability. They also allow to test if animals are immunized by non-infecting challenges, as immunization leads to a smaller and smaller fraction of animals becoming infected in the course of the challenge experiment. The same is true if there are differences in susceptibility between animals. As I show below, one typically finds evidence for both, immunization and susceptibility differences, and further analysis is needed to disentangle the two effects. Formally, I used simple stochastic models and maximum likelihood estimation to estimate the infection probability, and to study how it varies across challenge repeats and animals.
I found that there is no evidence for immunization in any of the studies. There is also no evidence for variation in susceptibility, except for one recent experiment conducted with the strain SIVsmE660
I analyzed repeated low-dose challenge data from seven studies
The step functions for each study terminate at the maximum number of challenges applied in this study, or when all animals are infected.
Monkey species | Number of monkeys (ctl/vac) | Challenge virus | Challenge route | Challenge frequency | Challenge dose | Maximum number of challenges | Treatment | Efficacy |
|
|
pigtailed macaques | 14/16 | SHIV-SF162P3 | rectal | weekly | up to 26 | DNA/MVA vaccine | 64% | |
|
rhesus macaques | 8/8 | SIVmac239 | rectal | weekly |
|
8 | DNA/Ad5 vaccine | NS |
|
rhesus macaques | 8/8 | SIVsmE660 | rectal | 3 weeks | 5 | DNA/Ad5 vaccine | NS |
|
|
rhesus macaques | 18/24 | SHIV-SF162P3 | rectal | weekly | 14 | FTC/TDF pre-exposure prophylaxis | 74–87% | |
|
rhesus macaques | 15/12 | SIVmac239 | rectal | weekly | 8 | RhCMV/SIV vaccine | ND |
|
|
rhesus macaques | 28/33 | SIVmac239 | rectal | weekly | 25 | RhCMV/SIV & DNA/Ad5 vaccines | NS |
|
|
rhesus macaques | 20/20 | SIVmac251 | rectal | weekly | 1 |
12 | DNA/Ad5 vaccine | NS |
|
rhesus macaques | 43/43 | SIVsmE660 | rectal | weekly | 1 |
12 | DNA/Ad5 vaccine | 50% |
“Tissue culture infectious dose 50”;
“focus forming units”;
“animal infectious dose at which 50% become infected”;
the efficacy of the intervention on susceptibility to challenge;
not done;
not significantly different from 0%.
In the most recent study
First, I assessed if there is evidence in the repeated low-dose challenge data for immunization in the sense that challenges, which do not give rise to infection, reduce suceptibility of the host. To this end, I first fit a stochastic model (the
The red dashed lines indicates the maximum of the likelihood.
|
|
(95% CI) | |
Ellenberger06v | −32.3 | 0.20 | (0.12, 0.31) |
Wilson06jv | −19.3 | 0.22 | (0.11, 0.37) |
Wilson09jv | −13.5 | 0.25 | (0.11, 0.44) |
GarciaLerma08pm | −42.1 | 0.20 | (0.13, 0.3) |
Hansen09nm | −30.1 | 0.24 | (0.14, 0.36) |
Hansen11n | −74.9 | 0.16 | (0.11, 0.21) |
Letvin11stm.SIVmac251 | −27.7 | 0.50 | (0.35, 0.65) |
Letvin11stm.SIVsmE660 | −98.0 | 0.16 | (0.12, 0.22) |
In a second step, I fit the immunization model to the data. This model assumed that the susceptibility to the challenge decreased with challenge repeats (see
Irrespective of the way I implemented immunization, the immune priming model fails to outperform the geometric infection model statistically, except for the SIVsmE660 challenge data of Letvin et al 2011 (in which susceptibility difference between the monkeys have been established), and Wilson et al 2006.
The red dots indicate the maximum of the likelihood.
immunization |
|
||||
Ellenberger06v |
|
−32.1 | 0.14 (0.03, 0.38) | 0.22 (0.12, 0.35) | 0.51 |
|
−30.8 | 0.31 (0.15, 0.5) | 0.13 (0.05, 0.26) | 0.09 | |
|
−31.1 | 0.28 (0.15, 0.45) | 0.12 (0.04, 0.27) | 0.12 | |
incremental | −31.9 | 0.25 (0.12, 0.41) | −0.01 (−0.03, 0.02) | 0.36 | |
Wilson06jv |
|
−17.1 | 0.00 (0, 0.21) | 0.28 (0.14, 0.45) | 0.03 * |
|
−17.1 | 0.06 (0, 0.25) | 0.33 (0.16, 0.55) | 0.04 * | |
|
−19.0 | 0.17 (0.06, 0.36) | 0.29 (0.1, 0.54) | 0.43 | |
incremental | −16.5 | 0.00 (0, 0.21) | 0.09 (0.02, 0.14) | 0.02 * | |
Wilson09jv |
|
−13.0 | 0.13 (0, 0.45) | 0.31 (0.13, 0.56) | 0.30 |
|
−13.5 | 0.27 (0.09, 0.52) | 0.22 (0.04, 0.54) | 0.81 | |
|
−13.5 | 0.26 (0.1, 0.48) | 0.20 (0.01, 0.63) | 0.77 | |
incremental | −13.5 | 0.25 (0.04, 0.54) | 0.00 (−0.14, 0.15) | 1.00 | |
GarciaLerma08pm |
|
−42.1 | 0.22 (0.07, 0.44) | 0.20 (0.12, 0.31) | 0.84 |
|
−41.2 | 0.28 (0.15, 0.45) | 0.16 (0.07, 0.27) | 0.18 | |
|
−41.1 | 0.27 (0.15, 0.42) | 0.14 (0.06, 0.27) | 0.15 | |
incremental | −41.8 | 0.24 (0.13, 0.38) | −0.01 (−0.03, 0.01) | 0.43 | |
Hansen09nm |
|
−30.0 | 0.27 (0.09, 0.52) | 0.22 (0.12, 0.37) | 0.75 |
|
−29.4 | 0.31 (0.15, 0.5) | 0.17 (0.07, 0.33) | 0.24 | |
|
−30.1 | 0.24 (0.12, 0.4) | 0.23 (0.09, 0.43) | 0.90 | |
incremental | −30.0 | 0.26 (0.12, 0.45) | −0.01 (−0.06, 0.04) | 0.69 | |
Hansen11n |
|
−74.5 | 0.21 (0.09, 0.39) | 0.14 (0.09, 0.21) | 0.37 |
|
−74.4 | 0.20 (0.11, 0.32) | 0.14 (0.08, 0.21) | 0.32 | |
|
−74.9 | 0.16 (0.09, 0.26) | 0.15 (0.09, 0.23) | 0.87 | |
incremental | −74.3 | 0.18 (0.11, 0.26) | −0.00 (−0.01, 0) | 0.27 | |
Letvin11stm.SIVmac251 |
|
−26.9 | 0.60 (0.38, 0.79) | 0.40 (0.21, 0.62) | 0.20 |
|
−27.5 | 0.54 (0.35, 0.71) | 0.42 (0.17, 0.69) | 0.49 | |
|
−27.6 | 0.48 (0.32, 0.65) | 0.57 (0.23, 0.87) | 0.68 | |
incremental | −27.7 | 0.52 (0.32, 0.71) | −0.01 (−0.12, 0.11) | 0.81 | |
Letvin11stm.SIVsmE660 |
|
−97.9 | 0.14 (0.06, 0.26) | 0.17 (0.12, 0.23) | 0.63 |
|
−95.6 | 0.24 (0.15, 0.34) | 0.12 (0.07, 0.18) | 0.03 * | |
|
−95.6 | 0.22 (0.15, 0.31) | 0.11 (0.06, 0.18) | 0.03 * | |
incremental | −96.4 | 0.21 (0.14, 0.29) | −0.01 (−0.02, 0) | 0.07 | |
Letvin11stm.SIVsmE660 |
|
−54.5 | 0.15 (0.05, 0.31) | 0.32 (0.22, 0.43) | 0.08 |
(permissive TRIM5 alleles) |
|
−56.1 | 0.28 (0.17, 0.41) | 0.26 (0.15, 0.40) | 0.83 |
|
−56.0 | 0.29 (0.18. 0.40) | 0.24 (0.12,0.40) | 0.65 | |
incremental | −55.9 | 0.29 (0.18,0.42) | −0.01 (−0.03,0.02) | 0.55 | |
Letvin11stm.SIVsmE660 |
|
−34.5 | 0.13 (0.02, 0.34) | 0.07 (0.03, 0.13) | 0.51 |
(restrictive TRIM5 alleles) |
|
−33.1 | 0.17 (0.06, 0.32) | 0.05 (0.02, 0.11) | 0.06 |
|
−34.1 | 0.12 (0.05, 0.24) | 0.06 (0.02, 0.13) | 0.25 | |
incremental | −34.7 | −0.00 (−0.01, 0.01) | 0.66 |
CI abbreviates confidence interval. The last column gives the
A notable exception are the data by Wilson et al, J Virol 2006.
I also found a significant improvement of the fit of the immune priming models over the geometric infection model for the SIVsmE660 challenge data of Letvin et al 2011. In particular, the immune priming model with an approximately two-fold drop in susceptibility after the second or third challenge (
To assess if there is any evidence for differences in susceptibility between animals, I followed the same statistical approach as in the previous subsection: I compared the fit of the geometric infection model to that of a model, in which the susceptibilities are allowed to vary from animal to animal (the
The likelihoods as a function of the two parameters of this model are shown in
The red dots indicate the maximum of the likelihood.
|
|
(95% CI) |
|
(95% CI) | ||
Ellenberger06v | −31.8 | 0.27 | (0.13, 0.48) | 0.019 | (0, 0.094) | 0.32 |
Wilson06jv | −19.3 | 0.22 | (0.11, 0.37) | 0.000 | (0, 0.031) | 1.00 |
Wilson09jv | −13.5 | 0.25 | (0.11, 0.53) | 0.000 | (0, 0.123) | 1.00 |
GarciaLerma08pm | −41.6 | 0.26 | (0.14, 0.46) | 0.018 | (0, 0.088) | 0.32 |
Hansen09nm | −30.0 | 0.27 | (0.14, 0.5) | 0.011 | (0, 0.106) | 0.67 |
Hansen11n | −74.3 | 0.19 | (0.11, 0.31) | 0.007 | (0, 0.043) | 0.27 |
Letvin11stm.SIVmac251 | −27.6 | 0.54 | (0.35, 0.76) | 0.019 | (0, 0.11) | 0.70 |
Letvin11stm.SIVsmE660 | −96.0 | 0.23 | (0.15, 0.34) | 0.022 | (3e−04, 0.065) | 0.04 * |
CI abbreviates confidence interval. The last column gives the
For the dataset for which susceptibility difference have been established (Letvin11stm.SIVsmE660), I found significant levels of inter-animal susceptibility differences (
The absence of evidence must not be confused with the evidence of absence. The non-significant results in the previous two subsections could simply be due to low sample sizes. To address this possibility, I conducted a power analysis. I simulated experiments using the same number of animals and challenges as in the experimental data, assuming immunization effects or inter-animal susceptibility differences of various sizes. I then analyzed these simulated data to test for immunization or heterogeneous susceptibility (see
For the immunization model, I defined the effect size as the relative reduction of susceptibility after the first challenge,
Power estimates are shown as a function of the effect size. The
How likely is it that I missed a significant effect in all of the studies simultaneously? A power analysis, in which I simulated each study repeatedly assuming study-specific model parameters (see
For the heterogeneous susceptibility model, the effect size was defined as the variance of the susceptibility distribution. Depending on the variance, the shape of the susceptibility distribution can be hump-shaped, monotonously falling (or rising), or U-shaped. The maximum variance depends on the mean of the distribution. For example, for a mean infection probability of 0.2 — the most common estimate obtained by fitting the geometric infection model to the various datasets — this maximum is 0.16. For Letvin11stm.SIVmac251, however, the mean infection probability is 0.5, and the maximum possible variance is 0.25.
Again, one can ask how probable it is that we missed a significant effect in all of the studies simultaneously. The probability not to detect susceptbility differences in any of the early studies is lower than 5% for susceptibility differences larger than
Using challenge data that were generated in the context of preclinical HIV vaccine studies in non-human primates, I investigated if low-dose challenges immunize the animal hosts. Potential immunization has been raised as an argument against the repeated low-dose challenge approach, which could impair its statistical power advantage. I also studied if there is evidence for susceptibility differences between animals. Formally, the analysis involved fitting simple stochastic models to the challenge data. To establish immunization or heterogeneity in susceptibility, the fits of models that accounted for such effects were compared statistically to fits of a model that ignored them.
For none of the datasets, I found evidence for immunization. There is also no evidence for differences in susceptibilities, except in the SIVsmE660 challenge data presented in
It is important to emphasize that the evidence for susceptibility differences in this dataset is not based on information of the TRIM5 alleles the animals carry. The inference only relies on the distribution of the number of challenges across animals. Hence, the method I am presenting allows the identification of heterogeneity in susceptibility from the challenge data alone and does not rely on measuring traits that modulate susceptibility. Such a factor ignorant method is important tool as considerable uncertainties about the determinants of susceptibility remain.
The SIVsmE660 challenge data presented in
In general it is difficult to disentangle the two effects. The difference between immunization and host heterogeneity is too subtle to be detected with the sample sizes of the challenge data I analyzed, and depends sensitively on the quantitative details of immunization effects and heterogeneity. However, as the SIVsmE660-challenged animals of the study by Letvin et al had been classified with respect to their susceptibility, I could test for immunization within these subgroups. As I did not find any evidence for immunization in each susceptibility class, I concluded that the immunization effect in the pooled data is misidentified.
The lack of evidence for immunization does, of course, not prove that there is no such effect. It may simply result from the low sample sizes in these studies. To go beyond this plain caveat, I conducted a power analysis that quantifies the probability that an effect was missed.
The study by Wilson et al (2009) may provide a likely case of too low power. This study used the same challenge strain (SIVsmE660) as the data by Letvin et al 2011, in which susceptibility differences between animals have been established. The animals involved in study by Wilson et al (2009) were, to my knowledge, not monitored for their TRIM5 alleles, but it is conceivable that some animals differed in their susceptibility for this reason. The power of this study, however, was the lowest among all the studies. To detect the same level of heterogeneity as I found in Letvin11stm.SIVsmE660 (
While sample sizes were clearly an issue in the study by Wilson et al (2009), especially the later studies
The lack of evidence for immunization by non-infecting challenges in the majority of the studies constitutes a crucial validation of the repeated low-dose challenge approach. Only if challenges do not immunize, one can safely assume that infection probabilities are independent. According to my analysis, there is no evidence against the assumption of independence. While we cannot exclude immunization effects of small size, the analysis presented in this paper provides evidence against at least very strong immunization effects. This suggests that the repeated low-dose challenge approach increases statistical power as we and others have previously predicted
Independently from the findings I present in this paper, the statistical power is further corroborated by the increasing number of studies that have used this approach successfully. For example, Ellenberger et al, Virol 2006
The power analysis also suggests that the susceptibility distribution among the experimental animals is, with high probability, not U-shaped. This is also very relevant to how vaccine efficacies are estimated statistically, and how many animals have to be involved in a preclinical study. If the susceptibility distribution were U-shaped, the animal population would essentially fall into two classes: almost completely susceptible and almost completely resistant. Any effect of a vaccine would be confined to the susceptible subpopulation, thus effectively decreasing the sample size.
But even in the case in which the susceptibility distribution is not U-shaped, yet susceptibilities still vary from animal to animal, some of the standard assumptions made when estimating vaccine efficacies from repeated low-dose challenge experiments are violated. While some studies consider animal-to-animal variation in the effect of the vaccine
Beyond the context of assessing HIV vaccines or prophylaxis, repeated low-dose challenge data provide insights into the natural transmission of HIV. It is extra-ordinarily challenging to assess how the rate of HIV acquisition depends on the exposure history. The reason for this difficulty is that, on logistic grounds, individuals cannot be monitored frequently enough to generate exposure and acquisition data with the level of detail required to establish the role of exposure history.
For example, the Rakai cohort
Consequently, most mathematical models that forecast the epidemiological spread of HIV neglect exposure history and assume that hosts retain no memory of previous exposures. The findings in this paper provide limited support for this assumption. The support is only limited because of issues relating to statistical power mentioned above, but also because the doses used in repeated low-dose challenge experiments are still much higher than those transmitted naturally. To definitively rule out any impact of exposure history it will be necessary to conduct experiments in which the challenge dose is further reduced and the frequency is systematically varied from more often than daily to less often than weekly. Some immunization effect may not be detectable if hosts are exposed weekly, as was done in most of the studies I analyzed in the present paper.
There is a group of HIV exposed individuals — sex workers from Kenya and Uganda — who remain uninfected despite frequent exposure to the virus. These highly exposed seronegative (HESN) individuals are hypothesized to be immunized by frequent exposures to the virus
The repeated low-dose challenge of monkeys much better reflects the frequent exposure of the HESN individuals, although the doses used in repeated low-dose challenge experiments are still high when compared to the doses to which humans are exposed. (They are termed “low” to distinguish them from the very high doses normally used in non-human primate challenge studies.) Therefore, if frequent exposure by itself were sufficient to lead to resistance, at least partial immunization should be observed in the challenge experiments. The fact that I failed to find any immunization effect suggests that there is more to the resistance of HESN individuals than high and frequent exposure. It is conceivable that the exposure frequency or dose is required to start at a low level and increase over time. The exposure route may also be relevant: in the studies I analyzed the challenge was performed rectally, while HESN individuals are exposed vaginally. A last possibility is that the frequency of challenges in the most of the experiments of one week is too low to kick off the immunizing mechanism, which render HESN individuals resistant. In any case, the hypotheses about resistance in HESN individuals will have to be refined by specifying the routes of infection as well as the ranges of exposure dose and frequency that can lead to resistance.
An important conclusion from the analysis presented in this paper is that the challenge data in every study — with the exception of the one by Letvin et al using SIVsmE660 as a challenge strain — are consistent with the geometric infection model. This means there is no evidence that animals differ in their susceptibilities. As a consequence, there is no justification to divide the animals in these studies into those that become infected early versus those become infected late in the challenge schedule. Neither is there any justification to compare these two groups immunologically, virologically or genetically. Approaches, such as the statistical comparison between the fit of geometric infection model with a fit of the heterogeneous susceptibility model, are required to establish susceptibility differences and to provide a solid statistical foundation for comparisons between animals with low and high susceptibility.
I selected repeated low-dose challenge data from seven previously published studies. These studies are: Ellenberger et al, Virol 2006;
The number of animals involved in the studies ranged from 16 to 86. The maximum number of challenges ranges from 8 to 26. Challenges were given rectally with a frequency of one week. Rectal challenges are the preferred route in such experiments as they can be performed on male animals and are relevant for human transmission. The involvement of female animals in preclinical studies is rare as they are required to maintain the colonies.
I analyzed only challenge data of the control animals involved in the studies listed in
In some studies the challenge dose was increased after a certain number of challenges. I ignored the data generated with increased doses. The reason for this is that the challenge with increased doses pertained to only few animals, and would therefore be only marginally informative. Moreover, incorporating these data would have forced me to introduce an additional susceptibility parameter into my models, which — due to the low sample size — could not be reliably estimated.
The dataset by Letvin et al (2011) involving challenges with the viral strain SIVsmE660 will serve as a control for our approach to establishing susceptibility differences. For this strain, genetic correlates of susceptibility have been identified (see
The challenge data consist of two pieces of information for each animal. The first is the number of challenges,
I constructed stochastic models and used them in combination with the challenge data to infer parameters characterizing the probability of animals becoming infected upon challenge with the virus. In the next subsection, I describe these models.
The simplest model I considered assumes that a challenge results in infection with a probability constant across animals and challenge repeats (see
Assume we have conducted a repeated challenge experiment with
The probability of infection
After a challenge that did not give rise to infection the host's susceptibility may be reduced due to its immunization by the unsuccessful challenge (see
For challenge data
While this formulation allows any pattern of change of the infection probability
The 95% confidence interval for the estimate of each model parameter was calculated as the likelihood ratio confidence region of the parameter using the profile likelihoods for this parameter
In this model, I assumed that the susceptibilities of each animal in the trial differ (see
To obtain the likelihood for one animal, which has been challenged
It is useful to re-parametrize this function since both parameters,
As for the immune priming model, the 95% confidence interval for the estimate of each model parameter was calculated as the likelihood ratio confidence region of the parameter using the profile likelihoods for this parameter
To test for immunization by repeated challenges or for differences in the susceptibilities of animals to infection, I first fit the geometric infection model, and then the immune priming and heterogeneous susceptibility models. The model fits were then compared by a likelihood ratio test. I applied a significance level of 0.05.
To determine the statistical power of the model fitting and comparison, I simulated data that conform to the immune priming or heterogeneous susceptibility models. In these simulations, I chose numbers of animals and maximum numbers of challenge repeats consistent with each experimental study.
In the case of the immune priming model, I set an animal's susceptibility at the first challenge is
The simulated data were then analyzed and significance was assessed. Power was determined as the fraction of simulated experiments, in which a significant immunization effect or heterogeneous susceptibilities could be established. In accordance with the comparison of the model fits to the experimental data, I applied a significance level of 0.05. If a simulated dataset could not be fitted (due to convergence problems of the fitting routine) it was excluded from the analysis.
The likelihoods, the model fitting and comparison, and the power analysis were implemented in the R language of statistical computing
(GZ)
(PDF)
I would like to thank Helen Alexander, Victor Garcia, Carsten Magnus, George Shirreff, Tanja Stadler, and Alexandra Trkola and two anonymous reviewers for valuable comments and criticism of this work.