Nonlinear Distribution and Uptake

The Fundamental Problem for Epidemiological Model Assumptions

Introduction–A Forword

The intrinsic motivation for this article:

Yesterday, I watched a hearing that particularly drew my attention to the argumentation of Prof. Klaus Stöhr, who relied primarily on individual epidemiological assumptions and presented them almost as a shield, centered around the statement: “The data show that…” The question arises whether it is scientifically justified, as a serious researcher who should be aware of the limitations and fallibility of epidemiological assumptions, to use precisely these limitations as the foundation for an argument of certainty after six years of large scale population exposure. This article aims to examine this question from the perspective of supramolecular assemblies.

It will be a speculative thought experiment, grounded in the properties of LNPs that we clearly and evidently delineated in our hypothesis.¹ (Hopefully, the paper will be published soon by APSB; the third round of editorial reviews is complete, and it finally feels like we’re in the home stretch.)

Before proceeding, it should be first established what the current evidence base actually shows, and equally importantly, what it does not. Judging from the studies that have received the greatest attention, the available evidence appears to support the safety profile of modRNA.²^,³^,⁴ (Not every study will be referenced; rather, representative examples are provided for illustrative purposes.)

In contrast, studies reporting controversial findings ⁵^,⁶ or results that diverge from the prevailing conclusion that vaccines are “safe and effective” are subjected to the level of critical scrutiny that, in principle, should be applied to all observational research. Their limitations are rigorously identified and acknowledged, including potential confounding, selection bias, incomplete datasets, population heterogeneity, and other methodological constraints.

Every observational association, whether suggesting that vaccination is protective or that a greater number of doses is associated with an increased risk, deserves the same methodological treatment. A hazard ratio does not constitute proof of causality, confounding may operate in either direction, and the authors themselves typically acknowledge these limitations in the discussion or limitations section regardless of the direction of the observed association.

However, this principle of consistently evaluating study limitations appears to be applied only incompletely, often in a highly abbreviated manner, or not at all when assessing studies whose findings support vaccine safety. This leads to a further distinction concerning the visibility and distribution of scientific criticism rather than the methodological quality of the studies themselves.

An example for making this concrete:

A study by Kakeya et al.⁷ was published in a relatively small Japanese journal and subsequently received a highly critical public response from experts in infection control, including university professors, who challenged its methodology and interpretation in considerable detail. Such direct post publication criticism became an integral part of how the study was perceived within the scientific community.

By contrast, as example, CDC MMWR studies were published by the same institution that develops and issues vaccination recommendations. In this context, the institution functions simultaneously as data custodian, research producer, and public health authority. This institutional position may create a different environment for post publication critique. Independent scientific responses that publicly characterize CDC publications as methodologically flawed, misleading, or potentially detrimental to public trust appear to be comparatively uncommon. This observation should not be interpreted as evidence that the underlying methodology is necessarily stronger. Rather, it may reflect differences in the institutional visibility and authority of the respective publications, as well as differences in the mechanisms through which criticism is expressed and disseminated.

Accordingly, the central issue is not that one study is inherently more robust than the other. Instead, it is that different studies may be exposed to different degrees of public methodological scrutiny. Institutional position and scientific authority may influence how prominently criticism is communicated and amplified, independently of the methodological strengths and weaknesses of the research itself. This raises the possibility of an institutional selection bias in which prevailing policy priorities influence not only the direction of research but also the subsequent academic discourse. Because major public institutions exert considerable influence over research funding, scientific networks, and public health narratives, studies consistent with established positions may be evaluated within a different institutional context than studies that challenge those positions.

On the one hand, this suggests an asymmetry in the visibility of scientific criticism rather than an asymmetry in methodological rigor. On the other hand, when examining the methodological quality of the evidence base itself, a further limitation becomes apparent. The interpretation of epidemiological safety signals depends not only on the availability of studies, but also on the methodological robustness and heterogeneity of the studies being synthesized. Recent evaluations of COVID 19 vaccine safety literature have highlighted substantial methodological variation, including limitations in study design and evidence synthesis.⁸ Furthermore, an assessment of systematic intervention reviews on vaccination reported that 110 out of 120 reviewed systematic reviews were rated as being of critically low methodological quality.⁹

Epidemiology

Epidemiology, despite often being presented as a robust method for identifying potential risks associated with newly introduced medications at the population level, faces inherent methodological limitations.¹⁰ To begin, it is necessary to start from the perspective of infections, as epidemiology is primarily concerned with the dynamics of infectious disease, even though it is subject to the same principles when monitoring vaccines, pharmaceuticals, and novel medical treatments.

1. Limited and Imperfect Data Availability

Epidemiological models are only as reliable as the data on which they are based. At the beginning of an outbreak, dependable information on case numbers, dates of infection, and the true number of infected individuals is often unavailable. Reporting delays, differences in testing strategies, and both underreporting and overreporting further contribute to substantial uncertainty in the available data.

2. Uncertainty in Biological Parameters

Many key model parameters, such as the incubation period, the duration of infectiousness, the probability of transmission, and the proportion of asymptomatic infections, are poorly understood during the early stages of an outbreak. These uncertainties can strongly affect model predictions, and different combinations of parameter values may produce similar results.

3. Simplifying Model Assumptions

Every mathematical model represents a simplification of reality. Many models assume that individuals interact randomly or that all people are equally susceptible to infection and equally infectious once infected. In reality, contact patterns vary according to age, occupation, household composition, geographic location, and social behavior. Such simplifying assumptions may therefore limit the explanatory and predictive value of epidemiological models.

4. Changes in Human Behavior

Population behavior changes continuously throughout an epidemic, either voluntarily or in response to public health policies. Reductions in social contacts, mask wearing, vaccine uptake, and pandemic fatigue all have a substantial impact on disease transmission. Because these behavioral changes are dynamic and difficult to predict, they are also difficult to represent accurately in mathematical models.

5. The Influence of Public Health Interventions

Nonpharmaceutical interventions, including lockdowns, school closures, and contact restrictions, as well as vaccination programs, substantially alter the course of an epidemic. However, the isolated effect of individual interventions is often difficult to determine because multiple measures are typically implemented simultaneously and may interact with one another.

6. Uncertainty of Model Predictions

Mathematical models do not generate exact forecasts. Instead, they produce scenarios based on a particular set of assumptions. Even small changes in input data or parameter values can lead to substantially different outcomes. Consequently, model predictions should always be presented together with measures of uncertainty and should not be interpreted as precise forecasts of future epidemic dynamics.

7. Model Validation

Models must be continuously evaluated against newly available data and updated whenever necessary. Because pathogens, population behavior, and intervention strategies change over time, a model that performs well during the early stages of an epidemic may lose accuracy as the situation evolves.

Crépey et al. conclude that mathematical models are indispensable tools in modern epidemiology. However, their usefulness depends critically on the quality of the available data, the assumptions on which they are based, and the transparent communication of their inherent uncertainties. Models should therefore be understood as tools that support evidence based decision making rather than as machines capable of predicting the exact course of an epidemic.

Schuster et al.

Whereas Crépey et al. describe the structural and conceptual challenges of epidemic modeling at the level of entire models and modeling frameworks, including data limitations, parameter uncertainty, behavioral change, intervention effects, and large scale validation, Schuster et al.¹¹ (2022) focus on a more fundamental issue: the correct statistical specification of individual variables within a regression model.

More specifically, Schuster et al. primarily address the third challenge identified by Crépey et al., namely simplifying model assumptions. One of the most common yet least critically examined assumptions in epidemiological regression models is linearity, namely the assumption that an exposure variable has a constant effect on the outcome across its entire range of values. Schuster et al. demonstrate that this assumption is frequently unjustified and, if accepted without careful examination, can produce biased effect estimates. This is conceptually similar to the simplifying assumptions criticized by Crépey et al., such as homogeneous susceptibility or random mixing, which likewise reduce the validity of a model when the underlying reality is more complex than the model assumes.

The key difference lies in the available solutions. Crépey et al. describe most of their challenges as either inherently unresolved or only partially manageable through transparent communication of uncertainty. There is no straightforward statistical method that can fully account for unpredictable behavioral changes or disentangle overlapping intervention effects. By contrast, Schuster et al. propose a concrete and practical solution to their more narrowly defined problem. They recommend the use of spline functions instead of quadratic terms or categorization to model nonlinear relationships appropriately. Thus, whereas the uncertainties discussed by Crépey et al. largely remain intrinsic to the modeling process and can only be managed rather than eliminated, the form of model misspecification addressed by Schuster et al., namely an incorrect assumption of linearity, can in fact be corrected through an appropriate statistical approach.

Crépey et al. provide the overarching reminder that every epidemiological model is constrained by the quality of its assumptions and data, and that uncertainty can never be eliminated completely. Schuster et al. offer a concrete counterexample at the methodological level by identifying one specific, frequently overlooked modeling assumption, linearity, that can be corrected using a well established and readily applicable statistical technique. Epidemiological modeling therefore operates between these two extremes: structural limitations that are inherently difficult to resolve, as emphasized by Crépey et al., and specific methodological errors that can be avoided by selecting appropriate statistical tools, as demonstrated by Schuster et al.

LNPs–Supramolecular Assemblies

As demonstrated by the two limitation examples discussed above, namely the broader structural limitations of epidemiological models described by Crépey et al. and the specific problem of incorrect linearity assumptions addressed by Schuster et al., the interpretation of epidemiological data is highly dependent on the assumptions embedded within the chosen analytical framework. Averages derived from multiple individual linear regression models may therefore fail when applied to systems characterized by nonlinear dynamics. Such approaches implicitly assume that the relationship between an exposure and an outcome remains constant across the observed range, while biological and epidemiological processes often involve thresholds, saturation effects, feedback loops, time dependent changes, and interactions between multiple variables. By averaging across these complex relationships, linear models can conceal meaningful variation and produce estimates that represent a statistical average rather than the underlying mechanism. Thus, the limitation is not necessarily the statistical method itself, but the mismatch between the simplifying assumptions of the model and the complexity of the biological system being analyzed.

The LNP modRNA platform presents precisely this challenge. As extensively discussed and reviewed in our paper¹, LNP populations do not exhibit homogeneity.¹²^,¹³ This heterogeneity includes differences in cargo loading, where approximately three to four modRNA strands are considered indicative of a fully loaded LNP, whereas particles containing fewer strands or no encapsulated modRNA are classified as partially loaded or “unpackaged” LNPs.

One could, of course, argue that these represent only minor uncertainties. However, this is precisely where epidemiological interpretation appears to overlook the implications of such “small” uncertainties in the context of LNPs. The issue with frozen kinetic minima: the system cannot be characterized as a conventional molecular entity with a fixed composition and predictable behavior. These seemingly minor differences can disproportionately influence the overall behavior and biological properties of the nanoparticle construct that emerges from LNPs in vivo.

The critical questions that would have needed to be addressed before attempting to derive an epidemiological model from the numerous underlying probabilities should have been the following:

First, how was the biological reactivity of LNPs as supramolecular interfacial assemblies accounted for or excluded, particularly when more recent studies no longer treat LNPs merely as passive delivery vehicles and when comprehensive investigations of this specific safety dimension were lacking?

Second, how was dose equivalence established at the level of the actual nanoparticle system? More specifically, how was a comparable number of LNP particles per patient dose ensured, and how was the approximate number of intact RNA molecules per LNP controlled between individual vials and production batches?

This question is critical because, based on extensive LNP literature and several peer reviewed studies investigating batch related differences in adverse event profiles, LNP based products are inherently subject to distributional variability arising from their nanoscale composition and assembly process.¹⁴^,¹⁵^,¹⁶^,¹⁷

Therefore, the relevant question is not simply whether a nominal amount of modRNA was administered, but rather: what was the actual composition of the nanoparticle population delivered in each dose?

What data are being used to support assumptions regarding these parameters that may not have been considered?

To make the question more specific: how many LNP particles does an average modRNA injection dose from Pfizer and Moderna contain? What proportion of different LNP species is present within a typical dose? How many particles meet the definition of a fully loaded LNP within the field of LNP research, meaning particles containing approximately three to four intact modRNA strands? How many represent minimally loaded or unpackaged particles? Furthermore, how many modRNA molecules per particle remain structurally intact on average?

Finally, how were these parameters characterized and validated at the scale of industrial mass production? And how much of the genetic material was intact?¹⁸

These questions do not appear to have been systematically addressed within the development and regulatory evaluation framework of the COVID 19 mRNA platforms. Instead, LNPs were primarily evaluated as delivery vehicles, with a major focus placed on immunological outcomes, particularly measures of immune response such as antibody generation, alongside conventional population level safety monitoring. As discussed above, such evaluations are inherently dependent on the variables selected and the assumptions embedded within the analytical framework. Consequently, potential implications arising from LNP heterogeneity, particle level variation, and probabilistic variation in biological exposure may not have been incorporated as central variables within conventional epidemiological assessments.

This creates a fundamental challenge when applying conventional epidemiological modeling assumptions to such a system. Many analyses implicitly rely on relatively simple dose response relationships, where the administered quantity is assumed to represent a sufficiently stable proxy for biological exposure. However, for complex nanoparticle based platforms, this assumption may be insufficient. Biological effects may depend not only on the nominal dose, but also on particle composition, biodistribution, cellular uptake, intracellular processing, and substantial probabilistic variation between individual nanoparticle populations.

Epidemiology is highly valuable for identifying population level patterns and associations, but its ability to detect subtle, mechanistically complex, or delayed biological effects is limited when these effects do not produce immediate or clearly attributable population signals. Potential long term consequences would therefore require dedicated longitudinal analyses rather than relying exclusively on short term observational patterns.

Based on the fact that the history of gene therapy shows that biological systems can exhibit delayed consequences due to mechanisms such as clonal selection, cellular adaptation, and nonlinear amplification processes (especially in intracellular behavior). Initial molecular processes may go clinically unnoticed for extended periods of time before they become detectable through subsequent biological changes.¹⁹^,²⁰^,²¹

A further consideration is that long term biological consequences do not necessarily require direct genetic modification. Cellular behavior is not exclusively determined by changes at the genomic level, but also by regulatory signaling networks that control proliferation, metabolism, differentiation, stress responses, and cellular adaptation. As discussed in our paper, pathways such as MAPK and mTOR signaling represent regulatory platforms capable of integrating and amplifying upstream signals over time. Therefore, persistent alterations in cellular regulatory states may theoretically emerge from changes in signaling dynamics, even in the absence of permanent genomic integration.

In other words, the biological outcome of a molecular intervention may depend not only on whether genetic information is introduced, but also on whether fundamental cellular regulatory networks are altered in their activity, feedback regulation, or response thresholds.

This highlights a broader methodological issue: epidemiological models are constrained by the variables they are designed to measure. When relevant biological mechanisms are not represented within the model framework, they may remain invisible regardless of the statistical power of the analysis.

(Addendum 28.06.) Discussion

Following the physicochemical characteristics of LNPs, it is assumed that the underlying exposure cannot be adequately described by a single linear relationship, but rather by a complex, non-linear probability distribution with numerous potential downstream outcomes. The associated probabilities vary as a function of individuum, LNP-probabilities, dose, and between individual batches (batch effects), while also changing across distinct temporal phases. At the same time, the available epidemiological datasets are globally heterogeneous, incomplete, of varying precision, and potentially influenced by systematic reporting or ascertainment biases. Nevertheless, conventional linear regression models are predominantly applied, resulting in inconsistent findings, with some studies identifying statistically significant associations while others do not.

The underlying dynamics include, in particular:

Time dependency

• Acute

• Intermediate

• Late

Additional methodological limitations

• Missing not at random

• Different reporting probabilities

• Politically motivated underreporting

• Different diagnostic standards

• Different follow-up durations

Under these assumptions, the following methodological conclusion appears plausible: If the true data-generating process is highly non-linear, time-dependent, hierarchical, and only incompletely observed, then simple linear regression models may exhibit low statistical power or produce biased estimates. Under such conditions, contradictory findings during the first years are to be expected, and more sophisticated approaches. From a hierarchical modeling perspective, Bayesian models with time-dependent components may better capture the underlying uncertainty and heterogeneity.

This does not automatically imply the following: The methodological limitations described above do not automatically imply that a clinically relevant long-term causal effect exists but has merely remained undetected. The same limitations may also generate spurious associations or lead studies to reach conflicting conclusions even when the true effect is small or absent.

A Bayesian modeling framework is particularly well suited to this scenario because it can simultaneously account for multiple sources of uncertainty. Rather than providing only a single point estimate, a Bayesian model can produce statements of the form:

Given the uncertainty associated with batch, dose, measurement error, and follow-up, the true risk is likely to lie between X and Y.

Furthermore, Bayesian models can:

• Represent hierarchical structures (Country → Clinic → Batch → Individual)

• Handle missing data probabilistically

• Integrate multiple data sources

• Explicitly incorporate prior knowledge

• Estimate posterior probabilities rather than relying exclusively on p-values

Under the assumptions described above, it would therefore be reasonable to expect that no consistent epidemiological patterns may emerge during the first 5–10 years, particularly if the following conditions are present simultaneously:

• Long latency periods

• Small to moderate effect sizes

• Substantial heterogeneity between subgroups

• Different exposure patterns

• Measurement error

• Incomplete follow-up

• Inappropriate model assumptions

However, the core limitation lies not in the complete absence of individual mechanistic models, but rather in the lack of an integrated, multiscale systems biology framework capable of linking the relevant biological levels. Existing approaches have largely focused on isolated components, such as formulation-specific behavior, pharmacokinetics, biodistribution, or immunological reactions. In other words, no broadly applicable multiscale model has yet been established that links the physicochemical properties of LNPs with cellular uptake mechanisms, cell-type-specific internalization patterns, tissue distribution, temporal dynamics, and variability at the population level.

Fundamental determinants (including the mechanisms governing entry into the cell, the spectrum of cell populations capable of LNP uptake, the probabilistic structure of biodistribution pathways, and the boundary conditions influencing these processes) remain insufficiently characterized in an integrated manner. Consequently, current modeling approaches lack the mechanistic depth required to reliably predict LNP behavior across individuals, formulations, dosages, and time scales.

Summary

This article presents a conceptual framework examining the limitations of applying conventional epidemiological models to complex biological systems such as LNP based modRNA platforms. While epidemiological approaches are essential tools for identifying population level associations, their interpretive power depends on the assumptions, variables, and biological processes represented within the model. The discussion highlights general limitations of epidemiological modeling, including uncertainty in data quality, parameter estimation, behavioral dynamics, intervention effects, and simplifying assumptions, before extending this framework to the challenge of modeling supramolecular nanoparticle systems.

In conclusion, it can be stated that the RKI/WHO’s own internal critique of the data sets, which highlighted serious quality deficiencies, continues to be ignored by the RKI, the WHO, and all health authorities. This methodological uncertainties, which are acknowledged even within the system but is not communicated transparently confirms the postulated critiques. As for my intrinsic motivation: It seems to be paying off, since Stöhr’s rhetoric, along the lines of “The data show…”, obscures reality and the literature that does not reflect a consensus and not a simple term of “The data show…”.

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