Leonid Kalachev

Quasi-Steady State Approximation approach is used to derive reduced models of facilitated diffusion for the cases where adjusted concentration of mediators located on the boundary separating two compartments is much smaller compared with the concentration of particles in these compartments transported across the boundary via carrier-mediated transport mechanism. The cases of a generic mediator kinetics scheme as well as of more realistic mediator reactions are studied for symmetric boundary permeable in both directions. It is shown that for the generic case, under condition of sinks and sources absence, the newly derived reduced facilitated diffusion formula can be transformed to a model of an ordinary diffusion between compartments following Fick's Law. However, for the more realistic kinetics case, such transformation is not possible due to the presence of non-linearity in the corresponding reduced model. For semi-permeable boundary, i.e., when the particles can be transported across the boundary only in one direction, from one compartment to another, but not in the opposite direction, the derived formulas can be further reduced to produce, as a special case, the classical Michaelis - Menten - Henri kinetics approximation. The derived approximate formulas are intended for use in multiple medical, neuroscience, biological, and other applications.

Drug dosage determination and potential drug interference when multiple medical compounds must be administered simultaneous is an important long-standing problem both in practical pharmacokinetics and in theoretical drug design modeling. Very simple, and mostly linear, models are currently used to describe drug distribution in a body, drug function, and drug elimination. Many of the processes involved in drug delivery occur on vastly different time scales. This fact and, in particular, the presence of fast forward and reverse drug binding to blood proteins, is used in this paper to produce the reduced models describing time dependent drug dynamics during intravenous drug delivery, i.e., when the drug is administered directly in patient's vein via catheter. In addition, the questions on whether the drug dosage must be adjusted in the presence of protein binding compared to the case of drugs which do not bind, as well as what happens when two administered drugs participate in competing protein binding reactions are addressed. The singularly perturbed models derived under natural assumptions are analyzed using the boundary function method approach.

Under-reporting and, thus, uncertainty around the true incidence of health events is common in all public health reporting systems. While the problem of under-reporting is acknowledged in epidemiology, the guidance and methods available for assessing and correcting the resulting bias are obscure. We aim to design a simple modification to the Susceptible – Infected – Removed (SIR) model for estimating the fraction or proportion of reported infection cases. The suggested modification involves rescaling of the classical SIR model producing its mathematically equivalent version with explicit dependence on the reporting parameter (true proportion of cases reported). We justify the rescaling using the phase plane analysis of the SIR model system and show how this rescaling parameter can be estimated from the data along with the other model parameters. We demonstrate how the proposed method is cross-validated using simulated data with known disease cases and then apply it to two empirical reported data sets to estimate the fraction of reported cases in Missoula County, Montana, USA, using: (1) flu data for 2016–2017 and (2) COVID-19 data for fall of 2020. We establish with the simulated and COVID-19 data that when most of the disease cases are presumed reported, the value of the additional reporting parameter in the modified SIR model is close or equal to one, so that the original SIR model is appropriate for data analysis. Conversely, the flu example shows that when the reporting parameter is close to zero, the original SIR model is not accurately estimating the usual rate parameters, and the re-scaled SIR model should be used. This research demonstrates the role of under-reporting of disease data and the importance of accounting for under-reporting when modeling simulated, endemic, and pandemic disease data. Correctly reporting the “true” number of disease cases will have downstream impacts on predictions of disease dynamics. A simple parameter adjustment to the SIR modeling framework can help alleviate bias and uncertainty around crucial epidemiological metrics (e.g.: basic disease reproduction number) and public health decision making.

Classical infectious disease models during epidemics have widespread usage, from predicting the probability of new infections to developing vaccination plans for informing policy decisions and public health responses. However, it is important to correctly classify reported data and understand how this impacts estimation of model parameters. The COVID-19 pandemic has provided an abundant amount of data that allow for thorough testing of disease modelling assumptions, as well as how we think about classical infectious disease modelling paradigms. We aim to assess the appropriateness of model parameter estimates and prediction results in classical infectious disease compartmental modelling frameworks given available data types (infected, active, quarantined, and recovered cases) for situations where just one data type is available to fit the model. Our main focus is on how model prediction results are dependent on data being assigned to the right model compartment. We first use simulated data to explore parameter reliability and prediction capability with three formulations of the classical Susceptible-Infected-Removed (SIR) modelling framework. We then explore two applications with reported data to assess which data and models are sufficient for reliable model parameter estimation and prediction accuracy: a classical influenza outbreak in a boarding school in England and COVID-19 data from the fall of 2020 in Missoula County, Montana, USA. We demonstrated the magnitude of parameter estimation errors and subsequent prediction errors resulting from data misclassification to model compartments with simulated data. We showed that prediction accuracy in each formulation of the classical disease modelling framework was largely determined by correct data classification versus misclassification. Using a classical example of influenza epidemics in an England boarding school, we argue that the Susceptible-Infected-Quarantined-Recovered (SIQR) model is more appropriate than the commonly employed SIR model given the data collected (number of active cases). Similarly, we show in the COVID-19 disease model example that reported active cases could be used inappropriately in the SIR modelling framework if treated as infected. We demonstrate the role of misclassification of disease data and thus the importance of correctly classifying reported data to the proper compartment using both simulated and real data. For both a classical influenza data set and a COVID-19 case data set, we demonstrate the implications of using the “right” data in the “wrong” model. The importance of correctly classifying reported data will have downstream impacts on predictions of number of infections, as well as minimal vaccination requirements.

We present a simple model of experimental setup for in vitro study of drug release from drug eluting stents and drug propagation in artificial tissue samples representing blood vessels. The model is further reduced using the assumption on vastly different characteristic diffusion times in the stent coating and in the artificial tissue. The model is used to derive a relationship between the times at which the measurements have to be taken for two experimental platforms, with corresponding artificial tissue samples made of different materials with different drug diffusion coefficients, to properly compare the drug release characteristics of drug eluting stents.