The relationship between compartment models and their stochastic counterparts:A comparative study with examples of the COVID-19 epidemic modeling

Deterministic compartment models(CMs)and stochastic models,including stochastic CMs and agent-based models,are widely utilized in epidemic modeling.However,the relationship between CMs and their corresponding stochastic models is not well understood.The present study aimed to address this gap by conducting a comparative study using the susceptible,exposed,infectious,and recovered(SEIR)model and its extended CMs from the coronavirus disease 2019 modeling literature.We demonstrated the equivalence of the numerical solution of CMs using the Euler scheme and their stochastic counterparts through theoretical analysis and simulations.Based on this equivalence,we proposed an efficient model calibration method that could replicate the exact solution of CMs in the corresponding stochastic models through parameter adjustment.The advancement in calibration techniques enhanced the accuracy of stochastic modeling in capturing the dynamics of epidemics.However,it should be noted that discrete-time stochastic models cannot perfectly reproduce the exact solution of continuous-time CMs.Additionally,we proposed a new stochastic compartment and agent mixed model as an alternative to agent-based models for large-scale population simulations with a limited number of agents.This model offered a balance between computational efficiency and accuracy.The results of this research contributed to the comparison and unification of deterministic CMs and stochastic models in epidemic modeling.Furthermore,the results had implications for the development of hybrid models that integrated the strengths of both frameworks.Overall,the present study has provided valuable epidemic modeling techniques and their practical applications for understanding and controlling the spread of infectious diseases.

A Reevaluation of Prazosin Pharmacokinetics in a Two-Compartment Model, the Apparent Volume of Distribution and Comparative Simulations in the One-Compartment Model

Published clinical data of Prazosin were reevaluated pharmacokinetically using explicit solutions to drug concentration as a function of total time for IV bolus injection, intermittent intravenous infusion and oral routes of administration in an open two-compartment model. In a novel way, the apparent volume of distribution was estimated from a two-compartment model and found to be close to the total body water suggesting that Prazosin is distributed in all tissues both extracellularly and intracellularly. In addition, extracting the value of the apparent volume of distribution from a two-compartment model allowed comparative simulations in the one-compartment model. It is shown that dosage calculations of Prazosin intermittent infusion can be safely performed using the simpler one-compartment model equations. Lastly, several additional time-dependent pharmacokinetic parameters e.g., the peak time in the central and peripheral compartment and non-steady state and steady state peak concentration and AUC were determined using series equations for all three routes of administration, as a function of dose number and total time upon multiple drug administrations in the two-compartment model. It is also the first time that steady-state plasma drug concentration equations were derived in a two-compartment mammillary model.

Real-Time Analytical Solutions as Series Formulas and Heaviside off/on Switch Functions for Multiple Intermittent Intravenous Infusions in One- and Two-Compartment Models

Pharmacokinetic compartment models are the only models that can extract pharmacokinetic parameters from data collected in clinical studies but their estimates lack accuracy, explanations and physiological significance. The objective of this work was to develop particular solutions to drug concentration and AUC in the form of mathematical series and Heaviside functions for repetitive intermittent infusions in the one- and two-compartment models, as a function of dose number and total time using differential calculus. It was demonstrated that the central and peripheral compartment volumes determined from regression analysis of the aminoglycoside antibiotic Sisomicin concentration in plasma represent the actual physiological body fluid volumes accessible by the drug. The drug peak time and peak concentration in the peripheral compartment were also calculated as a function of dose number. It is also shown that the time of intercompartmental momentary distribution equilibrium can be used to determine the drug’s apparent volume of distribution within any dosing interval in multi-compartment models. These estimates were used to carry out simulations of plasma drug concentration with time in the one-compartment model. In conclusion, the two-compartment open mammillary pharmacokinetic model was fully explained for the aminoglycoside antibiotic sisomicin through the new concept of the apparent volume of distribution.

Estimating Hydrogeological Parameters in Covered Carbonate Rocks Using a Discrete-State Compartment Model and Environmental Tritium

Nowadays, isotope environmental technique tends to be used as a reconnaissance tool , both qualitative and quantitative, to calculate the aquifer parameters particularly in carbonate rock aquifers. But, the heterogeneous flow is still problematic when Lumped parameter Models are usually used to calculate the residence times and hydraulic parameters. However, Discrete State Compartment Model can provide a powerful model to heterogeneous medium. One such study was carried on in Dazha valley, where the environmental tritium was used as a tracer for determining hydrogeological parameters based on a discrete state compartment model

The Presence of Phases and the Inability of the Classical Compartment Models to Provide Pharmacokinetic Parameters of Physiological Significance for Lipophilic Drugs

The first biphasic open one-compartment pharmacokinetic model is described. Its analytical solutions to drug concentration were developed from parameters of an open two-compartment pharmacokinetic model. The model is used to explain the unusually large compartment volumes and apparent volumes of distribution of lipophilic drugs, as well as to identify which of the pharmacokinetic parameters of the classical compartment models are biologically relevant.

PERTURBATION SOLUTION FOR COMPARTMENT MODEL OF VARIABLE EXTRACELLULAR VOLUME

The two compartment model with variable extracellular volume is presented and solved by using both perturbation and analytical method. The computation for both creatinine and urea show that the perturbation solution is not only simple but also accurate enough and is a good substitute for the more exact analytical solution.

Analysis of mesoscale effects in high-shear granulation through a computational fluid dynamics-population balance coupled compartment model

There is a need for mesoscale resolution and coupling between flow-field information and the evolution of particle properties in high-shear granulation. We have developed a modelling framework that com- partmentalizes the high-shear granulation process based on relevant process parameters in time and space. The model comprises a coupled-flow-field and population-balance solver and is used to resolve and analyze the effects of mesoscales on the evolution of particle properties. A Diosna high-shear mixer was modelled with microcrystalline cellulose powder as the granulation material. An analysis of the flow-field solution and compartmentalization allows for a resolution of the stress and collision peak at the impeller blades. Different compartmentalizations showed the importance of resolving the impeller region, for aggregating systems and systems with breakage. An independent study investigated the time evolution of the flow field by changing the particle properties in three discrete steps that represent pow- der mixing, the initial granulation stage mixing and the late stage granular mixing. The results of the temporal resolution study show clear changes in collision behavior, especially from powder to granular mixing, which indicates the importance of resolving mesoscale phenomena in time and space.

A compartment model and numerical analysis of circulatory economy

This paper describes an industrial structure and its equation system of a circular economy for material circulation and builds a system dynamic model for resources recycling utilization based on Compartment Model Theory.A circulation multiplier and its computational formula are defined for measuring the efficiency of resources recycling utilization.The simulated results indicate that the resources recycling utilization can not only realize the amount accumulation of natural resources and improve the resources recycling efficiency but can minimize discharges into natural environment by means of adjustment to each compartment parameter in the circular economy.

A Tutorial on Common Differential Equations and Solutions Useful for Modeling Epidemics Like COVID-19: Linear and Non-Linear Compartmentation Models

Purpose: To review some of the basic models, differential equations and solutions, both analytic and numerical, which produce time courses for the fractions of Susceptible (S), Infectious (I) and Recovered (R) fractions of the population during the epidemic and/or endemic conditions. Methods: Two and three-compartment models with analytic solutions to the proposed linear differential equations as well as models based on the non-linear differential equations first proposed by Kermack and McKendrick (KM) [1] a century ago are considered. The equations reviewed include the ability to slide between so-called Susceptible-Infected-Recovered (SIR), Susceptible-Infectious-Susceptible (SIS), Susceptible-Infectious (SI) and Susceptible-Infectious-Recovered-Susceptible (SIRS) models, effectively moving from epidemic to endemic characterizations of infectious disease. Results: Both the linear and KM model yield typical “curves” of the infected fraction being sought “to flatten” with the effects of social distancing/masking efforts and/or pharmaceutical interventions. Demonstrative applications of the solutions to fit real COVID-19 data, including linear and KM SIR fit data from the first 100 days following “lockdown” in the authors’ locale and to the total number of cases in the USA over the course of 1 year with SI and SIS models are provided. Conclusions: COVID-19 took us all by surprise, all wondering how to help. Spreading a basic understanding of some of the mathematics used by epidemiologists to model infectious diseases seemed like a good place to start and served as the primary purpose for this tutorial.

COMPARTMENTAL MODELS FOR SEPARATION COLUMNS

A generalized compartmental modelling method for the derivation of reduced-order models for multicomponent staged separation columns is presented. In contrast to the one proposed by Benallou, this method uses dynamic state relationships, rather than steady state ones, such that superfluous composition variables resulting from the compartmental representation of staged separation columns may be eliminated. The accuracy of transient responses of the resulting model is, therefore, substantially improved. Furthermore, the phenomena of initial inverse responses, usually encountered in the development of reduced-order methods, has also been deleted completely. Simulations of a deethanizer have been conducted to evaluate this modelling method. Results indicate high efficiency and robustness in minimizing the dimensionality and computation time of the staged separation column model.

A compartment and metapopulation model of Rocky Mountain spotted fever in southwestern United States and northern Mexico

Rocky Mountain spotted fever(RMSF)is a fatal tick-borne zoonotic disease that has emerged as an epidemic in western North America since the turn of the 21st century.Along the US south-western border and across northern Mexico,the brown dog tick,Rhipicephalus sanguineus,is responsible for spreading the disease between dogs and humans.The widespread nature of the disease and the ongoing epidemics contrast with historically sporadic patterns of the disease.Because dogs are amplifying hosts for the Rickettsia rickettsii bacteria,transmission dynamics between dogs and ticks are critical for understanding the epidemic.In this paper,we developed a compartment metapopulation model and used it to explore the dynamics and drivers of RMSF in dogs and brown dog ticks in a theoretical region in western North America.We discovered that there is an extended lag—as much as two years—between introduction of the pathogen to a naïve population and epidemic-level transmission,suggesting that infected ticks could disseminate extensively before disease is detected.A single large city-size population of dogs was sufficient to maintain the disease over a decade and serve as a source for disease in surrounding smaller towns.This model is a novel tool that can be used to identify high risk areas and key intervention points for epidemic RMSF spread by brown dog ticks.

眼用药物药代动力学模型研究进展

视力与生活质量息息相关。临床上倾向于使用侵入性较小的外用和全身给药方式治疗青光眼等眼部常见疾病。眼部许多的生理生化屏障包括泪液周转、角膜渗透、血-眼屏障等,限制药物在眼部的渗透与分布,造成眼内药代动力学特征不明确,常用眼部房室模型来描述药物在眼内处置动力学。经典眼房室模型以角膜或玻璃体为中央室,将眼部其他组织整体视为外周室,而基于生理的药代动力学(PBPK)模型则引入眼部血流量变化、转运体对药物转运影响、血-眼屏障等因素,可提供更多药物在眼部的处置细节,有助于辅助眼用新药的开发和指导眼部疾病药物治疗。文章综述了不同给药方式时眼部用药的药代动力学特征,经典房室模型和PBPK模型及其在临床眼部用药方案设计中的应用。

A Bayesian model calibration framework for stochastic compartmental models with both time-varying and timeinvariant parameters

We consider state and parameter estimation for compartmental models having both timevarying and time-invariant parameters.In this manuscript,we first detail a general Bayesian computational framework as a continuation of our previous work.Subsequently,this framework is specifically tailored to the susceptible-infectious-removed(SIR)model which describes a basic mechanism for the spread of infectious diseases through a system of coupled nonlinear differential equations.The SIR model consists of three states,namely,the susceptible,infectious,and removed compartments.The coupling among these states is controlled by two parameters,the infection rate and the recovery rate.The simplicity of the SIR model and similar compartmental models make them applicable to many classes of infectious diseases.However,the combined assumption of a deterministic model and time-invariance among the model parameters are two significant impediments which critically limit their use for long-term predictions.The tendency of certain model parameters to vary in time due to seasonal trends,non-pharmaceutical interventions,and other random effects necessitates a model that structurally permits the incorporation of such time-varying effects.Complementary to this,is the need for a robust mechanism for the estimation of the parameters of the resulting model from data.To this end,we consider an augmented state vector,which appends the time-varying parameters to the original system states whereby the time evolution of the time-varying parameters are driven by an artificial noise process in a standard manner.Distinguishing between time-varying and time-invariant parameters in this fashion limits the introduction of artificial dynamics into the system,and provides a robust,fully Bayesian approach for estimating the timeinvariant system parameters as well as the elements of the process noise covariance matrix.This computational framework is implemented by leveraging the robustness of the Markov chain Monte Carlo algorithm permits the estimation of time-invariant parameters while nested nonlinear filters concurrently perform the joint estimation of the system states and time-varying parameters.We demonstrate performance of the framework by first considering a series of examples using synthetic data,followed by an exposition on public health data collected in the province of Ontario.

Approximations of Quasi-Stationary Distributions of the Stochastic <i>SVIR</i>Model for the Measles

In this paper, we analyze the quasi-stationary distribution of the stochastic SVIR (Susceptible, Vaccinated, Infected, Recovered) model for the measles. The quasi-stationary distributions, as discussed by Danoch and Seneta, have been used in biology to describe the steady state behaviour of population models which exhibit discernible stationarity before to become extinct. The stochastic SVIR model is a stochastic SIR (Susceptible, Infected, Recovered) model with vaccination and recruitment where the disease-free equilibrium is reached, regardless of the magnitude of the basic reproduction number. But the mean time until the absorption (the disease-free) can be very long. If we assume the effective reproduction number Rp < 1 or , the quasi-stationary distribution can be closely approximated by geometric distribution. β and δ stands respectively, for the disease transmission coefficient and the natural rate.

Establishment and Analysis on Material Flow Model in Argo-animal Husbandry Ecosystem

This paper analyzed the material flow situation in argo-animal husbandry ecosystem by compartment model. This model was an important mean for investigating the whole structural characteristics in ecosystem. Based on this analysis, characteristics of material cycle and integrity in the system were mastered. As an example of natural conditions in Yonghe Village, Shuangcheng Township, Shuangeheng Municipal, Heilongjang Province, the system of linear differential equations in system was established by extracting each compartment and investigating material flow and stability of this model was proved by Lyapunov linear theory. The result showed that this system could not be interfered by initial value in the state of present, input and output.

A Stochastic SVIR Model for Measles

In this article, we consider the construction of a SVIR (Susceptible, Vaccinated, Infected, Recovered) stochastic compartmental model of measles. We prove that the deterministic solution is asymptotically the average of the stochastic solution in the case of small population size. The choice of this model takes into account the random fluctuations inherent to the epidemiological characteristics of rural populations of Niger, notably a high prevalence of measles in children under 5, coupled with a very low immunization coverage.

Delayed Dynamics of SIR Model for COVID-19

This paper presents a new modified SIR model which incorporates appropriate delay parameters leading to a more precise prediction of COVID-19 real time data. The efficacy of the newly developed SIR model is proven by comparing its predictions to real data obtained from four counties namely Germany, Italy, Kuwait, and Oman. Two included delay periods for incubation and recovery within the SIR model produce a sensible and more accurate representation of the real time data. In the absence of the two-delay period () the dynamical behavior of the model will not correspond to today’s picture and lag the detection of the epidemic peak. The reproductive number R0 is defined for the model for values of recovery time delay of the infective case. The effect of recovery time may produce second wave, and/or an oscillation which could destabilize the behavior of the system and a periodic oscillation can arise due to Hopf bifurcation phenomenon.

Mathematical Modelling of the COVID-19 Epidemic in Northern Ireland in 2020

In this study, we investigate the dynamics of the COVID-19 epidemic in Northern Ireland from 1st March 2020 up to 25th December 2020, using several copies of a Susceptible-Exposed-Infectious-Recovered (SEIR) compartmental model, and compare it to a detailed publicly available dataset. We split the data into 10 time intervals and fit the models on the consecutive intervals to the cumulative number of confirmed positive cases on each interval. Using the fitted parameter estimates, we also provide estimates of the reproduction number. We also discuss the limitations and possible extensions of the employed model.

Surging footprints of mathematical modeling for prediction of transdermal permeability

In vivo skin permeation studies are considered gold standard but are difficult to perform and evaluate due to ethical issues and complexity of process involved. In recent past, a useful tool has been developed by combining the computational modeling and experimental data for expounding biological complexity. Modeling of percutaneous permeation studies provides an ethical and viable alternative to laboratory experimentation. Scientists are exploring complex models in magnificent details with advancement in computational power and technology. Mathematical models of skin permeability are highly relevant with respect to transdermal drug delivery, assessment of dermal exposure to industrial and environmental hazards as well as in developing fundamental understanding of biotransport processes.Present review focuses on various mathematical models developed till now for the transdermal drug delivery along with their applications.

Modeling Transmission Dynamics of COVID-19 in Nepal

A novel coronavirus disease (COVID-19) is an infectious viral disease caused by SARS-CoV-2. The disease was first reported in Wuhan, China, in December 2019, and it has been epidemic in more than 110 countries. The first case of COVID-19 was found in Nepal on 23 January, 2020. Now the number of confirmed cases is increasing day by day. Thus, the disease has become a major public health concern in Nepal. The propose of this study is to describe the development of outbreak of the disease and to predict the outbreak in Nepal. In the present work, the transmission dynamics of the disease in Nepal is analyzed mathematically with the help of SIR compartmental model. Reported data from June 1st to June 17th 2020 of Nepal are used to identify the model parameters. The basic reproduction number of COVID-19 outbreak in Nepal is estimated. Predictions of the peak epidemic time and the final size of the epidemic are made using the model. Our work predicts that, after 125 days from June 1 the infection will reach the peak. In this work, a good correlation between the reported data and the estimation given by our model is observed.