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.

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.

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

1 INTRODUCTIONRigorous dynamic models for multicomponent staged separation columns,based on thematerial and energy balance relationships for each of the stages,are usually of highdimensionality and high stiffness.Its solution needs very efficient integration algorithm.Although in the last decade a number of fast and efficient integration algorithms havebeen developed ,it is still extremely time-consuming to solve dynamic models

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.

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.

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.

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.

Mathematical Modeling of Malaria Transmission Dynamics: Case of Burundi

Burundi, a country in East Africa with a temperate climate, has experienced in recent years a worrying growth of the Malaria epidemic. In this paper, a deterministic model of the transmission dynamics of malaria parasite in mosquito and human populations was formulated. The mathematical model was developed based on the SEIR model. An epidemiological threshold, R0, called the basic reproduction number was calculated. The disease-free equilibrium point was locally asymptotically stable if R0 < 1 and unstable if R0 > 1. Using a Lyapunov function, we proved that this disease-free equilibrium point was globally asymptotically stable whenever the basic reproduction number is less than unity. The existence and uniqueness of endemic equilibrium were examined. With the Lyapunov function, we proved also that the endemic equilibrium is globally asymptotically stable if R0 > 1. Finally, the system of equations was solved numerically according to Burundi’s data on malaria. The result from our model shows that, in order to reduce the spread of Malaria in Burundi, the number of mosquito bites on human per unit of time (σ), the vector population of mosquitoes (Nv), the probability of being infected for a human bitten by an infectious mosquito per unit of time (b) and the probability of being infected for a mosquito per unit of time (c) must be reduced by applying optimal control measures.

五种群体药动学分析工具计算结果的比较

目的:比较群体药动学分析工具Phoenix NLME、Monolix、R语言nlmixr包和CPhaMAS云平台所计算的结果与金标准NONMEM的符合程度。方法:基于一房室模型模拟50个稀疏采样数据集和二房室模型模拟50个密集采样数据集,分别使用上述五种分析工具计算,比较群体典型值、个体变异和个体药动学参数。结果:CPhaMAS和Phoenix NLME计算的群体典型值、个体变异与NONMEM的匹配程度最高,其次为nlmixr,Monolix最低,但Monolix和nlmixr的算法可能更稳健。清除率和分布容积的对应性优于吸收速率常数。除了Monolix计算的吸收速率常数和nlmixr计算的房室间清除率,所有分析工具计算的个体药动学参数相关系数均大于0.99。结论:上述四种群体药动学分析工具计算结果与NONMEM的结果高度相关。

多成分药物代谢网络动力学数学模型建立及参数分析

目的:建立多成分药物代谢网络动力学数学模型,并对参数进行初步分析。方法:采用线性动力学方法建模,用拉氏变换法求解,研究多成分代谢网络动力学数学模型与参数的关系,并对多成分代谢网络动力学模型中各参数的求算进行初步分析。结果:药物多成分代谢网络动力学遵循一级线性乳突模型,c_i为e的幂指数多项式形式,与总室模型相似,各参数(转运常数)由其与各室间常数构成矩阵的特征值α_i的关系方程组算得。其动力学参数可在总室模型的基础上求算。结论:多成分代谢网络动力学各参数与总室一样可求得。

沈阳张士污灌区镉循环的分室模型与污染防治对策研究

对沈阳张士污灌区大气、水、土壤和生物各分室中Ｃｄ的库存量和各分室间Ｃｄ的循环通量进行了研究，建立了土壤－水稻系统中Ｃｄ流分室模型。该模型表明，Ⅱ闸和Ⅲ闸地区土壤分室中Ｃｄ的库存增量均为正值。可见，该污灌区近年来Ｃｄ污染程度由于污水灌溉仍然处于加剧的趋势。研究最后指出，该污灌区Ｃｄ的污染防治对策主要是减少污水灌溉量、进一步降低灌溉水中Ｃｄ的浓度和通过富Ｃｄ植物收获带走Ｃｄ．其中，以野生苋的收获带走Ｃｄ最有意义，它可使Ⅱ闸和Ⅲ闸地区Ｃｄ的下降速率分别依次达到２９．４９和２２．８１ｍｇ／（ｍ￣２·ａ）。

氚水在模拟水稻田中的消长动力学（续）

对进入水稻田中的氚水在连作晚稻阶段的迁移、输运动态过程的研究表明，滞留于早稻田中的氚水继续为晚稻摄取；与早稻情况一样，氚水中的氚仍以自由水氚和结合态氚形式滞留在晚稻田各分室中；晚稻田水和土壤中的氚浓度均随时间单调地减少；晚稻地上部及根中的自由水氚及结合态氚浓度在采样期间也基本上随时间降低．