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.

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.

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.

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.

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.

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.

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

Numerical simulation of flow regions in red mud separation thickener's feedwell by analysis of residence-time distribution

The residence-time distribution （RTD） and the compartment model were applied to characterizing the flow regions in red mud separation thickener’s feedwells. Combined with the experimental work, validated mathematical model as well as three-dimensional computational fluid dynamics （CFD） model was established to analyze the flow regions of feedwells on an industrial scale. The concept of RTD, although a well-known method for the characterization of mixing behavior in conventional mixers and reactors, is still a novel measure for the characterization of mixing in feedwells. Numerical simulation results show that the inlet feed rate and the aspect ratio of feedwells are the most critical parameters which affect the RTD of feedwell. Further simulation experiments were then carried out. Under the optimal operation conditions, the volume fraction of dead zone can reduce by10.8% and an increasement of mixing flow volume fraction by 6.5% is also observed. There is an optimum feed inlet rate depending on the feedwell design. The CFD model in conjunction with the RTD analysis then can be used as an effective tool in the design, evaluation and optimization of thickener feedwell in the red mud separation.

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.

Nitrogen cycling of atmosphere-plant-soft system in the typical Calamagrostis angustifolia wetland in the Sanjiang Plain,Northeast China

The nitrogen （N） distribution and cycling of atmosphere-plant-soil system in the typical meadow Calamagrostis angustifolia wetland （TMCW） and marsh meadow Calamagrostis angustifolia wetland （MMCW） in the Sanjiang plain were studied by a compartment model. The results showed that the N wet deposition amount was 0.757 gN/（m^2·a）, and total inorganic N （TIN） was the main body （0.640 gN/（m^2·a））. The ammonia volatilization amounts of TMCW and MMCW soils in growing season were 0.635 and 0.687 gN/m^2, and the denitrification gaseous lost amounts were 0.617 and 0.405 gN/m^2, respectively. In plant subsystem, the N was mainly stored in root and litter. Soil organic N was the main N storage of the two plant-soil systems and the proportions of it were 93.98% and 92.16%, respectively. The calculation results of N turnovers among compartments of TMCW and MMCW showed that the uptake amounts of root were 23.02 and 28.18 gN/（m^2·a） and the values of aboveground were 11.31 and 6.08 gN/（m^2·a）, the re-translocation amounts from aboveground to root were 5.96 and 2.70 gN/（m^2·a）, the translocation amounts from aboveground living body to litter were 5.35 and 3.38 gN/（m^2·a）, the translocation amounts from litter to soil were larger than 1.55 and 3.01 gN/（m^2·a）, the translocation amounts from root to soil were 14.90 and 13.17 gN/（m^2·a）, and the soil （0-15 cm） N net mineralization amounts were 1.94 and 0.55 gN/（m^2·a）, respectively. The study of N balance indicated that the two plant-soil systems might be situated in the status of lacking N, and the status might induce the degradation of C. angustifolia wetland.

Temporal patterns of storage and flux of N and P in young Teak plantations of tropical moist deciduous forest,India

Teak （Tectona grandis Linn. f.） ranks among the top five tropical hardwood species and is being promoted for use in plantations in its non-native range due to its high economic value. However, there is a general lack of data on ecosystem functioning of teak plantations. We aimed at understanding storage and flux of nutrients related to young plantations of teak. Cycling of nitrogen （N） and phosphorus （P） in a chronosequence of plantations （1, 5, 11, 18, 24 and 30 years） was studied in the Moist Deciduous Forest Region of North India with the objective of investigating the nutrient cycling pattern at younger age since the current trend of harvesting age of the species in several tropical countries is being drastically reduced for quick return from this high value crop. Standing state, nutrient uptake, nutrient return and nutrient retransloca-tion in these plantations were estimated by tree harvesting and chemical analysis methods. The range of total standing nutrient across all these plantations was 20.3 to 586.6 kg？ha-1 for N and 5.3 to 208.8 kg？ha-1 for P. Net uptake of N ranged from 19.4 to 88.9 kg？ha-1？a-1 and P from 3.8 to 18.1 kg？ha-1？a-1. Retranslocation of N and P among all the stands ranged from 8.7 to 48.0 kg？ha-1？a-1 and 0.01 to 3.5 kg？ha-1？a-1, respectively. Range of total nutrient return was 25.8 to 91.3 kg？ha-1？a-1 for N and 2.7 to 10.1 kg？ha-1？a-1 for P. N and P use efficiency was between 107.4 and 192.5 g dry organic matter （OM） g-1N, and 551.9 and 841.1 g OM g-1P, respec-tively. The turnover time ranged from 2.04-13.17 years for N and be-tween 2.40-22.66 years for P. Quantity of N and P in the soil nutrient pool ranged from 2566.8 to 4426.8 kg？ha-1 and 372 to 520 kg？ha-1, re-spectively. Storage and flux of components in different plant parts of different aged plantations were assessed and depicted in compartment models. Percentage storage in soil, litter and vegetation ranged from 82% to 99%, 0.6% to 2.4% and 0.5% to 15% for N, respectively, and from 63% to 98%, 0.5% to 2% and 1% to 35% for P, respectively. This infor-mation could be useful in managing external nutrient manipulation to crops of different ages for optimum biomass production or carbon se-questration.

Ecosystem Productivity and Energy Flow of Three-Hardwood Forest

The main characteristics of energy environment, energy products, primary productivity and basic process ofenergy flow for three-hardwood forest(Juglans mandshurica, Fraxinus mandshurica, and Phellodendron amurense) werestudied. The research was mainly hased on the thcory and method of community energetics, dealing with fixed position,quantitative test and expcrimental analysis. The time-space dynamics of sun-radiation in three-hardwood forest were measured and the energy compartment model was set up. his rescarch work provided a scientitic basis for the exploitation, utilization and management of three-hardtwood forest.

Parameter optimization of pharmacokinetics based on artificial immune network

A new method for parameter optimization of pharmacokinetics based on an artificial immune network named PKAIN is proposed. To improve local searching ability of the artificial immune network, a partition-based concurrent simplex mutation is developed. By means of evolution of network cells in the PKAIN artificial immune network, an optimal set of parameters of a given pharmacokinetic model is obtained. The Laplace transform is applied to the pharmacokinetic differential equations of remifentanil and its major metabolite, remifentanil acid. The PKAIN method is used to optimize parameters of the derived compartment models. Experimental results show that the twocompartment model is sufficient for the pharmacokinetic study of remifentanil acid for patients with mild degree of renal impairment.

Sorption and desorption kinetics of phthalates and phenol on water/sediment interface

The sorption and desorption kinetics of dimethyl phthalate (DMP), diethyl phthalate (DEP), dibutyl phthalate (DBP) and phenol on water and sediment interface were studied using two compartment model in this paper. The results showed that the sorption coefficients of DMP, DEP, DBP and phenol measured by batch equilibrium method were 16.79, 24.55, 132 and 0.65μg 1-1/n · g -1 · ml -1/n , the sorption and desorption kinetic constants of DMP, DEP, DBP, phenol were 0.0248, 0.0357, 0.0727, 0.014ml·cm -2 ·h -1 and 0.000512, 0.000754, 0.00127, 0 000899h -1 at static condition respectively; and the sorption and desorption kinetics constants of above chemicals were 0 279, 0.382, 0.496, 0.0904ml·cm -2 ·h -1 and 0.0442, 0.0031, 0.00116, 0.00247h -1 at flow water condition respectively.

On the Origin of the Apparent Volume of Distribution and Its Significance in Pharmacokinetics

The apparent volume of distribution was defined for the first time as the phase volume that can hold the total amount of a substance at the measured phase substance concentration, in a system composed of two immiscible media that are in contact under conditions of constant phase volumes, at equilibrium. Its value is not affected by the total system solute mass and it only depends on the total system volume, the phase volumes and the affinity of the solute for the two phases in the system. Using this new concept of the apparent volume of distribution, we were able to demonstrate that under certain conditions compartment volumes in multi-compartment and multi-phasic pharmacokinetic models represent the actual physiological volumes of body fluids accessible by drugs. The classical pharmacokinetic models are now fully explained and can be used to provide accurate estimation of the pharmacokinetic parameters for hydrophilic drugs. In contrast, in the absence of tissue-plasma partition coefficients, lipophilic drugs that do not follow a one-compartment model are unlikely to be adequately described with classical multi-compartment pharmacokinetic models.

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.

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.