Susceptible-infected-removed Model Research Articles

Influence of Fractional Order on the Behavior of a Normalized Time-Fractional SIR Model

In this paper, we propose a novel normalized time-fractional susceptible–infected–removed (SIR) model that incorporates memory effects into epidemiological dynamics. The proposed model is based on a newly developed normalized time-fractional derivative, which is similar to the well-known Caputo fractional derivative but is characterized by the property that the sum of its weight function equals one. This unity property is crucial because it helps with evaluating how the fractional order influences the behavior of time-fractional differential equations over time. The normalized time-fractional derivative, with its unity property, provides an intuitive understanding of how fractional orders influence the SIR model’s dynamics and enables systematic exploration of how changes in the fractional order affect the model’s behavior. We numerically investigate how these variations impact the epidemiological dynamics of our normalized time-fractional SIR model and highlight the role of fractional order in improving the accuracy of infectious disease predictions. The appendix provides the program code for the model.

Exact solution for a discrete-time SIR model

We propose a nonstandard finite difference scheme for the Susceptible–Infected–Removed (SIR) continuous model. We prove that our discretized system is dynamically consistent with its continuous counterpart and we derive its exact solution. We end with the analysis of the long-term behavior of susceptible, infected and removed individuals, illustrating our results with examples. In contrast with the SIR discrete-time model available in the literature, our new model is simultaneously mathematically and biologically sound.

A combined neural ODE-Bayesian optimization approach to resolve dynamics and estimate parameters for a modified SIR model with immune memory

We propose a novel hybrid approach that integrates Neural Ordinary Differential Equations (NODEs) with Bayesian optimization to address the dynamics and parameter estimation of a modified time-delay-type Susceptible-Infected-Removed (SIR) model incorporating immune memory. This approach leverages a neural network to produce continuous multi-wave infection profiles by learning from both data and the model. The time-delay component of the SIR model, expressed through a convolutional integral, results in an integro-differential equation. To resolve these dynamics, we extend the NODE framework, employing a Runge-Kutta solver, to handle the challenging convolution integral, enabling us to fit the data and learn the parameters and dynamics of the model. Additionally, through Bayesian optimization, we enhance prediction accuracy while focusing on long-term dynamics. Our model, applied to COVID-19 data from Mexico, South Africa, and South Korea, effectively learns critical time-dependent parameters and provides accurate short- and long-term predictions. This combined methodology allows for early prediction of infection peaks, offering significant lead time for public health responses.

Numerical solutions and simulations of the fractional COVID‐19 model via Pell–Lucas collocation algorithm

The aim of this study is to present the evolution of COVID‐19 pandemic in Turkey. For this, the SIR (Susceptible, Infected, Removed) model with the fractional order derivative is employed. By applying the collocation method via the Pell–Lucas polynomials (PLPs) to this model, the approximate solutions of model with fractional order derivative are obtained. Hence, the comments are made about the susceptible population, the infected population, and the recovered population. For the method, firstly, PLPs are expressed in matrix form for a selected number of . With the help of this matrix relationship, the matrix forms of each term in the SIR model with the fractional order derivative are constituted. For implementation and visualization, we utilize MATLAB. Moreover, the outcomes for the Runge–Kutta method (RKM) are obtained using MATLAB, and these results are compared with the results obtained with the Pell–Lucas collocation method (PLCM). From all simulations, it is concluded that the presented method is effective and reliable.

A simple modification to the classical SIR model to estimate the proportion of under-reported infections using case studies in flu and COVID-19

BackgroundUnder-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. ObjectiveWe aim to design a simple modification to the Susceptible – Infected – Removed (SIR) model for estimating the fraction or proportion of reported infection cases. MethodsThe 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. ResultsWe 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. ConclusionsWe 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.

Applying The Sir Model To A Restricted Population

Abstract OBJECTIVE Using data collected from the Johns Hopkins COVID-19 Repository, we investigate the reliability of the SIR (Susceptible-Infected-Removed) model. SUMMARY OF BACKGROUND DATA Modern pandemic responses have evolved into effective defenses when implemented correctly but present issues in epidemiological modeling efforts. Intervention strategies such as quarantining and masking limit population size (N), which affects the accuracy of the modeling population-based rate systems especially in high-transmissible diseases. METHODS We begin by reviewing the SIR model retroactively to the initial SARS-Cov-2 Wuhan strain. We compared the parameters available in the published literature (N = 2,717,000, β = 2, γ = 1/14) to the best-fitting SIR-yielded values by minimizing the root-mean-squared error function. Subsequently, we evaluated its predictive capabilities on the Delta variant using early surge data which was later compared against a retroactive analysis. RESULTS Using a least squares error best-fit analysis allowed us to retroactively define remarkably accurate model parameters for the Wuhan waves. Parameters including N=730, β = 0.46, γ = 0.043 in the first wave and N=11200, β = 0.198, γ = 0.07 in the second reflected effective intervention strategies. We show it is an effective predictive tool regarding the Delta variant, yielding parameters N = 50,900, β = 0.87, γ = 1/3.7 which proved accurate when compared to parameters from a full retroactive analysis (N = 60,000, β = 0.94, γ = 1/3.6). CONCLUSION The similarity of the yielded parameters in our results supports the SIR Model’s utility in epidemiologically monitoring of high-transmissibility, low-mortality outbreaks vis-à-vis various containment measures.

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

We consider state and parameter estimation for compartmental models having both time-varying 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 time-invariant 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.

Understanding How Parasites from Farmed Fish May Influence Wild Fish Declines Using Epidemiological Modelling

Beside various fields of its applications, in this study epidemiological modelling was used to understand how parasites from farmed fish may cause wild fish declines. Two separate strategic models were constructed addressing the transmission of micro-parasites and macro-parasites between farmed and wild fish: A SIR (Susceptible-Infective-Removed) model for micro-parasite infections and a compartmental density-dependent model for macro-parasite infestations. The results indicated that parasites originated in wild fish populations, after infecting farmed fish can cause epizootics. Subsequently, these parasites can be transmitted from farmed to wild fish and might have negative impact on the dynamics of wild fish populations. Sensitivity analysis of the basic model parameters in both models showed that model parameters, which are influenced by abiotic factors and allow passive manipulation, such as pathogen specific transmission rate (β), pathogen specific transmission rate between infected farmed and susceptible wild fish (δ), the rate of production of infective stages by an adult parasite (λ) and transmission rate between host and parasite infective stages (β) are more sensitive compared to model parameters which encompass chemical control and fallowing. This emphasizes the importance of the preventive medicine rather than intervention procedures in aquaculture aiming at eradicating epizootics caused by parasites and protecting wild fish stocks.

A SIR-UC EPIDEMIC MODEL: THE ANALYSIS OF SUSCEPTIBLE-INFECTED-REMOVED (SIR) EPIDEMIC MODEL WITH THE COVERAGE OF HEALTH INSURANCE (UNCOVERED AND COVERED INDIVIDUALS)

Susceptible-Infected-Removed (SIR) model is a widely used epidemic model that simulates the spread of infectious diseases within a population. It classifies individuals into susceptible, infected, and removed states, with the number of individuals in each state being time-dependent variables denoted by S(t), I(t), and R(t), respectively. The model considers direct contact transmission between infected and susceptible individuals. In developed countries, some people cannot afford medical treatment. In contrary, the recovery rate of infected individual in the population is directly proportional to the number of people receiving medical treatment. Affordable health insurance increases the number of people receiving medical treatment thus insurance should be considered aspect in epidemic model. The main purpose of this research is to analyze the effect of insurance on the SIR epidemic model. This research classifies individuals in both S(t) and I(t) based on their insurance coverage status. This model assumes permanent immunity for R(t), thus it is unnecessary to classify individuals in this state based on their insurance coverage status because they do not spread the disease nor have potential to be re-infected. Numerical simulation is organized to find the effect of insurance in SIR model by analyzing the equilibrium point. The result based on the equilibrium point suggests that the insurance in SIR epidemic model: (1) decrease the I(t) because it accelerate the recovery rate; (2) decrease theR(t) because there is less infected people for recovery; (3) increase the S(t) because there is less infected people to transmit the disease, compared to the SIR model without the insurance.

The probability of epidemic burnout in the stochastic SIR model with vital dynamics

We present an approach to computing the probability of epidemic "burnout," i.e., the probability that a newly emergent pathogen will go extinct after a major epidemic. Our analysis is based on the standard stochastic formulation of the Susceptible-Infectious-Removed (SIR) epidemic model including host demography (births and deaths) and corresponds to the standard SIR ordinary differential equations (ODEs) in the infinite population limit. Exploiting a boundary layer approximation to the ODEs and a birth-death process approximation to the stochastic dynamics within the boundary layer, we derive convenient, fully analytical approximations for the burnout probability. We demonstrate-by comparing with computationally demanding individual-based stochastic simulations and with semi-analytical approximations derived previously-that our fully analytical approximations are highly accurate for biologically plausible parameters. We show that the probability of burnout always decreases with increased mean infectious period. However, for typical biological parameters, there is a relevant local minimum in the probability of persistence as a function of the basic reproduction number [Formula: see text]. For the shortest infectious periods, persistence is least likely if [Formula: see text]; for longer infectious periods, the minimum point decreases to [Formula: see text]. For typical acute immunizing infections in human populations of realistic size, our analysis of the SIR model shows that burnout is almost certain in a well-mixed population, implying that susceptible recruitment through births is insufficient on its own to explain disease persistence.

Taking cues from machine learning, compartmental and time series models for SARS-CoV-2 omicron infection in Indian provinces

SARS-CoV-2, the virus responsible for COVID-19, posed a significant threat to the world. We analyzed COVID-19 dissemination data in the top ten Indian provinces by infection incidences using the Susceptible-Infectious-Removed (SIR) model, an Autoregressive Integrated Moving Average (ARIMA) time series model, a machine learning model based on the Random Forest, and distribution fitting. Outbreaks are expected to continue if the Basic Reproduction Number (R0) > 1, and infection waves are anticipated to end if the R0 < 1, as determined by the SIR model. Different parametric probability distributions are also fitted. Data collected from December 12, 2021, to March 31, 2022, encompassing data from both before and during the implementation of strict control measures. Based on the estimates of the model parameters, health agencies and government policymakers can develop strategies to combat the spread of the disease in the future, and the most effective technique can be recommended for real-world application for other outbreaks of COVID-19. The best method out of these could be also implemented further on the epidemiological data of other similar infectious agents.

Predicting longitudinal brain atrophy in Parkinson's disease using a Susceptible-Infected-Removed agent-based model.

Parkinson's disease is a progressive neurodegenerative disorder characterized by accumulation of abnormal isoforms of alpha-synuclein. Alpha-synuclein is proposed to act as a prion in Parkinson's disease: In its misfolded pathologic state, it favors the misfolding of normal alpha-synuclein molecules, spreads trans-neuronally, and causes neuronal damage as it accumulates. This theory remains controversial. We have previously developed a Susceptible-Infected-Removed (SIR) computational model that simulates the templating, propagation, and toxicity of alpha-synuclein molecules in the brain. In this study, we test this model with longitudinal MRI collected over 4 years from the Parkinson's Progression Markers Initiative (1,068 T1 MRI scans, 790 Parkinson's disease scans, and 278 matched control scans). We find that brain deformation progresses in subcortical and cortical regions. The SIR model recapitulates the spatiotemporal distribution of brain atrophy observed in Parkinson's disease. We show that connectome topology and geometry significantly contribute to model fit. We also show that the spatial expression of two genes implicated in alpha-synuclein synthesis and clearance, SNCA and GBA, also influences the atrophy pattern. We conclude that the progression of atrophy in Parkinson's disease is consistent with the prion-like hypothesis and that the SIR model is a promising tool to investigate multifactorial neurodegenerative diseases over time.

A critical review on applications of the Avrami equation beyond materials science.

The Johnson-Mehl-Avrami-Kolmogorov (JMAK) formalization, often referred to as the Avrami equation, was originally developed to describe the progress of phase transformations in material systems. Many other transformations in the life, physical and social sciences follow a similar pattern of nucleation and growth. The Avrami equation has been applied widely to modelling such phenomena, including COVID-19, regardless of whether they have a formal thermodynamic basis. We present here an analytical overview of such applications of the Avrami equation outside its conventional use, emphasizing examples from the life sciences. We discuss the similarities that at least partially justify the extended application of the model to such cases. We point out the limitations of such adoption; some are inherent to the model itself, and some are associated with the extended contexts. We also propose a reasoned justification for why the model performs well in many of these non-thermodynamic applications, even when some of its fundamental assumptions are not satisfied. In particular, we explore connections between the relatively accessible verbal and mathematical language of everyday nucleation- and growth-based phase transformations, represented by the Avrami equation, and the more challenging language of the classic SIR (susceptible-infected-removed) model in epidemiology.

Implications of the delayed feedback effect on the stability of a SIR epidemic model

A basic mathematical model in epidemiology is the SIR (Susceptible–Infected–Removed) model, which is commonly used to characterize and study the dynamics of the spread of some infectious diseases. In humans, the time scale of a disease can be short and not necessarily fatal, but in some animals (for example, insects) this same short time scale can make the disease fatal if we take into account their life expectancy. In this work, we will see how a positive feedback effect (decrease of the susceptible population at small densities) in a SIR model can cause a qualitative characterization of the dynamics defined by the original SIR model. Finally, we will also show with numerical simulations how a delay in the feedback effect causes very interesting qualitative changes of the system with epidemiological significance.

Do more stringent policies reduce daily COVID-19 case counts? Evidence from Canadian provinces

The enactment of COVID-19 policies in Canada falls under provincial jurisdiction. This study exploits time-series variation across four Canadian provinces to evaluate the effects of stricter COVID-19 policies on daily case counts. Employing data from this time-period allows an evaluation of the efficacy of policies independent of vaccine impacts. While both OLS and IV results offer evidence that more stringent Non-Pharmaceutical Interventions (NPIs) can reduce daily case counts within a short time-period, IV estimates are larger in magnitude. Hence, studies that fail to control for simultaneity bias might produce confounded estimates of the efficacy of NPIs. However, IV estimates should be treated as correlations given the possibility of other unobserved determinants of COVID-19 spread and mismeasurement of daily cases. With respect to specific policies, mandatory mask usage in indoor spaces and restrictions on business operations are significantly associated with lower daily cases. We also test the efficacy of different forecasting models. Our results suggest that Gradient Boosted Regression Trees (GBRT) and Seasonal Autoregressive-Integrated Moving Average (SARIMA) models produce more accurate short-run forecasts relative to Vector Auto Regressive (VAR), and Susceptible–Infected–Removed (SIR) epidemiology models. Forecasts from SIR models are also inferior to results from basic OLS regressions. However, predictions from models that are unable to correct for endogeneity bias should be treated with caution.

Revisiting classical SIR modelling in light of the COVID-19 pandemic.

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.

How to promote the diffusion of green behavior among contractors? Analysis and simulation using the SIR model

To promote the green development of the construction industry, improve resource utilization, and mitigate the environmental pollution caused by engineering projects, this study identifies the key paths and influencing factors of behavioral diffusion through the analysis of green behavior diffusion among contractors based on the behavioral decisions of the main participants. The study aims to improve positive influences among contractors with respect to the sustainable development of construction. Using the SIR (susceptible-infected-removed) model, we reconsider contractors of different states; construct state transformation paths for potential adopters, adopters, and abandoners of green behaviors among contractors; and analyze the factors that influence the diffusion of green behaviors among contractors to simulate the effects of different paths of behavior diffusion. The results show that two paths, adoption and recovery rates, have a positive influence on the diffusion of green behavior, while three other paths have a negative influence. The identified factors exhibit two types of influence, promotion and hindrance, involving both intra-firm and government regulations, and are associated with other stakeholders. This study promotes the diffusion of green behavior among contractors, which allows contractors to gain a competitive advantage in advance and has positive implications for the implementation of environmentally friendly concepts in the construction industry.

Optimal Isolation Strategies in an SIR Model with Erlang–Distributed Infectious Period

In classical epidemic models, a major simplification consists in assuming that the infectious period is exponentially distributed. Here, we first attempt to investigate the consequences of relaxing this assumption on the performances of time–variant disease control strategies by using optimal control theory. In the framework of a basic susceptible–infected– removed (SIR) model, an Erlang distribution of the infectious period is considered and optimal isolation strategies are searched for. The objective functional takes into account the cost of the isolation efforts and the sanitary costs due to the incidence of the epidemic outbreak. Applying the Pontryagin's minimum principle, we prove that the problem admits only bang–bang solutions with at most two switches. Finally, by means of numerical simulations, we show how the shape of the optimal solutions is affected by the different distributions of the infectious period, and by the relative weight of the two cost components.

Implementation of an Adaptable COVID-19 Utilization and Resource Visualization Engine (CURVE) to Depict In-Hospital Resource Forecasts Over Time

We developed an interactive web-based, decision support application that can adapt to the rapid pace of change in region-specific pandemic related variables and knowledge, thereby providing timely, accurate insights to inform a large healthcare system’s proactive response to COVID-19 hospital resource planning. We designed the COVID-19 Utilization and Resource Visualization Engine (CURVE) app to be adaptable to real-time changes as the pandemic evolved, enabling decisions to be supported by contemporary local data and accurate predictive models. To demonstrate this flexibility, we sequentially implemented a Susceptible-Infected-Removed (SIR) model that incorporates social-distancing and imperfect detection (SIR-D2), an extended-state-space Bayesian SIR model (eSIR), and a time-series model (ARIMA). CURVE improves upon other pandemic forecasting solutions by providing adaptable decision support that generates locally calibrated forecasts aligned to health system specific data to guide COVID-19 pandemic planning. The app additionally enables systematic monitoring of forecast model performance and realignment that keeps pace with the pandemic’s volatile spread and behavior. CURVE provides a flexible pandemic decision support framework that places the most accurate, locally relevant information in front of decision makers to enable health systems to be proactive and prepared.

Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay

Abstract This article is concerned with the nonlinear stability of traveling waves of a delayed susceptible-infective-removed (SIR) epidemic model with nonlocal dispersal, which can be seen as a continuity work of Li et al. [Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput. 247 (2014), 723–740]. We prove that the traveling wave solution is exponentially stable when the initial perturbation around the traveling wave is relatively small in a weighted norm. The time decay rate is also obtained by weighted-energy estimates.