Susceptible-infected-recovered Research Articles

Modeling tuberculosis transmission dynamics in Kazakhstan using SARIMA and SIR models

Tuberculosis (TB) is a highly contagious disease that remains a global concern affecting numerous countries. Kazakhstan has been facing considerable challenges in TB prevention and treatment for decades. This study aims to understand TB transmission dynamics by developing and comparing statistical and deterministic models: Seasonal Autoregressive Integrated Moving Average (SARIMA) and the basic Susceptible-Infected-Recovered (SIR). TB data from 2014 to 2019 were collected from the Unified National Electronic Health System (UNEHS) using retrospective quantitative analysis. SARIMA models were able to capture seasonal variations, with Model 2 exhibiting superior predictive accuracy. Both models showed declining TB incidence and revealed a notable predictive performance evaluated by statistical metrics. In addition, the SIR model calculated the basic reproduction number (\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R_0}$$\\end{document}) below 1, indicating a receding epidemic. Models proved the capability of each to accurately capture trends (SARIMA) and provide mathematical insights (SIR) into TB transmission dynamics. This study contributes to the general knowledge of TB transmission dynamics in Kazakhstan thus laying the foundation for more comprehensive studies on TB and control strategies.

Exponential series approximation of the SIR epidemiological model

IntroductionThe SIR (Susceptible-Infected-Recovered) model is one of the simplest and most widely used frameworks for understanding epidemic outbreaks.MethodsA second-order dynamical system for the R variable is formulated using an infinite exponential series expansion, and a recursion relation is established between the series coefficients. A numerical approximation scheme for the R variable is also developed.ResultsThe proposed numerical method is compared to a double exponential (DE) nonlinear approximate analytic solution, which reveals two coupled timescales: a relaxation timescale, determined by the ratio of the model’s time constants, and an excitation timescale, dictated by the population size. The DE solution is applied to estimate model parameters for a well-known epidemiological dataset—the boarding school flu outbreak.DiscussionFrom a theoretical standpoint, the primary contribution of this work is the derivation of an infinite exponential, Dirichlet, series for the model variables. Truncating the series yields a finite approximation, known as a Prony series, which can be interpreted as a sequence of coupled exponential relaxation processes, each with a distinct timescale. This apparent complexity can be approximated well by the DE solution, which appears to be of main practical interest.

A Novel Nonlinear SAZIQHR Epidemic Transmission Model: Mathematical Modeling, Simulation, and Optimal Control

Abstract This study presents a seven-compartmental nonlinear epidemic model to explore the dynamics and control of the epidemic. The model extends the traditional susceptible-infected-recovered (SIR) framework by including additional compartments for aware, infodemic, hospitalized, and quarantined populations. It also incorporates three Holling type II saturated nonlinear incidence rates to better represent the disease transmission process. The "infodemic population" refers to those spreading misinformation about the disease, its spread, control, and treatment. The study examines disease dynamics in the absence of a vaccine and identifies two key equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). It is found that the DFE is locally asymptotically stable when the basic reproduction number (R_0) is below one and becomes unstable when R_0 exceeds one. The model's behavior at R_0=1 is analyzed using center manifold theory, revealing a forward bifurcation. Additionally, the existence of a positive endemic equilibrium is confirmed for R_0>1, with no backward bifurcation observed when R_0≤1. The EE is also shown to be locally asymptotically stable when R_0 is greater than one. Furthermore, the study formulates and mathematically analyzes an optimal control problem. Finally, numerical simulations are presented to support the analytical results.

Stability and Optimality Criteria for an SVIR Epidemic Model with Numerical Simulation

The mathematical modeling of infectious diseases plays a vital role in understanding and predicting disease transmission, as underscored by recent global outbreaks; to delve deep into the dynamic of infectious disease considering latent period presciently is inevitable as it bridges the gap between realistic nature and mathematical modeling. This study extended the classical Susceptible–Infected–Recovered (SIR) model by incorporating vaccination strategies during incubation. We introduced multiple time delays to an account incubation period to capture realistic disease dynamics better. The model is formulated as a system of delay differential equations that describe the transmission dynamics of diseases such as polio or COVID-19, or diseases for which vaccination exists. Critical aspects of the study include proving the positivity of the model’s solutions, calculating the basic reproduction number (R0) using next-generation matrix theory, and identifying disease-free and endemic equilibrium points. The local stability of these equilibria is then analyzed using the Routh–Hurwitz criterion. Due to the complexity introduced by the delay components, we examine the stability by studying the roots of a fourth-degree exponential polynomial. The effects of educational campaigns and vaccination efficacy are also investigated as control measures. Furthermore, an optimization problem is formulated, based on Pontryagin’s maximum principle, to minimize the number of infections and associated intervention costs. Numerical simulations of the delay differential equations are conducted, and a modified Runge–Kutta method with delays is used to solve the optimal control problem. Finally, we present a few simulation results to illustrate the analytical findings.

A fractional order SIR model describing hesitancy to the COVID-19 vaccination

This study introduces a SIR (Susceptible-Infectious-Recovered) model using fractional derivatives to assess the population's hesitancy to the COVID-19 vaccination campaign in Portugal. Leveraging the framework developed by Angstmann [1], our approach incorporates fractional derivatives to best describe the nuanced dynamics of the vaccination process. We begin by examining the qualitative properties of the proposed model. To substantiate the inclusion of fractional derivatives, empirical data along with statistical criteria are applied. Numerical simulations are performed to compare both integer and fractional order models. An epidemiological interpretation for the fractional order of the model is provided, in the context of a vaccination campaign.

Epidemics on critical random graphs with heavy-tailed degree distribution

We study the susceptible-infected-recovered (SIR) epidemic on a random graph chosen uniformly over all graphs with certain critical, heavy-tailed degree distributions. We prove process level scaling limits for the number of individuals infected on day h on the largest connected components of the graph. The scaling limits contain non-negative jumps corresponding to some vertices of large degree. These weak convergence techniques allow us to describe the height profile of the α-stable continuum random graph (Goldschmidt et al., 2022; Conchon–Kerjan and Goldschmidt, 2023), extending results known in the Brownian case (Miermont and Sen, 2022). We also prove model-independent results that can be used on other critical random graph models.

Estimating the parameters of epidemic spread on two-layer random graphs: a classical and a neural network approach

In this paper, we propose and analyse various methods for estimating the infection parameter τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} in an SIR (susceptible-infected-recovered) epidemic spread model simulated on two-layer random graphs with a preferential attachment component. The statistics that can be used for the estimates is the number of susceptible and infected vertices. Our classical approach is based on the maximum likelihood method, while the machine learning approach uses a graph neural network (GNN). The underlying graph has a layer consisting of disjoint cliques (representing e.g. households) and a random second layer with a preferential attachment structure. Our simulation study reveals that the estimation is poorer at the beginning of the epidemic, for larger preferential attachment parameter of the graph, and for larger τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}. As for the neural networks, we find that dense networks offer better training datasets. Moreover, GNN perfomance is measured better using the l2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_2$$\\end{document} loss function rather than cross-entropy.

Extended Floor Field Cellular Automata Model for Pedestrian Emergency Evacuation Considering Panic Emotion

Panic emotion appears frequently in an emergency evacuation, affecting evacuation efficiency and pedestrian safety. Individual diversity in emotion propagation causes pedestrians to behave differently. To better describe the emotion propagation in the movement of pedestrians, we analyze the characteristics of the emotions of different pedestrians in emergencies. Considering pedestrian heterogeneity, we divide pedestrians into three types (conservative, moderate, and radical) according to the strength of the expression of individual emotions. Based on an analysis of the characteristics of the three types of pedestrian, an improved susceptible-infected-recovered (SIR) model is built to describe the pedestrian emotion propagation process. In addition, a conflict avoidance method that considers pedestrian speed is proposed to demonstrate pedestrian motion more realistically. Based on this foundation, a cellular automata (CA) model that integrates emotion propagation and conflict avoidance techniques is employed to replicate the process of pedestrian emergency evacuation. The simulation results suggest that a higher proportion of pedestrians with initial panic emotions and a radical behavior type leads to a prolonged evacuation time. Many pedestrians in emergency evacuations gather at the exit in a short time (approximately 37 time steps), the density then reaches a high-level (approximately 5.5 pedestrians/m2) for a period of time, and the duration increases with the increase in the number of pedestrians. Methods for guiding pedestrians to evacuate in a rational manner are worthy of promotion and research.

Fiscal policy in the face of the health crisis: A simulation using a hybrid DSGE‐SIR model

AbstractThis study employs a dynamic stochastic general equilibrium (DSGE) model, integrated with a Susceptible‐Infected‐Recovered (SIR) epidemiological framework, to assess the macroeconomic impacts of fiscal policy during the COVID‐19 pandemic in Morocco. Calibrated with Moroccan COVID‐19 data, the model links epidemiological dynamics to macroeconomic variables, offering a detailed analysis of fiscal interventions. The primary objective is to evaluate the effectiveness of various fiscal measures, including government spending shocks, consumption tax cuts, and labor tax reductions, in stimulating economic activity and supporting households and businesses impacted by the pandemic. The results indicate that government spending shocks significantly stimulated economic activity and employment, but also led to increased public debt and inflationary pressures, thereby illustrating the inherent trade‐offs. Consumption tax cuts, intended to boost demand, had mixed effects on inflation; while prices for some goods declined, higher demand caused price increases in others. Labor tax reductions, aimed at enhancing employment, generated varied effects on labor supply and contributed to rising public debt due to lower tax revenues. The study underscores the necessity of balanced fiscal strategies to achieve both immediate economic recovery and long‐term fiscal sustainability, highlighting the critical role of well‐calibrated fiscal policies in mitigating the economic consequences of pandemics.

Quantifying indirect and direct vaccination effects arising in the SIR model.

Vaccination campaigns have both direct and indirect effects that act to control an infectious disease as it spreads through a population. Indirect effects arise when vaccinated individuals block disease transmission in any infection chain they are part of, and this in turn can benefit both vaccinated and unvaccinated individuals. Indirect effects are difficult to quantify in practice but, in this article, working with the susceptible-infected-recovered (SIR) model, they are analytically calculated in important cases, through pivoting on the final size formula for epidemics. Their relationship to herd immunity is also clarified. The analysis allows us to identify the important distinction between quantifying the indirect effects of vaccination at the 'population level' versus the 'per capita' level, which often results in radically different conclusions. As an example, our analysis unpacks why the population-level indirect effect can appear significantly larger than its per capita analogue. In addition, we consider a recently proposed epidemiological non-pharmaceutical intervention (by the means of recovered individuals) used over the COVID-19 pandemic, referred to as 'shielding', and study its impact on our mathematical analysis. The shielding scheme is extended to take advantage of vaccination including imperfect vaccination.

SIR epidemics in interconnected networks: threshold curve and phase transition

For simplicity of mathematical modeling of epidemic spreading, the assumption is that hosts have identical rates of disease-causing contacts. However, in the real world, the scenario is different. The network-based framework allows us to capture the complex interdependencies and structural heterogeneity present in real-world systems. We examine two distinct scenarios involving the dynamics of susceptible-infected-recovered (SIR) in interconnected networks. In the first part, we show how the epidemic threshold of a contact network changes as a result of being coupled with another network for a fixed infection strength. The model employed in this work considers both the contact networks and interconnections as generic. We have depicted the epidemic threshold curve for interconnected networks, considering the assumption that the infection could be initially present in either one or both of the networks. If the normalized infection strengths are above the threshold curve, the infection spreads, whereas if the normalized infection strengths are below the threshold curve, the disease does not spread. This is true for any level of interconnection. In the second part, we investigate the spillover phenomenon, where the disease in a novel host population network comes from a reservoir network. We have observed a clear phase transition when the number of links or the inter-network infection rate exceeds a certain threshold, keeping all other parameters constant. We observe two regimes for spillover: a major spillover region and a minor spillover region based on interpopulation links (fraction of links between two networks) and inter-network infection strength (infection rate between reservoir and host network). If the interpopulation links and inter-network infection strength are in the major spillover region, the spillover probability is high, while if the former parameters are in the minor spillover region, the spillover probability is low. When the number of infected individuals within a reservoir network is nearly equal, and the inter-network infection strength remains constant, the threshold number of links required to achieve the spillover threshold condition varies based on the network topology. Overall, this work contributes to the understanding of SIR dynamics in interconnected networks and sheds light on the behavior of epidemics in complex systems.

Spurious self-feedback of mean-field predictions inflates infection curves.

The susceptible-infected-recovered (SIR) model and its variants form the foundation of our understanding of the spread of diseases. Here, each agent can be in one of three states (susceptible, infected, or recovered), and transitions between these states follow a stochastic process. The probability of an agent becoming infected depends on the number of its infected neighbors, hence all agents are correlated. The simplest mean-field theory of the same stochastic process, however, assumes that the agents are statistically independent. This leads to a self-feedback effect in the approximation: when an agent infects its neighbors, this infection may subsequently travel back to the original agent at a later time, leading to a self-infection of the agent which is not present in the underlying stochastic process. We here compute the first-order correction to the mean-field assumption from a systematic expansion, called dynamical TAP theory. This correction, which takes fluctuations up to second order in the interaction strength into account, cancels the self-feedback effect, leading to smaller infection rates. The correction significantly improves predictions compared to mean-field theory. In particular, it captures how sparsity dampens the spread of the disease: this indicates that reducing the number of contacts is more effective than predicted by mean-field models. We further apply the expansion to variants of the SIR model, such as the SIRS model, in which the immunity of an individual to the disease wanes over time. We find that up to the second order, the correction terms in the SIR and SIRS model are equivalent, meaning that fluctuations partially cancel the self-feedback effect even when self-feedback is in principle allowed.

Effects of Intervention Timing on Health-Related Fake News: Simulation Study.

Fake health-related news has spread rapidly through the internet, causing harm to individuals and society. Despite interventions, a fenbendazole scandal recently spread among patients with lung cancer in South Korea. It is crucial to intervene appropriately to prevent the spread of fake news. This study investigated the appropriate timing of interventions to minimize the side effects of fake news. A simulation was conducted using the susceptible-infected-recovered (SIR) model, which is a representative model of the virus spread mechanism. We applied this model to the fake news spread mechanism. The parameters were set similarly to those in the digital environment, where the fenbendazole scandal occurred. NetLogo, an agent-based model, was used as the analytical tool. Fake news lasted 278 days in the absence of interventions. As a result of adjusting and analyzing the timing of the intervention in response to the fenbendazole scandal, we found that faster intervention leads to a shorter duration of fake news (intervention at 54 days = fake news that lasted for 210 days; intervention at 16 days = fake news that lasted for 187 days; and intervention at 10 days = fake news that lasted for 157 days). However, no significant differences were observed when the intervention was performed within 10 days. Interventions implemented within 10 days were effective in reducing the duration of the spread of fake news. Our findings suggest that timely intervention is critical for preventing the spread of fake news in the digital environment. Additionally, a monitoring system that can detect fake news should be developed for a rapid response.

MULTI-AGENT SIMULATION MODEL OF INFECTIOUS DISEASE SPREAD

The aim of the research is to develop a multi-agent simulation model for predicting the spread of infectious diseases, particularly COVID-19. In the context of the COVID-19 pandemic, there emerged an urgent need to create tools for forecasting and analyzing the dynamics of epidemics, as well as for evaluating the effectiveness of management decisions. The use of mathematical models in this process allows for an adequate description of the infection spread dynamics, which is essential for making informed decisions. The article discusses traditional approaches to epidemic modeling, such as the predator-prey model and the compartmental SIR (Susceptible-Infectious-Recovered) model. The predator-prey model describes the interaction between two species in an ecosystem using differential equations, which allows for modeling population dynamics. The compartmental SIR model divides the population into three groups: susceptible, infected, and recovered, which enables the analysis of the spread of infectious diseases. However, these models have limitations, particularly due to assumptions about population homogeneity and constant parameters. To more accurately model complex epidemic processes, a multi-agent simulation model was developed. In this model, agents interact within a defined area, mimicking real conditions of infection spread. Agents are divided into three classes: healthy, infected, and recovered. The movement of agents is modeled using random walk in a two-dimensional space, taking into account the possibility of contact between them, which can lead to infection. Infected agents transition to the recovered class after a certain period of illness and can no longer be infected. Modeling results showed that the multi-agent model allows for more accurate prediction of infection spread dynamics. Numerous experiments were conducted, demonstrating the model's adequacy in replicating the infection process, peak infection rates, and recovery periods. The influence of various parameters, such as the duration of illness, on the epidemic dynamics was investigated. The obtained results confirm that considering individual characteristics and behavioral traits of agents improves the accuracy of modeling. This allows the multi-agent simulation model to be used for developing effective control strategies and predicting the spread of infectious diseases, which can be useful for making management decisions in real pandemic conditions.

Interdisciplinary applications of finite group theory - From epidemic modeling to financial market analysis

Finite group theory has a long history. Many mathematicians have done researches on it. As time goes on, it becomes a useful tool for human as it has a lot of applications. Finite group theory becomes more than just mathematical. As a result, its applications on epidemic modeling and financial market analysis are introduced respectively. These focus on SIR (Susceptible-Infective-Recovered) model and stock markets. Whats more, there are some other applications in different areas. What we came to the conclusion is that wired groups have developed in many fields, not only in financial markets and infectious disease models, but also in many other fundamental disciplines, chemistry, physics, computer science. They are going to be showed in the following essay.

A method for characterizing disease emergence curves from paired pathogen detection and serology data

Abstract Wildlife disease surveillance programs and research studies track infection and identify risk factors for wild populations, humans and agriculture. Often, several types of samples are collected from individuals to provide more complete information about an animal's infection history. Methods that jointly analyse multiple data streams to study disease emergence and drivers of infection via epidemiological process models remain underdeveloped. Joint‐analysis methods can more thoroughly analyse all available data, more precisely quantifying epidemic processes, outbreak status, and risks. We contribute a paired data modelling approach that analyses multiple samples from individuals. We use ‘characterization maps’ to link paired data to epidemiological processes through a hierarchical statistical observation model. Our approach can provide both Bayesian and frequentist estimates of epidemiological parameters and state. Our approach can also incorporate test sensitivity and specificity, and we propose model fit diagnostics. We motivate our approach through the need to use paired pathogen and antibody detection tests to estimate parameters and infection trajectories for the widely applicable susceptible, infectious, recovered (SIR) model. We contribute general formulas to link characterization maps to arbitrary process models and datasets and an extended SIR model that better accommodates paired data. We find via simulation that paired data can more efficiently estimate SIR parameters than unpaired data, requiring samples from 5 to 10 times fewer individuals. We use our method to study SARS‐CoV‐2 in wild white‐tailed deer (Odocoileus virginianus) from three counties in the United States. Estimates for average infectious times corroborate captive animal studies. The estimated average cumulative proportion of infected deer across the three counties is 73%, and the basic reproductive number (R0) is 1.88. Wildlife disease surveillance programs and research studies can use our methods to jointly analyse paired data to estimate epidemiological process parameters and track outbreaks. Paired data analyses can improve precision and accuracy when sampling is limited. Our methods use general statistical theory to let applications extend beyond the SIR model we consider and to more complicated examples of paired data. The methods can also be embedded in larger hierarchical models to provide landscape‐scale risk assessment and identify drivers of infection.

Passive and active field theories for disease spreading

The worldwide COVID-19 pandemic has led to a significant growth of interest in the development of mathematical models that allow to describe effects such as social distancing measures, the development of vaccines, and mutations. Several of these models are based on concepts from soft matter theory. Considerably less well investigated is the reverse direction, i.e. how results from epidemiological research can be of interest for the physics of colloids and polymers. In this work, we consider the susceptible-infected-recovered (SIR)-dynamical density functional theory (DDFT) model, a combination of the SIR model from epidemiology with DDFT from nonequilibrium soft matter physics, which allows for an explicit modeling of social distancing. We extend the SIR-DDFT model both from an epidemiological perspective by incorporating vaccines, asymptomaticity, reinfections, and mutations, and from a soft matter perspective by incorporating noise and self-propulsion and by deriving a phase field crystal (PFC) model that allows for a simplified description. On this basis, we investigate via computer simulations how epidemiological models are affected by the presence of non-reciprocal interactions. This is done in a numerical study of a zombie outbreak.

The futility of being selfish in vaccine distribution

We study vaccine budget-sharing strategies in the SIR (Susceptible-Infected-Recovered) model given a structured community network to investigate the benefit of sharing vaccine across communities. The network studied comprises two communities, one of which controls vaccine budget and may share it with the other. Different scenarios are considered regarding the connectivity between communities, infection rates and the unvaccinated fraction of the population. Properties of the SIR model facilitates the use of dynamic message passing (DMP) and optimal control methods to investigate preventive and reactive budget-sharing scenarios. Our results show a large set of budget-sharing strategies in which the sharing community benefits from the reduced global infection rates with no detrimental impact on its local infection rate.

An SIR‐based Bayesian framework for COVID‐19 infection estimation

AbstractEstimating the COVID‐19 infection fatality rate, inferring the latent incidence and predicting the future epidemic evolution are critical to public health surveillance, but often challenging due to limited data availability or quality. Recently, a Bayesian framework combining time series deconvolution of deaths with a parametric Susceptible–Infectious–Recovered (SIR) model was proposed by Irons and Raftery, 2021. We assess the parameter identifiability of the model using the profile likelihood approach and simulations, when only the time series of deaths and seroprevalence survey data are available. The robustness of the model to the more complex but also more realistic Susceptible–Exposed–Infectious–Recovered (SEIR)‐based epidemics is evaluated through simulations; the influence of potential biases in the serosurveys on the inference is also investigated. We use a stationary first‐order autoregressive prior to account for the variability of transmission rate over time. The results suggest that the model is relatively robust to SEIR‐based epidemics, especially when the reproductive number is low, given sufficient information from serosurveys or priors. However, the lack of parameter identifiability under limited data availability cannot be neglected. We apply the model to infer the COVID‐19 infections in Ontario and Quebec, Canada during the Omicron era.

Estimating Cross-Border Mobility from the Difference in Peak Timing: A Case Study of Poland–Germany Border Regions

Human mobility contributes to the fast spatiotemporal propagation of infectious diseases. During an outbreak, monitoring the infection on either side of an international border is crucial as cross-border migration increases the risk of disease importation. Due to the unavailability of cross-border mobility data, mainly during pandemics, it becomes difficult to propose reliable, model-based strategies. In this study, we propose a method for estimating commuting-type cross-border mobility flux between any pair of regions that share an international border from the observed difference in their infection peak timings. Assuming the underlying disease dynamics are governed by a Susceptible–Infected–Recovered (SIR) model, we employ stochastic simulations to obtain the maximum likelihood cross-border mobility estimate for any pair of regions. We then investigate how the estimate of cross-border mobility flux varies depending on the transmission rate. We further show that the uncertainty in the estimates decreases for higher transmission rates and larger observed differences in peak timing. Finally, as a case study, we apply the method to some selected regions along the Poland–Germany border that are directly connected through multiple modes of transportation and quantify the cross-border fluxes from the COVID-19 cases data from 20 February to 20 June 2021.