Susceptible-infected-recovered Model Research Articles

Optimal control for an SIR model with limited hospitalised patients

This paper analyses the optimal control of infectious disease propagation using a classic susceptible-infected-recovered (SIR) model characterised by permanent immunity and the absence of available vaccines. The control is performed over a time-dependent mean reproduction number, in order to minimise the cumulative number of ever-infected individuals (recovered), under different constraints. We consider constraints on non-pharmaceutical interventions ranging from partial lockdown to non-intervention, as well as the social and economic costs associated with such interventions, and the capacity limitations of intensive care units that limits the number of infected individuals to a maximum allowed value. We rigorously derive an optimal quarantine strategy based on necessary optimality conditions. The obtained optimal strategy is of a boundary-bang type, comprising three phases: an initial phase with no intervention, a second phase maintaining the infected population at its maximum possible value, and a final phase of partial lockdown applied over a single interval. The optimal policy is further refined by optimising the transition times between these phases. We show that these results are in excellent agreement with the numerical solution of the problem.

Infectious Disease Modeling

Infectious diseases pose a persistent challenge to public health worldwide. Recent global health crises, such as the COVID-19 pandemic and Ebola outbreaks, have underscored the vital role of infectious disease modeling in guiding public health policy and response. Infectious disease modeling is a critical tool for society, informing risk mitigation measures, prompting timely interventions, and aiding preparedness for healthcare delivery systems. This article synthesizes the current landscape of infectious disease modeling, emphasizing the integration of statistical methods in understanding and predicting the spread of infectious diseases. We begin by examining the historical context and the foundational models that have shaped the field, such as the SIR (susceptible, infectious, recovered) and SEIR (susceptible, exposed, infectious, recovered) models. Subsequently, we delve into the methodological innovations that have arisen, including stochastic modeling, network-based approaches, and the use of big data analytics. We also explore the integration of machine learning techniques in enhancing model accuracy and responsiveness. The review identifies the challenges of parameter estimation, model validation, and the incorporation of real-time data streams. Moreover, we discuss the ethical implications of modeling, such as privacy concerns and the communication of risk. The article concludes by discussing future directions for research, highlighting the need for data integration and interdisciplinary collaboration for advancing infectious disease modeling.

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 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.

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.

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.

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.

Impact of navigation apps on congestion and spread dynamics on a transportation network

In recent years, the widespread adoption of navigation apps by motorists has raised questions about their impact on local traffic patterns. Users increasingly rely on these apps to find better, real-time routes to minimize travel time. This study uses microscopic traffic simulations to examine the connection between navigation app use and traffic congestion. The research incorporates both static and dynamic routing to model user behavior. Dynamic routing represents motorists who actively adjust their routes based on app guidance during trips, while static routing models users who stick to known fastest paths. Key traffic metrics, including flow, density, speed, travel time, delay time, and queue lengths, are assessed to evaluate the outcomes. Additionally, we explore congestion propagation at various levels of navigation app adoption. To understand congestion dynamics, we apply a susceptible–infected–recovered (SIR) model, commonly used in disease spread studies. Our findings reveal that traffic system performance improves when 30–60% of users follow dynamic routing. The SIR model supports these findings, highlighting the most efficient congestion propagation-to-dissipation ratio when 40% of users adopt dynamic routing, as indicated by the lowest basic reproductive number. This research provides valuable insights into the intricate relationship between navigation apps and traffic congestion, with implications for transportation planning and management.

Nearly Instantaneous Time-Varying Reproduction Number for Contagious Diseases-a Direct Approach Based on Nonlinear Regression.

While the world recovers from the COVID-19 pandemic, another outbreak of contagious disease remains the most likely future risk to public safety. Now is therefore the time to equip health authorities with effective tools to ensure they are operationally prepared for future events. We propose a direct approach to obtain reliable nearly instantaneous time-varying reproduction numbers for contagious diseases, using only the number of infected individuals as input and utilisingthe dynamics of the susceptible-infected-recovered (SIR) model. Our approach is based on a multivariate nonlinear regression model simultaneously assessing parameters describing the transmission and recovery rate as a function of the SIR model. Shortly after start of a pandemic, our approach enables estimation of daily reproduction numbers. It avoids numerous sources of additional variation and provides a generic tool for monitoring the instantaneous reproduction numbers. We use Norwegian COVID-19 data as case study and demonstrate that our results are well aligned with changes in the number of infected individuals and the change points following policy interventions. Our estimated reproduction numbers are notably less volatile, provide more credible short-time predictions for the number of infected individuals, and are thus clearly favorable compared with the results obtained by two other popular approaches used for monitoring a pandemic. The proposed approach contributes to increased preparedness to future pandemics of contagious diseases, as it can be used as a simple yet powerful tool to monitor the pandemics, provide short-term predictions, and thus support decision making regarding timely and targeted control measures.

Unveiling Influence in Networks: A Novel Centrality Metric and Comparative Analysis through Graph-Based Models.

Identifying influential actors within social networks is pivotal for optimizing information flow and mitigating the spread of both rumors and viruses. Several methods have emerged to pinpoint these influential entities in networks, represented as graphs. In these graphs, nodes correspond to individuals and edges indicate their connections. This study focuses on centrality measures, prized for their straightforwardness and effectiveness. We divide structural centrality into two categories: local, considering a node's immediate vicinity, and global, accounting for overarching path structures. Some techniques blend both centralities to highlight nodes influential at both micro and macro levels. Our paper presents a novel centrality measure, accentuating node degree and incorporating the network's broader features, especially paths of different lengths. Through Spearman and Pearson correlations tested on seven standard datasets, our method proves its merit against traditional centrality measures. Additionally, we employ the susceptible-infected-recovered (SIR) model, portraying virus spread, to further validate our approach. The ultimate influential node is gauged by its capacity to infect the most nodes during the SIR model's progression. Our results indicate a notable correlative efficacy across various real-world networks relative to other centrality metrics.

Using an epidemiological model to explore the interplay between sharing and advertising in viral videos

How to exploit social networks to make internet content spread rapidly and consistently is an interesting question in marketing management. Although epidemic models have been employed to comprehend the spread dynamics of internet content, such as viral videos, the effects of advertising and individual sharing on information dissemination are difficult to distinguish. This gap forbids us to evaluate the efficiency of marketing strategies. In this paper, we modify a classic mean-field SIR (susceptible–infected–recovered) model, incorporating the influences of sharing and advertising in viral videos. We mathematically analyze the global stability of the system and propose an agent-based modeling approach to evaluate the efficiency of sharing and advertising. We further provide a case study of music videos on YouTube to show the validity of our model.

Temporal logics for compartmental models

Abstract This paper introduces two logic frameworks for the study of SIR (Susceptible-Infected-Recovered) and SIRS compartmental epidemic models, one based on Linear Temporal Logic and the other on Computation Tree Logic. We provide a short literature overview on compartmental models and other related works using logics, and then define our logics with their respective axiomatizations, and demonstrate their soundness and completeness proofs.

GDLC: A new Graph Deep Learning framework based on centrality measures for intrusion detection in IoT networks

The increasing growth of the Internet of Things (IoT) with the diverse and dynamic nature of devices made detecting and preventing network intrusions more important and challenging. As new and sophisticated cyber-attacks are being used, there is an increasing need for advanced intrusion detection systems that can adapt to emerging threats. The majority of existing methods in the literature lack a comprehensive consideration of complex network features namely the centrality measures. Previous works overlooked the potential insights that can be obtained from the structured relationships within network traffic data. In this study, we propose an approach called Graph Deep Learning framework based on Centrality measures (GDLC). Our approach includes an algorithm to dynamically select the most appropriate centrality measures according to the network’s topological properties created from traffic data. After that, they are integrated with Artificial Intelligence (AI) techniques, specifically the deep learning models: CNN, LSTM, and GRU. We tested our methodology on multiple publicly available cybersecurity datasets, having different network structures and sizes. First, we employ the Susceptible Infectious Recovered (SIR) model to validate the importance of the added centrality measures in identifying the most influential nodes that can be the subject of intrusion attacks. Then, through extensive experimentation and evaluation, we test the effectiveness of our approach in improving the accuracy of the Network Intrusion Detection System (NIDS). The obtained results indicate a significant enhancement in detection rates, that can reach up to 7.7%. This improvement demonstrates the practical value of our proposed methodology and highlights its high capacity to adapt to varying network structures. Integrating AI and complex network features is a promising approach for enhancing the capabilities of the Network Intrusion Detection System, contributing to a more resilient cybersecurity framework.

Analyzing epidemics and designing policies with a modified SIR model

This research paper aims to study the impact of COVID-19 on the economy, where we have extended the standard epidemiological model, the Susceptible–Infected–Recovered (SIR) model. In the economy’s dynamics, we have incorporated aggregate demand and aggregate supply and further set up a system of three-dimensional nonlinear differential equations, which is more likely the Labor–Infected–Quarantine (LIQ) model, an extension of the SIR model. Here, we have derived the steady-state condition and analyzed the model’s stability. In addition, we have done a comparative static analysis of parameters with some policy implications. We aim to achieve the twin objective, i.e., controlling the pandemic and reviving the economy under the set of the policy mix.

Development and Impact Analysis of a Multi-Pandemic Real-Time Data Dashboard: A Comparative Analytical Approach

This research presents the development and implementation of a Multi Pandemic Real-Time Data Dashboard, designed to facilitate enhanced comparative analysis across various global pandemics. Utilizing a combination of real-time data acquisition, statistical modeling, and predictive analytics, the dashboard provides a dynamic platform for analyzing pandemic trends and outcomes. Our methodology involved integrating diverse data sources, employing mathematical models such as the SIR (Susceptible, Infected, Recovered) model and machine learning algorithms for data processing and trend prediction. Key findings indicate that the dashboard effectively identifies patterns in pandemic spread and impact, offering valuable insights for public health decision-making. The research underscores the significance of advanced data analytics in managing public health crises, highlighting the dashboard’s potential in aiding proactive pandemic response and policy formulation.