Susceptible-infected-recovered Model Research Articles

Study of lump and periodic waves along with rogue waves and breathers for mathematical modeling of biological dynamical system

Globally, infectious diseases pose a significant threat. Notable cases include influenza, Hepatitis B and HIV. COVID-19, caused by the coronavirus, has been the subject of recent discussion due to its great transmissibility. This study explores the Susceptible–Infectious–Recovered (SIR) epidemic model, explaining several analytical approaches to comprehend the nonlinear incident rates and geographic dispersion. The model investigates phenomena such as M-shaped solitons, homoclinic breather-like solitons, lump waves (LWs), lump solution (LS) with one or two kinks, rogue waves (RWs), periodic waves (PWs) and periodic cross-kink waves. These solutions aid in understanding the disease spread and informing containment strategies by identifying optimal outbreak control methods. This work studies localized wave solutions in nonlinear wave equations, known as soliton solutions including the LS. RWs are characterized by unexpectedly large amplitude and rapid profile changes, drawing attention for their erratic features. PWs exhibit periodic repetitions over time and space. Besides, some type of solitons with special waveforms are the periodic cross-kink, M-shaped and homoclinic breather-like solitons. Graphical representation visualizes the behavior of these effective waves, providing insights into virus spread in specific regions over time. SIR models help identify optimal strategies for controlling outbreaks. The work adds to our understanding of epidemic dynamics by illuminating how the movement of susceptible and infected individuals affects the spread of disease.

SIR Fractional Order of Simulated Covid-19 Cases using Adams Bashforth-Moulton Method

Fractional order derivative has been widely used in many different areas such as bioengineering, fluid mechanics, circuits systems, biomathematics, and biomedicine. This study introduces the system of the fractional differential equation on SIR (Susceptible-Infected-Recovered) model to simulate the COVID-19 in Malaysia. The fractional derivative is described in Caputo sense and solved by the Adams Bashforth Moulton method. The Runge-Kutta method is used to prove and validate the numerical results obtained. The graphical representations of the simulation with difference fractional order have been presented. The derivative order, with values more than 0.5 are acceptable and reliable.

A novel higher-order Deffuant–Weisbuch networks model incorporating the Susceptible Infected Recovered framework

The literature on opinion dynamics lacks consideration of the changing states of nodes during the process of opinion interaction, as observed in the literature on social networks. To address this gap, this paper proposes an adapted three-dimensional Deffuant–Weisbuch model based on the Susceptible Infected Recovered (SIR) model, which explores the impact of node state changes on opinion interaction. We incorporate the SIR model to describe changes in node state and improve the trust threshold by taking into account nodes’ emotional factor, cognitive ability, and state of the arts. The model is also modified using the opinion natural evolution coefficient and accounts for dynamic changes in social network structure. We validate our model with different network types and structures, and verify our results using the Louvain algorithm and index of centrality. Our model reveals that under different network types and structures, multiple opinion cliques are formed, and the opinions of ordinary nodes tend to converge around those of opinion leaders, indicating the guiding role of opinion leaders in shaping public opinion.

Investigating disaster response for resilient communities through social media data and the Susceptible-Infected-Recovered (SIR) model: A case study of 2020 Western U.S. wildfire season

Effective disaster response is critical for communities to remain resilient and advance the development of smart cities. Responders and decision-makers would benefit from reliable, timely measures of the issues impacting their communities during a disaster, and social media offers a potentially rich data source. Social media can reflect public concerns and behaviors during a disaster, offering valuable insights for decision-makers to understand evolving situations and optimize resource allocation. We used Bidirectional Encoder Representations from Transformers (BERT) topic modeling to cluster topics from Twitter data. Then, we conducted a temporal-spatial analysis to examine the distribution of these topics across different regions during the 2020 western U.S. wildfire season. Our results show that Twitter users mainly focused on three topics: “health impact,” “damage,” and “evacuation.” We used the Susceptible-Infected-Recovered (SIR) theory to explore the magnitude and velocity of topic diffusion on Twitter. The results displayed a clear relationship between topic trends and wildfire propagation patterns. The estimated parameters obtained from the SIR model in selected cities revealed that residents exhibited a high level of several concerns during the wildfire. Our study offers a quantitative approach to measure disaster response and support community resilience enhancement.

A temporal approach to online discussion during disasters: Applying SIR infectious disease model to predict topic growth and examining effects of temporal distance

Discussions on social media during major disasters are robust and often have multiple frames of reference. Temporal perspectives, however, are still lacking in current understandings of social-mediated discussions during disasters and crises, but incorporating temporal perspectives can significantly enhance environmental scanning efforts as prescribed in the issues management framework. The purpose of the current research is twofold: to apply and validate the SIR (Susceptible-Infectious-Recovered) model to examine topics’ growth over time on social media and to understand how future orientation of social media users (an indicator of temporal distance) affects their construal of a disaster through supervised machine learning. We based our analysis on Twitter discussions during the Texas winter storm in 2021. Results of the study show great fit of the SIR model for topic growth, and that temporal distance affects users’ construal of the event in line with core predictions of construal level theory. Theoretical, methodological, and practical implications on social-mediated discussions related to climate change-induced and -intensified disasters and issues management are discussed.

Dynamical vaccination behavior with risk perception and vaccination rewards.

Vaccination is the most effective way to control the epidemic spreading. However, the probability of people getting vaccinated changes with the epidemic situation due to personal psychology. Facing various risks, some people are reluctant to vaccinate and even prefer herd immunity. To encourage people to get vaccinated, many countries set up reward mechanisms. In this paper, we propose a disease transmission model combining vaccination behaviors based on the SIR (Susceptible-Infected-Recovered) model and introduce three vaccination mechanisms. We analyze the impact of the infection rate and the recovery rate on the total cost and the epidemic prevalence. Numerical simulations fit with our intuitive feelings. Then, we study the impact of vaccination rewards on the total social cost. We find that when vaccination rewards offset vaccination costs, both the total cost and the epidemic prevalence reach the lowest levels. Finally, this paper suggests that encouraging people to get vaccinated at the beginning of an epidemic has the best effect.

COVID-19 in Saudi Arabia: Real Data and Simulation for Impact Prediction - A Case Study

The start of the COVID-19 pandemic in early 2020 has caused trouble all over the world. This contagious disease is mostly spread by people, and it spreads much faster than other flu viruses that have been found before. Although vaccines have been discovered and are functional, it will still be the greatest challenge to conquer this disease. To effectively respond to this unprecedented crisis and save human lives from other infectious diseases in the future, it is crucial to better understand how the virus is transmitted from one host to another and how future zones of contagion can be anticipated. Several waves of infection have hit nations worldwide for almost the last four years, and governments have implemented necessary measures to tackle the spread of the virus. However, mathematical modeling has emerged as a powerful tool to inform decision-making, allowing for the prediction of COVID-19’s effects. In this research article, we investigate the impact of COVID-19 in Saudi Arabia using the three most commonly used mathematical models: the classic SIR (Susceptible-Infected-Recovered) model, the extended SEIR (Susceptible-Exposed-Infected-Recovered) model, and the advanced fractional-order models using freely available real recorded data for research. By incorporating actual data from Saudi Arabia and utilizing three simulation techniques, we strive to provide valuable insights into the dynamics of the pandemic and aid in the formulation of effective strategies to control its spread in Saudi Arabia.

Dynamic identification of important nodes in complex networks by considering local and global characteristics

Abstract By combining centrality measures and community detection, a better insight into the nature of the evolution of important nodes in complex networks is obtained. Meanwhile, the dynamic identification of important nodes in complex networks can be enhanced by considering both local and global characteristics. Local characteristics focus on the immediate connections and interactions of a node within its neighbourhood, while global characteristics take into account the overall structure and dynamics of the entire network. Nodes with high local centrality in dynamic networks may play crucial roles in local information spreading or influence. On the global level, community detection algorithms have a significant impact on the overall network structure and connectivity between important nodes. Hence, integrating both local and global characteristics offers a more comprehensive understanding of how nodes dynamically contribute to the functioning of complex networks. For more comprehensive analysis of complex networks, this article identifies important nodes by considering local and global characteristics (INLGC). For local characteristic, INLGC develops a centrality measure based on network constraint coefficient, which can provide a better understanding of the relationship between neighbouring nodes. For global characteristic, INLGC develops a community detection method to improve the resolution of ranking important nodes. Extensive experiments have been conducted on several real-world datasets and various performance metrics have been evaluated based on the susceptible–infected–recovered model. The simulation results show that INLGC provides more competitive advantages in precision and resolution.

Optimal intertemporal curative drug expenses: The case of hepatitis C in France

We study intertemporal tradeoffs that health authorities face when considering the control of an epidemic using innovative curative medical treatments. We set up a dynamically controlled susceptible–infected–recovered (SIR) model for an epidemic in which patients can be asymptomatic, and we analyze the optimality conditions of the sequence of cure expenses decided by health authorities at the onset of the drug innovation process. We show that analytical conclusions are ambiguous because of their dependence on parameter values. As an application, we focus on the case study of hepatitis C, the treatment for which underwent a major upheaval when curative drugs were introduced in 2014. We calibrate our controlled SIR model using French data and simulate optimal policies. We show that the optimal policy entails some front loading of the intertemporal budget. The analysis demonstrates how beneficial intertemporal budgeting can be compared to non-forward-looking constant budget allocation.

Exploring the complex dynamics of a diffusive epidemic model: Stability and bifurcation analysis.

The recent pandemic has highlighted the need to understand how we resist infections and their causes, which may differ from the ways we often think about treating epidemic diseases. The current study presents an improved version of the susceptible-infected-recovered (SIR) epidemic model, to better comprehend the community's overall dynamics of diseases, involving numerous infectious agents. The model deals with a non-monotone incidence rate that exhibits psychological or inhibitory influence and a saturation treatment rate. It has been identified that depending on the measure of medical resources and the effectiveness of their supply, the model exposes both forward and backward bifurcations where two endemic equilibria coexist with infection-free equilibrium. The model also experiences local and global bifurcations of codimension two, including saddle-node, Hopf, and Bogdanov-Takens bifurcations. Additionally, the stability of equilibrium points is investigated. For a spatially extended SIR model system, we have shown that cross-diffusion allows S and I populations to coexist in a habitat. Also, the Turing instability requirements and Turing bifurcation regime are derived. The relationship between distinct role-playing model parameters and various pattern formations like spot and stripe patterns is validated by carrying out in-depth numerical simulations. The findings in the vicinity of the endemic equilibrium solution demonstrate the significance of positive and negative valued cross-diffusion coefficients in regulating the genesis of spatial patterns in susceptible as well as diseased individuals. The discussion of the findings of epidemiological ramifications concludes the manuscript.

SIR model with social gatherings

AbstractWe introduce an extension to Kermack and McKendrick’s classic susceptible–infected–recovered (SIR) model in epidemiology, whose underlying mechanism of infection consists of individuals attending randomly generated social gatherings. This gives rise to a system of ordinary differential equations (ODEs) where the force of the infection term depends non-linearly on the proportion of infected individuals. Some specific instances yield models already studied in the literature, to which the present work provides a probabilistic foundation. The basic reproduction number is seen to depend quadratically on the average size of the gatherings, which may be helpful in understanding how restrictions on social gatherings affect the spread of the disease. We rigorously justify our model by showing that the system of ODEs is the mean-field limit of the jump Markov process corresponding to the evolution of the disease in a finite population.

Possible counter-intuitive impact of local vaccine mandates for vaccine-preventable infectious diseases.

We modeled the impact of local vaccine mandates on the spread of vaccine-preventable infectious diseases, which in the absence of vaccines will mainly affect children. Examples of such diseases are measles, rubella, mumps, and pertussis. To model the spread of the pathogen, we used a stochastic SIR (susceptible, infectious, recovered) model with two levels of mixing in a closed population, often referred to as the household model. In this model, individuals make local contacts within a specific small subgroup of the population (e.g., within a household or a school class), while they also make global contacts with random people in the population at a much lower rate than the rate of local contacts. We considered what would happen if schools were given freedom to impose vaccine mandates on all of their pupils, except for the pupils that were exempt from vaccination because of medical reasons. We investigated first how such a mandate affected the probability of an outbreak of a disease. Furthermore, we focused on the probability that a pupil that was medically exempt from vaccination, would get infected during an outbreak. We showed that if the population vaccine coverage was close to the herd-immunity level, then both probabilities may increase if local vaccine mandates were implemented. This was caused by unvaccinated pupils possibly being moved to schools without mandates.

The disutility of compartmental model forecasts during the COVID-19 pandemic.

During the COVID-19 pandemic, several forecasting models were released to predict the spread of the virus along variables vital for public health policymaking. Of these, the susceptible-infected-recovered (SIR) compartmental model was the most common. In this paper, we investigated the forecasting performance of The University of Texas COVID-19 Modeling Consortium SIR model. We considered the following daily outcomes: hospitalizations, ICU patients, and deaths. We evaluated the overall forecasting performance, highlighted some stark forecast biases, and considered forecast errors conditional on different pandemic regimes. We found that this model tends to overforecast over the longer horizons and when there is a surge in viral spread. We bolstered these findings by linking them to faults with the SIR framework itself.

Viable control of COVID-19 spread with vaccination

The rapid and widespread transmission of COVID-19 has necessitated the development and implementation of effective control measures. Vaccination has emerged as a key tool in combating the pandemic. This article introduces a novel approach to modeling the dynamics of COVID-19 transmission by integrating vaccination strategies into the susceptible-infected-recovered (SIR) framework using viability theory. We have defined a set of constraints including a guaranteed level of vaccination, we analyze the impact of different vaccination rates on curbing the spread of the virus. Our findings reveal the significant role of vaccination in reducing transmission and offer valuable insights into optimizing vaccination campaigns. The viability-based SIR model provides a comprehensive framework for policymakers and healthcare professionals to devise targeted strategies and allocate resources effectively in the battle against COVID-19.

Combatting Malaysia's Dengue Outbreaks with Auto-Dissemination Mosquito Traps: A Hybrid Stochastic-Deterministic SIR Model

Classical mosquito control methods (e.g. chemical fogging) struggle to sustain long-term reductions in mosquito populations to combat vector-borne diseases like dengue. The Mosquito Home System (MHS) is an auto-dissemination mosquito trap, that kills mosquito larvae before they hatch into adult mosquitoes. A novel hybrid stochastic-deterministic model is presented, that successfully predicts the effect of deploying MHSs within high-rise flats in Selangor, Malaysia. Stochastic SIR (Susceptible-Infected-Recovered) equations (flats) are paired with an existing deterministic SIR model (wider Kuala Lumpur population). Model predictions provide excellent agreement with data from a 44 week MHS trial within the flats. The stochastic model is validated as a powerful tool for predicting short- and long-term impacts of deploying this style of trap within similar environments. Significant, sustainable reductions in mosquito populations are predicted when the MHS is active: with a mean of 9 (95% Uncertainty Range (UR): 1; 30) during the 44 week trial period, compared to 35 (95% UR: 1; 234) dengue cases with no MHSs. Long-term predictions for endemic equilibrium show MHSs significantly narrow the mosquito population distribution and reduce dengue prevalence: from a mean of 5 (95% UR: 0; 52) (no MHS), to 1 (95% UR: 0; 8) dengue cases annually (with MHS).

Geographic expansion of Fasciola hepatica (Linnaeus, 1758) due to changes in land use and cover in Brazil

Fasciolosis is caused by parasites of the genus Fasciola, affecting animals and humans worldwide. In South America, the disease is a result of infection with Fasciola hepatica and although animal infections are more frequently reported, the full extent of the impact on human health due to underdiagnosis remains uncertain. This study analyzed changes in land use and the distribution of F. hepatica in bovine livers in Brazil over 18 years. Data on land use and land cover were collected from the Mapbiomas Project. Data on 414,481,963 slaughtered cattle and condemned livers due to F. hepatica infection were obtained from 4,433 municipalities. Joinpoint analysis was used to study the time series, and the Susceptible-Infected-Recovered (SIR) model was utilized to explore the behavior of F. hepatica infection. In the North, pasture areas significantly increased (P = 0.000001), while forested areas decreased (P = 0.000001). The midwestern and northern regions concentrated the highest number (>290 million) of cattle slaughtered in Brazil. More than 2 million bovine livers were infected by F. hepatica. The infected cattle originated from 194 municipalities in 2002, increasing to 747 in 2020. We consider that the changes in land use and intense cattle transportation may have caused the expansion of F. hepatica. The SIR model analyzed the spread of the disease looking at all six biomes: Caatinga, Amazon Forest, Cerrado, Pantanal, Atlantic Forest, and Pampa. Moreover, this infection not only threatens the health of animals but is also a major concern to biodiversity and vulnerable human communities in South America. Emblematic biomes such as the Amazon basin already face challenges with logging, desertification, and loss of biodiversity. Therefore, strategies for mitigating infection should include controlling illegal pasture areas, establishing health inspections of animal transport, quarantine of newly arrived animals, and livestock zoning, as well as clear One Health policies.

An agent-based nested model integrating within-host and between-host mechanisms to predict an epidemic

The COVID-19 pandemic has remarkably heightened concerns regarding the prediction of communicable disease spread. This study introduces an innovative agent-based modeling approach. In this model, the quantification of human-to-human transmission aligns with the dynamic variations in the viral load within an individual, termed “within-host” and adheres to the susceptible–infected–recovered (SIR) process, referred to as “between-host.” Variations in the viral load over time affect the infectivity between individual agents. This model diverges from the traditional SIR model, which employs a constant transmission probability, by incorporating a dynamic, time-dependent transmission probability influenced by the viral load in a host agent. The proposed model retains the time-integrated transmission probability characteristic of the conventional SIR model. As observed in this model, the overall epidemic size remains consistent with the predictions of the standard SIR model. Nonetheless, compared to predictions based on the classical SIR process, notable differences existed in the peak number of the infected individuals and the timing of this peak. These nontrivial differences are induced by the direct correlation between the time-evolving transmission probability and the viral load within a host agent. The developed model can inform targeted intervention strategies and public health policies by providing detailed insights into disease spread dynamics, crucial for effectively managing epidemics.

Novel class of susceptible–infectious–recovered models involving power-law interactions

It is shown that the ordinary SIR (Susceptible–Infectious–Recovered) epidemic model exhibits features that are common to a class of compartmental models with power-law interactions. Within this class of theoretical models, the standard SIR model emerges as a singular non-integrable model. Various integrable models, whose solutions are defined explicitly or implicitly in terms of elementary functions, are discovered within the same class. A Hamiltonian dynamics with position-depending forces underlies a sub-class of these models. The general class of models is very flexible and capable of describing epidemics characterized by a finite or indefinite lifespan. In the last case, the compartment population distributions evolve in time exhibiting exponential or power-law tails.

The susceptible–infected–recovered model as a tool to motivate and teach differential equations: an analytic approach

Since the introduction by Kermack and McKendrick in 1927, the Susceptible–Infected–Recovered (SIR) epidemic model has been a foundational model to comprehend and predict the dynamics of infectious diseases. Almost for a century, the SIR model has been modified and extended to meet the needs of different characteristics of various infectious diseases including recent COVID-19 breakouts, which stimulates the advance in mathematical epidemiology theory significantly. Some of the mathematical ideas and techniques developed are also relevant to motivate teaching various topics in differential equations by connecting students' life experiences with current pandemics to meaningful classroom learning activities. Here, various pedagogically relevant topics from the SIR model are provided for undergraduate differential equation class, which includes (1) compartmental modelling and mass action kinetic modelling, (2) conservation rule and model reduction, (3) introduction to phase plane by removing time variable to derive trajectory equations, (4) transformation of equations to equivalent forms, (5) transformation of second-order system to second-order ODE, (6) deriving an analytic solution to the SIR equation by solving Bernoulli's equation, (7) deriving an analytic solution to the SIR equation from trajectory equations and (8) deriving an analytic solution to the SIR equation from exponential substitutions.

Epidemic modeling using differential equations with implementation in R

The COVID-19 pandemic, like past historical events such as the Vietnam War or 9/11, will shape a generation. Mathematics educators can seize this unprecedented opportunity to teach the principles of mathematical modeling in epidemiology. Compartmental epidemiological models, such as the SIR (susceptible-infected-recovered), are widely used by researchers and educators to study and teach infectious disease dynamics, particularly for COVID-19. The SIR model is considered a fundamental mathematical model for the spread of epidemic disease and employs Ordinary Differential Equations to determine the number of individuals in each compartment of a population at a given time. In this note, we demonstrate how to build a basic SIR model. We show how to simulate, plot, and interpret the results using numerical simulation with R, a popular programming language widely used in data analysis and scientific research.