Susceptible-infected-recovered Model Research Articles

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

<p>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.</p>

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

<p>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.</p>

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.

Stochastic Modeling of a Measles Outbreak in Brazil

Development of mathematical models and its numerical implementations are essential tools in epidemiological modeling. Susceptible-Infected-Recovered (SIR) compartmental model, proposed by Kermack and McKendrick, in 1927, is a widely used deterministic model which serves as a basis for more involved mathematical models. In this work, we consider two stochastic versions of the SIR model for analysing a measles outbreak in Ilha Grande, Rio de Janeiro, in 1976; Continuous Time Markov Chain and Stochastic Differential Equations. The SIR Continuous Time Markov Chain model is used to extract specific information from the measles outbreak, obtaining results in excellent agreement with the reported epidemic values. Numerical simulations are performed in Python.

Large-deviation properties of SIR model incorporating protective measures

We simulate spreads of diseases for the susceptible–infected–recovered (SIR) model on contact networks with a particular focus on incorporated protective measures such as face masks. We consider the small-world network model. By using a large-deviation approach, in particular the Wang–Landau algorithm, we obtained the full probability density function (pdf) of the cumulative number C of infected people over the full range of its support. In this way we are able to reach probabilities as small as 10−50. We obtain distinct characteristics in the pdf such as non-analyticities induced by the onset of the protective measures. Still, the results indicate that the mathematical large-deviation principle also holds for this extended SIR model, meaning that the size-dependence enters in P(C) in a simple fashion and the distribution is determined by the so-called rate function. We observe different phases in the pdf, which we investigate by analyzing the corresponding infection courses, i.e. time series, and and their correlations to the observed values of C.

Agent Based Modeling of the Spread of Social Unrest Using Infectious Disease Models

Prior research suggests that the timing and location of social unrest may be influenced by similar unrest activities in another nearby region, potentially causing a spread of unrest activities across space and time. In this paper, we model the spread of social unrest across time and space using a novel approach, grounded in agent-based modeling (ABM) . In it, regions (geographic polygons) are represented as agents that transition from one state to another based on changes in their environment. Our approach involves (1) creating a vector for each region/agent based on socio-demographic, infrastructural, economic, geographic, and environmental (SIEGE) factors, (2) formulating a neighborhood distance function to identify an agent's neighbors based on geospatial distance and SIEGE proximity, (3) designing transition probability equations based on two distinct compartmental models—i.e., the Susceptible-Infected-Recovered (SIR) and the Susceptible-Infected-Susceptible (SIS) models, and (4) building a ground truth for evaluating the simulations. We use ABM to determine the individualized probabilities of each region/agent to transition from one state to another. The models are tested using the districts of three states in India as agents at a monthly scale for 2016-2019. For ground truth of unrest events, we use the Armed Conflict Location and Event Data (ACLED) dataset. Our findings include that (1) the transition probability equations are viable, (2) the agent-based modeling of the spread of social unrest is feasible while treating regions as agents (Brier's score < 0.25 for two out of three regions), and (3) the SIS model performs comparatively better than the SIR model.

Optimal vaccination strategies for COVID-19 in population migration between two regions scenario

ABSTRACT Population movements had a significant impact on the spread of COVID-19, and vaccination is considered the most effective means for humans to face viral infections. This study identifies the optimal control strategy for COVID-19 prevention and control, and explores the impact of short-term and long-term migration on the optimal proportion of vaccine allocation between two regions. We proposed to establish the SIR (Susceptible-Infectious-Recovered) model and determine the stability by calculating the disease free equilibrium and Jacobi matrix of the model. We then established the vaccine optimization model, solved the optimal vaccine distribution strategy by gradient descent method and explored the impact of short-term and long-term migration on the optimal vaccine allocation ratio. The stability analysis revealed that the virus could not be eliminated only by reducing the migration rates and infection rates. we introduced the vaccine methods and obtained the optimal vaccine allocation ratio in Shenzhen and Hong Kong as p 1 : p 2 = 0.000341 : 0.001739 , and the daily vaccination rate we need to impose in each region as p 1 : p 2 = 0.00068 : 0.001901 . The presence or absence of short-term migration had no greater impact on the distribution of the vaccine, whereas R v with long-term migration had a greater effect than no migration. We found that migration rates could not eliminate the outbreak in both regions and that adopting an effective vaccine distribution strategy could be more effective in eliminating the outbreak. And for different allocation scenarios with limited vaccine supply, we obtained the optimal allocation most favorable to control the epidemic.

A Proposed Analytical and Numerical Treatment for the Nonlinear SIR Model via a Hybrid Approach

This paper re-analyzes the nonlinear Susceptible–Infected–Recovered (SIR) model using a hybrid approach based on the Laplace–Padé technique. The proposed approach is successfully applied to extract several analytic approximations for the infected and recovered individuals. The domains of applicability of such analytic approximations are addressed. In addition, the present results are validated through various comparisons with the Runge–Kutta numerical method. The obtained analytical results agree with the numerical ones for a wide range of numbers of contacts featured in the studied model. The efficiency of the present analysis reveals that it can be implemented to deal with other systems describing real-life phenomena.

A real-world data validation of the value of early-stage SIR modelling to public health

Performance of Susceptible-Infected-Recovered (SIR) model in the early stage of a novel epidemic may be hindered by data availability. Additionally, the traditional SIR model may oversimplify the disease progress, and knowledge about the virus and transmission is limited early in the epidemic, resulting in a greater uncertainty of such modelling. We aimed to investigate the impact of model inputs on the early-stage SIR projection using COVID-19 as an illustration to evaluate the application of early infection models. We constructed a modified SIR model using discrete-time Markov chain to simulate daily epidemic dynamics and estimate the number of beds needed in Wuhan in the early stage of COVID-19 epidemic. We compared eight scenarios of SIR projection to the real-world data (RWD) and used root mean square error (RMSE) to assess model performance. According to the National Health Commission, the number of beds occupied in isolation wards and ICUs due to COVID-19 in Wuhan peaked at 37,746. In our model, as the epidemic developed, we observed an increasing daily new case rate, and decreasing daily removal rate and ICU rate. This change in rates contributed to the growth in the needs of bed in both isolation wards and ICUs. Assuming a 50% diagnosis rate and 70% public health efficacy, the model based on parameters estimated using data from the day reaching 3200 to the day reaching 6400 cases returned a lowest RMSE. This model predicted 22,613 beds needed in isolation ward and ICU as on the day of RWD peak. Very early SIR model predictions based on early cumulative case data initially underestimated the number of beds needed, but the RMSEs tended to decline as more updated data were used. Very-early-stage SIR model, although simple but convenient and relatively accurate, is a useful tool to provide decisive information for the public health system and predict the trend of an epidemic of novel infectious disease in the very early stage, thus, avoiding the issue of delay-decision and extra deaths.