Basic SIR Model Research Articles

ANALYSIS OF MEASLES DISEASE IN INDIVIDUAL USING BASIC SIR MODEL

In this paper, we examine the measles pandemic using basic SIR model. We investigate endemic equilibrium points, disease-free equilibrium points, reproduction number as well as basic reproduction numbers in order to examine the effects of various partition transitions. It is proposed that the measles model is both locally and globally asymptotically stable at the disease-free equilibrium point. Additionally we have analysis the spreading of disease inside human body and its recovery time. The illustration is carried out by Euler method

Vaccination Strategies based on a Mathematical Model of Epidemics Considering the Age Structure of the Population

During the COVID-19 pandemic, it is important to promote the skills needed for analyzing the disease course, including determining the relevance of vaccinations, especially among people who are unfamiliar with computer programming. This paper describes the basic epidemiological model (SIR), its extensions that allow vaccinations, and the emergence of renewed waves of disease growth. It also discusses a literature model, extended SEIRD, which includes a more detailed division of the population into susceptible, latent, symptomatic, and asymptomatic infected, recovered, and dead in eight age groups. Modifying the SEIRD model as shown on the basic SIR model, we analyzed five vaccination strategies, considering the limited vaccine supply, the number of vaccinations performed per day, and their effectiveness. The analysis was performed for a group of one million people, using the parameters of the model characteristic of the COVID-19 pandemic and Sweden's generational structure. We analyzed in terms of reducing both the number of deaths and the incidence of symptomatic infections, which represent the main burden of healthcare.

Application of Susceptible-Infected-Removed Model with Vital Dynamics for COVID-19 Outbreak in Malaysia

In late 2019, the unique severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), also known as COVID-19, first emerged in Wuhan City, Hubei Province, China and quickly spread throughout the world. Until June 30, 2022, a total of 4,566,055 cases of COVID-19 have been reported in Malaysia, with 35,765 deaths and 4,500,856 recovered cases. This study aims to generalise a deterministic SIR model with vital dynamics for understanding the proliferation of infectious diseases. The SIR model with vital dynamics is more realistic in mimicking reality than the basic SIR model because it can determine the dynamic behaviours of COVID-19 over a more extended period. The SIR model utilises vital dynamics with unequal birth and death rates. Furthermore, the SIR model with vital dynamics is rescaled with the total time-varying population and analysed according to its epidemic condition. The results indicated that the number of infected individuals would peak about 10 - 15 days and reach their steady state about 25 - 60 days. The findings of this research may help policymakers establish, plan, and implement effective COVID-19 pandemic response strategies.

Optimal Incentives to Mitigate Epidemics: A Stackelberg Mean Field Game Approach

Motivated by the models of epidemic control in large populations, we consider a Stackelberg mean field game model between a principal and a mean field of agents whose states evolve in a finite state space. The agents play a noncooperative game in which they control their rates of transition between states to minimize an individual cost. The principal influences the nature of the resulting Nash equilibrium through incentives to optimize its own objective. We analyze this game using a probabilistic approach. We then propose an application to an epidemic model of SIR type in which the agents control the intensities of their interactions, and the principal is a regulator acting with nonpharmaceutical interventions. To compute the solutions, we propose an innovative numerical approach based on Monte Carlo simulations and machine learning tools for stochastic optimization. We conclude with numerical experiments illustrating the impact of the agents' and the regulator's optimal decisions in two specific models: a basic SIR model with semiexplicit solutions and a more complex model with a larger state space.

Short-Term Prediction Methodology of COVID-19 Infection in South Korea

The purpose of this study is to predict the short-term trend of the COVID-19 pandemic and give insights into effective response strategies. Based on the basic SIR model, a compartment method for modeling the course of an epidemic, the short-term infection change ratio md, is derived. The number of infected people can be predicted using this ratio. We calculated different md values on a weekly basis. As we tested different combinations of md, the prediction from the combination of md based on a week and md based on 4 weeks was found to be statistically reliable. According to our regression analysis, our approach has an explanatory power of 96%. However, this method could only predict 1 week ahead of current data. Thus, we use LSTM, a deep learning method applied for time series data, to forecast the trend 4 weeks ahead. The forecasted trends show that the number of infected people in South Korea will reach its peak a week after the writing of this work and start to gradually decline after that.

How seasonal variations in birth and transmission rates impact population dynamics in a basic SIR model

The changing climate is expected to alter the timings of key events in species life-histories. These shifts are likely to have important consequences for infectious disease dynamics, as the distribution and abundance of host species will lead to a different environment for parasites. Previous work has shown how seasonality in single host traits - most commonly the reproduction rate or transmission rate - can lead to an array of complex epidemiological dynamics, including chaos and multiple-stable states, with changes to the timing and amplitude of the seasonal peaks often driving drastic changes in behaviour. However, more than one life-history trait is likely to be seasonal, and changing environmental conditions may impact each of them in different ways, yet there have been few studies of host-parasite dynamics that include more than one seasonal trait. Here we examine a Susceptible-Infected-Recovered epidemiological model in which both reproduction and transmission exhibit seasonal fluctuations. We examine how the amplitude and timing of these seasonal peaks impact disease dynamics. We show that the relative timing of the two events is key, with the most stable dynamics when births peak a few months before transmission. We also show that chaotic dynamics become more likely when transmission in particular has a high amplitude, and when baseline transmission and virulence are high. Our results emphasise the importance of seasonality and timing of host life-history events to disease dynamics.

Estimation of the reproduction number of influenza A(H1N1)pdm09 in South Korea using heterogeneous models

BackgroundThe reproduction number is one of the most crucial parameters in determining disease dynamics, providing a summary measure of the transmission potential. However, estimating this value is particularly challenging owing to the characteristics of epidemic data, including non-reproducibility and incompleteness.MethodsIn this study, we propose mathematical models with different population structures; each of these models can produce data on the number of cases of the influenza A(H1N1)pdm09 epidemic in South Korea. These structured models incorporating the heterogeneity of age and region are used to estimate the reproduction numbers at various terminal times. Subsequently, the age- and region-specific reproduction numbers are also computed to analyze the differences illustrated in the incidence data.ResultsIncorporation of the age-structure or region-structure allows for robust estimation of parameters, while the basic SIR model provides estimated values beyond the reasonable range with severe fluctuation. The estimated duration of infectious period using age-structured model is around 3.8 and the reproduction number was estimated to be 1.6. The estimated duration of infectious period using region-structured model is around 2.1 and the reproduction number was estimated to be 1.4. The estimated age- and region-specific reproduction numbers are consistent with cumulative incidence for corresponding groups.ConclusionsNumerical results reveal that the introduction of heterogeneity into the population to represent the general characteristics of dynamics is essential for the robust estimation of parameters.

Modeling and controlling the spread of epidemic with various social and economic scenarios.

We propose a dynamical model for describing the spread of epidemics. This model is an extension of the SIQR (susceptible-infected-quarantined-recovered) and SIRP (susceptible-infected-recovered-pathogen) models used earlier to describe various scenarios of epidemic spreading. As compared to the basic SIR model, our model takes into account two possible routes of contagion transmission: direct from the infected compartment to the susceptible compartment and indirect via some intermediate medium or fomites. Transmission rates are estimated in terms of average distances between the individuals in selected social environments and characteristic time spans for which the individuals stay in each of these environments. We also introduce a collective economic resource associated with the average amount of money or income per individual to describe the socioeconomic interplay between the spreading process and the resource available to infected individuals. The epidemic-resource coupling is supposed to be of activation type, with the recovery rate governed by the Arrhenius-like law. Our model brings an advantage of building various control strategies to mitigate the effect of epidemic and can be applied, in particular, to modeling the spread of COVID-19.

Analytical mechanics allows novel vistas on mathematical epidemic dynamics modeling

This contribution aims to shed light on mathematical epidemic dynamics modelling from the viewpoint of analytical mechanics. To set the stage, it recasts the basic SIR model of mathematical epidemic dynamics in an analytical mechanics setting. Thereby, it considers two possible re-parameterizations of the basic SIR model. On the one hand, it is proposed to re-scale time, while on the other hand, to transform the coordinates, i.e.\ the independent variables. In both cases, Hamilton's equations in terms of a suited Hamiltonian as well as Hamilton's principle in terms of a suited Lagrangian are considered in minimal and extended phase and state space coordinates, respectively. The corresponding Legendre transformations relating the various options for the Hamiltonians and Lagrangians are detailed. Ultimately, this contribution expands on a multitude of novel vistas on mathematical epidemic dynamics modelling that emerge from the analytical mechanics viewpoint. As result, it is believed that interesting and relevant new research avenues open up when exploiting in depth the analogies between analytical mechanics and mathematical epidemic dynamics modelling.

How to control the immigration of infectious individuals for a region?

Immigration has contributed to the increase of certain infectious diseases in some regions. To inspect the impact of controlling immigration of infectious individuals on the transmission of infectious diseases in a region, we construct a basic SIR model with switching imported infectious population. There is no disease-free equilibrium and no basic reproduction number for this system. Based on the global dynamical analysis for the switching system, we know that a proper control of immigration of infectious individuals can control the infectious population for a region at a certain low level.

Computation of epidemic final size distributions.

We develop a new methodology for the efficient computation of epidemic final size distributions for a broad class of Markovian models. We exploit a particular representation of the stochastic epidemic process to derive a method which is both computationally efficient and numerically stable. The algorithms we present are also physically transparent and so allow us to extend this method from the basic SIR model to a model with a phase-type infectious period and another with waning immunity. The underlying theory is applicable to many Markovian models where we wish to efficiently calculate hitting probabilities.

STUDYING THE IDENTIFIABILITY OF EPIDEMIOLOGICAL MODELS USING MCMC

Studying different theoretical properties of epidemiological models has been widely addressed, while numerical studies and especially the calibration of models, which are often complicated and loaded with a high number of unknown parameters, against measured data have received less attention. In this paper, we describe how a combination of simulated data and Markov Chain Monte Carlo (MCMC) methods can be used to study the identifiability of model parameters with different type of measurements. Three known models are used as case studies to illustrate the importance of parameter identifiability: a basic SIR model, an influenza model with vaccination and treatment and a HIV–Malaria co-infection model. The analysis reveals that calibration of complex models commonly studied in mathematical epidemiology, such as the HIV–Malaria co-dynamics model, can be difficult or impossible, even if the system would be fully observed. The presented approach provides a tool for design and optimization of real-life field campaigns of collecting data, as well as for model selection.

GLOBAL DYNAMICS OF A MULTI-GROUP EPIDEMIC MODEL WITH GENERAL RELAPSE DISTRIBUTION AND NONLINEAR INCIDENCE RATE

In this paper, we investigate a class of multi-group epidemic models allowing heterogeneity of the host population and that has taken into consideration with general relapse distribution and nonlinear incidence rate. We establish that the global dynamics are completely determined by the basic reproduction number R0. The proofs of the main results utilize the persistence theory in dynamical systems, Lyapunov functionals and a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach. Biologically, the disease (with any initial inoculation) will persist in all groups of the population and will eventually settle at a constant level in each group. Furthermore, our results demonstrate that heterogeneity and nonlinear incidence rate do not alter the dynamical behaviors of the basic SIR model. On the other hand, the global dynamics exclude the existence of Hopf bifurcation leading to sustained oscillatory solutions.

Towards a Characterization of Behavior-Disease Models

The last decade saw the advent of increasingly realistic epidemic models that leverage on the availability of highly detailed census and human mobility data. Data-driven models aim at a granularity down to the level of households or single individuals. However, relatively little systematic work has been done to provide coupled behavior-disease models able to close the feedback loop between behavioral changes triggered in the population by an individual's perception of the disease spread and the actual disease spread itself. While models lacking this coupling can be extremely successful in mild epidemics, they obviously will be of limited use in situations where social disruption or behavioral alterations are induced in the population by knowledge of the disease. Here we propose a characterization of a set of prototypical mechanisms for self-initiated social distancing induced by local and non-local prevalence-based information available to individuals in the population. We characterize the effects of these mechanisms in the framework of a compartmental scheme that enlarges the basic SIR model by considering separate behavioral classes within the population. The transition of individuals in/out of behavioral classes is coupled with the spreading of the disease and provides a rich phase space with multiple epidemic peaks and tipping points. The class of models presented here can be used in the case of data-driven computational approaches to analyze scenarios of social adaptation and behavioral change.

EXPOSURE RISK ASSESSMENT BY COUPLED ANALYSIS OF UNSTEADY CONCENTRATION DISTRIBUTION AND SIR MODEL IN ENCLOSED SPACES

The indoor environment can play a significant role in the transmission and exposure of various contaminants. In some emerging aerial infections, such as influenza virus, tuberculosis virus, and other biological, the airborne route of transmission is thought to be important to evaluate exposure health risk. In this paper, first, we have presented the relationship between basic SIR (MK) model and Wells-Riley model, and introduced the analytical procedure of coupled analysis of CFD based prediction of unsteady contaminant concentration distribution and basic SIR model to predict exposure risk of residents in large enclosed space. The results of sensitivity analysis that changed ventilation rate and other parameters of infections showed non-uniform distribution in enclosed spaces and strong dependence on unsteady contaminant distribution.

Optimal control of epidemics with limited resources

We extend the existing work on the time-optimal control of the basic SIR epidemic model with mass action contact rate. Previous results have focused on minimizing an objective function that is a linear combination of the cost associated with using control and either the outbreak size or the infectious burden. We instead, provide analytic solutions for the control that minimizes the outbreak size (or infectious burden) under the assumption that there are limited control resources. We provide optimal control policies for an isolation only model, a vaccination only model and a combined isolation-vaccination model (or mixed model). The optimal policies described here contain many interesting features especially when compared to previous analyses. For example, under certain circumstances the optimal isolation only policy is not unique. Furthermore the optimal mixed policy is not simply a combination of the optimal isolation only policy and the optimal vaccination only policy. The results presented here also highlight a number of areas that warrant further study and emphasize that time-optimal control of the basic SIR model is still not fully understood.

Effects of predation on host–pathogen dynamics in SIR models

The integration of infectious disease epidemiology with community ecology is an active area of research. Recent studies using SI models without acquired immunity have demonstrated that predation can suppress infectious disease levels. The authors recently showed that incorporating immunity (SIR models) can produce a “hump”-shaped relationship between disease prevalence and predation pressure; thus, low to moderate levels of predation can boost prevalence in hosts with acquired immunity. Here we examine the robustness of this pattern to realistic extensions of a basic SIR model, including density-dependent host regulation, predator saturation, interference, frequency-dependent transmission, predator numerical responses, and explicit resource dynamics. A non-monotonic relationship between disease prevalence and predation pressure holds across all these scenarios. With saturation, there can also be complex responses of mean host abundance to increasing predation, as well as bifurcations leading to unstable cycles (epidemics) and pathogen extinction at larger predator numbers. Firm predictions about the relationship between prevalence and predation thus require one to consider the complex interplay of acquired immunity, host regulation, and foraging behavior of the predator.

Pulse vaccination strategy in the SIR epidemic model.

Theoretical results show that the measles 'pulse' vaccination strategy can be distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination. Using the SIR epidemic model we showed that under a planned pulse vaccination regime the system converges to a stable solution with the number of infectious individuals equal to zero. We showed that pulse vaccination leads to epidemics eradication if certain conditions regarding the magnitude of vaccination proportion and on the period of the pulses are adhered to. Our theoretical results are confirmed by numerical simulations. The introduction of seasonal variation into the basic SIR model leads to periodic and chaotic dynamics of epidemics. We showed that under seasonal variation, in spite of the complex dynamics of the system, pulse vaccination still leads to epidemic eradication. We derived the conditions for epidemic eradication under various constraints and showed their dependence on the parameters of the epidemic. We compared effectiveness and cost of constant, pulse and mixed vaccination policies.

Pulse vaccination strategy in the SIR epidemic model

Theoretical results show that the measles `pulse' vaccination strategy can be distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination. Using the SIR epidemic model we showed that under a planned pulse vaccination regime the system converges to a stable solution with the number of infectious individuals equal to zero. We showed that pulse vaccination leads to epidemics eradication if certain conditions regarding the magnitude of vaccination proportion and on the period of the pulses are adhered to. Our theoretical results are confirmed by numerical simulations. The introduction of seasonal variation into the basic SIR model leads to periodic and chaotic dynamics of epidemics. We showed that under seasonal variation, in spite of the complex dynamics of the system, pulse vaccination still leads to epidemic eradication. We derived the conditions for epidemic eradication under various constraints and showed their dependence on the parameters of the epidemic. We compared effectiveness and cost of constant, pulse and mixed vaccination policies.