Classical SIR Model Research Articles

Global dynamics of SIR model with switched transmission rate.

We propose a new epidemiological model, based on the classical SIR model, taking additionally into account a switching prevention strategy. The model has two distinct thresholds that determine the beginning and the end of an intervention and two different transmission rates. We study the global dynamics of the proposed two-dimensional model.

Paradoxical phenomena and chaotic dynamics in epidemic models subject to vaccination

An alternative to the constant vaccination strategy could be the administration of a large number of doses on immunization days with the aim of maintaining the basic reproduction number to be below one. This strategy, known as pulse vaccination, has been successfully applied for the control of many diseases especially in low-income countries. In this paper, we analytically prove (without being computer-aided) the existence of chaotic dynamics in the classical SIR model with pulse vaccination. To the best of our knowledge, this is the first time in which a theoretical proof of chaotic dynamics is given for an epidemic model subject to pulse vaccination. In a realistic public health context, our analysis suggests that the combination of an insufficient vaccination coverage and high birth rates could produce chaotic dynamics and an increment of the number of infectious individuals.

On a quarantine model of coronavirus infection and data analysis

Attempts to curb the spread of coronavirus by introducing strict quarantine measures apparently have different effect in different countries: while the number of new cases has reportedly decreased in China and South Korea, it still exhibit significant growth in Italy and other countries across Europe. In this brief note, we endeavour to assess the efficiency of quarantine measures by means of mathematical modelling. Instead of the classical SIR model, we introduce a new model of infection progression under the assumption that all infected individual are isolated after the incubation period in such a way that they cannot infect other people. Disease progression in this model is determined by the basic reproduction numberR0(the number of newly infected individuals during the incubation period), which is different compared to that for the standard SIR model. IfR0> 1, then the number of latently infected individuals exponentially grows. However, ifR0< 1 (e.g.due to quarantine measures and contact restrictions imposed by public authorities), then the number of infected decays exponentially. We then consider the available data on the disease development in different countries to show that there are three possible patterns: growth dynamics, growth-decays dynamics, and patchy dynamics (growth-decay-growth). Analysis of the data in China and Korea shows that the peak of infection (maximum of daily cases) is reached about 10 days after the restricting measures are introduced. During this period of time, the growth rate of the total number of infected was gradually decreasing. However, the growth rate remains exponential in Italy. Arguably, it suggests that the introduced quarantine is not sufficient and stricter measures are needed.

Nash SIR: An Economic-Epidemiological Model of Strategic Behavior During a Viral Epidemic

This paper develops a Nash-equilibrium extension of the classic SIR model of infectious-disease epidemiology ("Nash SIR"), endogenizing people's decisions whether to engage in economic activity during a viral epidemic and allowing for complementarity in social-economic activity. An equilibrium epidemic is one in which Nash equilibrium behavior during the epidemic generates the epidemic. There may be multiple equilibrium epidemics, in which case the epidemic trajectory can be shaped through the coordination of expectations, in addition to other sorts of interventions such as stay-at-home orders and accelerated vaccine development. An algorithm is provided to compute all equilibrium epidemics.

How Testing Reduces Infections and Prevents Herd Immunity: An SIR Model with Testing

In this paper, I incorporate infection testing into the classic SIR model of epidemics and analytically illustrate how testing can reduce the reproduction number and the total number of infections in an epidemic. Testing identifies infected individuals and reduces their contact rate, and therefore, reduces the reproduction number of the disease, total infections and even prevents herd immunity. Costs of testing can be kept low if initially sufficiently many tests are conducted. Moreover, when testing is implemented as a sustainable suppression policy, other temporary suppression policies become complementary to the sustainable suppression policy - testing - and can reduce total infections over the epidemic.

Analysis of the impact of online social networks addiction on the propagation of rumors

The growing popularity of online social networks (OSNs) has emerged phenomenon of addiction to OSNs, known as OSN-addiction, which has widely attracted researchers to study its impact on individuals. The OSN-addiction can lead to the dissemination of information without being convinced of its reliability. Moreover, the last couple of years have witnessed the appearance of OSNs that allow users to obtain profits of their content, such as YouTube. They create the ambition to have an extra income where individuals will seek for content that offers more views. As a result, these OSNs increase the degree of OSN-addiction in which it can significantly affect the propagation of rumors since they spread faster and broader than usual information. Therefore, this paper studies the effect of OSN-addiction on the propagation of rumors with the epidemic model. Thus, we propose a novel model that extends of the classical SIR model by integrating the role of addicted individuals into the spreading process based on psychological studies. A theoretical analysis is presented for homogeneous and heterogeneous networks, demonstrating a significant influence of OSN-addiction on the propagation of rumors. Finally, experiments were conducted on homogeneous networks, heterogeneous networks and real large-scale network (YouTube). The results confirm the theoretical analysis and illustrate the significant influence of the addicted individuals to amplify the impact of the propagation of a rumor whereas in particular cases it also accelerates the propagation.

It’s contagious: calculation and analysis of xylem vulnerability to embolism by a mechanistic approach based on epidemic modeling

Vulnerability of xylem to embolism is expressed by vulnerability curves (VC) depicting accumulating loss of hydraulic conductivity against mounting pressure stress. VCs are obtained experimentally by curve fitting, providing important benchmark data on vulnerability of a species, organ, or other specific xylem categories. Since embolism spread by air seeding represents principally an epidemic process, a method based on epidemic modeling is introduced in this contribution which allows rapid and efficient calculation of continuous VC directly from mechanistic equations. The approach is based on the classic SIR model which considers the population to be divided into susceptibles S (functional), infectives I (embolized), and “removed” R individuals (embolized conduits which have exhausted their capacity to pass gas to their neighbors). It is also possible to consider xylem which is composed of different conduit groups (differing in interconnectivity and/or conduit vulnerability). The approach is able to reconstruct all sorts of VC shapes on the basis of two key parameters of embolism, conduit vulnerability and interconnectivity, and the amount of native embolism (initial proportion of I and R conduits). The effect of I and R conduits on VC is different. High initial I leads to non-sigmoidal VC, whereas high initial R promotes sigmoidal VC. Heterogeneous conduit populations being composed of conduit groups of different sizes, interconnectivities, and conduit vulnerabilities can also lead to non-sigmoidal VC with complex shapes.

Space dependent models for studying the spread of some diseases

In this paper we present some extensions of the classical SIR model with non-symmetric spatial dependence. SIR-type models usually describe the epidemic in a population, which is split into three categories, namely the susceptibles (S), the infected (I) and the recovered (R). The proposed model yields a system of partial integro-differential equations. Two methods handling the integrals of the equations are presented. We give numerical examples which show that the discrete models preserve the basic qualitative properties of the original biological process, and these results are supported by the oretical theorems.

Epidemic Spreading in Urban Areas Using Agent-Based Transportation Models

Human mobility is a key element in the understanding of epidemic spreading. Thus, correctly modeling and quantifying human mobility is critical for studying large-scale spatial transmission of infectious diseases and improving epidemic control. In this study, a large-scale agent-based transport simulation (MATSim) is linked with a generic epidemic spread model to simulate the spread of communicable diseases in an urban environment. The use of an agent-based model allows reproduction of the real-world behavior of individuals’ daily path in an urban setting and allows the capture of interactions among them, in the form of a spatial-temporal social network. This model is used to study seasonal influenza outbreaks in the metropolitan area of Zurich, Switzerland. The observations of the agent-based models are compared with results from classical SIR models. The model presented is a prototype that can be used to analyze multiple scenarios in the case of a disease spread at an urban scale, considering variations of different model parameters settings. The results of this simulation can help to improve comprehension of the disease spread dynamics and to take better steps towards the prevention and control of an epidemic.

A mathematical model of Hepatitis B transmission in Turkey

Hepatitis B infection is one of the serious viral infections that is treating the global health. Turkey has an intermediate endemicity for hepatitis B. In this study, a classical SIR model for hepatitis B virus (HBV) transmission is proposed and analyzed. Based on the available data of Republic of Turkey Ministry of Health, associated parameters are estimated and the fitted model is shown by appropriate simulations. The basic reproductive number is obtained by using the estimated parameters. Finally, we discuss the sensitivity of parameters and the effect of changes of parameters in the spread of disease.

Disease spreading on populations structured by groups

Epidemiological modeling usually relies on contact between two individuals as the method for a contagious infection spread over the population. Although for some diseases the contact is indeed between only two individuals (HIV), many contagious diseases are favored to spread inside a group of individuals. For instance, consider flu: usually infected persons spread the disease on public transport, inside a room in their jobs or their homes. All these situations are related to groups of individuals that may get a disease due to the presence of an infected individual in the group. Here we use a population model structured by groups, where individuals move inside a neighborhood forming groups, where the disease may be transmitted. The population model is based on evolutionary graph theory and modeled by probabilistic cellular automata, and the disease is modeled by the classical SIR-model. A mean-field approach is presented in terms of ordinary differential equations, and population parameters are used to analyze the disease dynamic. The major results of this paper are: (1) the number of groups that an individual belongs and the lattice temperature can estimate the disease spread strength inside the population and; (2) when infected individuals do not move, this is not enough to control the disease spreading.

A multiscale approach for spatially inhomogeneous disease dynamics

In this paper we introduce an agent-based epidemiological model that generalizes the classical SIR model by Kermack and McKendrick. We further provide a multiscale approach to the derivation of a macroscopic counterpart via the mean-field limit. The chain of equations acquired via the multiscale approach are investigated, analytically as well as numerically. The outcome of these results provide strong evidence of the models' robustness and justifies their applicability in describing disease dynamics, in particularly when mobility is involved.

State estimators for some epidemiological systems.

We consider a class of epidemiological models that includes most well-known dynamics for directly transmitted diseases, and some reduced models for indirectly transmitted diseases. We then propose a simple observer that can be applied to models in this class. The error analysis of this observer leads to a non-autonomous error equation, and a new bound for fundamental matrices is also presented. We analyse and implement this observer in two examples: the classical SIR model, and a reduced Bailey-Dietz model for vector-borne diseases. In both cases we obtain arbitrary exponential convergence of the observer. For the latter model, we also applied the observer to recover the number of susceptible using dengue infection data from a district in the city of Rio de Janeiro.

Vaccination threshold size and backward bifurcation of SIR model with state-dependent pulse control

Depending on the potential susceptible human size, we consider the state-dependent integrated infectious disease control strategies including vaccination, isolation and treatment. Correspondingly, we propose a state-dependent pulse SIR model, in which whether the control measures implemented or not depends on the threshold size of susceptible population. By defining the Poincaré map, we first investigate the existence and global stability of the semi-trivial (or disease free) periodic solution, and the threshold condition is proposed. Further, by employing bifurcation theories of the one-parameter family of maps related to the Poincaré map, we then focus on the bifurcation with respect to the key parameters. The main results reveal that backward bifurcation via transcritical bifurcation or pitchfork bifurcation can occur for all the interesting parameters including isolation rate, vaccination rate, threshold susceptible population size and birth rate. The complex relationships between the basic reproduction number of classical SIR model and the threshold condition of the model with state-dependent pulse control depict that the control strategies related to the four parameters should be carefully designed, otherwise the paradoxical effects could occur and the gains cannot make up for losses. For example, too small vaccination rate will result in an increasing of threshold condition and the number of infected population. Therefore, our results suggest that when the state-dependent feedback control strategy is implemented for infectious disease control, the effective and optimal control program should take the population dynamics, the threshold susceptible population size, vaccination and isolation or treatment rate into consideration.

Endemic SIR model in random media with applications

We consider an averaging principle for the endemic SIR model in a semi-Markov random media. Under stationary conditions of a semi- Markov media we show that the perturbed endemic SIR model converges to the classic endemic SIR model with averaged coefficients. Numerical toy examples and their interpretations are also presented for two-state Markov and semi-Markov chains. We also discuss two numerical examples involving real data: 1) Dengue Fever Disease (Indonesia and Malaysia (2009)) and 2) Cholera Outbreak in Zimbabwe (2008-2009). Novelty of the paper consists in studying of an endemic SIR model in semi-Markov random media and in implementations and interpretations of the results through numerical toy examples and discussion of numerical examples with real data.

Endemic SIR Model in Random Media with Applications

We consider an averaging principle for the endemic SIR model in a semi-Markov random media. Under stationary conditions of a semi- Markov media we show that the perturbed endemic SIR model converges to the classic endemic SIR model with averaged coefficients. Numerical toy examples and their interpretations are also presented for two-state Markov and semi-Markov chains. We also discuss two numerical examples involving real data: 1) Dengue Fever Disease (Indonesia and Malaysia (2009)) and 2) Cholera Outbreak in Zimbabwe (2008-2009). Novelty of the paper consists in studying of an endemic SIR model in semi-Markov random media and in implementations and interpretations of the results through numerical toy examples and discussion of numerical examples with real data.

Diffusion, Seeding, and the Value of Network Information

Identifying the optimal set of individuals to first receive information (‘seeds’) in a social network is a widely-studied question in many settings, such as diffusion of information, spread of microfinance programs, and adoption of new technologies. Numerous studies have proposed various network-centrality based heuristics to choose seeds in a way that is likely to boost diffusion. Here we show that, for the classic SIR model of diffusion and some of its generalizations, randomly seeding s + x individuals can prompt a larger diffusion than optimally targeting the best s individuals, for a small x. We prove our results for large classes of random networks, and verify them in several small, real-world networks. Our results identify practically relevant settings under which collecting and analyzing network data to boost diffusion is not cost-effective.

Modeling individual fear factor with optimal control in a disease-dynamic system

This paper proposes a new, information-transmission-based behavior-switch that applies the individual fear factor (IFF) to describe how information regarding current disease epidemics can cause human behavior change in a disease-dynamic system. This research is a first attempt to mathematically model how an individual's emotions influence behavior. The approach can be used to study the relationship of information dissemination (e.g., broadcasting, public health education, news media, etc.) and human behaviors during disease outbreaks. The expression of IFF and a mathematical IFF model that combines human behaviors with a classic SIR model is presented, and an optimal strategy that reduces the number of infected individuals and financial loss due to switch behaviors is proposed. In particular, model stability is analyzed and corresponding necessary conditions are determined. This novel modeling approach shows that information transmission influence individual fear, resulting in a variety of human behaviors and leading to numerous disease consequences.

Modeling dengue data from Semarang, Indonesia

Dengue fever is a viral mosquito-borne infection which in recent years has become a major international public health concern, a leading cause of illness and death in tropical and subtropical regions. Models from mathematical epidemiology, like the classical SIR-model and its variants, are used to describe the spread of dengue in a given population. Based on data of hospitalized dengue cases for the city of Semarang, Northern Java, Indonesia, we identify certain parameters in a simplified IR-model. In a second step, we connect those model parameters to available meteorological data, like precipitation.

SHIR competitive information diffusion model for online social media

Abstract In online social media, opinion divergences and differentiations generally exist as a result of individuals’ extensive participation and personalization. In this paper, a Susceptible–Hesitated–Infected–Removed (SHIR) model is proposed to study the dynamics of competitive dual information diffusion. The proposed model extends the classical SIR model by adding hesitators as a neutralized state of dual information competition. It is both hesitators and stable spreaders that facilitate information dissemination. Researching on the impacts of diffusion parameters, it is found that the final density of stiflers increases monotonically as infection rate increases and removal rate decreases. And the advantage information with larger stable transition rate takes control of whole influence of dual information. The density of disadvantage information spreaders slightly grows with the increase of its stable transition rate, while whole spreaders of dual information and the relaxation time remain almost unchanged. Moreover, simulations imply that the final result of competition is closely related to the ratio of stable transition rates of dual information. If the stable transition rates of dual information are nearly the same, a slightly reduction of the smaller one brings out a significant disadvantage in its propagation coverage. Additionally, the relationship of the ratio of final stiflers versus the ratio of stable transition rates presents power characteristic.