Classical SIR Model Research Articles

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.

Effective Approaches to Control the Epidemic Based on the Improved SIR Model

<p class="zhengwen">In this paper, we have established a SECADI model on the basis of the traditional epidemic model and under the consideration of factors such as the spread of the disease, the quantity of the medicine in need, the medicine production speedetc. We have improved the crowd classification standard and the spread styledifferential equation model in classical SIR model. We distinguished the crowd into six categories, including the susceptible, the exposed, the curable, the advanced, the dead and the immune, and we established integrated transformation relationships between them after taking control measures through qualitative and quantitative method, and then derive the adequate epidemic differential equation model before taking controls. We applied the method of computer simulation to solve the model, worked out uncertain parameters with the method of parameter identification, and we verified the validity and accuracy of the SECADI model. Meanwhile, we calculated with the actual data of Ebola in the epidemic area in WesternAfrica, simulated the evolution of the epidemic, analyze and offered effective approaches to control the epidemic situation. We further discussed development directions of this model in the end.</p>

Stochastic Modelling of the Spatial Spread of Influenza in Germany

In geographical epidemiology, disease counts are typically available in discrete spatial units and at discrete time-points. For example, surveillance data on infectious diseases usually consists of weekly counts of new infections in pre-defined geographical areas. Similarly, but on a different timescale, cancer registries typically report yearly incidence or mortality counts in administrative regions.A major methodological challenge lies in building realistic models for spacetime interactions on discrete irregular spatial graphs. In this paper we will discuss an observation-driven approach, where past observed counts in neighboring areas enter directly as explanatory variables, in contrast to the parameterdriven approach through latent Gaussian Markov random fields (Rue and Held, 2005) with spatio-temporal structure. The main focus will lie on the demonstration of the spread of influenza in Germany, obtained through the design and simulation of a spatial extension of the classical SIR model (Hufnagel et al., 2004).

Epidemic characteristics of two classic models and the dependence on the initial conditions.

The epidemic characteristics, including the epidemic final size, peak, and turning point, of two classical SIR models with disease-induced death are investigated when a small initial value of the infective population is released. The models have mass-action (i.e. bilinear), or density dependent (i.e. standard) incidence, respectively. For the two models, the conditions that determining whether the related epidemic characteristics of an epidemic outbreak appear are explicitly determine by rigorous mathematical analysis. The dependence of the epidemic final size on the initial values of the infective class is demonstrated. The peak, turning point (if it exists) and the corresponding time are found. The obtained results suggest that their basic reproduction numbers are one factor determining the epidemic characteristics, but not the only one. The characteristics of the two models depend on the initial values and proportions of various compartments as well. At last, the similarities and differences of the epidemic characteristics between the two models are discussed.

Modeling the Effect of Human Mobility on Dengue Transmission

Dengue is one of the most prevalent viruses transmitted by mosquitoes where increasing incidence and severity claim severe social burden. This virus is common throughout the tropics and subtropics. Dengue is a virus which propagated during the day and thus the mobility of humans can cause it to spread quickly. In this paper, we introduce mobility of humans between two neighboring areas into a mathematical model for the transmission of dengue. Dengue transmission is modeled using the classical SIR model. Simulations have been carried under four cases to compare the impact of human mobility to propagate the dengue disease. These cases are based on existence of dengue and human mobility direction regarding neighboring area. Numerical simulations are carried out using Matlab routine ode 45.

A social computing approach to rumour spreading with consideration of illusory truth effect and the latency reverse phenomenon

Illusory truth effect impacts the process of rumour spreading on the internet. A cell automata model with consideration of illusory truth effect and the social computing approach is presented in this paper. Firstly, the formal description of illusory truth effect is given; illusory truth effect factor is used to describe the intension of it. Then, we extend the classic SIR model and build the cell automata model. Comparative study with the real data has proved the validity of this model. Our results show that the illusory truth effect factor has significantly impact on the rumour spreading process, the latency reverse phenomenon and spread avalanche phenomenon is observed in the rumour spreading process, which enhance the range and intension of the rumour spreading. Using the latency reverse phenomenon, the avalanche spread of rumour can be predicted so that we can intervene in time. The results are meaningful to the theory and practice of rumour spreading management.

改进的微博传播模型的影响因素分析

微博作为社会网络信息传播的热门平台，对舆论传播与监督具有重要作用。本文基于经典SIR 传播模型，以微博信息在易传播者、传播者和沉默者之间的状态转换为基础，建立了基于认证用户和绝对沉默者的微博网络传播模型，并通过仿真分析了微博传播的影响因素。研究表明认证用户节点和普通节点的传播率越高，停止传播率越低或者绝对沉默率越高，则微博信息传播达到稳定状态越快，对实际控制社交平台的信息传播具有一定的理论分析依据和实践指导意义。 Microblog is a popular platform for the information dissemination of social networks, which plays an important role for communication and supervision of public voice. Based on the classic SIR model and the process of microblog information dissemination on the state transformation between user nodes, including easy disseminators, disseminators and silencers, the improved microblog dissemination model with certified users and absolute silencers was built. The factors affecting the spread of microblog information were analyzed and simulated. Researches show that the higher the transmission rate of authenticated users and the normal nodes, the lower the stop transmission rate, and the higher the rate of absolute silence, the sooner time to reach steady state of the microblog information dissemination. It has some theoretical basis and practical significance for actually controlling information dissemination of the popular social platform.

埃博拉疫情控制模型

本模型旨在模拟埃博拉病毒爆发后，在相关地区现有的防护和医疗条件下，模拟埃博拉疫情传播情况，并对病毒的传播趋势进行预测，为相关决策部门提供一种询证决策依据。选取传染病SIR动力学模型。首先，我们构造的埃博拉病毒疫情的SIR-L模型是基于经典的SIR传染病模型和logistic模型，来模拟和预测埃博拉病毒的蔓延趋势，然后我们预测得出未来一周的新增病例，几内亚、利比里亚和塞拉利昂分别是88例、28例和238例。其次，通过分析我们得到的预测数据，结合各个国家人口规模和地理位置，根据重心法和灰色系统理论，选择出各国的最佳配送中心。里弗塞斯城、博城、玛木分别为利比里亚，塞拉利昂和几内亚的配送中心。我们选择疫情最严重的塞拉利昂作为生产国，博城作为生产中心。最后，基于最优配送中心和生产中心，参考现代一些先进的物流系统和现代企业物流系统运营经验，我们设计了应急物资分配三级模型。即以生产中心作为网络第一级，三个国家的配送中心作为网络第二级节点，各国中每个县的城镇中心作为第三级节点。我们以埃博拉疫情初期配送时间最短，疫情后期救援体系成本最低为原则，设计了物资分配网络体系。 This paper aims to analyze the cases of Ebola in West Africa, forecast the future trend of the spread of the virus, and provide some consulting opinions for related departments to make decisions. Firstly, we construct the SIR-L model for Ebola epidemics based on the classical infectious diseases SIR model and the Logistic model to simulate and predict the spread trends of the virus. We get that the Ebola has crossed the peak period in the three main countries, but the cases of the Ebola are still at a high level. Then we forecast the new cases in the next week: 88, 28 and 238 cases for Guinea, Liberia, and Sierra Leone, respectively. Secondly, by analyzing the data that we forecast, combing with the population size and geographical location of each country, a dynamical model is established based on gravity center method and gray theory, which are utilized to select the optimal distribution center. It reveals that River Cess, Bo, Mamou are the optimal distribution centers of Liberia, Sierra Leone, and Guinea, respectively. Sierra Leone, the hardest-hit area, is se-lected as the production country, and Bo is selected as the production center. Finally, based on the optimal distribution center and production center, we develop the distribution model of emergency supplies in three layers, which refers to some of advanced sophisticated logistics systems and operational experiences of modern enterprise logistics systems. That is to say, the production center is considered as the first layer of the network; the distribution center of the three countries is the second layer; and the town center of each country’s county is the third layer. We design a distribution network system based on the principle that the delivery time of the Ebola is the shortest in the beginning, and the cost of the rescue system is the lowest in the later period.

Optimal isolation strategies of emerging infectious diseases with limited resources

A classical deterministic SIR model is modified to take into account of limited resources for diagnostic confirmation/medical isolation. We show that this modification leads to four different scenarios (instead of three scenarios in comparison with the SIR model) for optimal isolation strategies, and obtain analytic solutions for the optimal control problem that minimize the outbreak size under the assumption of limited resources for isolation. These solutions and their corresponding optimal control policies are derived explicitly in terms of initial conditions, model parameters and resources for isolation (such as the number of intensive care units). With sufficient resources, the optimal control strategy is the normal Bang-Bang control. However, with limited resources the optimal control strategy requires to switch to time-variant isolation at an optimal rate proportional to the ratio of isolated cases over the entire infected population once the maximum capacity is reached.

Endemic SIR Model in Random Media

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. Novelty of the paper lies in the study of an endemic SIR model in semi-Markov random media.

MODELING THE IMPACT OF REHABILITATION, AMELIORATION AND RELAPSE ON THE PREVALENCE OF DRUG EPIDEMICS

Substance abuse remains a global menace in spite of recurrent warnings, seizures, social and pharmacological effects associated with addiction to drugs. In this paper, we use a mathematical model which is a combination of the classical SIS and SIR models to investigate the dynamics of substance abuse. Initiation into drug use is based on contact of those at risk (the susceptible population) with drug users at different levels of drug use. We evaluate the threshold number and use it to analyze the model. We show that when this threshold number is less than unity, the drug-free steady state is globally asymptotically stable and when this threshold number is greater than unity the drug-persistent steady state is also globally stable. The impact of amelioration, rehabilitation and re-initiation on drug epidemics is investigated. Amelioration in presence of quitting for light users is observed to reduce the prevalence of substance abuse and this is supported by numerical simulations. The results show that both prevention and treatment/rehabilitation are necessary strategies for reduction of drug epidemics. Our recommendation is that preventive strategies should be directed toward reducing the contact rate and treatment should be combined with psychotherapy to accelerate quitting and reduce re-initiation.

Mixed SI (R) epidemic dynamics in random graphs with general degree distributions

Analytical description of disease propagation on random networks has become one of the most productive fields in recent years, yet more complex contact patterns and dynamics have been resorted to numerical study. In this paper, an epidemic model is defined in which each individual, once infected, has chances to recover from infection at certain rate. The chance is represented by a parameter q∈[0,1]. This model can be viewed as an interpolation between classical SI model (q=0) and SIR model (q=1). We develop a low-dimensional system of non-linear ordinary differential equations to model the mixed susceptible-infected (-recovered) SI (R) epidemics on random network with general degree distributions. Both the final size of epidemics and the time-dependent behaviors are derived within this simple framework. In addition, we present the exact transmissibility and the epidemic threshold for this model.

How much complexity is needed to describe the fluctuations observed in dengue hemorrhagic fever incidence data?

Different extensions of the classical single-strain SIR model for the host population, motivated by modeling dengue fever epidemiology, have reported a rich dynamic structure including deterministic chaos which was able to mimic the large fluctuations of disease incidences. A comparison between the basic two-strain dengue model, which already captures differences between primary and secondary infections including temporary cross-immunity, with the four-strain dengue model, that introduces the idea of competition of multiple strains in dengue epidemics shows that the difference between first and secondary infections drives the rich dynamics more than the detailed number of strains to be considered in the model structure. Chaotic dynamics were found to happen in the same parameter region of interest, for both the two and the four-strain models, able to describe the fluctuations observed in empirical data and shows a qualitatively good agreement between empirical data and model simulation. The predictability of the system does not change significantly when considering two or four strains, i.e. both models present a positive dominant Lyapunov exponent giving approximately the same prediction horizon in time series. Since the law of parsimony favors the simplest of two competing models, the two-strain model would be the better candidate to be analyzed, as well the best option for estimating all initial conditions and the few model parameters based on the available incidence data.

Towards uncertainty quantification and inference in the stochastic SIR epidemic model

In this paper we address the problem of estimating the parameters of Markov jump processes modeling epidemics and introduce a novel method to conduct inference when data consists on partial observations in one of the state variables. We take the classical stochastic SIR model as a case study. Using the inverse-size expansion of van Kampen we obtain approximations for the first and second moments of the state variables. These approximate moments are in turn matched to the moments of an inputed Generic Discrete distribution aimed at generating an approximate likelihood that is valid both for low count or high count data. We conduct a full Bayesian inference using informative priors. Estimations and predictions are obtained both in a synthetic data scenario and in two Dengue fever case studies.

Bifurcation analysis of a family of multi-strain epidemiology models

In this paper we use bifurcation theory to analyze three multi-strain compartmental models, motivated by dengue fever epidemiology. The models are extensions of the classical SIR model where the notion of two different strains is needed to describe primary and secondary infections. Ferguson et al. formulate and analyzed in Ferguson et al. (1999) [1] a two-strain model with Antibody Dependent Enhancement (ade), where the pre-existing antibodies to previous dengue infection cannot neutralize but rather enhance the new infection. No cross-immunity is assumed and therefore co-infection is possible and moreover individuals can belong to multiple compartments. In the modeling approach proposed by Billings et al. in (2007) [2] cross-infection is not possible as long as an individual is primarily infected. As a result all compartments are distinct. In the third model Aguiar et al. (2011) [9] starting from the Billings et al. model, temporary cross-immunity is introduced by additional compartments for the recovered from the primary infection. The study of the different models gives insight into how different long-term dynamic behavior originate. We use the same parameter set for all models, except the duration of the cross-immunity period in the last model which will be a bifurcation parameter. Another bifurcation parameter is the ade factor. Besides endemic equilibria and periodic solutions there is also chaotic behavior originating via different routes. Numerical bifurcation analysis, Lyapunov exponent calculation and simulation techniques (quantitative results) as well as symbolic analysis (qualitative results) are used to unravel the different types of long-term dynamics. When the cross-immunity period is moderately long or short the predictions show, besides similar chaotic dynamics as in the two models without temporal cross-immunity, a new chaotic attractor with a different origin.

Global analysis on delay epidemiological dynamic models with nonlinear incidence

In this paper, we derive and study the classical SIR, SIS, SEIR and SEI models of epidemiological dynamics with time delays and a general incidence rate. By constructing Lyapunov functionals, the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium is shown. This analysis extends and develops further our previous results and can be applied to the other biological dynamics, including such as single species population delay models and chemostat models with delay response.

A new delay-SIR model for pulse vaccination

This paper introduces a new model for disease outbreaks. This model describes the disease evolution through a system of nonlinear differential equations with distributed-delay. The main difference between classical SIR-model resides in the fact that the recovery rate of the population is expressed as a distributed-delay term modeling the time spent being sick by infected people. This model is identified to fit realistic measurements which shows the effectiveness of the model. Finally, we develop an optimal campaign vaccination strategy based on recent results on the impulsive control of time-delay systems.

Generalization of the Kermack-McKendrick SIR Model to a Patchy Environment for a Disease with Latency

In this paper, with the assumptions that an infectious disease has a fixed latent pe- riod in a population and the latent individuals of the population may disperse, we reformulate an SIR model for the population living in two patches (cities, towns, or countries etc.), which is a generalization of the classic Kermack-McKendrick SIR model. The model is given by a system of delay differential equations with a fixed delay accounting for the latency and non-local terms caused by the mobility of the individuals during the latent period. We analytically show that the model preserves some properties that the classic Kermack-McKendrick SIR model possesses: the disease always dies out, leaving a certain portion of the susceptible population untouched (called final sizes). Although we can not determine the two final sizes, we are able to show that the ratio of the final sizes in the two patches is totally determined by the ratio of the dispersion rates of the susceptible individuals between the two patches. We also explore numerically the patterns by which the disease dies out, and find that the new model may have very rich patterns for the disease to die out. In particular, it allows multiple outbreaks of the disease before it goes to extinction, strongly contrasting to the classic Kermack-McKendrick SIR model.

Generalization of the Kermack-McKendrick SIR Model to a Patchy Environment for a Disease with Latency

In this paper, with the assumptions that an infectious disease has a fixed latent period in a population and the latent individuals of the population may disperse, we reformulate an SIR model for the population living in two patches (cities, towns, or countries etc.), which is a generalization of the classic Kermack-McKendrick SIR model. The model is given by a system of delay differential equations with a fixed delay accounting for the latency and non-local terms caused by the mobility of the individuals during the latent period. We analytically show that the model preserves some properties that the classic Kermack-McKendrick SIR model possesses: the disease always dies out, leaving a certain portion of the susceptible population untouched (called final sizes). Although we can not determine the two final sizes, we are able to show that the ratio of the final sizes in the two patches is totally determined by the ratio of the dispersion rates of the susceptible individuals between the two patches. We also explore numerically the patterns by which the disease dies out, and find that the new model may have very rich patterns for the disease to die out. In particular, it allows multiple outbreaks of the disease before it goes to extinction, strongly contrasting to the classic Kermack-McKendrick SIR model.

An SEIQR model for childhood diseases.

It has been shown that the inclusion of an isolated class in the classical SIR model for childhood diseases can be responsible for self-sustained oscillations. Hence, the recurrent outbreaks of such diseases can be caused by autonomous, deterministic factors. We extend the model to include a latent class (i.e. individuals who are infected with the disease, but are not yet able to pass the disease to others) and study the resulting dynamics. The existence of Hopf bifurcations is shown for the model, as well as a homoclinic bifurcation for a perturbation to the model. For historical data on scarlet fever in England, our model agrees with the epidemiological data much more closely than the model without the latent class. For other childhood diseases, our model suggests that isolation is unlikely to be a major factor in sustained oscillations.