Academic literature on the topic 'Modèle Susceptible-Infected-Recovered (SIR)'

This article analyzes the dynamic of an extended SEIR model for the spread of COVID-19 considering a system of 7 differential equations whose stages are susceptible, exposed, infected, quarantined, recovered, dead and vaccinated. The necessary and sufficient conditions are determined for non-negativity, delimitation, existence and uniqueness of the solution of the model, local stability of the equilibrium points and the next generation matrix method. The simulations made in Python complement the qualitative analysis of the mathematical model to conclude the behavior of the virus spread over time; the information shown in this work could also be useful for the development of new prevention measures.

In the present paper, the SIR epidemiological model was used to study the behavior of the spread of the COVID-19 Pandemic in the Tacna Region. To determine the parameters of the model, the information published through social networks by the Regional Health Directorate of the Tacna Region of Peru was used, which was systematized in an EXCEL matrix and then exported to process the information in the System of Scientific Computing Mathematica. As a result, the graphs corresponding to the model referred to the Susceptible, Infected, Recovered and Deceased individuals from the COVID-19 Pandemic in the Tacna Region were obtained and then the graphs were interpreted in the time interval of the study.

Climate variables influence the occurrence, growth, and distribution of Vibrio cholerae in the aquatic environment. Together with socio-economic factors, these variables affect the incidence and intensity of cholera outbreaks. The current pandemic of cholera began in the 1960s, and millions of cholera cases are reported each year globally. Hence, cholera remains a significant health challenge, notably where human vulnerability intersects with changes in hydrological and environmental processes. Cholera outbreaks may be epidemic or endemic, the mode of which is governed by trigger and transmission components that control the outbreak and spread of the disease, respectively. Traditional cholera risk assessment models, namely compartmental susceptible-exposed-infected-recovered (SEIR) type models, have been used to determine the predictive spread of cholera through the fecal–oral route in human populations. However, these models often fail to capture modes of infection via indirect routes, such as pathogen movement in the environment and heterogeneities relevant to disease transmission. Conversely, other models that rely solely on variability of selected environmental factors (i.e., examine only triggers) have accomplished real-time outbreak prediction but fail to capture the transmission of cholera within impacted populations. Since the mode of cholera outbreaks can transition from epidemic to endemic, a comprehensive transmission model is needed to achieve timely and reliable prediction with respect to quantitative environmental risk. Here, we discuss progression of the trigger module associated with both epidemic and endemic cholera, in the context of the autochthonous aquatic nature of the causative agent of cholera, V. cholerae, as well as disease prediction.

A classic susceptible–infected–recovered (SIR) model with nonlinear state-dependent feedback control is proposed and investigated in which integrated control measures, including vaccination, treatment and isolation, are applied once the number of the susceptible population reaches a threshold level. The interventions are density dependent due to limitations on the availability of resources. The existence and global stability of the disease-free periodic solution (DFPS) are addressed, and the threshold condition is provided, which can be used to define the control reproduction number Rc for the model with state-dependent feedback control. The DFPS may also be globally stable even if the basic reproduction number R0 of the SIR model is larger than one. To show that the threshold dynamics are determined by the Rc, we employ bifurcation theories of the discrete one-parameter family of maps, which are determined by the Poincaré map of the proposed model, and the main results indicate that under certain conditions, a stable or unstable interior periodic solution could be generated through transcritical, pitchfork, and backward bifurcations. A biphasic vaccination rate (or threshold level) could result in an inverted U-shape (or U-shape) curve, which reveals some important issues related to disease control and vaccine design in bioengineering including vaccine coverage, efficiency, and vaccine production. Moreover, the nonlinear state-dependent feedback control could result in novel dynamics including various bifurcations.

AbstractMechanism-driven models based on transmission dynamics and statistic models driven by public health data are two main methods for simulating and predicting emerging infectious diseases. In this paper, we intend to combine these two methods to develop a more comprehensive model for the simulation and prediction of emerging infectious diseases. First, we combine a standard epidemic dynamic, the susceptible–exposed–infected–recovered (SEIR) model with population migration. This model can provide a biological spread process for emerging infectious diseases. Second, to determine suitable parameters for the model, we propose a data-driven approach, in which the public health data and population migration data are assembled. Moreover, an objective function is defined to minimize the error based on these data. Third, based on the proposed model, we further develop a swarm-optimizer-assisted simulation and prediction method, which contains two modules. In the first module, we use a level-based learning swarm optimizer to optimize the parameters required in the epidemic mechanism. In the second module, the optimized parameters are used to predicate the spread of emerging infectious diseases. Finally, various experiments are conducted to validate the effectiveness of the proposed model and method.

Cette thèse explore la prise de décision décentralisée dans les dynamiques épidémiques et de marketing viral en utilisant la théorie des jeux afin d'évaluer son efficacité. La thèse commence par une revue des outils mathématiques, mettant l'accent sur la théorie des graphes/jeux. Dans la suite de ce manuscrit, l'analyse de jeu épidémiologique et de compétition en marketing viral est établie. Notamment, dans le chapitre 2 où il est présenté un jeu épidémique en réseau dans lequel chaque joueur (région ou pays) cherche à trouver un compromis entre les pertes socio-économiques et sanitaires, tout en prenant en compte des contraintes telles que la disponibilité des unités de soins intensifs (USI). L'équilibre de Nash et l'équilibre de Nash généralisé sont analysés, et l'impact de la décentralisation sur l'efficacité est mesuré à l'aide de paramètres tels que le prix de l'anarchie (PoA) et le prix de la connectivité (PoC). Une application pratique du jeu à un scénario de Covid-19 est également illustrée. Le chapitre 3 étend l'analyse du chapitre 2 en incorporant la dynamique des opinions dans le contrôle décentralisé d'une épidémie en réseau. L'analyse se concentre sur l'existence et l'unicité de l'équilibre de Nash généralisé (GNE), et un algorithme pour atteindre le GNE est proposé. Les simulations identifient les scénarios où la décentralisation est acceptable en termes d'efficacité globale et soulignent l'importance de la dynamique des opinions dans les processus de prise de décision. Finalement, le chapitre 4 explore un modèle de duopole de Stackelberg dans le contexte des campagnes de marketing viral. L'objectif est de caractériser la stratégie d'allocation optimale des budgets publicitaires entre les régions pour maximiser la part de marché. Des stratégies d'équilibre sont déduites et des conditions pour un résultat de type "le gagnant rafle tout" sont établies. Les résultats théoriques sont complétés par des simulations numériques et un exemple illustrant la caractérisation de l'équilibre. Cette thèse offre des perspectives précieuses sur l'efficacité de la prise de décision décentralisée dans les dynamiques épidémiques et de marketing viral. Les résultats ont des implications pour la gestion des soins de santé, la concurrence commerciale et d'autres domaines connexes
This thesis investigates decentralized decision-making in epidemic and viral marketing dynamics. The mathematical framework of game theory is exploited to design and assess the effectiveness of decentralized strategies. The thesis begins with a review of mathematical tools, emphasizing graph theory and game theory. Chapter 2 presents a networked epidemic game where each player (region or country) seeks to implement a tradeoff between socio-economic and health looses, incorporating constraints such as intensive care unit (ICU) availability. Nash equilibrium and Generalized Nash equilibrium are analyzed, and the influence of decentralization on global efficiency is measured using metrics like the Price of Anarchy (PoA) and the Price of Connectedness (PoC). The practical application of the game to a Covid-19 scenario is illustrated. Chapter 3 extends the analysis of Chapter 2 by incorporating opinion dynamics into the decentralized control of a networked epidemic. A new game model is introduced, where players represent geographical aera balancing socio-economic and health losses; the game is built to implement features of practical interests and to possess some mathematical properties (e.g., posynomiality) which makes its analysis tractable. The analysis focuses on the existence and uniqueness of the Generalized Nash Equilibrium (GNE), and an algorithm for computing the GNE is proposed. Numerical simulations quantify the efficiency loss induced by decentralization in the presence and absence of opinion dynamics. The results identify scenarios where decentralization is acceptable in terms of global efficiency measures and highlight the importance of opinion dynamics in decision-making processes. Chapter 4 explores a Stackelberg duopoly model in the context of viral marketing campaigns. The objective is to characterize the optimal allocation strategy of advertising budgets across regions to maximize market share. A relatively simple Equilibrium strategies are derived, and conditions for a "winner takes all" outcome are established. Theoretical findings are complemented by numerical simulations and an example illustrating equilibrium characterization.This thesis offers valuable insights into the effectiveness of decentralized decision-making in the context of epidemic and viral marketing dynamics. The findings have implications for healthcare management, business competition, and related fields

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SIS (susceptible-infected-susceptible) and SIR (susceptible-infectedremoved) epidemiological models based on probabilistic cellular automaton (PCA) are used in order to simulate the temporal evolution of the number of people infected by dengue in the city of Rio de Janeiro in 2007, and to predict the cases of infection in 2008. In the PCA, three different sizes of lattices and two kinds of neighborhoods are utilized, and each time step of simulation is equivalent to one week of real time. A genetic algorithm (GA) is employed to identify the probabilities of the state transition S→I, in order to reproduce the historical series of 2007 related to this disease propagation. These probabilities depend on the number of infected neighbors. Time-varying and constant probabilities are taken into account. These models based on PCA and GA were able of satisfactorily fitting the data from 2007 and making a good prediction for 2008 (mainly about the total number of cases registered during 2008).
Usam-se modelos epidemiológicos SIS (suscetível-infectado-suscetível) e SIR (suscetível-infectado-removido) baseados em autômato celular probabilista (ACP) a fim de simular a evolução temporal do número de pessoas infectadas por dengue, na cidade do Rio de Janeiro em 2007, e de prever os casos de infecção em 2008. No ACP, utilizam-se reticulados de três tamanhos diferentes e dois tipos de vizinhanças, e cada passo de tempo da simulação equivale a uma semana de tempo real. Emprega-se um algoritmo genético (AG) para identificar os valores das probabilidades da transição de estados S→I, de modo a reproduzir a série histórica de 2007 relacionada à propagação dessa doença. Essas probabilidades dependem do número de vizinhos infectados. Probabilidades variantes e invariantes no tempo são consideradas. Esses modelos baseados em ACP e AG foram capazes de fazer um ajuste satisfatório dos dados de 2007 e de fornecerem uma boa previsão para 2008, (principalmente no que diz respeito ao número total de casos registrados em 2008).

Social networks play an important role in disseminating a piece of information in a population. Companies advertising a newly launched product, movie promotion, political campaigns, social awareness campaigns by governments, charity campaigns by NGOs and crowd funding campaigns by entrepreneurs are a few examples where an entity is interested in disseminating a piece of information in a target population, possibly under resource constraints. In this thesis we model information diffusion in a population using various epidemic models and study optimal campaigning strategies to maximize the reach of information. In the different problems considered in this thesis, information epidemics are modeled as the Susceptible-Infected, Susceptible-Infected-Susceptible, Susceptible-Infected-Recovered and Maki Thompson epidemic processes; however, we modify the models to incorporate the intervention made by the campaigner to enhance information propagation. Direct recruitment of individuals as spreaders and providing word-of-mouth incentives to the spreaders are considered as two intervention strategies (controls) to enhance the speed of information propagation. These controls can be implemented by placing advertisements in the mass media, announcing referral/cash back rewards for introducing friends to a product or service being advertised etc. In the different problems considered in this thesis, social contacts are modeled with varying levels of complexity---population is homogeneously mixed or follows heterogeneous mixing. The solutions to the problems which consider homogeneous mixing of individuals identify the most important periods in the campaign duration which should be allocated more resources to maximize the reach of the message, depending on the system parameters of the epidemic model (e.g., epidemics with high and low virulence). When a heterogeneous model is considered, apart from this, the solution identifies the important classes of individuals which should be allocated more resources depending upon the network considered (e.g. Erdos-Renyi, scale-free) and model parameters. These classes may be carved out based on various centrality measures in the network. If multiple strategies are available for campaigning, the solution also identifies the relative importance of the strategies depending on the network type. We study variants of the optimal campaigning problem where we optimize different objective functions. For some of the formulated problems, we discuss the existence and uniqueness of the solution. Sometimes our formulations call for novel techniques to prove the existence of a solution.

Social networks play an important role in disseminating a piece of information in a population. Companies advertising a newly launched product, movie promotion, political campaigns, social awareness campaigns by governments, charity campaigns by NGOs and crowd funding campaigns by entrepreneurs are a few examples where an entity is interested in disseminating a piece of information in a target population, possibly under resource constraints. In this thesis we model information diffusion in a population using various epidemic models and study optimal campaigning strategies to maximize the reach of information. In the different problems considered in this thesis, information epidemics are modeled as the Susceptible-Infected, Susceptible-Infected-Susceptible, Susceptible-Infected-Recovered and Maki Thompson epidemic processes; however, we modify the models to incorporate the intervention made by the campaigner to enhance information propagation. Direct recruitment of individuals as spreaders and providing word-of-mouth incentives to the spreaders are considered as two intervention strategies (controls) to enhance the speed of information propagation. These controls can be implemented by placing advertisements in the mass media, announcing referral/cash back rewards for introducing friends to a product or service being advertised etc. In the different problems considered in this thesis, social contacts are modeled with varying levels of complexity---population is homogeneously mixed or follows heterogeneous mixing. The solutions to the problems which consider homogeneous mixing of individuals identify the most important periods in the campaign duration which should be allocated more resources to maximize the reach of the message, depending on the system parameters of the epidemic model (e.g., epidemics with high and low virulence). When a heterogeneous model is considered, apart from this, the solution identifies the important classes of individuals which should be allocated more resources depending upon the network considered (e.g. Erdos-Renyi, scale-free) and model parameters. These classes may be carved out based on various centrality measures in the network. If multiple strategies are available for campaigning, the solution also identifies the relative importance of the strategies depending on the network type. We study variants of the optimal campaigning problem where we optimize different objective functions. For some of the formulated problems, we discuss the existence and uniqueness of the solution. Sometimes our formulations call for novel techniques to prove the existence of a solution.

Epidemiology is the population dynamics of host–parasite systems. The spread of an infective disease is analysed with several tools. The SIR model (susceptible, infected, recovered hosts) is a standard model, with the basic reproductive number (R ₀) as a characteristic. Diseases, in general, spread if R ₀ > 1, which suggests a threshold size for host populations, and also for endemic maintenance or periodic outbreaks. Furthermore, spatial heterogeneity or the distribution of infections among hosts affects an epidemic. Individual-based models can follow the fate of infections more closely. Network analysis provides insights into transmission and contact rates. Models also describe the epidemics of vectored diseases, or of macroparasitic infections. Molecular epidemiology uses genetic markers or genomes to follow the spread of an infectious disease; phylodynamics reconstructs transmission chains, especially for viral diseases. Immunoepidemiology studies how immune defences affect an epidemic and identifies immunological markers for the study of infectious disease dynamics.

COVID-19 pandemic was identified in Wuhan, China in 2019 and has spread at a tremendous rate affecting all countries over the world. Understanding the spreading disease is crucial; hence, the dynamic behaviour of the disease can be predicted. This paper is aimed to model the COVID-19 outbreak by extending the deterministic susceptible-infected-recovered-death (DSIRD) into a stochastic SIRD (SSIRD) model. Infectious rate parameter of the DSIRD model is perturbed with Brownian motion to reflect the uncertainty of the COVID-19 outbreak. Fourth order stochastic Runge-Kutta (SRK4) method is used to simulate the model. Parameter estimation is estimated using the Markov Chain Monte Carlo (MCMC) method. The simulated results for three ASEAN countries of Malaysia, Indonesia and Singapore indicate that SSIRD model is consistent with the infected COVID-19 data;hence, shows the model is adequate in explaining the behaviour of the infectious disease.

In this chapter, we present a mathematical framework that provides a new insight for understanding the spread of traffic congestions in an urban network system. In particular, we consider a dynamical model, based on the well-known susceptible-infected-recovered (SIR) model from mathematical epidemiology, with small random perturbations, that describes the process of traffic congestion propagation and dissipation in an urban network system. Here, we provide the asymptotic probability estimate based on the Freidlin-Wentzell theory of large deviations for certain rare events that are difficult to observe in the simulation of urban traffic network dynamics. Moreover, the framework provides a computational algorithm for constructing efficient importance sampling estimators for rare event simulations of certain events associated with the spread of traffic congestions in the dynamics of the traffic network.

The COVID-19 pandemic has caused health system collapse and led governments to implement non-pharmaceutical interventions, such as wearing a mask and social distancing. In this study, the SEIR (susceptible-exposed-infected-recovered) model is proposed for assessing the effects of social distancing and wearing a mask on the prediction of COVID-19 transmission dynamic in Minas Gerais – Brazil. This work presents a theoretical-computational study and the mathematical modeling simulations of COVID-19 transmission dynamics. The model describes eight population groups: susceptible, confined, exposed, asymptomatic, symptomatic, hospitalized, recovered, and dead. The mask-wearing is inferred by the following parameters: mask aerosol reduction (M_red ), mask availability (M_ava), and proper mask-wearing (M_cov). Different scenarios are simulated for evaluating the effect of the parameters on the pandemic evolution. Simulations demonstrate a reduction of around 99% compared to the no-mask-wearing scenario when masks are available for 80% of the population. Professional masks, such as N95 and FFP2 (M_red=97%), reduce by 98,9% of the number of deaths. The proper mask-wearing shows a significant impact on pandemic development, by reducing considerably M_cov it could even overcome the total number of deaths and infections than those in a no-mask-wearing scenario, if the social distancing measures were not intensified. Wearing a mask is extremely efficient and required in the fight against the COVID-19 pandemic. A combination of social distancing and wearing a mask, if properly performed, allows controlling the pandemic more efficiently, minimizing the total and the daily number of deaths and infections, and avoiding a greater health system overload.

Compartmental models have proven to be useful in various forms of epidemics: Ebola, dengue fever, swine fever, flu, avian flu, etc. Such models have experienced a great development especially because of the multitude of practical applications, but also due to the many possibilities of software implementation. Although there are a few software tools, for example, EpiModel, they have limitations in modelling and use of language employed in the simulation. Measles can be described as a model with 4 compartments, SEIR, but for a better knowledge of all geographic implications in terms of vaccination, we propose a MSEIR model (S = susceptible, I = infected, R = recovered, vaccinated, E = exposed, M = maternally immune). The educational software created in MATLAB, can generate a graphical system, with specific interactions between compartments, starting with initial values set by the user as a result of specific epidemiological studies. After defining the compartments and their links, a generator will build the system of equations which models the compartmental system, solvable through numerical methods specific to measles. Optionally, the user will be able to view the system of differential equations using symbolic calculation. The solutions will be displayed graphically and will allow the user to compare the epidemic evolution with or without vaccine, as well as optimal policy for vaccination using the Jacobian matrix for determination of the endemic equilibrium. Graphical user interface is intuitive, educational, with the possibility of amending the transfer rate between compartments, parameters and initial values. The reproduction number R0 can be defined to be used in order to analyse the influence of it for fit the model with experimental data. The main novelty in this paper is composed mainly by a graphical interface an easy-to-use by student and researcher for in create abilities to use a compartmental model and at the same time, able to provide sufficient arguments in the decision making process and health policies. The other innovative approach is the possibility to define by user a symbolic equation for a compartmental model meanwhile the too will translate in mathematical one that can be solved by differential equation using numerical methods.