190 research outputs found

    A survey on Lyapunov functions for epidemic compartmental models

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    In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, and provide a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey–predator or rumor spreading

    Key Epidemic Parameters of the SIRV Model Determined from Past COVID-19 Mutant Waves

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    Monitored infection and vaccination rates during past past waves of the coronavirus are used to infer a posteriori two-key parameter of the SIRV epidemic model, namely, the real-time variation in (i) the ratio of recovery to infection rate and (ii) the ratio of vaccination to infection rate. We demonstrate that using the classical SIR model, the ratio between recovery and infection rates tends to overestimate the true ratio, which is of relevance in predicting the dynamics of an epidemic in the presence of vaccinations

    Dynamical analysis of a fractional SIR model with treatment and quarantine

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    We propose a fractional SIR model with treatment and quarantine policies, whose dynamics is described by the Caputo fractional derivative. Local stability of the equilibrium points is studied, and the threshold value R0 is found. Finally, some numerical simulations are presented for different values of the parameters.publishe

    Global dynamics for a class of reaction–diffusion multigroup SIR epidemic models with time fractional-order derivatives

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    This paper investigates the global dynamics for a class of multigroup SIR epidemic model with time fractional-order derivatives and reaction–diffusion. The fractional order considered in this paper is in (0; 1], which the propagation speed of this process is slower than Brownian motion leading to anomalous subdiffusion. Furthermore, the generalized incidence function is considered so that the data itself can flexibly determine the functional form of incidence rates in practice. Firstly, the existence, nonnegativity, and ultimate boundedness of the solution for the proposed system are studied. Moreover, the basic reproduction number R0 is calculated and shown as a threshold: the disease-free equilibrium point of the proposed system is globally asymptotically stable when R0 ≤ 1, while when R0 > 1, the proposed system is uniformly persistent, and the endemic equilibrium point is globally asymptotically stable. Finally, the theoretical results are verified by numerical simulation

    Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

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    In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations (FDEs) as approximations of some type of fractional nonlinear birth and death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While ordinary differential equations (ODEs) are commonly used to model epidemics, FDEs are more flexible in fitting empirical data and theoretically offer improved model predictions. The question arises whether, in practice, the benefits of using FDEs over ODEs outweigh the added computational complexities. While important differences in transient dynamics were observed, the FDE only outperformed the ODE in one of out three data sets. In general, FDE modeling approaches may be worth it in situations with large refined data sets and good numerical algorithms

    On fuzzy and crisp solutions of a novel fractional pandemic model

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    Understanding disease dynamics is crucial for accurately predicting and effectively managing epidemic outbreaks. Mathematical modeling serves as an essential tool in such understanding. This study introduces an advanced susceptible, infected, recovered, and dead (SIRD) model that uniquely considers the evolution of the death parameter, alongside the susceptibility and infection states. This model accommodates the varying environmental factors influencing disease susceptibility. Moreover, our SIRD model introduces fractional changes in death cases, which adds a novel dimension to the traditional counts of susceptible and infected individuals. Given the model’s complexity, we employ the Laplace-Adomian decomposition method. The method allows us to explore various scenarios, including non-fuzzy non-fractional, non-fuzzy fractional, and fuzzy fractional cases. Our methodology enables us to determine the model’s equilibrium positions, compute the basic reproduction number, confirm stability, and provide computational simulations. Our study offers insightful understanding into the dynamics of pandemic diseases and underscores the critical role that mathematical modeling plays in devising effective public health strategies. The ultimate goal is to improve disease management through precise predictions of disease behavior and spread.ANCD -Agenția Națională pentru Cercetare și Dezvoltare(UIDP/00013/2020

    On a Controlled Se(Is)(Ih)(Iicu)AR Epidemic Model with Output Controllability Issues to Satisfy Hospital Constraints on Hospitalized Patients

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    An epidemic model, the so-called SE(Is)(Ih)(Iicu)AR epidemic model, is proposed which splits the infectious subpopulation of the classical SEIR (Susceptible-Exposed-Infectious-Recovered) model into four subpopulations, namely asymptomatic infectious and three categories of symptomatic infectious, namely slight infectious, non-intensive care infectious, and intensive care hospitalized infectious. The exposed subpopulation has four different transitions to each one of the four kinds of infectious subpopulations governed under eventually different proportionality parameters. The performed research relies on the problem of satisfying prescribed hospitalization constraints related to the number of patients via control interventions. There are four potential available controls which can be manipulated, namely the vaccination of the susceptible individuals, the treatment of the non-intensive care unit hospitalized patients, the treatment of the hospitalized patients at the intensive care unit, and the transmission rate which can be eventually updated via public interventions such as isolation of the infectious, rules of groups meetings, use of face masks, decrees of partial or total quarantines, and others. The patients staying at the non-intensive care unit and those staying at the intensive care unit are eventually, but not necessarily, managed as two different hospitalized subpopulations. The controls are designed based on output controllability issues in the sense that the levels of hospital admissions are constrained via prescribed maximum levels and the measurable outputs are defined by the hospitalized patients either under a joint consideration of the sum of both subpopulations or separately. In this second case, it is possible to target any of the two hospitalized subpopulations only or both of them considered as two different components of the output. Different algorithms are given to design the controls which guarantee, if possible, that the prescribed hospitalization constraints hold. If this were not possible, because the levels of serious infection are too high according to the hospital availability means, then the constraints are revised and modified accordingly so that the amended ones could be satisfied by a set of controls. The algorithms are tested through numerically worked examples under disease parameterizations of COVID-19.This research received funding from the Spanish Institute of Health Carlos III through Grant COV 20/01213, the Spanish Government and the European Commission through Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and the Basque Government for Grant IT1207-19
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