Global stability of an SIR epidemic model with information dependent vaccination

Buonomo, Bruno; d’Onofrio, Alberto; Lacitignola, Deborah

doi:10.1016/j.mbs.2008.07.011

Cited by 216 publications

(156 citation statements)

References 35 publications

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“…The global stability of EE, suggested by numerical sumulations, is nontrivial and it has been investigated in [456] by means of the Li-Muldowney geometric theorem, which extends the Dulac-Bendixon theorem.…”

Section: Memory-triggered Oscillationsmentioning

confidence: 99%

Statistical physics of vaccination

et al. 2016

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Historically, infectious diseases caused considerable damage to human societies, and they continue to do so today. To help reduce their impact, mathematical models of disease transmission have been studied to help understand disease dynamics and inform prevention strategies. Vaccination-one of the most important preventive measures of modern times-is of great interest both theoretically and empirically. And in contrast to traditional approaches, recent research increasingly explores the pivotal implications of individual behavior and heterogeneous contact patterns in populations. Our report reviews the developmental arc of theoretical epidemiology with emphasis on vaccination, as it led from classical models assuming homogeneously mixing (mean-field) populations and ignoring human behavior, to recent models that account for behavioral feedback and/or population spatial/social structure. Many of the methods used originated in statistical physics, such as lattice and network models, and their associated analytical frameworks. Similarly, the feedback loop between vaccinating behavior and disease propagation forms a coupled nonlinear system with analogs in physics. We also review the new paradigm of digital epidemiology, wherein sources of digital data such as online social media are mined for high-resolution information on epidemiologically relevant individual behavior. Armed with the tools and concepts of statistical physics, and further assisted by new sources of digital data, models that capture nonlinear interactions between behavior and disease dynamics offer a novel way of modeling real-world phenomena, and can help improve health outcomes. We conclude the review by discussing open problems in the field and promising directions for future research.

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Section: Memory-triggered Oscillationsmentioning

confidence: 99%

Statistical physics of vaccination

et al. 2016

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“…For detailed calculations one can see Chakraborty et al [34], Li and Muldowney [35], Bunomo et al [36], Martin [37] etc.…”

Section: Global Stabilitymentioning

confidence: 99%

An ecological perspective on marine reserves in prey–predator dynamics

Chakraborty

Das²,

Kar

2013

J Biol Phys

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This paper describes a prey-predator type fishery model with prey dispersal in a two-patch environment, one of which is a free fishing zone and other is a protected zone. The existence of possible steady states, along with their local stability, is discussed. A geometric approach is used to derive the sufficient conditions for global stability of the system at the positive equilibrium. Relative size of the reserve is considered as control in order to study optimal sustainable yield policy. Subsequently, the optimal system is derived and then solved numerically using an iterative method with Runge-Kutta fourth-order scheme. Numerical simulations are carried out to illustrate the importance of marine reserve in fisheries management. It is noted that the marine protected area enables us to protect and restore multi-species ecosystem. The results illustrate that dynamics of the system is extremely interesting if simultaneous effects of a regulatory mechanism like marine reserve is coupled with harvesting effort. It is observed that the migration of the resource, from protected area to unprotected area and vice versa, is playing an important role towards the standing stock assessment in both the areas which ultimately control the harvesting efficiency and enhance the fishing stock up to some extent.

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“…As a way to merge the significant features of the foundational and more advanced SIR and SIRS models [13,14] along with the SEIR and SEIRS models [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], the focus of the work is to develop and implement an extension of the SEIRS model. Fundamentally, the new SEIRS model is now a more advanced generalization of the previous models and incorporates vital dynamics with unequal birth and death rates, vaccinations for both newborns and non-newborns, and temporary immunity for describing the spread of infectious diseases.…”

Section: Introductionmentioning

confidence: 99%

Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity

Trawicki

2017

Mathematics

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Abstract:In this paper, the author proposes a new SEIRS model that generalizes several classical deterministic epidemic models (e.g., SIR and SIS and SEIR and SEIRS) involving the relationships between the susceptible S, exposed E, infected I, and recovered R individuals for understanding the proliferation of infectious diseases. As a way to incorporate the most important features of the previous models under the assumption of homogeneous mixing (mass-action principle) of the individuals in the population N, the SEIRS model utilizes vital dynamics with unequal birth and death rates, vaccinations for newborns and non-newborns, and temporary immunity. In order to determine the equilibrium points, namely the disease-free and endemic equilibrium points, and study their local stability behaviors, the SEIRS model is rescaled with the total time-varying population and analyzed according to its epidemic condition R 0 for two cases of no epidemic (R 0 ≤ 1) and epidemic (R 0 > 1) using the time-series and phase portraits of the susceptible s, exposed e, infected i, and recovered r individuals. Based on the experimental results using a set of arbitrarily-defined parameters for horizontal transmission of the infectious diseases, the proportional population of the SEIRS model consisted primarily of the recovered r (0.7-0.9) individuals and susceptible s (0.0-0.1) individuals (epidemic) and recovered r (0.9) individuals with only a small proportional population for the susceptible s (0.1) individuals (no epidemic). Overall, the initial conditions for the susceptible s, exposed e, infected i, and recovered r individuals reached the corresponding equilibrium point for local stability: no epidemic (DFE X DFE ) and epidemic (EE X EE ).

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Global stability of an SIR epidemic model with information dependent vaccination

Cited by 216 publications

References 35 publications

Statistical physics of vaccination

Statistical physics of vaccination

An ecological perspective on marine reserves in prey–predator dynamics

Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity

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