Time to extinction of infectious diseases through age-dependent branching models

González, Miguel; Martı́nez, Rodrigo; Slavtchova-Bojkova, Maroussia

doi:10.1007/978-3-642-11156-3_17

Cited by 3 publications

(13 citation statements)

References 9 publications

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“…In the present paper, we extend the results in González et al [13,14] in several directions that are both practically and theoretically important. First, we assume that the spread of infection is modelled as a CMJ branching process.…”

Section: Introductionsupporting

confidence: 80%

“…González et al [13,14] studied properties of the time to extinction of an epidemic given that a fraction c of individuals is vaccinated, when the number of infectious individuals in the population is modelled by a continuous-time BHBP and a (more general) continuous-time SBP, respectively. In an earlier work, De Serres et al [10] used a discretetime Bienaymé-Galton-Watson branching process to study the spread of an infectious disease under various control measures, specifically to estimate the effective (i.e., postcontrol) value of R 0 from observations on size and durations of small outbreaks.…”

Section: Introductionmentioning

confidence: 99%

“…Second, we consider more general vaccination processes. In González et al [13,14] it was assumed that the fraction of the population that is vaccinated remained constant with time. We now allow this fraction to be an arbitrary but specified function of time, thus capturing for example the setting in which people are vaccinated as the disease spreads.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic monotonicity and continuity properties of functions defined on Crump–Mode–Jagers branching processes, with application to vaccination in epidemic modelling

Ball

González²,

Martínez³

et al. 2014

Bernoulli

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This paper is concerned with Crump-Mode-Jagers branching processes, describing spread of an epidemic depending on the proportion of the population that is vaccinated. Births in the branching process are aborted independently with a time-dependent probability given by the fraction of the population vaccinated. Stochastic monotonicity and continuity results for a wide class of functions (e.g., extinction time and total number of births over all time) defined on such a branching process are proved using coupling arguments, leading to optimal vaccination schemes to control corresponding functions (e.g., duration and final size) of epidemic outbreaks. The theory is illustrated by applications to the control of the duration of mumps outbreaks in Bulgaria.

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Section: Introductionsupporting

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stochastic monotonicity and continuity properties of functions defined on Crump–Mode–Jagers branching processes, with application to vaccination in epidemic modelling

Ball

González²,

Martínez³

et al. 2014

Bernoulli

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“…We have concluded that the above motivation is appealing for the use of the SBP. Moreover we applied this process to model outbreaks of the avian influenza virus in Vietnam showing that such treatment of the data may be adequate (see González et al, 2010b ).…”

Section: Branching Model Of Epidemic Spreadmentioning

confidence: 99%

“…Specifically, we have proposed to model the number of infectious individuals in the population depending on the vaccination level by means of Bellman–Harris branching processes (BHBP). To further elucidate the method of our modeling and to make the model more realistic, later on we have considered Sevast'yanov's branching processes (SBP; see González et al, 2010b ), which actually generalize BHBP. Both kinds of branching processes are particular cases of the general branching process (see Jagers, 1975 ), also called Crump–Mode–Jagers branching process, which is the most adequate model to fit infectious diseases following SIR scheme (see Ball and Donnelly, 1995 ).…”

Section: Introductionmentioning

confidence: 99%

Age-Dependent Branching Processes for Surveillance of Vaccine-Preventable Diseases with Incubation Period

Slavtchova-Bojkova¹,

González²,

Martı́nez³

2010

Front. Psychiatry

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The purpose of this paper is to review the recent results of the authors in the area of infectious disease modeling by means of branching stochastic processes. This is a new approach involving age-dependent branching models, which turned out to be more appropriate and flexible for describing the spread of an infection in a given population, than discrete time ones. Concretely, Bellman–Harris and Sevast'yanov's branching processes are investigated. It is justified that the proposed models are proper candidates as models of infectious diseases with incubation period like measles, mumps, avian flu, etc. It is worth to notice that in general the developed methodology is applicable to the diseases that follow the so-called SIR (susceptible–infected–removed) scheme in terms of epidemiological models. Two policies of extra-vaccination level are proposed and compared on the ground of simulation examples.

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Synchronization and extinction in a high-infectivity spatial SIRS with long-range links

Arceo-May¹,

Moukarzel²

2019

J. Stat. Mech.

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A numerical study of synchronization and extinction is done for a SIRS model with fixed infective and refractory periods, in the regime of high infectivity, on one-and two-dimensional networks for which the connectivity probability decays as r −α with distance. In both one and two dimensions, a long-lasting synchronized state is reached when α < d but not when α > d. Three dynamical stages are identified for small α, respectively: a short period of initial synchronization, followed by a long oscillatory stage of random duration, and finally a third phase of rapid increase in synchronization that invariably leads to dynamical extinction. For large α, the second stage is not synchronized, but is instead a long-lasting endemic state of incoherent activity. Dynamical extinction is in this case still preceded by a short third stage of rapidly intensifying synchronized oscillations. A simple model of noise-induced escape from a potential barrier is introduced, that explains the main characteristics of the observed three-stage dynamical structure before extinction. This model additionally provides specific predictions regarding the size-scaling of the different timescales for the observed dynamical stages, which are found to be consistent with our numerical results.

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Time to extinction of infectious diseases through age-dependent branching models

Cited by 3 publications

References 9 publications

Stochastic monotonicity and continuity properties of functions defined on Crump–Mode–Jagers branching processes, with application to vaccination in epidemic modelling

Stochastic monotonicity and continuity properties of functions defined on Crump–Mode–Jagers branching processes, with application to vaccination in epidemic modelling

Age-Dependent Branching Processes for Surveillance of Vaccine-Preventable Diseases with Incubation Period

Synchronization and extinction in a high-infectivity spatial SIRS with long-range links

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