Gronwall–Bellman Type Integral Inequalities in Measure Spaces

Horváth, László

doi:10.1006/jmaa.1996.0311

Cited by 10 publications

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“…Here, we clarify how we can then deduce that Writing , the inequality ( 54 ) implies As earlier, consider for any , so that A general version of Grönwall’s inequality (Horváth 1996 , Theorem 3.1(d)), which does not require f to be known to be continuous a priori, then implies that for any , from which ( 55 ) readily follows. Another case we need is where Then we have and ( 55 ) follows by applying again Horvàth ( 1996 , Theorem 3.1(d)).…”

Section: A Discretising Cumulative Incidence Under the Bellman–harris...mentioning

confidence: 92%

Unifying incidence and prevalence under a time-varying general branching process

et al. 2023

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Renewal equations are a popular approach used in modelling the number of new infections, i.e., incidence, in an outbreak. We develop a stochastic model of an outbreak based on a time-varying variant of the Crump–Mode–Jagers branching process. This model accommodates a time-varying reproduction number and a time-varying distribution for the generation interval. We then derive renewal-like integral equations for incidence, cumulative incidence and prevalence under this model. We show that the equations for incidence and prevalence are consistent with the so-called back-calculation relationship. We analyse two particular cases of these integral equations, one that arises from a Bellman–Harris process and one that arises from an inhomogeneous Poisson process model of transmission. We also show that the incidence integral equations that arise from both of these specific models agree with the renewal equation used ubiquitously in infectious disease modelling. We present a numerical discretisation scheme to solve these equations, and use this scheme to estimate rates of transmission from serological prevalence of SARS-CoV-2 in the UK and historical incidence data on Influenza, Measles, SARS and Smallpox.

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Section: A Discretising Cumulative Incidence Under the Bellman–harris...mentioning

confidence: 92%

Unifying incidence and prevalence under a time-varying general branching process

et al. 2023

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“…Estimates for the above integrals have been studied in [13,14] under a particular condition (see [13], p. 184, Condition (C)), where the area of integration ( ) might be (0, ) or [0, ) rather than (0, ]. Below we propose an extension of Theorem 2.1 given in [13]. We shall distinguish two cases: = 0 and ≠ 0, We start with the following observation.…”

Section: Methodsmentioning

confidence: 99%

Filippov lemma for measure differential inclusion

Fryszkowski

Sadowski

2021

Mathematische Nachrichten

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In this work we propose a Filippov-type lemma for the differential inclusion () ∈ (, ()), (0) = 0 , (0.1) where ∶ [0, ] × ℝ ⇝ ℝ is a given multifunction and is a finite Borel signed measure on [0, ] (possibly atomic). By a solution of (0.1) we mean a function ∶ [0, ] ⟶ ℝ such that (0) = 0 and () = 0 + ∫ () () () f or > 0, where (⋅) is a-integrable function such that () ∈ (, ()) for-almost every ∈ [0, ] and () stands for either (0, ] for each ∈ or [0,). Such setting leads to at least two nonequivalent notions of a solution to (0.1) and therefore we formulate two different Filippov-type inequalities (Theorems 2.1 and 2.2). These two concepts coincide in case of the Lebesgue measure. The purpose of our considerations is to cover a class of impulsive control systems, a class of stochastic systems and differential systems on time scales.

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“…In particular, very general results on Gronwall-Bellmann type inequalities for general measure spaces were obtained by Horváth [10,11,12] (see also Győri and Horváth [8]). However, in the special case of the homogeneous linear integral inequality considered in (1.1), our condition (M) is still weaker than the conditions imposed in [10,Theorem 3.1].…”

Section: Introductionmentioning

confidence: 99%

“…While most of the extant results stay within the realm of ordinary Riemann integration, also other integrals are considered in the literature: Riemann-Stieltjes integrals [13,5], modified Stieltjes integrals [18], abstract Stieltjes integrals [9,14], Lebesgue-Stieltjes integrals [15,16,4], and integrals on general measure spaces. In particular, very general results on Gronwall-Bellmann type inequalities for general measure spaces were obtained by Horváth [10,11,12] (see also Győri and Horváth [8]). However, in the special case of the homogeneous linear integral inequality considered in (1.1), our condition (M) is still weaker than the conditions imposed in [10,Theorem 3.1].…”

Section: Introductionmentioning

confidence: 99%

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Minimal conditions for implications of Gronwall–Bellman type

Herdegen

Herrmann

2017

Journal of Mathematical Analysis and Applications

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A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. Abstract Gronwall-Bellman type inequalities entail the following implication: if a sufficiently integrable function satisfies a certain homogeneous linear integral inequality, then it is nonpositive. We present a minimal (necessary and sufficient) condition on the Borel measure underlying the integrals for this implication to hold. The condition is also a necessary prerequisite for any nontrivial bound on solutions to inhomogeneous linear integral inequalities of GronwallBellman type.

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Gronwall–Bellman Type Integral Inequalities in Measure Spaces

Cited by 10 publications

References 0 publications

Unifying incidence and prevalence under a time-varying general branching process

Unifying incidence and prevalence under a time-varying general branching process

Filippov lemma for measure differential inclusion

Minimal conditions for implications of Gronwall–Bellman type

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