Mathematical analysis of a model describing the invasion of bacteria in burn wounds

Hilhorst, Danielle; King, John R.; Röger, Matthias

doi:10.1016/j.na.2006.01.009

Cited by 20 publications

(24 citation statements)

References 9 publications

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“…Surprisingly enough, for k > k 0 the minimal speed of travelling waves is identical to the minimal speed of travelling waves for the Stefanlike limit problem that was formulated in [HKR07]. Our analysis is based on two main facts.…”

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confidence: 84%

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Travelling-wave analysis of a model describing tissue degradation by bacteria

Hilhorst¹,

King²,

Röger³

2007

Eur. J. Appl. Math

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Abstract. We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the 'degradation-rate parameter' solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave-speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range, nonlinear selection of the minimal speed occurs.

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confidence: 84%

“…One noteworthy aspect of (1.1), (1.2) is the convergence of solutions to the solution of a Stefan-like free boundary problem as the degradation rate k tends to infinity. This large-degradation-rate limit was identified by a formal asymptotic analysis in [KKC + 03] and was proved in [HKR07]. Reaction-diffusion systems of the general form…”

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confidence: 90%

Travelling-wave analysis of a model describing tissue degradation by bacteria

Hilhorst¹,

King²,

Röger³

2007

Eur. J. Appl. Math

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“…. Mathematical models of stall and infected wounds with focus on host tissue degradation by bacteria or quorum-sensing by bacteria have also been considered [12,13,14,15,16,31]. These papers provide complex models expressed as deterministic reaction-diffusion equations that must be solved numerically.…”

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confidence: 99%

The Effect of Bacteria on Epidermal Wound Healing

Agyingi

Maggelakis

Ross

2010

Math. Model. Nat. Phenom.

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Abstract. Epidermal wound healing is a complex process that repairs injured tissue. The complexity of this process increases when bacteria are present in a wound; the bacteria interaction determines whether infection sets in. Because of underlying physiological problems infected wounds do not follow the normal healing pattern. In this paper we present a mathematical model of the healing of both infected and uninfected wounds. At the core of our model is an account of the initiation of angiogenesis by macrophage-derived growth factors. We express the model as a system of reaction-diffusion equations, and we present results of computations for a version of the model with one spatial dimension.

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“…Note that the monotonicity properties of F are used here, in establishing the sign condition (2.33). See also [7,Prop. 4] and [10,Prop.…”

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confidence: 99%

Self-similar fast-reaction limits for reaction–diffusion systems on unbounded domains

Crooks

Hilhorst

2016

Journal of Differential Equations

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We present a unified approach to characterising fast-reaction limits of systems of either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains, motivated by models of fast chemical reactions where either one or both reactant(s) is/are mobile. For appropriate initial data, solutions of four classes of problems each converge in the fast-reaction limit k → ∞ to a self-similar limit profile that has one of four forms, depending on how many components diffuse and whether the spatial domain is a half or whole line. For fixed k, long-time convergence to these same self-similar profiles is also established, thanks to a scaling argument of Kamin. Our results generalise earlier work of Hilhorst, van der Hout and Peletier to a much wider class of problems, and provide a quantitative description of the penetration of one substance into another in both the fast-reaction and long-time regimes.

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Mathematical analysis of a model describing the invasion of bacteria in burn wounds

Cited by 20 publications

References 9 publications

Travelling-wave analysis of a model describing tissue degradation by bacteria

Travelling-wave analysis of a model describing tissue degradation by bacteria

The Effect of Bacteria on Epidermal Wound Healing

Self-similar fast-reaction limits for reaction–diffusion systems on unbounded domains

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