Run for Your Life - Note on the Asymptotic Speed of Propagation of an Epidemic

O DIEKMANN

doi:10.1016/0022-0396(79)90080-9

Run for Your Life - Note on the Asymptotic Speed of Propagation of an Epidemic

O DIEKMANN

Sub Mathematical Modeling

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We study the large-time behaviour of the solution of a nonlinear integral equation of mixed Volterra-Fredholm type describing the spatio-temporal development of an epidemic. For this model it is known that there exists a minimal wave speed c₀ (i.e., travelling wave solutions with speed c exist if |c| > c₀ and do not exist if |c| < c₀). In this paper we show that c₀ is the asymptotic speed of propagation (i.e., for any c₁, c₂with 0 < c₁ < c₀ < c₂ the solution tends to zero uniformly in the region |x| ≥ c₂t, whereas it is bounded away from zero uniformly in the region |x| ≤ c₁t for t sufficiently large).

Original language	English
Pages (from-to)	58-73
Number of pages	16
Journal	Journal of Differential Equations
Volume	33
Issue number	1
DOIs	https://doi.org/10.1016/0022-0396(79)90080-9
Publication status	Published - 1979

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title = "Run for Your Life - Note on the Asymptotic Speed of Propagation of an Epidemic",

abstract = "We study the large-time behaviour of the solution of a nonlinear integral equation of mixed Volterra-Fredholm type describing the spatio-temporal development of an epidemic. For this model it is known that there exists a minimal wave speed c0 (i.e., travelling wave solutions with speed c exist if |c| > c0 and do not exist if |c| < c0). In this paper we show that c0 is the asymptotic speed of propagation (i.e., for any c1, c2with 0 < c1 < c0 < c2 the solution tends to zero uniformly in the region |x| ≥ c2t, whereas it is bounded away from zero uniformly in the region |x| ≤ c1t for t sufficiently large).",

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doi = "10.1016/0022-0396(79)90080-9",

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AU - DIEKMANN, O

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N2 - We study the large-time behaviour of the solution of a nonlinear integral equation of mixed Volterra-Fredholm type describing the spatio-temporal development of an epidemic. For this model it is known that there exists a minimal wave speed c0 (i.e., travelling wave solutions with speed c exist if |c| > c0 and do not exist if |c| < c0). In this paper we show that c0 is the asymptotic speed of propagation (i.e., for any c1, c2with 0 < c1 < c0 < c2 the solution tends to zero uniformly in the region |x| ≥ c2t, whereas it is bounded away from zero uniformly in the region |x| ≤ c1t for t sufficiently large).

AB - We study the large-time behaviour of the solution of a nonlinear integral equation of mixed Volterra-Fredholm type describing the spatio-temporal development of an epidemic. For this model it is known that there exists a minimal wave speed c0 (i.e., travelling wave solutions with speed c exist if |c| > c0 and do not exist if |c| < c0). In this paper we show that c0 is the asymptotic speed of propagation (i.e., for any c1, c2with 0 < c1 < c0 < c2 the solution tends to zero uniformly in the region |x| ≥ c2t, whereas it is bounded away from zero uniformly in the region |x| ≤ c1t for t sufficiently large).

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JO - Journal of Differential Equations

JF - Journal of Differential Equations

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ER -

Run for Your Life - Note on the Asymptotic Speed of Propagation of an Epidemic

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