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The uniqueness of Atkinson and Reuter's epidemic waves

Published online by Cambridge University Press:  24 October 2008

Andrew D. Barbour
Andrew D. Barbour
Affiliation:
Gonville and Caius College, Cambridge
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Extract

Atkinson and Reuter(1) consider travelling wave solutions for the deterministic epidemic, with or without removals, spreading along the line. In the case where there are no removals, they reformulate the problem in terms of the solutions X(·) to the integral equation

which satisfy X(− ∞) = − ∞, X( + ∞) = 0, X(u) < 0 for u ∈ (− ∞, ∞), where

is the left hand tail of the contact distribution, and where ʗ > 0 is the velocity of the wave corresponding to X. They show that no solution is possible unless

converges for λ > 0 sufficiently small, and that any solution X must satisfy

for some C > 0. They then prove the following existence theorems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 1 , July 1977 , pp. 127 - 130
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Atkinson, C. and Reuter, G. E. H.Deterministic epidemic waves. Math. Proc. Cambridge Philos. Soc. 80 (1976), 315–330.CrossRefGoogle Scholar
(2)Brown, K. J. and Carr, J.Deterministic epidemic waves of critical velocity. Math. Proc. Cambridge Philos. Soc. 81 (1977), 431–433.CrossRefGoogle Scholar