The uniqueness of Atkinson and Reuter's epidemic waves

Abstract

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 equationwhich satisfy X(− ∞) = − ∞, X( + ∞) = 0, X(u) < 0 for u ∈ (− ∞, ∞), whereis 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 unlessconverges for λ > 0 sufficiently small, and that any solution X must satisfyfor some C > 0. They then prove the following existence theorems.