Saddle point approximations in n-type epidemics and contact birth processes

Spreading speed and traveling waves for a multi-type SIS epidemic model

“…For linear epidemic models, Radcliffe and Rass [10,11] presented the saddle point method to determine the value ofc under the assumption thatc exists and is positive. The main idea is to solve the linear system by the method of the Laplace transformation and its inverse, and then to estimate the value ofc via some analytic techniques.…”

“…Suppose that P ij (θ ) = R e θu p ij (u) du is analytic in some region of the right half of the complex plane, and > 0 is the minimum of the corresponding abscissae of convergence. By the saddle point method, as applied to system (5.2), we have the following result (see [10] and [11,Theorem 7.1]). By assumption (H1), we see that = ∞ in our current case.…”

“…In Section 4, we obtain the existence of monotone traveling waves with wave speed c c * by the method of upper and lower solutions and a limiting argument, and the nonexistence of traveling waves with wave speed c ∈ (0, c * ) by a result in [6]. Section 5 is devoted to the comparison of our results for nonlinear system (1.1) with those obtained in [10,11] for the linear system associated with (1.1). It turns out that the asymptotic speed of spread coincides with the speed of first spread.…”

“…, n}, then (1.1) is an open system in the sense that the ith population has the combined death/emigration rate μ i − β i . The saddle point method for linear approximations was used in [10,11] to obtain the speed of first spread of infection. However, as remarked by Rass and Radcliffe in recent monograph [11, p. 205], there are at present no exact results for the asymptotic speed of propagation of infection in SIS epidemic models such as (1.1).…”

On an upper bound for the spreading speed

“…Relying on this conjecture, the asymptotic rate of spread has been studied for various nonlinear models by comparison or asymptotic approximation methods, e.g. [37,39,33,6,41,36,29].…”