Fast traveling waves for an epidemic model

Asymptotic solutions are investigated for the travelling wave consisting of infectives I(x-ct) propagating at speed c into a region of uninfected susceptibles S=S+, on the basis that S+ is large. In the moving frame, three domains are identified. In the narrow leading frontal region, the infectives terminate relatively abruptly. Conditions ahead (increasing x) of the front control the speed c of the front advance. In the trailing region (decreasing x), the number of infectives decay relatively slowly. Our asymptotic development focuses on the dependence of I on S in the central region. Then, the apparently simple problem is complicated by the presence of both algebraic and logarithmic dependencies. Still, we can construct an asymptotic expansion to a high order of accuracy that embeds the trailing region solution. A proper solution in the frontal region is numerical, but here the central region solution works well too. We also investigated numerically the evolution from an initial state to a travelling wave. Following the decay of transients, the speed adopted by the wave is fast, though the slowest of those admissible. The asymptotic solutions are compared with the numerical solutions and display excellent agreement.