On a discrete epidemic model

Abstract

We investigate the global asymptotic behavior of solutions of the difference equation xn+1 = (1 - ∑j=0x-1 xn-j) (1 - e-Axn, n ∈ ℕ0, where A ∈ (0, ∞), k ∈ {2, 3,...}, and the initial values x-k+1, x-k+2,..., x0 are arbitrary negative numbers. Asymptotics of some positive solutions of the equation are also found.