On a modification of a discrete epidemic model

In this paper under some conditions on the constants A,B∈(0,∞)A,B∈(0,∞) we study the existence of positive solutions, the existence of a unique nonnegative equilibrium and the convergence of the positive solutions to the nonnegative equilibrium of the system of difference equations xn+1=(1−yn−yn−1)(1−e−Ayn),yn+1=(1−xn−xn−1)(1−e−Bxn) where A,B∈(0,∞)A,B∈(0,∞) and the initial values x−1,x0,y−1,y0x−1,x0,y−1,y0 are positive numbers which satisfy the relations x0+x−1<1,y0+y−1<1,1−y0>(1−x0−x−1)(1−e−Bx0),1−x0>(1−y0−y−1)(1−e−Ay0).