This paper is concerned with the analysis of interconnected systems where positive subsystems are connected by a nonnegative interconnection matrix with communication delays. Recently, we have shown that, under mild assumptions on the positive subsystems and the interconnection matrix, the delay interconnected system has stable poles only except for a simple pole at the origin, and the output of the interconnected system converges to a scalar multiple of a prescribed positive vector. This result is effectively used for formation control of multi-agent systems with positive dynamics, where the desired formation is basically achieved irrespective of the length of the delays. However, the rate of convergence varies according to the length of the delays and hence quantitative evaluation of the convergence rate is an important issue. For the quantitative evaluation of the convergence rate, in this paper, we provide an efficient method for the computation of the lower bounds of the second largest real part of the (infinitely many) poles of the delay interconnected positive systems.