The existence of all kinds of solutions for integral equations becomes important research topics in thestudy fields of integral equations． Since this kind of equation was established about epidemic problem early，almostperiodic type solutions were discussed for the kind of equation with the constant delay by relevant literature． Theexistence of almost periodic solution is investigated for this kind of integral equation with the delay by constant intodependence on argument by some fixed point theorem on Hilbert projective metric in this paper， ． Thus，this kind ofintegral equation can be used more widely．