Abstract: In this paper the fixed point index problem for a class of positive operators with boundary control conditions is discussed, and some sufficient conditions for the fixed pointindex to be equal to 1 or 0 are given. Moreover, a general fixed point theorem of expansions and compressions for cone is obtained, which generalizes and improves the corresponding results of [3,8,9]. As an application, we utilize the results presented above to study the existence conditions of positive solutions of nonlinear integral equations modelling infectious diseases.