Memristive technologies are attractive due to their nonvolatility, high density, low power, nanoscale geometry, nonlinearity, binary/multiple memory capacity, and negative differential resistance. For memristive devices, a model corresponding with practical behavioral characteristics is highly favorable for the realization of its neuromorphic system and applications. In this paper, we propose a novel memristor model based on the Ag/TiOx nanobelt/Ti configuration, which can reflect three different states (i.e. original stage, transition stage, and resistive switching state) of the physical memristor with a satisfactory fitting precision (greater than 99.88%). Meanwhile, this work gives (1) an insight onto the electrical characteristics of the memristor model under different humidity conditions; (2) the influence of the water molecular concentration on the memristor behavior, which is of importance for the memristor fabrication and subsequent applications. For verification purposes, the proposed three-state switchable memristor is applied into the memristor-based logic implementation. The experimental results demonstrate that the constructed circuit is able to realize basic Boolean logic operations with fast response speed and high efficiency.