Recently, the authors extended the interval method into the time domain and proposed a new mathematical model for time-varying uncertainty quantification, namely, the "interval process model". Interval process uses a lower bound and an upper to describe the imprecision of a time-variant parameter at any time point rather than the precise probability distribution, and hence compared with the traditional stochastic process it shows some advantages in uncertainty quantification such as easy to understand, convenient to use, small dependence on samples, etc. This paper firstly gives the conceptions of limit and continuity of interval process, based on which the differential and integral of interval process are defined. Secondly, the middle point function, auto-covariance function and cross-covariance function of the differential and integral of interval process are deduced, which thus enriches the theory of interval process model. Thirdly, the above conceptions are applied to the vibration analysis of structures under uncertain excitations. The variation bounds of the velocity and acceleration responses are deduced for both of the linear single degree of freedom (SDOF) vibration system and the multiple degree of freedom (MDOF) vibration system under uncertain excitations.