We propose an efficient model-based sufficient dimension reduction method to detect interactions. We introduce a new class of multivariate adaptive varying index models (MAVIM) to investigate nonlinear interaction effects of the grouped covariates on multivariate response variables. Grouping the covariates through linear combinations in the MAVIM accommodates weak individual interaction effects as long as their joint interaction effects are strong enough to be detectable. This is the first attempt in the area of sufficient dimension reduction which reduces the dimension of the covariates in a model-based fashion. We estimate the joint interaction effects by a weighted profile least squares method, which is numerically stable and computationally fast. The resultant profile least squares estimate is root-$n$ consistent and asymptotically normal. We also investigate how to choose an optimal weight to improve the estimation efficiency. We determine the structural dimension with a BIC-type criterion, and establish its consistency. The effectiveness of our proposed method is illustrated through comprehensive simulation studies and an analysis of Framingham heart study.