We introduce a new class of stochastic models for polymer stresses which offers a blending of continuum mechanics, network theory and reptation theory. The stochastic dynamics of the model involve two independent Gaussian stochastic processes, Q(1) and Q(2). Associated with each random vector, ei, is a random variable, Si, that describes the vector's survival time during which it evolves according to a deterministic equation of motion. The expression for the stress tensor is an ensemble average of f(1)(Q(1)(2),Q(2)(2))Q(1)Q(1) + f(2) (Q(1)(2), Q(2)(2))1Q(2)Q(2,) where the f(i) are scalar functions of Q(1)(2) = Q(1).Q(1) and Q(2)(2) = Q(2).Q(2). The relationship between this new class of models and the class of factorized Rivlin-Sawyers integral models is indicated, and simulation models from this new class are used to predict theological behavior of three low-density-polyethylene melts. We find that the steady-state sheat data of all three melts, and the time-dependent elongational viscosity of one of the melts, can be predicted well by models with the same f(i), but different probability densities for S-i which are obtained from the different relaxation spectra.