To draw inferences from a sample to the target population, where the sample is not a random sample of the target population, various generalizability and transportability methods can be considered. Many of these modern approaches rely on a structural positivity assumption, such that all relevant covariate patterns in the target population are also observed in the secondary population of which the data is random sample of. Strict eligibility criteria, particularly in the context of randomized trials, may lead to violations of this positivity assumption. To address this concern, common methods are to restrict the target population, restrict the adjustment set, or extrapolate from a statistical model. Instead of these approaches, which all have concerning limitations, we propose a synthesis, or combination, of statistical (e.g., g-methods) and mathematical (e.g., microsimulation, mechanistic) models. Briefly, a statistical model is fit for the regions of the parameter space where positivity holds, and a mathematical model is used to fill-in, or impute, the nonpositive regions. For estimation, we propose two augmented inverse probability weighting estimators; one based on estimating the parameters of a marginal structural model, and the other based on estimating the conditional average causal effect. The standard approaches and the proposed synthesis method are illustrated with a simulation study and an applied example on the effect of antiretroviral therapy on CD4 cell count. The proposed synthesis method sheds light on a way to address challenges associated with the positivity assumption for transporting and causal inference more generally.