Accurate representation of fire sprinkler spray enables quantitative engineering analysis of fire suppression performance. Increasingly, fire sprinkler systems are evaluated using computer fire models in which sprinkler spray is simulated with Lagrangian particles. However, limited guidance exists as to how to predict the formation of complex, spatio-stochastic fire sprinkler spray or how to accurately represent the dispersion of spray in terms of Lagrangian particles. The current work predicts the fire sprinkler spray generated by a canonical axisymmetric sprinkler using a Deflection Atomization Dispersion (DAD) framework, developed as a predictive modeling approach generalizable to typical fire sprinklers. In a DAD framework, spray evolution is divided into three stages: deflection of the water jet by the sprinkler deflector, atomization of the resulting thin fluid sheets into an initial spray, and dispersion of the initial spray into far-field spray. Deflection is described as a free-surface flow and is modeled deterministically using a boundary integral method (BIM). Atomization of the deflected fluid sheet is described by linear-stability theory to develop scaling laws relating sheet characteristics to statistically distributed, spatially resolved initial spray characteristics including breakup radius, volume flux, drop size, and drop velocity. The resulting initial spray is then described by a multivariate probability distribution function that varies over the predicted initialization surface. This function is stochastically sampled to generate Lagrangian particles representative of the near-field spray and the dispersion of these Lagrangian particles is in turn simulated in FireFOAM (an open source computational fluid dynamics fire model) to predict the far-field spray. Modeled results are compared to highly resolved near- and far-field measurements of axisymmetric sprinkler sprays generated by the Spatially-Resolved Spray Scanning System (4S). The end results shows agreement across all three stages of modeling with less than 10\% error when compared to experimental measurements. Further, the newly implemented model shows a stronger ability to capture spray induced airflow when compared to a baseline model. This work is the first to predict sprinkler spray dispersion entirely from sprinkler deflector geometry and operating pressure.