Exhaust plumes produced by launch vehicles generate acoustic and gravity waves in the upper atmosphere. These waves in turn drive travelling ionospheric disturbances which can be observed, for example, in measurements of total electron content (TEC) from navigation satellite signals. We used the Global Ionosphere Thermosphere Model (GITM) to model acoustic wave propagation and ionospheric interaction. GITM boundary conditions incorporated perturbations computed using a ray tracing model and analytic source to represent the rocket exhaust plume. This model accounted for neutral winds and non-linear aging. The launch of Jason-3 on a SpaceX Falcon 9 rocket was simulated using these methods. Based on the resulting pressure perturbations and times of arrival, wave propagation within GITM was compared with results of the ray tracing simulation. TEC estimates from GITM were compared with those from previous studies using navigation satellite signals. Sensitivity to changes in the thermosphere and ionosphere state was investigated by varying the solar flux index in the simulation. The vertical distributions of changes in ion velocity drifts and electron density within the ionosphere were examined, indicating that the largest changes occurred around the F2 layer peak.