Helium-ion induced deposition (He-IBID) is a small-scale manufacturing technique that uses a helium microscope to focus a helium-ion beam onto an adsorbed precursor gas, causing nanometer-scaled depositions onto a substrate. However, due to local depletion of the precursor gas, the amount of deposition can not be expected to be proportional to the beam current. The mechanisms of this growth need to be well investigated to understand the limitations and capabilities of this technique. Most studies look at simple pillar growth with a stationary beam, and most models exclude the surface diffusion of precursor molecules. We attempt to model the growth of horizontally grown pillars using a horizontally moving ion beam. According to experiments with this type of growth \cite{HammerheadAFMProbes}, precursor surface diffusion is a key mechanism, and thus past studies may not be sufficient. We use develop a continuum model based on solving differential equations. One benefit over the more common Monte Carlo methods is that the effect of the various mechanisms are more explicitly written in differential equation form. Another benefit is that the simulation is differentiable, lending itself to better application of new differentiable programming techniques. One difficulty, however, is in the accurate modelling of the energy-dependent interactions with a source that has varying electron energies. We derive a system of ordinary differential equations to model a simplified two-dimensional approach and an approximation for an effective secondary electron flux to use in decomposition rate calculations. We investigate the available literature to find realistic values for the physical and operational parameters required to calculate solutions. Through numerical integration we find solutions and show how the model estimates pillar width changes when these parameters are altered. We use numerical integration to solve the ODE system, and present the results. We find a nucleation boundary---dependent on the beam movement speed and current---below which pillars cannot be grown, consistent with experiment \cite{HammerheadAFMProbes}. We find that the assumptions made in the surface diffusion calculation are too extreme, and that likely solving a PDE system of the full surface in three-dimensional space will be required for accurate results.