The most metal-poor, high redshift damped Lyman-alpha systems (DLAs) provide a window to study some of the first few generations of stars. In this paper, we present a novel model to investigate the chemical enrichment of the near-pristine DLA population. This model accounts for the mass distribution of the enriching stellar population, the typical explosion energy of their supernovae, and the average number of stars that contribute to the enrichment of these DLAs. We conduct a maximum likelihood analysis of these model parameters using the observed relative element abundances ([C/O], [Si/O], and [Fe/O]) of the 11 most metal-poor DLAs currently known. We find that the mass distribution of the stars that have enriched this sample of metal-poor DLAs can be well-described by a Salpeter-like IMF slope at M > 10 M_sun and that a typical metal-poor DLA has been enriched by < 72 massive stars (95 per cent confidence), with masses < 40 M_sun. The inferred typical explosion energy (E_exp = 1.8^{+0.3}_{-0.2}x10^51 erg) is somewhat lower than that found by recent works that model the enrichment of metal-poor halo stars. These constraints suggest that some of the metal-poor DLAs in our sample may have been enriched by Population II stars. Using our enrichment model, we also infer some of the typical physical properties of the most metal-poor DLAs. We estimate that the total stellar mass content is log10(M_*/M_sun) = 3.5^{+0.3}_{-0.4} and the total gas mass is log10(M_gas/M_sun) = 7.0^{+0.3}_{-0.4} for systems with a relative oxygen abundance [O/H] ~ -3.0.
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