Long-duration Gamma-Ray Bursts (GRBs) are linked to the collapse of massive stars and their hosts are exclusively identified as active, star-forming galaxies. Four long GRBs observed at high spectral resolution at redshift 1.5 ≤ z ≤ 4 allowed the determination of the elemental abundances for a set of different chemical elements. In this paper, for the first time, by means of detailed chemical evolution models taking into account also dust production, we attempt to constrain the star formation history of the host galaxies of these GRBs from the study of the measured chemical abundances measured in their interstellar medium. We are also able to provide constraints on the age and on the dust content of GRB hosts. Our results support the hypothesis that long duration GRBs occur preferentially in low metallicity, star-forming galaxies. We compare the specific star formation rate (SSFR), namely the star formation rate (SFR) per unit stellar mass, predicted for the hosts of these GRBs with observational values for GRB hosts distributed across a large redshift range. Our models predict a decrease of the SSFR with redshift, consistent with the observed decrease of the comoving cosmic SFR density between z ~ 2 and z = 0. On the other hand, observed GRB hosts seem to follow an opposite trend in the SSFR versus redshift plot, with an increase of the SSFR with decreasing redshift. Future SSFR determination in larger samples of GRB hosts will be important to understand whether this trend is real or due to some selection effect. Finally, we compare the SSFR of GRB 050730 host with values derived with a sample of Quasar damped Lyman alpha (DLA) systems. Our results indicate that the abundance pattern and the SSFRs of the host galaxies of this GRB are basically compatible with the ones determined for a sample of Quasar DLA systems, suggesting similar chemical evolution paths.
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