We study the population of asymptotic giant branch (AGB) stars in the Small Magellanic Cloud (SMC) by means of full evolutionary models of stars of mass 1 M⊙ ≤ M ≤ 8 M⊙, evolved through the thermally pulsing phase. The models also account for dust production in the circumstellar envelope. We compare Spitzer infrared colours with results from theoretical modelling. We show that ˜75 per cent of the AGB population of the SMC is composed by scarcely obscured objects, mainly stars of mass M ≤ 2 M⊙ at various metallicity, formed between 700 Myr and 5 Gyr ago; ˜70 per cent of these sources are oxygen-rich stars, while ˜30 per cent are C-stars. The sample of the most obscured AGB stars, accounting for ˜25 per cent of the total sample, is composed almost entirely by carbon stars. The distribution in the colour-colour ([3.6] - [4.5], [5.8] - [8.0]) and colour-magnitude ([3.6] - [8.0], [8.0]) diagrams of these C-rich objects, with a large infrared emission, traces an obscuration sequence, according to the amount of carbonaceous dust in their surroundings. The overall population of C-rich AGB stars descends from 1.5-2 M⊙ stars of metallicity Z = 4 × 10-3, formed between 700 Myr and 2 Gyr ago, and from lower metallicity objects, of mass below 1.5 M⊙, 2-5 Gyr old. We also identify obscured oxygen-rich stars (M ˜ 4-6 M⊙) experiencing hot bottom burning. The differences between the AGB populations of the SMC and LMC are also commented.