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The pumping of 22.2-GHz H$_2$O masers in the circumstellar envelopes ofasymptotic giant branch stars has been simulated numerically. The physicalparameters adopted in the calculations correspond to those of the circumstellarenvelope around IK Tau. The one-dimensional plane-parallel structure of thegas-dust cloud is considered. The statistical equilibrium equations for theH$_2$O level populations and the thermal balance equations for the gas-dustcloud are solved self-consistently. The calculations take into account 410rotational levels belonging to the five lowest vibrational levels of H$_2$O.The stellar radiation field is shown to play an important role in the thermalbalance of the gas-dust cloud due to the absorption of emission inrotational-vibrational H$_2$O lines. The dependence of the gain in the 22.2-GHzmaser line on the gas density and H$_2$O number density in the gas-dust cloudis investigated. Gas densities close to the mean density of the stellar wind,10$^7-10^8$ cm$^{-3}$, and a high relative H$_2$O abundance, more than10$^{-4}$, have been found to be the most likely physical conditions in masersources.

H2O Maser Emission in Circumstellar Envelopes around AGB Stars: Physical Conditions in Gas-Dust Clouds