Vacuum stability is one of the key issues in the design of particle accelerators, especially high-energy super proton-proton colliders. The synchrotron radiation generated by proton beams in the bending area will desorb and crack the gas molecules which have adsorbed on the wall of the cold bore. The collision or scattering between the proton beam and the desorbed gas molecules may result in the degradation of the beam quality and the reduction of beam life time, and even the collapse of the beam. Usually a copper coated stainless steel beam screen is installed in the cold bore to intercept synchrotron radiation and reduce gas desorption. Based on the design parameters of the Super Proton-Proton Collider, in this paper the source of gas in the beam screen is analyzed. By considering the photon-induced desorption process and the gas molecule cracking process, the gas dynamic model in the beam screen is established. Moreover, the calculation of the evolution of the gas density in the beam screen with the beam operating time is carried out, and the effect of TiZrV non-evaporable getter film coated beam screen on the dynamic gas density is explored. The results show that H₂ is the main desorption gas in the beam, the next is CO, while the molecular density of CO₂ and CH₄ are limited by molecular cracking. The maximum gas density in the beam screen appears at the initial stage of operation, and the gas density decreases with time going by. In order to strengthen adsorption and reduce desorption, TiZrV coated beam screen is discussed in this paper. In the case of TiZrV coated stainless steel beam screen, the maximum equivalent H₂ density is about two order of magnitude lower than in the case of copper coated stainless steel beam screen. The non-evaporable getter(NEG) for beam screen material can significantly improve vacuum performance. The calculation results can qualitatively reflect the dynamic vacuum evolution in the beam screen during the beam operation and provide a reference for designing vacuum systems.