Modeling the removal of primary and secondary pollutants from the atmosphere of a city by rain

In this paper, we have proposed and analyzed a nonlinear mathematical model for the removal of primary and secondary pollutants from the atmosphere of an industrial city by rain. To model the phenomenon, it is assumed that the atmosphere consists of five nonlinearly interacting phases i.e. the raindrops phase, the primary pollutants phase, the secondary pollutants phase and absorbed phases of these pollutants in the raindrops. The dynamics of these phases is assumed to be governed by nonlinear differential equations with source, interaction, recycle and removal terms. The model is analyzed using stability theory of differential equations. It is shown that these pollutants can be washed out from the atmosphere completely by rain in the case of instantaneous emission of primary pollutants. However, when the primary pollutant is emitted at a constant rate, it is found that both the primary and secondary pollutants can still be washed out from the atmosphere under some appropriate conditions and the remaining equilibrium amount would depend upon the rate of emission of primary pollutants, rate of formation of secondary pollutants, rate of raindrops formation and different removal parameters. The equilibrium levels of these pollutants are much smaller after rain than its corresponding value before rain. A numerical study of the model is also performed to investigate the influence of certain key parameters on the dynamics of model system.