Browsing by Author "Amita Tripathi"

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A COMPREHENSIVE INVESTIGATION INTO DETERMINISTIC AND STOCHASTIC MODELS CONCERNING THE EFFECTS OF SEEDING ON RAINFALL AND ATMOSPHERIC POLLUTION
(World Scientific, 2025) Amita Tripathi; Sayan Mandal; Pankaj Kumar Tiwari; Arvind Kumar Misra; Maia Martcheva
This study introduces a mathematical model aimed at evaluating the potential influence of aerosol introduction into the atmosphere for inducing rainfall and managing atmospheric pollution. By expanding on the proposed model, we incorporate stochastic elements to encompass environmental white noises that impact the system’s dynamics. Both mathematical and numerical methods are employed to analyze the system’s behavior. In the context of the deterministic model, we examine the solutions’ positivity and boundedness, identify feasible equilibria, and scrutinize the stability characteristics both locally and globally. The analysis of the stochastic system encompasses discussions regarding the existence of a unique solution, its ultimate boundedness, and the conditions that prompt the establishment of a unique stationary distribution characterized by ergodic properties. Our simulations illustrate that augmenting cloud formation rates and externally introduced aerosols can amplify rainfall while mitigating atmospheric pollution levels. Minor intensities of white noise do not alter the system’s behavior, whereas significant intensities result in high-amplitude oscillations of the system’s variables. We explore the effects of white noise intensities using histograms and stationary distributions, highlighting long-term rainfall trends in a noisy environment. © 2025 World Scientific Publishing Company.
A mathematical model for the removal of pollutants from the atmosphere through artificial rain
(Taylor and Francis Ltd., 2022) Amita Tripathi; A.K. Misra; J.B. Shukla
To reduce the pollution from the atmosphere or polluted cities like the capital city Delhi of India, use of artificial rain is a solution. In this paper, we have proposed and analyzed a nonlinear mathematical model to reduce the pollution level by rain making. In the proposed model five variables are considered, namely; (i) number density of water vapor, (ii) number density of cloud drops, (iii) number density of raindrops, (iv) cumulative concentration of aerosols, and (v) concentration of pollutant particles suspended in the region of consideration. The effect of environmental fluctuations has been studied with the help of Lyapunov functionals. The model is analyzed in the presence of white noise and proved that if rain persists, the pollutants can be totally washed out. It has been observed that the environmental disturbances are not much favorable in such experiments as the presence of environmental disturbance may destabilize the system. It is found that to remove pollutants completely, it is necessary to prevent the formation of pollutants. The simulation is performed to support the analytical findings. © 2021 Taylor & Francis Group, LLC.
A stochastic model for making artificial rain using aerosols
(Elsevier B.V., 2018) A.K. Misra; Amita Tripathi
In this paper, a nonlinear deterministic mathematical model along with its stochastic version for artificial rain is proposed and analyzed. We have considered three dynamical variables in the modeling process; namely (i) density of cloud droplets, (ii) density of raindrops, and (iii) concentration of mixture of conducive aerosols. It is assumed that the cloud droplets are continuously formed in the atmosphere at a constant rate but its conversion into raindrops does not take place in the same proportion. The artificially introduced aerosols increase the rate of formation of raindrops from cloud droplets. These aerosols are introduced in the regional atmosphere at a rate proportional to the density of cloud droplets. The proposed model is analyzed using stability theory of differential equations in deterministic as well as stochastic environment. Numerical simulation is performed to see the effect of important parameters on the process leading to rainfall. © 2018 Elsevier B.V.
An optimal control model for cloud seeding in a deterministic and stochastic environment
(John Wiley and Sons Ltd, 2020) Arvind Kumar Misra; Amita Tripathi
To promote artificial rain in India and other such developing countries, in this article, we have proposed and analyzed a nonlinear mathematical model for cloud seeding by considering that aerosols are introduced proportional to the density of water vapors present in the atmosphere. The model is analyzed using Lyapunov's stability theory of differential equations. To reduce the cost of cloud seeding, an optimal control strategy is designed by incorporating four control parameters. We have shown the existence and uniqueness of solution of this optimal control problem, using Pontryagins Maximum Principle. To minimize the cost in making artificial rain, the optimal control problem provides the strategy for the rate of introduction of aerosols in the atmosphere. To capture the effects of environmental noise on control strategies, the model in deterministic framework is converted into stochastic framework. In this regard, the Hamilton-Jacobi-Bellman equation for stochastic control cost function has been formed. The stochastic analysis implies that the control strategy is effective in reducing the cost and increasing rainfall. © 2020 John Wiley & Sons, Ltd.
Estimating the risk of pandemic avian influenza
(Taylor and Francis Ltd., 2021) Amita Tripathi; Harish Chandra Dhakal; Khagendra Adhikari; Ramesh Chandra Timsina; Lindi M. Wahl
Outbreaks of highly pathogenic strains of avian influenza (HPAI) cause high mortality in avian populations worldwide. When spread from avian reservoirs to humans, HPAI infections cause mortality in about 50% of human infections. Cases of human-to-human transmission of HPAI are relatively rare, and have, to date, only been reported in situations of close contact. These transmissions have resulted in isolated clusters of human HPAI infections, but have not yet caused a pandemic. Given the large number of human H5N1 HPAI infections to date, none of which has resulted in a pandemic, we estimate an upper bound on the probability of H5N1 pandemic emergence. We use this estimate to provide the likelihood of observing such a pandemic over the next decade. We then develop a more accurate parameter-based estimate of the emergence probability and predict the likelihood that, through rare mutations, an H5N1 influenza pandemic will emerge over the same time span. © 2021 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
Impacts of transpiration of agricultural crops and seeding on rainfall: Implications from a mathematical model
(World Scientific, 2022) Amita Tripathi; Pankaj Kumar Tiwari; Arvind Kumar Misra; Yun Kang
As the source of replenishment, rainfall has an extensive impact because its variability shapes biologically efficient pulses of soil moisture recharge across layers from rainfall events. In this paper, a mathematical model is proposed to explore the importance of transpiration from agricultural crops and aerosols on the pattern of rainfall. For the system without seeding, the simulation results show destabilizing roles of parameters related to formation of cloud drops due to transpiration of agricultural crops, formation of raindrops due to cloud drops and growth of agricultural crops due to rain. The model without seeding is extended to its stochastic counterpart to encapsulate the uncertainty observed in some important parameters. We observe the variability in the system's variables and found their distributions at certain fixed times, which explore the importance of stochasticity in the system. Our findings show that transpiration through agricultural crops plays an important role in cloud formation, and thus, affects the effectiveness of different rainfall events. Moreover, the combined actions of transpiration and seeding are much more beneficial in producing rain. Finally, we see the behavior of system by considering seasonal variations of some rate parameters. © 2022 World Scientific Publishing Company.
Mathematical Models of how the Transpiration Affects Rainfall through Agriculture Crops
(Springer Science and Business Media Deutschland GmbH, 2021) Arvind Kumar Misra; Amita Tripathi
The water vapor in the atmosphere is mainly formed through the vaporization of water from oceans, open land water sources, and transpiration through forests and agricultural crops. In the present study, our aim was to formulate a mathematical model to see the effect of transpiration from agricultural crops on rainfall. In the model formulation, it is assumed that clouds are formed in the atmosphere through the transpiration of agricultural crops proportional to the crops apart from the oceans and other sources (assumed constant). It is observed that in the presence of agriculture crops, the average rainfall increases. In the presence of clouds, cloud seeding techniques (use of conducive aerosols) are used to stimulate rainfall, which increases the agricultural crops with saturated type functional form. To capture the impact of environmental fluctuations, the stochastic version of the proposed model is also studied. The analysis of the model reveals that cloud seeding not only increases the rainfall, but it also increases the agricultural crops and clouds in the atmospheric environment. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
Modelling and analysis of the effects of aerosols in making artificial rain
(Springer Science and Business Media Deutschland GmbH, 2016) A.K. Misra; Amita Tripathi; Ram Naresh; J.B. Shukla
To overcome the water crisis for irrigation and other purposes, in this paper, we propose a non-linear mathematical model for artificial rain making by considering five dependent variables namely, water vapor density, densities of cloud droplets of small and large sizes, density of rain drops and cumulative concentration of mixture of aerosols of different sizes. It is assumed that these aerosols are conducive to the process of rain making, i.e. (a) the formation of small size cloud droplets from water vapors through the processes of nucleation and condensation, (b) changing them into large size cloud droplets through the processes of condensation, agglomeration, etc., and (c) changing these large cloud droplets into rain drops. The proposed model is analyzed using stability theory of differential equations. It is found that only one equilibrium is feasible and sufficient conditions for stability of such equilibrium are obtained. It is shown that the intensity of rainfall increases as the cumulative concentration of externally introduced aerosols in the atmosphere increases. Analysis reveals that for the continuous rainfall, it is necessary that water vapors must be continuously formed in the atmosphere. The numerical simulation of the model supports the analytical results. © 2016, Springer International Publishing Switzerland.
Stochastic stability of aerosols-stimulated rainfall model
(Elsevier B.V., 2019) A.K. Misra; Amita Tripathi
With increasingly erratic nature of monsoon, especially in Indian Ocean region clamors for artificial rain have gathering pace recently. For artificial rain, aerosols are used which act as cloud condensation nuclei (CCN) in the formation of cloud droplets from water vapors. In this paper, a nonlinear mathematical model is proposed and analyzed to increase the intensity of rain in the rain deficient regions using aerosols. To stimulate rainfall two kinds of aerosols are used. We have considered three different stages of water involved in the process of precipitation namely, (i) the density of water vapors, (ii) the density of cloud droplets, and (iii) the density of raindrops. Previously proposed models consist of a set of ordinary differential equations (ODE), whereas the use of stochastic differential equations (SDE) depicts more realistic situation as they include environmental disturbances. First, we propose a deterministic model and then convert it into a stochastic model by introducing white noise terms. We have found that the deterministic model is globally asymptotically stable without any condition, while the stability of stochastic model solely depends on the intensity of white noise. © 2019 Elsevier B.V.