Browsing by Author "J.B. Shukla"

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A mathematical model for the depletion of forestry resources due to population and population pressure augmented industrialization
(World Scientific Publishing Co. Pte Ltd, 2014) A.K. Misra; Kusum Lata; J.B. Shukla
In this paper, a nonlinear mathematical model is proposed and analyzed to study the depletion of forestry resources caused simultaneously by population and population pressure augmented industrialization. The control of population pressure, using economic efforts is also considered in the modeling process. It is assumed that cumulative biomass density of forestry resources and the density of population follow logistic models. It is further assumed that the density of population and the level of industrialization increase as the cumulative biomass density of forestry resources increases. The cumulative density of economic efforts, which are applied to control the population pressure, is considered to be proportional to the population pressure. The model analysis shows that as the population pressure increases, the level of industrialization increases leading to decrease in the cumulative biomass density of forestry resources. It is found that if population pressure is controlled by using some economic efforts, the decrease in cumulative biomass density of forestry resources can be made much less than the case when no control is applied. It is also noted that if the population pressure augmented industrialization increases without control, the forestry resources may become extinct. © 2014 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 modeling study on the role of fungi in removing inorganic pollutants
(2013) Ashish Goyal; Rashmi Sanghi; A.K. Misra; J.B. Shukla
In this paper, a non-linear mathematical model for removing an inorganic pollutant such as chromium from a water body using fungi is proposed and analyzed. It is assumed that the inorganic pollutant is discharged in a water body with a constant rate, which is depleted due to natural factors as well as by fungal absorption using dissolved oxygen in the process. The model is analyzed by using stability theory of differential equations and simulation. The analysis shows that the inorganic pollutant can be removed from the water body by fungal absorption, the rate of removal depends upon the concentration of inorganic pollutant, the density of fungal population and various interaction processes. The simulation analysis of the model confirms the analytical results. It is noted here this theoretical result is qualitatively in line with the experimental observations of one of the authors (Sanghi). © 2013 Elsevier Inc.
A nonlinear mathematical model to study the interactions of hot gases with cloud droplets and raindrops
(2009) Shyam Sundar; Ram Naresh; A.K. Misra; J.B. Shukla
In this paper, a nonlinear mathematical model is proposed and analyzed to study the interactions of hot gases with cloud droplets as well as with raindrops and their removal by rain from the stable atmosphere. The atmosphere, during rain, is assumed to consist of five nonlinearly interacting phases i.e. the vapour phase, the phase of cloud droplets, the phase of raindrops, the phase of hot gaseous pollutants and the absorbed phase of hot gases in the raindrops (if it exists). It is further assumed that these phases undergo ecological type growth and nonlinear interactions. The proposed model is analyzed using stability theory of differential equations and by numerical simulation. It is shown that the cumulative concentration of gaseous pollutants decreases due to rain and its equilibrium level depends upon the density of cloud droplets, the rate of formation of raindrops, emission rate of pollutants, the rate of falling absorbed phase on the ground, etc. It is noted here that if gases are very hot, cloud droplets are not formed and rain may not take place. In such a case gaseous pollutants may not be removed from the atmosphere due to non-occurrence of rain. © 2008 Elsevier Inc. All rights reserved.
Effect of global warming on sea level rise: A modeling study
(Elsevier B.V., 2017) J.B. Shukla; Maitri Verma; A.K. Misra
Global mean sea level has been rising in response to global warming since the past few decades and is anticipated to potentially affect the coastal population. The main driver of global warming is the enhanced concentration of the heat-trapping gas carbon dioxide in the atmosphere. In this paper, we propose a nonlinear mathematical model to study the effect of an increase in the anthropogenic carbon dioxide emissions on sea level rise and its effect on the human population. The long-term behavior of the proposed system is analyzed using stability theory of differential equations. The model analysis shows that an increase in the anthropogenic emission rate of carbon dioxide leads to increase in the equilibrium levels of surface temperature and sea water level. Further, it is found that the increase in anthropogenic emission rate of carbon dioxide and melting rate of ice sheets lead to decrease in the equilibrium level of human population as a result of crowding caused by the decrease in the total inhabitable land area due to sea level rise. Numerical simulations are carried out to illustrate the effect of key parameters on the dynamics of the system. © 2017 Elsevier B.V.
Effects of population and population pressure on forest resources and their conservation: A modeling study
(Kluwer Academic Publishers, 2014) A.K. Misra; Kusum Lata; J.B. Shukla
In this paper, a nonlinear mathematical model is proposed and analyzed to study the depletion of forest resources caused by population and the corresponding population pressure. It is assumed that the cumulative density of forest resources and the density of populations follow logistic models with prey-predator type nonlinear interaction terms. It is considered that the carrying capacity of forest resources decreases by population pressure, the main focus of this paper. A conservation model is also proposed to control the population pressure by providing some economic incentives to people, the amount of which is assumed to be proportional to the population pressure. The model is analyzed by using stability theory of differential equations and numerical simulation. The model analysis shows that as the density of population or population pressure increases, the cumulative density of forest resources decreases, and the resources may become extinct if the population pressure becomes too large. It is also noted that by controlling the population pressure, using some economic incentives, the density of forest resources can be maintained at an equilibrium level, which is population density dependent. The simulation analysis of the model confirms analytical results. © 2013 Springer Science+Business Media Dordrecht.
How artificial rain can be produced? A mathematical model
(2010) J.B. Shukla; A.K. Misra; Ram Naresh; Peeyush Chandra
A non-linear mathematical model for rain making from water vapor in the atmosphere is proposed and analyzed. The model considers the process of artificial rain by introducing two kinds of aerosol particles conducive to nucleation of cloud droplets and formation of rain drops. The model analysis shows that, for uninterrupted rain, the water vapor in the atmosphere must be formed continuously with the required rate of rainfall. It is shown further that the intensity of rainfall increases as the concentrations of externally introduced aerosols, as well as the density of water vapor in the atmosphere, increase. Numerical simulation is also performed to see the effect of various parameters on the process of artificial rain making leading to rainfall. © 2009 Elsevier Ltd. All rights reserved.
Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases
(2011) A.K. Misra; Anupama Sharma; J.B. Shukla
In this paper, a non-linear mathematical model for the effects of awareness programs on the spread of infectious diseases such as flu has been proposed and analyzed. In the modeling process it is assumed that disease spreads due to the contact between susceptibles and infectives only. The growth rate of awareness programs impacting the population is assumed to be proportional to the number of infective individuals. It is further assumed that due to the effect of media, susceptible individuals form a separate class and avoid contact with the infectives. The model is analyzed by using stability theory of differential equations. The model analysis shows that the spread of an infectious disease can be controlled by using awareness programs but the disease remains endemic due to immigration. The simulation analysis of the model confirms the analytical results. © 2010 Elsevier Ltd.
Modeling and analysis of the depletion of organic pollutants by bacteria with explicit dependence on dissolved oxygen
(Rocky Mountain Mathematics Consortium, 2014) A.K. Misra; P.K. Tiwari; Ashish Goyal; J.B. Shukla
In this paper, a nonlinear mathematical model is proposed and analyzed to study the depletion of dissolved oxygen caused by interactions of organic pollutants with bacteria in a water body, such as lake. The system is assumed to be governed by three dependent variables, namely, the cumulative concentration of organic pollutants, the density of bacteria and the concentration of dissolved oxygen. In the model, the coefficient of interaction of organic pollutants with bacteria depends upon the concentration of dissolved oxygen nonlinearly and explicitly, which is the main focus of this paper, has never been studied before. The stability theory of differential equations is used to analyze the model and to confirm the analytical results numerical simulation is performed. The model analysis shows that if the coefficient of interaction mentioned above depends upon dissolved oxygen explicitly, the decrease in its concentration is more than the case when the interaction does not depend on dissolved oxygen and consequently the depletion of organic pollutants is also more in such a case. © 2014 Wiley Periodicals, Inc.
Modeling and analysis of the removal of an organic pollutant from a water body using fungi
(Elsevier Inc., 2014) Ashish Goyal; Rashmi Sanghi; A.K. Misra; J.B. Shukla
In this paper, a non linear mathematical model for removing an organic pollutant such as a dye from a water body is proposed and analyzed. In the modeling process four variables are considered, namely, (i) the concentration of the dye, (ii) the density of fungus population, (iii) the concentration of a nutrient and (iv) the concentration of dissolved oxygen (DO). It is assumed that an organic pollutant is present in water with given concentration or discharged with a constant rate in water. It is assumed further that fungus population is kept alive and active due to supply of a nutrient. It is considered that nutrient and oxygen are supplied to the water body from outside with constant rates. The model is analyzed by using the stability theory of differential equations. The model analysis shows that organic pollutant can be removed from the water body by fungus population and the level of degradation depends upon the concentration of organic pollutant, the density of fungal population and the interaction processes involved.The simulation analysis of the proposed model confirms the analytical results. It is also found that these results are qualitatively in line with the experimental observations of one of the authors (Sanghi). © 2014 Elsevier Inc.
Modeling the depletion of a renewable resource by population and industrialization: Effect of technology on its conservation
(Rocky Mountain Mathematics Consortium, 2011) J.B. Shukla; Kusum Lata; A.K. Misra
In this paper, a nonlinear mathematical model is proposed and analyzed to study the depletion of a renewable resource by population and industrialization with resource-dependent migration. The effect of technology on resource conservation is also considered. In the modeling process, four variables are considered, namely, density of a renewable resource, population density, density of industrialization, and technological effort. Both the growth rate and carrying capacity of resource biomass, which follows logistic model, are assumed to be simultaneously depleted by densities of population and industrialization but it is conserved by technological effort. It is further assumed that densities of population and industrialization increase due to increase in the density of renewable resource. The growth rate of technological effort is assumed to be proportional to the difference of carrying capacity of resource biomass and its current density. The model is analyzed by using the stability theory of differential equations and computer simulation. The model analysis shows that the biomass density decreases due to increase in densities of population and industrialization. It decreases further as the resource-dependent industrial migration increases. But the resource may never become extinct due to population and industrialization, if technological effort is applied appropriately for its conservation. Copyright © 2011 Wiley Periodicals, Inc.
Modeling the desalination of saline water by using bacteria and marsh plants
(2011) J.B. Shukla; Rashmi Sanghi; Ashish Goyal; A.K. Misra
On the planet earth, large amount of saline water (in seas and some lakes) is present and a concerted effort is needed to desalinate this water by using cheaper methods already existing in nature. Keeping this in view, a non-linear model for desalination of saline water is proposed and analyzed by using bacteria (halophiles) and marsh plants (salt grass). The system is assumed to be governed by four dependent variables namely the salt concentration, the density of halophile bacteria, the biomass density of marsh plants and the concentration of dissolved oxygen. The density of halophiles and biomass density of marsh plants are assumed to follow logistic models and their growth rates and carrying capacities increase due to interactions with salt in water. The model is analyzed by using the theory of differential equations as well as by using numerical simulation. The analysis shows that salt concentration can become very small if densities of halophiles and marsh plants are very large. The analytical result is confirmed by numerical simulation. © 2011 Elsevier B.V.
Modeling the effects of aerosols to increase rainfall in regions with shortage
(Springer-Verlag Wien, 2013) J.B. Shukla; Shyam Sundar; A.K. Misra; Ram Naresh
It is well known that the emissions of hot gases from various power stations and other industrial sources in the regional atmosphere cause decrease in rainfall around these complexes. To overcome this shortage, one method is to introduce artificially conducive aerosol particles in the atmosphere using aeroplane to increase rainfall. To prove the feasibility of this idea, in this paper, a nonlinear mathematical model is proposed involving five dependent variables, namely, the volume density of water vapour, number densities of cloud droplets and raindrops, and the concentrations of small and large size conducive aerosol particles. It is assumed that two types of aerosol particles are introduced in the regional atmosphere, one of them is of small size CCN type which is conducive to increase cloud droplets from vapour phase, while the other is of large size and is conducive to transform the cloud droplets to raindrops. The model is analyzed using stability theory of differential equations and computer simulation. The model analysis shows that due to the introduction of conducive aerosol particles in the regional atmosphere, the rainfall increases as compared to the case when no aerosols are introduced in the atmosphere of the region under consideration. The computer simulation confirms the analytical results. © 2013 Springer-Verlag Wien.
Modeling the role of dissolved oxygen-dependent bacteria on biodegradation of organic pollutants
(World Scientific Publishing Co. Pte Ltd, 2014) J.B. Shukla; Ashish Goyal; P.K. Tiwari; A.K. Misra
In this paper, a nonlinear mathematical model is proposed and analyzed to study the role of dissolved oxygen (DO)-dependent bacteria on biodegradation of one or two organic pollutant(s) in a water body. In the case of two organic pollutant(s), it is assumed that the one is fast degrading and the other is slow degrading and both are discharged into the water body from outside with constant rates. The density of bacteria is assumed to follow logistic model and its growth increases due to biodegradation of one or two organic pollutant(s) as well as with the increase in the concentration of DO. The model is analyzed using the stability theory of differential equations and by simulation. The model analysis shows that the concentration(s) of one or both organic pollutant(s) decrease(s) as the density of bacteria increases. It is noted that for very large density of bacteria, the organic pollutant(s) may be removed almost completely from the water body. It is found that simulation analysis confirms the analytical results. The results obtained in this paper are in line with the experimental observations published in literature. © 2014 World Scientific Publishing Company.
Modeling the role of government efforts in controlling extremism in a society
(John Wiley and Sons Ltd, 2015) Ashish Goyal; J.B. Shukla; A.K. Misra; Ajai Shukla
People having extreme idealogies affect the process in a region using fear of terror acts, money power, and the word of mouth communication network to change individuals to their way of thinking. This forces government to divert its limited financial resources for controlling extremism and thus affecting development. In this paper, therefore, a nonlinear mathematical model is proposed to study the dynamics of extremism governed by four dependent variables, namely, number of people in the general population having no extreme ideology, number of extreme ideologists, number of isolated ideologists (prisoners), and the cumulative density of government efforts and their interactions. The model is analyzed using the stability theory of differential equations and computer simulation. The analysis shows that if appropriate level of government efforts is applied on extremists, the spread of their ideology can be controlled in the general population. A numerical study of the model is also carried out to investigate the effects of certain parameters on the spread of extremism confirming the analytical results. Copyright © 2014 John Wiley & Sons, Ltd.
Modeling the spread of an infectious disease with bacteria and carriers in the environment
(2011) J.B. Shukla; Vishal Singh; A.K. Misra
An SIS model with immigration for the spread of an infectious disease with bacteria and carriers in the environment is proposed and analyzed. It is assumed that susceptibles get infected directly by infectives as well as by their contacts with bacteria discharged by infectives in the environment. The growth rate of density of bacteria is assumed to be proportional to the density of infectives and decreases naturally as well as by bacterial interactions with susceptibles and carriers. The carrier population density is considered to follow the logistic model and grows due to conducive human population density related factors. It is assumed further that the number of bacteria transported by carriers to susceptibles is proportional to densities of both bacteria and carriers. The model study shows that the spread of the infectious disease increases due to growth of bacteria and carriers in the environment and disease becomes more endemic due to immigration. © 2011 Elsevier Ltd. All rights reserved.
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.
Modelling the Effect of Washout of Hot Pollutants for Increasing Rain in an Industrial Area
(Springer Science and Business Media Deutschland GmbH, 2023) Shyam Sundar; A.K. Misra; Ram Naresh; J.B. Shukla
In areas where hot gases and nanoparticulate materials are released uncontrollably from pollutants generating sources which are human population density dependent, the rainfall is significantly affected. When these hot gases and nanoparticulate materials mix with clouds, the cloud droplets evaporate, causing no rain or less intense rain. For increasing rain, the abatement of these pollutants is necessary and the washout of these pollutants by water drops is one of the effective removal mechanisms. Taking this into account, in this paper, a nonlinear mathematical model is proposed to analyse the effect of hot pollutants on rainfall and its control by washout of hot pollutants at sources of emission using spraying water droplets. To model the phenomenon, five nonlinearly interacting variables are considered namely densities of water vapours, clouds droplets, raindrops, the cumulative concentration of hot pollutants and the density of water drops introduced in the atmosphere to washout the hot pollutants. On analysing the model, using the stability theory of differential equations, it is found that the rain fall decreases as the cumulative concentration of hot pollutants increases. The spraying of water drops at the sources of emissions in the industrial area reduces the cumulative concentration of hot pollutants, which ultimately stimulates an increase in the density of raindrops in the atmosphere. Numerical simulation is also done to validate analytical results. © 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
Stability analysis and optimal control of an epidemic model with awareness programs by media
(Elsevier Ireland Ltd, 2015) A.K. Misra; Anupama Sharma; J.B. Shukla
The impact of awareness campaigns and behavioral responses on epidemic outbreaks has been reported at times. However, to what extent does the provision of awareness and behavioral changes affect the epidemic trajectory is unknown, but important from the public health standpoint. To address this question, we formulate a mathematical model to study the effect of awareness campaigns by media on the outbreak of an epidemic. The awareness campaigns are treated as an intervention for the emergent disease. These awareness campaigns divide the whole populations into two subpopulation; aware and unaware, by inducing behavioral changes amongst them. The awareness campaigns are included explicitly as a separate dynamic variable in the modeling process. The model is analyzed qualitatively using stability theory of differential equations. We have also identified an optimal implementation rate of awareness campaigns so that disease can be controlled with minimal possible expenditure on awareness campaigns, using optimal control theory. The control setting is investigated analytically using optimal control theory, and the numerical solutions illustrating the optimal regimens under various assumptions are also shown. © 2015 Elsevier Ireland Ltd.