Department of Mathematics

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Subject: search.filters.subject.Stability

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Now showing 1 - 10 of 27
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    Dynamics and stability analysis of enzymatic cooperative chemical reactions in biological systems with time-delayed effects
    (Elsevier, 2024-09) Sharma, Bhupendra Kumar
    The mathematical modeling and dynamic analysis of time-delayed enzymatic chemical reactions in biological systems are presented in this research. The objective is to examine the function of time lags in these reactions and to get a complete knowledge of the behavior of biological systems in a reaction to modifications in the quantity present of reactants and products. The model, which is based on delay differential equations, includes a time delay term to account for the lag between changes in the concentration of reactants, reaction rate constants and product responses. The findings give insight into how enzymatic processes behave dynamically and how stability is impacted by time lags, oscillation and general efficiency of the system. These results have significant importance for our comprehension of how biological processes are regulated and for the creation of biological control structures
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    The impact of radio-chemotherapy on tumour cells interaction with optimal control and sensitivity analysis
    (Elsevier, 2024-03) Dubey, Balram; Dubey, Uma S.
    Oncologists and applied mathematicians are interested in understanding the dynamics of cancer-immune interactions, mainly due to the unpredictable nature of tumour cell proliferation. In this regard, mathematical modelling offers a promising approach to comprehend this potentially harmful aspect of cancer biology. This paper presents a novel dynamical model that incorporates the interactions between tumour cells, healthy tissue cells, and immune-stimulated cells when subjected to simultaneous chemotherapy and radiotherapy for treatment. We analysed the equilibria and investigated their local stability behaviour. We also study transcritical, saddle–node, and Hopf bifurcations analytically and numerically. We derive the stability and direction conditions for periodic solutions. We identify conditions that lead to chaotic dynamics and rigorously demonstrate the existence of chaos. Furthermore, we formulated an optimal control problem that describes the dynamics of tumour-immune interactions, considering treatments such as radiotherapy and chemotherapy as control parameters. Our goal is to utilize optimal control theory to reduce the cost of radiotherapy and chemotherapy, minimize the harmful effects of medications on the body, and mitigate the burden of cancer cells by maintaining a sufficient population of healthy cells. Cost-effectiveness analysis is employed to identify the most economical strategy for reducing the disease burden. Additionally, we conduct a Latin hypercube sampling-based uncertainty analysis to observe the impact of parameter uncertainties on tumour growth, followed by a sensitivity analysis. Numerical simulations are presented to elucidate how dynamic behaviour of model is influenced by changes in system parameters. The numerical results validate the analytical findings and illustrate that a multi-therapeutic treatment plan can effectively reduce tumour burden within a given time frame of therapeutic intervention.
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    The impact of social media advertisements and treatments on the dynamics of infectious diseases with optimal control strategies
    (Elsevier, 2024-05) Dubey, Uma S.; Dubey, Balram
    The dissemination of public health information through television and social media posts is essential for informing the public about the transmission of contagious diseases, which is crucial in preventing the spread of various infectious diseases. In this paper, we propose a non-linear mathematical model to assess the effect of advertisements through social media in creating awareness and limiting treatment on spreading infectious diseases. These initiatives may alter population behaviour and divide the susceptible population into subgroups. In addition, to comprehend these dynamics better, we use half-saturation constant rates for media coverage and treatment. The model’s well-posedness and feasibility are evaluated. The possible biological equilibrium points are calculated. Local and global stability are carried out. The objective of our study is to produce the model’s bifurcation. Transcritical, Saddle–node, Hopf bifurcation of codimension 1 and Cusp, Generalized-Hopf (Bautin), and Bogdanov–Takens (BT) bifurcation of codimension 2 are studied for this purpose. Due to the limited medical resources and supply efficiency, the model exhibits backward bifurcation, resulting in bistability. Moreover, the occurrence condition for stability and direction of Hopf bifurcation is discussed. This model study demonstrates that the system is significantly influenced by the pace with which awareness programmes are implemented and that raising this value above a threshold may result in continuous oscillation. Sensitivity analysis employs the normalized forward sensitivity index of the basic reproduction number to provide a comprehensive understanding of the effect of various parameters on accelerating and limiting disease spread. Further, the minimum possible cost is determined by formulating an optimal control system based on sensitivity analysis and applying Pontryagin’s maximum principle. Methods of cost-effectiveness, such as ACER and ICER, are used to determine the most cost-effective control intervention strategy among all the strategies. Numerical simulations have been done to support all theoretical findings.
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    Fractional differential equation with movable boundary conditions
    (Taru Publication, 2024-03) Mathur, Trilok; Agarwal, Shivi
    In this research paper, we discuss the complex-valued solutions for the nonlinear fractional boundary value problem (FBVP) of complex order (δ = τ + ιa; 1 < τ ≤ 2, a ∈ R+) with movable boundary conditions. The fractional operators are taken in the sense of Riemann-Liouville (R-L) with complex order. By using the concept of Green’s function, the existence and uniqueness of solutions are established in this article. Also, we prove that the FBVP of complex order with movable boundary conditions is Ulam-Hyers Stable. Using illustrative examples, the results for this nonlinear FBVP have been shown.
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    Stability of Positive Solution to Fractional Logistic Equations
    (Division of Functional Equations, 2019) Dwivedi, Gaurav
    In this paper, we show the existence of a classical solution to a class of fractional logistic equations in an open bounded subset with smooth boundary. We use the method of sub- and super-solutions with variational arguments to establish the existence of a unique positive solution. We also establish the stability and nondegeneracy of the positive solution.
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    Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment
    (Science PC, 2016-10) Kulshrestha, Rakhee
    The mathematical model for analyzing the transmission dynamics of HIV/AIDS epidemic with treatment is studied by considering the three latent compartments for slow, medium and fast progresses of developing the AIDS. By constructing the system of differential equations for the different population groups namely susceptible, three types of latent individuals, symptomatic stage group and full blown AIDS individuals, the mathematical analysis is carried out in order to understand the dynamics of disease spread. By determining the basic reproduction number (R0), the model examines the two equilibrium points (i) the disease free equilibrium and (ii) the endemic equilibrium. It is established that if R0 <1, the disease free equilibrium is locally and globally asymptotically stable. The stability of endemic equilibrium has also been discussed.
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    Simultaneous effect of two toxicants on biological species: a mathematical model
    (World Scientific, 1996) Dubey, Balram
    In this paper, a mathematical model to study the simultaneous effect of two toxicants (one is more toxic than the other) on the growth and survival of a biological species is proposed. The cases of instantaneous spill, constant and periodic emissions of each of the toxicant into the environment are considered. It is shown that in the case of an instantaneous spill of each of the toxicant into the environment, the species after its initial decrease in density may recover to its original level after a period of time, the magnitude of which depends on the toxicity and washout rate of each of the toxicant. However, if both the toxicants are emitted with constant rates, the species in the habitat is doomed to extinction sooner than the case of a single toxicant having the same influx and washout rates as one of them, the extinction rate becoming faster with the increase in toxicity and emission rate of the other toxicant. It is also shown that for a small amplitude periodic emission of the toxicant with a constant mean, the stability behavior of the system is same as that of the case of the constant emission. It is found further through the model study that if suitable efforts are made to reduce the emission rate of each of the toxicant at the source and its concentration in the environment by some removal mechanism, an appropriate level of species density can be maintained.
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    A model for the allelopathic effect on two competing species
    (Elsevier, 2000-05) Dubey, Balram
    In this paper, a mathematical model is proposed and analysed to study the coexistence of two competing plant species in a finite habitat by assuming that each species produces a toxic substance affecting the other species. The diffusion of toxic substances is also considered in the model. It is shown that the usual existence criteria between two competing species in the absence of toxicant may be changed if each species produces toxicant in large amount affecting the other. In case of no diffusion criteria for local stability, instability and global stability of the system are obtained. In case of allelopathy, where one species produces toxicant and affects the other, it is found that the affected species may be driven to extinction. It is also found that diffusion has a stabilizing effect on the system.
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    A predator–prey interaction model with self and cross-diffusion
    (Elsevier, 2001-07) Dubey, Balram
    In this paper, a mathematical model for a predator–prey interaction with self and cross-diffusion is proposed and analysed. Criteria for local stability, instability and global stability are obtained. The effect of the critical wave length which can drive a system to instability is investigated. The effect of time-varying cross-diffusivity on the stability of the system is also examined.
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    Effects of industrialization and pollution on resource biomass: a mathematical model
    (Elsevier, 2003-09) Dubey, Balram
    In this paper, a mathematical model is proposed and analyzed to study the depletion of resource biomass (plant/tree) due to industrialization and pollution. Industrialization dependent, constant, instantaneous, and periodic emissions of pollutant into the environment are taken into consideration. Criteria for local stability, instability, and global stability of non-negative equilibria are obtained. Numerical simulations are carried out to investigate the dynamics of the system. It is found that in the case of small periodic influx of pollutant into the environment, the resource biomass has a periodic behavior if the depletion rate coefficient of environmental pollutant is small. However, if this coefficient increases beyond a threshold value, then resource biomass converges towards its equilibrium.