BITS Faculty Publications
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Item Modeling the effect of vaccinations, hospital beds, and treatments on the dynamics of infectious disease with delayed optimal control and sensitivity analysis(Springer, 2024-08) Dubey, Uma S.; Dubey, BalramImmunization plays a vital role in eradicating infectious diseases, typically requiring multiple doses at specific time intervals. This study focuses on developing and analyzing an infectious disease model governed by a six-dimensional system of ordinary differential equations, considering the impact of first and second vaccination doses along with hospital beds and treatment. The model’s qualitative behavior is analyzed, including conditions for positive solutions, the invariant region of the solution, equilibrium points, and their stability. When the basic reproduction number () is less than one (), the disease will be eradicated; conversely, an epidemic occurs when . Moreover, the transcritical bifurcation of the system is examined using the center manifold theory. Interestingly, backward bifurcation is discovered, and it indicates that the disease is not entirely eradicated even when . We have investigated different bifurcations like saddle-node, transcritical, and Hopf bifurcations of codimension 1, as well as Generalized-Hopf (GH), Cusp (CP), and Bogdanov–Takens (BT) bifurcations of codimension 2. Additionally, a delayed epidemiological model is explored, assuming a lag in vaccination among the susceptible population. A Hopf-bifurcation is observed near the endemic equilibrium point, linked to critical parameter values during the latent period. Moreover, the model is calibrated using the least-squares technique, incorporating coronavirus-infected case data and vaccination information from India and Italy’s mass vaccination program between March 1, 2021, and May 30, 2021. Global sensitivity analysis, utilizing the Partial Rank Correlation Coefficient (PRCC), identifies crucial parameters affecting threshold quantities after fitting the model. The study highlights the significance of critical parameters such as the effective transmission rate, rates of first and second-dose vaccinations, and recovery rate due to double-dose vaccination. Further, delayed optimal control measures are determined using Pontryagin’s maximal principle to mitigate infection, prevention, and treatment burdens. Numerical simulations are conducted to understand the effect of these delayed control measures on disease progression and demonstrate the insights obtained through analytical investigations. The study indicates that implementing all control strategies effectively reduces the disease burden among the population. Accurate estimation of vaccine efficacy is crucial for disease prevention, underlining the importance of well-planned vaccination strategies. Moreover, the numerical simulations validate all the theoretical findings, emphasizing the validity of this model in a real-world situation. Relying solely on vaccination might not swiftly or completely control the disease. Complementary pharmaceutical and non-pharmaceutical measures are necessary to combat the infection effectively. Further limitations on medical resources could lead to a backward bifurcation. Simulation results suggest that delaying the implementation of control measures could exacerbate epidemic situations.Item Impact of chemo-immunotherapy on tumour-immune interactions: a non-autonomous model of tumor necrosis factor and T cell dynamics(2025) Dubey, Uma S.; Dubey, BalramThis study explores the interaction between cancer cells, helper T cells, cytotoxic T cells, and tumour necrosis factors in chemotherapy and immunotherapy treatment in the microenvironment [1]. The goal is to analyze the connection of helper and cytotoxic T-cell levels with the anti-tumour immune response and the impact of various dosing regimens when combined with immunotherapy and chemotherapy. These protocols aim to shorten the interval between treatment cycles from three to two weeks or less to prevent tumour regrowth and maximize its cell elimination by treatment. Motivated by clinical trials, we thoroughly compare procedures involving two medications supplied sequentially or simultaneously in a non-autonomous system. We discussed the positivity and boundedness of the model. Further, we analyze the biologically valid equilibria and investigate their local stability properties, examining transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations numerically and analytically [2]. Furthermore, direction and stability conditions for periodic solutions are determined.Item Bifurcation and chaotic dynamics in a spatiotemporal epidemic model with delayed optimal control, stochastic process, and sensitivity analysis(AIP, 2025-03) Dubey, Uma S.; Dubey, BalramThis study introduces an epidemic model with a Beddington–DeAngelis-type incidence rate and Holling type II treatment rate. The Bedding- ton–DeAngelis incidence rate is used to evaluate the effectiveness of inhibitory measures implemented by susceptible and infected individuals. Moreover, the choice of Holling type II treatment rate in our model aims to assess the impact of limited treatment facilities in the context of disease outbreaks. First, the well-posed nature of the model is analyzed, and then, we further investigated the local and global stability analysis along with bifurcation of co-dimensions 1 (transcritical, Hopf, saddle-node) and 2 (Bogdanov–Takens, generalized Hopf) for the system. Moreover, we incorporate a time-delayed model to investigate the effect of incubation delay on disease transmission. We provide a rigorous demonstration of the existence of chaos and establish the conditions that lead to chaotic dynamics and chaos control. Additionally, sensitivity analysis is performed using partial rank correlation coefficient and extended Fourier amplitude sensitivity test methods. Furthermore, we delve into optimal control strategies using Pontryagin’s maximum principle and assess the influence of delays in state and control parame- ters on model dynamics. Again, a stochastic epidemic model is formulated and analyzed using a continuous-time Markov chain model for infectious propagation. Analytical estimation of the likelihood of disease extinction and the occurrence of an epidemic is conducted using the branching process approximation. The spatial system presents a comprehensive stability analysis and yielding criteria for Turing instability. Moreover, we have generated the noise-induced pattern to assess the effect of white noise in the populations. Additionally, a case study has been conducted to estimate the model parameters, utilizing COVID-19 data from Poland and HIV/AIDS data from India. Finally, all theo- retical results are validated through numerical simulations. This article extensively explores various modeling techniques, like deterministic, stochastic, statistical, pattern formation(noise-induced), model fitting, and other modeling perspectives, highlighting the significance of the inhibitory effects exerted by susceptible and infected populations.Item Stability and bifurcation analysis of an infectious disease model with different optimal control strategies(Elsevier, 2023-11) Dubey, Balram; Dubey, Uma S.This paper deals with the non-linear Susceptible–Infected–Hospitalized–Recovered model with Holling type II incidence rate, treatment with saturated type functional response for the prevention and control of disease with limited healthcare facilities. The well-posedness of the model is ensured with the help of the non-negativity and boundedness of the solution of the system. The feasibility of the model with DFE (Disease-free equilibrium) and EE (endemic equilibrium) is analysed. The local and global stability are discussed with the help of the computed basic reproduction number . At , we use the Centre manifold theory to analyse the transcritical bifurcation exhibited by the system. It is found that the disease is not eradicated even if due to the occurrence of backward bifurcation. The occurrence condition of Hopf bifurcation is obtained. The optimal control theory is used to analyse the effects of the minimum possible medical facilities, hospital beds, and awareness creation on the population dynamics. The Hamiltonian function is constructed with the extended optimal control model and solved by Pontryagin’s maximum principle to get the minimum possible expenditure. Different types of control strategies are shown by numerical simulation. The sensitivity analysis is discussed with the help of a crucial parameter that depends on the reproduction number. Further, the model is simulated numerically to support the theoretical studies. This paper emphasizes the significance of treatment intensity, the total number of hospital bed available and their occupancy rate as vital parameters for prevention of disease prevalence.Item An SIR Model with Nonlinear Incidence Rate and Holling Type III Treatment Rate(Digital Science, 2016-12) Dubey, Uma S.; Dubey, BalramWe propose a mathematical model with nonlinear incidence rate and treatment rate to study the dynamics of susceptible-infected-recovered population. We consider nonlinear incidence rate as Crowley-Martin type and nonlinear treatment rate as Holling type III (saturated treatment function). The global stability analysis of disease-free equilibrium point and endemic equilibrium point has been investigated using Lasalles’ invariance principle and Lyapunov function. A threshold value has been found to ensure the extinction or persistence of infection. The non-existence of periodic solutions have been shown using Dulac’s criterion. Numerical simulations are performed to validate these analytical findings.Item OPTIMAL CONTROL FOR THERAPEUTIC DRUG TREATMENT ON A DELAYED MODEL INCORPORATING IMMUNE RESPONSE(World Scientific, 2015) Dubey, Uma S.; Dubey, BalramMillions of people get infected every year by viral pathogens. Newly emergent diseases such as Ebola, Swine-flu, HIV/AIDS, etc. are spreading worldwide at an alarming rate. We introduced a delayed mathematical model with immune response and therapeutic drug treatment to understand the dynamics of pathogenimmune interaction. Here, we are considering the innate immune response and the two major component of the acquired immune response, namely, cytotoxic T lymphocytes (CTLs) and humoral immunity. This model also incorporates the absorption of pathogens i.e. loss of pathogens and its related mechanisms. Further, an optimal control model is formulated with two optimal controls i.e. maximization of uninfected cells count and minimization of cost of treatments. This is done by using the Pontryagins' Maximum Principle. Existence of non-negative equilibria is established and their stability behavior is studied using theory of ordinary differential equations. Further, numerical simulations are carried out to exemplify the qualitative results.Item A MATHEMATICAL MODEL FOR THE EFFECT OF TOXICANT ON THE IMMUNE SYSTEM(World Scientific, 2007) Dubey, Uma S.; Dubey, BalramIn this paper, a nonlinear mathematical model is proposed and analyzed to study the effect of environmental toxicant on the immune response of the body. Criteria for local stability, instability and global stability are obtained. It is shown that the immune response of the body decreases as the concentration of environmental toxicant increases, and certain criteria are obtained under which it settles down at its equilibrium level. In the absence of toxicant, an oscillatory behavior of immune system and pathogenic growth is observed. However, in the presence of toxicant, oscillatory behavior is not observed. These studies show that the toxicant may have a grave effect on our body's defense mechanism.Item MODELING THE INTERACTION BETWEEN AVASCULAR CANCEROUS CELLS AND ACQUIRED IMMUNE RESPONSE(World Scientific, 2008) Dubey, Uma S.; Dubey, BalramThis paper deals with the interaction between dispersed cancer cells and the major populations of the immune system, namely, the T helper cells, T Cytotoxic cells, B cells, and antibodies produced. The system is described by a set of five ordinary differential equations. Both local and global stability of the system has been investigated. It has been observed that under appropriate conditions this interaction is capable of controlling the growth of these cancer cells. The analytical findings are supported by numerical and computational analytical methods.Item MODELING EFFECTS OF TOXICANT ON UNINFECTED CELLS, INFECTED CELLS AND IMMUNE RESPONSE IN THE PRESENCE OF VIRUS(World Scientific, 2011) Dubey, Uma S.; Dubey, BalramIn this paper, two mathematical models are proposed and analyzed. The first one deals with the interaction of uninfected cells, infected cells, viruses and immune response within humans. The second one deals with the effects of environmental toxicant on the first model. In each case, sufficient conditions for local stability and global stability of the equilibria are obtained, computer simulations are performed and the result is biologically interpreted. It has been seen that the environmental toxicant has detrimental effects on healthy cells, infected cells as well as on the immune response of the body.Item MODELING AND ANALYSIS OF AN SEIR MODEL WITH DIFFERENT TYPES OF NONLINEAR TREATMENT RATES(World Scientific, 2013) Dubey, Uma S.; Dubey, BalramIn this study, an SEIR epidemic model is proposed for treatment of infectives considering the development of acquired immunity in recovered individuals. We employed two different types of treatment functions. Stability analysis for disease-free as well as endemic equilibria is performed. It is observed that the existence of unique endemic equilibrium depends on the basic reproductive number R0 as well as on treatment rate. Numerical simulations are performed on the proposed models to support and analyze theoretical findings.