Department of Biological Sciences
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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 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 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.Item Dynamics of an SIR Model with Nonlinear Incidence and Treatment Rate(AAM, 2015-12) Dubey, Uma S.; Dubey, BalramIn this paper, global dynamics of an SIR model are investigated in which the incidence rate is being considered as Beddington-DeAngelis type and the treatment rate as Holling type II (saturated). Analytical study of the model shows that the model has two equilibrium points (diseasefree equilibrium (DFE) and endemic equilibrium (EE)). The disease-free equilibrium (DFE) is locally asymptotically stable when reproduction number is less than one. Some conditions on the model parameters are obtained to show the existence as well as nonexistence of limit cycle. Some sufficient conditions for global stability of the endemic equilibrium using Lyapunov function are obtained. The existence of Hopf bifurcation of model is investigated by using Andronov-Hopf bifurcation theorem. Further, numerical simulations are done to exemplify the analytical studies.Item Modeling the intracellular pathogen-immune interaction with cure rate(Elsiever, 2016-09) Dubey, Uma S.; Dubey, BalramMany common and emergent infectious diseases like Influenza, SARS, Hepatitis, Ebola etc. are caused by viral pathogens. These infections can be controlled or prevented by understanding the dynamics of pathogen-immune interaction in vivo. In this paper, interaction of pathogens with uninfected and infected cells in presence or absence of immune response are considered in four different cases. In the first case, the model considers the saturated nonlinear infection rate and linear cure rate without absorption of pathogens into uninfected cells and without immune response. The next model considers the effect of absorption of pathogens into uninfected cells while all other terms are same as in the first case. The third model incorporates innate immune response, humoral immune response and Cytotoxic T lymphocytes (CTL) mediated immune response with cure rate and without absorption of pathogens into uninfected cells. The last model is an extension of the third model in which the effect of absorption of pathogens into uninfected cells has been considered. Positivity and boundedness of solutions are established to ensure the well-posedness of the problem. It has been found that all the four models have two equilibria, namely, pathogen-free equilibrium point and pathogen-present equilibrium point. In each case, stability analysis of each equilibrium point is investigated. Pathogen-free equilibrium is globally asymptotically stable when basic reproduction number is less or equal to unity. This implies that control or prevention of infection is independent of initial concentration of uninfected cells, infected cells, pathogens and immune responses in the body. The proposed models show that introduction of immune response and cure rate strongly affects the stability behavior of the system. Further, on computing basic reproduction number, it has been found to be minimum for the fourth model vis-a-vis other models. The analytical findings of each model have been exemplified by numerical simulations.