BITS Faculty Publications
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Item Spatio-temporal dynamics of an ecological model with Cosner's functional response and prey taxis in networked vs. non-networked environments(Springer, 2025-05) Dubey, Balram; Dubey, Uma S.This study examines a susceptible-infected-temporary-permanent-recovered (SITHR) epidemic model incorporating the Holling type II incidence rate to prevent and control the disease with optimal use of hospital beds. Initially, the well-posedness and feasibility of the model are analyzed, and then valid biological equilibrium points are calculated. Subsequently, the stability of these equilibrium points is assessed and the basic reproduction number is calculated as a threshold value that controls the dynamics of the disease. The proposed model undergoes several bifurcations, including transcritical (backward and forward), saddle-node, Hopf, and Bogdanov–Takens bifurcations. The normal form is derived to demonstrate the presence of a Bogdanov–Takens bifurcation. Furthermore, parameter estimation is conducted using COVID-19 data from Italy to refine the model’s accuracy and boost the reliability of the study’s predictions. Using the normalized forward sensitivity index (NFSI), a sensitivity analysis of parameters associated with the basic reproduction number is performed, and the partial rank correlation coefficient (PRCC) is calculated to locate the key parameters affecting disease transmission dynamics. Moreover, the system is expanded to incorporate time-dependent control variables to reduce the infected population and the cost associated with implementing these controls. The developed optimal control system is employed to build the Hamiltonian function, which is solved using Pontryagin’s maximum principle. Also, a cost-effectiveness analysis is performed to evaluate the economic efficiency of various intervention strategies. Beyond the deterministic framework, the study includes formulations for continuous-time Markov chains and stochastic differential equations to assess the impact of environmental noise on the system. Moreover, the Galton–Watson branching process determines the extinction threshold for the stochastic model and sets the parameters that govern disease extinction or persistence. Finally, numerical simulations are demonstrated to illustrate the impact of changes in system parameters on the dynamic behavior of the model. These findings will enhance preparedness and enable more efficient responses to health emergencies, leading to better patient care and less pressure on healthcare systems.Item Impact of hospital bed availability on infectious disease management: a stochastic and optimal control approach(Springer, 2025-05) Dubey, BalramThis study examines a susceptible-infected-temporary-permanent-recovered (SITHR) epidemic model incorporating the Holling type II incidence rate to prevent and control the disease with optimal use of hospital beds. Initially, the well-posedness and feasibility of the model are analyzed, and then valid biological equilibrium points are calculated. Subsequently, the stability of these equilibrium points is assessed and the basic reproduction number is calculated as a threshold value that controls the dynamics of the disease. The proposed model undergoes several bifurcations, including transcritical (backward and forward), saddle-node, Hopf, and Bogdanov–Takens bifurcations. The normal form is derived to demonstrate the presence of a Bogdanov–Takens bifurcation. Furthermore, parameter estimation is conducted using COVID-19 data from Italy to refine the model’s accuracy and boost the reliability of the study’s predictions. Using the normalized forward sensitivity index (NFSI), a sensitivity analysis of parameters associated with the basic reproduction number is performed, and the partial rank correlation coefficient (PRCC) is calculated to locate the key parameters affecting disease transmission dynamics. Moreover, the system is expanded to incorporate time-dependent control variables to reduce the infected population and the cost associated with implementing these controls. The developed optimal control system is employed to build the Hamiltonian function, which is solved using Pontryagin’s maximum principle. Also, a cost-effectiveness analysis is performed to evaluate the economic efficiency of various intervention strategies. Beyond the deterministic framework, the study includes formulations for continuous-time Markov chains and stochastic differential equations to assess the impact of environmental noise on the system. Moreover, the Galton–Watson branching process determines the extinction threshold for the stochastic model and sets the parameters that govern disease extinction or persistence. Finally, numerical simulations are demonstrated to illustrate the impact of changes in system parameters on the dynamic behavior of the model. These findings will enhance preparedness and enable more efficient responses to health emergencies, leading to better patient care and less pressure on healthcare systems.Item Exploring turing pattern formation in a seasonally forced predator-prey model incorporating fear effects and prey refuge(Springer, 2025-07) Dubey, BalramSeasonal variations critically influence species movement and migration, with profound implications for ecological stability as evidenced by numerous natural phenomena. In this work, we modify the traditional Lotka-Volterra model by incorporating three key mechanisms: predator-induced fear effects on prey reproduction and mortality, prey refuge dynamics, and periodic environmental fluctuations. For the autonomous system, we conduct a comprehensive stability analysis and uncover rich dynamics, including key bifurcation such as saddle-node, Hopf, and codimension-two bifurcations specifically Bogdanov-Takens and cusp bifurcations as well as global homoclinic bifurcations. Building upon the temporal case, we explore the non-autonomous dynamics, by including seasonal changes in the fear and refuge parameters, where we establish criteria for permanence and the existence of globally attractive periodic solutions, highlighting how seasonal forcing can lead to ecological collapse by crossing extinction thresholds. We further analyze a reaction-diffusion system under both autonomous and non-autonomous frameworks to investigate the spatial distribution of species. For non-autonomous cases with time-varying cross-diffusion and periodic reaction rates, we derive Turing instability conditions using comparison principles, expressed through inequalities involving time-varying parameters and their derivatives. The autonomous case recovers classical Turing conditions, validating our generalized approach. Numerical simulations quantify how fear intensity and refuge availability modulate pattern formation, while seasonality induces complex dynamics such as periodic oscillations, chaotic regimes, and bursting behaviors. This study highlights the profound impact of seasonal variations on ecological stability and pattern formation, offering valuable tools for understanding non-autonomous systems in ecological modelling.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 Role reversal in a stage-structured prey–predator model with fear, delay, and carry-over effects(AIP, 2023-09) Dubey, BalramThe present work highlights the reverse side of the same ecological coin by considering the counter-attack of prey on immature predators. We assume that the birth rate of prey is affected by the fear of adult predators and its carry-over effects (COEs). Next, we introduce two discrete delays to show time lag due to COEs and fear-response. We observe that the existence of a positive equilibrium point and the stability of the prey-only state is independent of fear and COEs. Furthermore, the necessary condition for the co-existence of all three species is determined. Our system experiences several local and global bifurcations, like, Hopf, saddle-node, transcritical, and homoclinic bifurcation. The simultaneous variation in the attack rate of prey and predator results in the Bogdanov–Takens bifurcation. Our numerical results explain the paradox of enrichment, chaos, and bi-stability of node-focus and node-cycle types. The system, with and without delay, is analyzed theoretically and numerically. Using the normal form method and center manifold theorem, the conditions for stability and direction of Hopf-bifurcation are also derived. The cascade of predator attacks, prey counter-attacks, and predator defense exhibit intricate dynamics, which sheds light on ecological harmonyItem Spatiotemporal dynamics of a multi-delayed prey–predator system with variable carrying capacity(AIP, 2023-11) Dubey, BalramThis paper presents the temporal and spatiotemporal dynamics of a delayed prey–predator system with a variable carrying capacity. Prey and predator interact via a Holling type-II functional response. A detailed dynamical analysis, including well-posedness and the possibility of coexistence equilibria, has been performed for the temporal system. Local and global stability behavior of the co-existence equilibrium is discussed. Bistability behavior between two coexistence equilibria is demonstrated. The system undergoes a Hopf bifurcation with respect to the parameter , which affects the carrying capacity of the prey species. The delayed system exhibits chaotic behavior. A maximal Lyapunov exponent and sensitivity analysis are done to confirm the chaotic dynamics. In the spatiotemporal system, the conditions for Turing instability are derived. Furthermore, we analyzed the Turing pattern formation for different diffusivity coefficients for a two-dimensional spatial domain. Moreover, we investigated the spatiotemporal dynamics incorporating two discrete delays. The effect of the delay parameters in the transition of the Turing patterns is depicted. Various Turing patterns, such as hot-spot, coldspot, patchy, and labyrinth, are obtained in the case of a two-dimensional spatial domain. This study shows that the parameter and the delay parameters significantly instigate the intriguing system dynamics and provide new insights into population dynamics. Furthermore, extensive numerical simulations are carried out to validate the analytical findings. The findings in this article may help evaluate the biological revelations obtained from research on interactions between the species.Item Complex dynamics of a predator–prey system with fear and memory in the presence of two discrete delays(Springer, 2023-11) Dubey, BalramIn this paper, we consider a two-species predator–prey model with fading memory, where the growth rate of prey species is subject to predation induced fear. Growth rate of predator species depends not only on the present density of prey but also on the past densities with diminishing impact. As the societal activities and behavioral practices influence carrying capacity of any species, we consider the density dependent carrying capacity of prey species instead of a constant. As fear on growth rate and societal activities on carrying capacity entail some time lags to show their effect, so we incorporate two delay parameters to corroborate this in the modeling phenomenon. Feasibility criteria of equilibria and their stability analysis are carried out. We observe that fear parameter and predation rate have destabilizing effect on the system’s dynamics, whereas parameter representing intensity of fading memory has stabilizing impact. We also distinguish stability and instability regions in different parametric planes. With increasing value of production factor from negative to positive, stability region decreases. The system also shows multiple stability switching phenomenon with respect to delay parameters. Solutions show chaotic behavior for a range of fear response delay both in the absence and presence of other delay parameter.Item Trade-off dynamics and chaotic behavior in nonautonomous prey-predator model with group defense(Springer, 2023-11) Dubey, BalramEcological “trade-off” is prioritising one trait over another. Predators put their lives at danger to pursue dangerous prey, and their injuries can reduce their chances of survival. Prey must “trade-off” between reproduction rate and safety, whereas predators must “trade-off” between food and safety. We present a two-dimensional prey and predator model that takes into account prey logistic growth rate and Monod-Haldane type functional response to reflect prey collective defense. We investigate the cost of fear in order to depict prey trade-off dynamics, and we change the predator’s mortality rate by incorporating a function that reflects predator loss as a result of encountering dangerous prey. Our model shows bistability and goes through transcritical bifurcation, saddle node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, Bautin bifurcation, Homoclinic bifurcation, and Limit point of cycle bifurcation. We investigated the effects of our critical parameters on both populations and discovered that predators become extinct if their loss of predator is too high due to encounters with dangerous prey, demonstrating how predators risk their own health for food. We find that fear can lead to global stability in a system by causing the stable and unstable limit cycles to collide. We also see that the degree of seasonality in the level of fear in the nonautonomous model might lead to chaos. Sensitivity analysis, the positivity of the maximal Lyapunov exponent, and the uneven distribution of points observed in the Poincaré map shown are the established signs of chaotic nature. We note that variations in intensity of seasonality in carry-over can cause a system to shift under many different periodic windows. The findings presented in this article may be beneficial in comprehending the biological insights derived from investigating prey-predator interactions.