Mahadevananda Mahavidyalaya

In this study, we develop and analyze a novel mathematical model for colon cancer progression in intestinal epithelial cells incorporating the role of CD8^+ immune response. While traditional numerical techniques are commonly employed for studying such biological systems, we employ a machine learning–driven physics-informed neural network (PINN) approach to address the nonlinear system of ordinary differential equations arising from the model. The PINN approach integrates biological domain knowledge (e.g., tumor growth kinetics and immune interactions) into the training process of a neural network, thereby combining data-driven learning with governing biological laws to estimate system dynamics and unknown parameters with improved accuracy and efficiency. A rigorous analytical investigation ensures positivity, boundedness, and local stability of equilibrium states, characterized by a threshold parameter analogous to the basic reproduction number ( ℛ_0 ) from infectious disease modeling. Sensitivity analysis highlights the critical influence of mutation rates and immune efficacy on long-term disease dynamics. In particular, an enhanced CD8^+ immune response significantly reduces cancerous cell populations while promoting healthy epithelial cell survival. PINN-based simulations not only validate the theoretical predictions but also reveal critical thresholds for treatment effectiveness and early detection. Overall, this hybrid mathematical machine learning framework provides a powerful tool for modeling cancer progression, offering insights for improving early intervention strategies and optimizing therapeutic and screening policies in colon cancer management.

HIV/AIDS remains a major global health concern, necessitating advanced therapeutic strategies to enhance immune response and suppress viral replication. This research formulates and examines a novel fractional-order mathematical model that integrates Nucleoside Reverse Transcriptase Inhibitors (NRTIs) and Protease Inhibitors (PIs) to evaluate their collective effectiveness in managing HIV infection. Unlike existing models, our framework incorporates memory effects via fractional calculus, explicitly accounts for macrophage-derived viral load as an external reservoir, and employs optimal control theory together with bifurcation analysis to investigate long-term treatment outcomes. In addition, the model is validated using real patient data, ensuring both biological relevance and practical applicability. We determine the biologically feasible steady states of the system and compute the basic reproductive ratio ( ℛ_0 ), which serves as a threshold parameter for infection persistence. Stability analysis is performed for each equilibrium point to derive conditions for disease eradication or persistence. Sensitivity analysis identifies key parameters influencing disease progression, and an optimal control strategy is derived using the Backward-Forward Runge-Kutta method to enhance CD 4^+ T-cell counts while reducing infected CD 4^+ cells and HIV viral load. Furthermore, the impact of drug efficacy parameters ( Π _R ) and ( Π _P ) is explored through extensive numerical simulations, demonstrating their role in mitigating viral replication and stabilizing the system. We also conduct a bifurcation analysis, including Hopf and transcritical bifurcations, to examine qualitative system dynamics. A novel Hopf bifurcation diagram illustrates the influence of fractional-order derivatives on oscillatory behavior, and a two-parameter bifurcation diagram reveals the combined effect of critical parameters on system stability. Our findings demonstrate that NRTIs and PIs effectively suppress oscillations and high infection rates, with combination therapy significantly reducing viral load and enhancing system stability. The fractional-order parameter ( α ) plays a crucial role in suppressing oscillations and ensuring long-term stability. Finally, a comparative study with existing literature highlights the originality and significance of our results, reinforcing the applicability of fractional-order modeling with real data in HIV/AIDS therapy research.

In this study, we develop and analyze a tri-trophic predator prey model that incorporates predator-induced fear in the prey population and wind-driven modulation of predation efficiency. The model represents interactions among insects (prey), birds (predators), and feral cats (top predators), where fear induces non-consumptive effects that suppress prey growth, while wind acts as an environmental disturbance influencing intermediate hunting efficiency. We establish the positivity, boundedness, and existence of biologically feasible equilibria and derive sufficient conditions for both local and global stability using Jacobian analysis and Lyapunov functions. Conditions for saddle-node, transcritical, and Hopf bifurcations are obtained, revealing critical thresholds that govern stability loss and the emergence of oscillatory dynamics. Comprehensive numerical investigations including bifurcation diagrams and iso-spike patterns reveal a wide range of dynamical behaviors, from steady states and periodic oscillations to chaos. The presence of chaos is confirmed through the computation of maximum Lyapunov exponents. To suppress chaotic oscillations, a linear feedback control strategy is designed around the interior equilibrium, and its effectiveness is verified analytically via eigenvalue analysis and numerically through time-series simulations and control-parameter bifurcation diagrams. Global sensitivity analysis based on Partial Rank Correlation Coefficients (PRCC) identifies the most influential parameters governing long-term system dynamics. In addition, a Physics-Informed Neural Network (PINN) framework is employed for robust parameter estimation, representing a novel application of PINNs to chaotic ecological systems. The results indicate that increased predation fear has a stabilizing effect by suppressing chaos, whereas strong wind flow tends to destabilize the system by promoting oscillatory and chaotic dynamics. PINNs method found four unknown parameters with only 0.51% error on average. The neural network predictions matched the true population dynamics with R2 = 0.99, meaning it explained 99% accuracy. This worked 100 to 500 times faster than traditional optimization methods. Overall, the study highlights the combined roles of behavioral responses, environmental disturbances, control strategies, and machine learning tools in shaping complex ecological dynamics.

A computational investigation into the formation of X3CH & ctdot;S and X3CH & ctdot;Se (X = F, Cl) H-bonds using DFT and MP2 levels of theory with dimethylsulfane (DMS) and thiirane, and dimethylselane and selenirane (DMSe) as the H-bond acceptors is reported. The electronic structure parameters indicate that fluoroform has a greater tendency of forming blue-shifting H-bonds while chloroform has a greater tendency of forming red-shifting H-bonds. This is further substantiated from Natural Bond Orbital calculations which reveal the predominance of n(Z)-> sigma*(X - H) hyperconjugation in the red-shifting H-bonds whereas repolarization of the C-H bond leads to the predominance of rehybridization effect for blue-shifting H-bonds. This study also attempts to underscore the discrepancy in predicting the nature (blue- or red-shifting) of the H-bonds for calculations at various levels of theory. The topological parameters corresponding to the H-bonds reveal a dominant closed-shell (electrostatic) interaction with both the H-bond donors, CHF3 and CHCl3.

We present a quantum chemical investigation into the nature of bifurcated intermolecular hydrogen-bonds (HBs) in symmetrical complexes of 1,8-naphthyridine-4,5(1H,8H)-dione (ND) with small molecules (ND:H2O, ND:H2S, ND:Furan, ND:Thiofuran, and ND:Pyridine). The natural bond orbital (NBO) analysis accounts for the formation of redshifting HBs (X-H⋅⋅⋅Y) in the complexes by hyperconjugative charge transfer from the filled lone-pair orbital of atom-Y to the antibonding σ ∗ ( X - H ) $\sigma ^{*}(\mathrm{X}-\mathrm{H})$ orbital. It is demonstrated that hyperconjugation outweighs the rehybridization effect underlying the formation of the HBs, which is in parity with the Bent rule. The quantum theory of atoms-in-molecules (QTAIM) is employed to rationalize the nature of the HBs. From the topological properties at the bond critical points (bcps) along the bond paths for the HBs, the nature of the HBs is identified as predominantly closed-shell (electrostatic) interaction in all the studied complexes. The noncovalent interaction (NCI) analysis reveals that the HB energies are relatively weak in the sense that they typically fall in the range of van der Waals interactions. The observed nonhydrogen bonding O⋅⋅⋅O NCIs in ND also fall in the range of van der Waals type of interactions, and are even stronger than the HB interactions for ND:H2S and ND:Thiofuran complexes.