Department of Mathematics
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Item Mathematical Modeling and Analysis of Pertussis Cough(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Syed Shan-E- Ahmad; CIIT/SP24-RMT-017/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 10078Mathematical modeling of dynamical systems often uses ordinary differential equations to represent how key variables evolve over time. Such models support prediction, inter-pretation, and decision-making in biology, engineering, economics, and environmental science by linking mechanisms to observable outcomes. Stability analysis is central in this setting: it determines whether equilibria persist under perturbations and clarifies long-term behavior, resilience, and the potential impact of interventions. We apply these tools to a pertussis (whooping cough) model with five compartments: susceptible, latent-immunity, exposed, infectious, and recovered. The formulation in-corporates waning vaccine-induced immunity, reinfection, and disease-induced mortal-ity, and it is proven to be positive, bounded, and forward invariant. The basic repro-duction number Ro is derived via the next-generation matrix method; sensitivity analy-sis shows transmission, progression to infectivity, recovery, and vaccination rates most strongly influence persistence. Local and global stability of the disease-free and endem-ic equilibria are established. A nonstandard finite difference scheme preserves quali-tative dynamics and supports simulations. Numerical experiments confirm the analy-sis and illustrate how parameters shape outbreak magnitude and duration. Finally, an integral sliding mode control acting on vaccination robustly suppresses exposure and infection and can drive elimination under parameter uncertainty and disturbances. Keywords: Mathematical modeling; pertussis; compartmental model; Vaccination; wan-ing immunity; basic reproduction number Ro; sensitivity analysisItem Optical and Thermal Properties of Quantum Corrected AdS Black Hole in Kiselev Spacetime(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Husnain Hafeez; CIIT/SP24-MSMATH-010/LHR; Dr. Rabia Saleem; LHR TP 10072In this work, we investigate the optical and thermodynamic properties of a quantum-corrected AdS black hole (BH) in Kiselev spacetime. The study focuses on the effects of key param-eters, including the quantum correction parameter a, the cosmological fluid parameter λ. the electric charge Q, and the entropy correction parameter ẞ, on the Hawking temperature, entropy, specific heat, Gibbs free energy, and BH shadow. The Hawking temperature gen-erally increases with the horizon radius, while the quantum and cosmological parameters enhance the corrected entropy and specific heat, indicating regions of local thermodynamic stability. Analysis of the Gibbs free energy reveals that small black holes (BHs) exhibit global instability, which transitions to stability for larger horizons. The BH shadow and photon sphere are found to expand with increasing a and A, and the infalling matter shows characteristic Doppler-induced darkening. These results highlight how quantum correction parameter a and surrounding cosmological fluid parameter A influence both the thermal behavior and observational features of AdS black holes, providing new insights beyond classical treatments.Item Application of the Bilinear Neural Network Approach to Kadomtsev-Petviashvili Model for Nonlinear Waves(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Numan Ahmed; CIIT/SP24-RMT-003/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 10066Nonlinear evolution equations (NLEEs) serve as fundamental models for wave phe-nomena in many physical systems, including fluid dynamics, plasma physics, optical pulse propagation, and atmospheric processes. They describe the spatiotemporal evolu-tion of waves in media where nonlinear interactions and dispersive effects act together. A prominent feature of such equations is the emergence of solitons, which are localized travelling waves capable of preserving their shape and speed over long distances. This remarkable property is generally attributed to a balance between nonlinear steepening and dispersive spreading. Solitons appear across a broad spectrum of physical contexts, ranging from hydrodynamics, nonlinear optics, and plasma physics to solid-state and lat-tice systems, where they often display particle-like behavior while retaining their wave nature. They have also been observed in more complex settings such as astrophysical, condensed-matter, nuclear, and other nonlinear media, highlighting their wide applica-bility and importance in both theoretical studies and practical applications of nonlinear wave dynamics. Different types of solitons exist based on their shape and behavior. Kink solitons rep-resent a transition from one state to another, similar to steps in a staircase. Breather solitons are localized waves that expand and contract over time. Rogue waves are ex-tremely large and sudden waves that can be much higher than normal waves. Lump solitons are localized in all directions and gradually disappear with distance. These d-ifferent wave structures help researchers better understand nonlinear wave behavior in real-world systems. This thesis presents a bilinear neural network modeling (BNNM) framework that cou-ples the Hirota bilinear approach with trainable neural parameters to obtain closed-form solutions of the (3+1)-dimensional generalized Kadomtsev-Petviashvili equationItem Machine Learning Approaches in Medical Diagnosis(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Muhammad Bilal; CIIT/SP24-RMT-001/LHR; Dr. Muhammad Yousaf Bhatti; LHR TP 10065Brain tumors are among the most life-threatening neurological conditions, characterized by abnormal and uncontrolled cell growth within brain tissues. Early identification of tumor type and stage is crucial for effective intervention, yet conventional diagnostic methods often struggle to detect tumors in their early or subtle stages. This study proposes advanced deep-learning models for multi-class classification of brain tumors and machine-learning approaches for predicting chemical properties of anti-cancer drugs to support new drugs development. A hybrid deep-learning model was developed for classifying three tumor categories glioma, Meningioma and pituitary tumors using MRI scans. the model was trained on the brain tumor MRI dataset and integrates ResNet50 and Efficient Net50 as its base ar-chitectures, combined through a novel triangular dense-layer fusion strategy to optimize multi-stage feature extraction. The proposed model achieved a test accuracy of 97%, recall of 97%, and an AUC score of 99%, demonstrating its effectiveness for early and accurate brain-tumor classification. In second phase of this research, machine-learning algorithms and topological indices were used to analyze molecular structure of thirteen anti-cancer drugs. Topological in-dices were computed using a python program, and true physicochemical properties were retrieved from the ChemSpider database via an automated script. QSPR and SHAP anal-ysis were performed to identify indices most predictive of each physicochemical property. Machine-learning models were then trained on these features to develop generalized pre-dictive models for key durg properties, providing a foundation for computationally assisted drug discovery. Overall, this thesis demonstrates the combined potential of deep learning for early tu-mor detection and machine-learning for anti-cancer drug analysis, ultimately contributing to improved diagnostic accuracy and future therapeutic developmentItem Effects of Modified Entropy on Black Hole Thermodynamics in Bumblebee Gravity(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Zainab Nazish; CIIT/SP24-RMT-019/LHR; Dr. Abdul Jawad; LHR TP 10080This thesis explores the thermodynamic behavior and stability of anti-de Sitter (AdS) black holes within the framework of bumblebee gravity, a theory in which Lorentz symmetry is spontaneously broken by a vector field. The study examines how the Lorentz-violating bumblebee parameter influences the event horizon structure and various thermodynamic properties in both standard and extended phase spaces. Thermal stability is analyzed through heat capacity and Helmholtz free energy. The results show that large bumblebee AdS black holes can simultaneously satisfy local and global stability conditions, with the stable regions strongly depending on the Lorentz-violating parameter. The research also investigates heat capacity at constant pressure in the extended phase space, where the cosmological constant is interpreted as thermodynamic pressure. In this case, the stability criteria are found not to be fulfilled, highlighting the significant role of Lorentz symmetry violation in black hole thermodynamics. In addition, the thesis studies the sparsity parameter and Hawking radiation emission rates for black hole solutions in bumblebee gravity using generalized entropy models, including Sharma–Mittal entropy and three-parameter entropy indices. The findings reveal that changing the sign of the cosmological constant significantly affects thermodynamic sparsity and entropy behavior. A negative cosmological constant reduces sparsity while increasing entropy, indicating a denser thermodynamic system. Conversely, a positive cosmological constant preserves higher thermodynamic sparsity. Furthermore, the emission spectrum becomes sharper for negative cosmological constant values, indicating enhanced low-energy emission within a confining AdS background, while a positive cosmological constant generally suppresses the emission rate.Item Yager’s Prioritized Model for Multi-Attribute Group Decision-Making Using p,q-Quasirung Orthopair Fuzzy Information(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Nisha Ashfaq; CIIT/SP24-RMT-014/LHR; Dr. Atiq-ur-Rehman; LHR TP 10075This study presents a multi-attribute group decision-making (MAGDM) approach based on Yager’s prioritized model under the framework of p,q-quasirung orthopair fuzzy sets (p,q-QROFSs). The proposed method effectively handles uncertainty and vagueness in decision-making environments while considering the priority relationships among decision attributes. Aggregation operators are developed to combine experts’ evaluations expressed in p,q-quasirung orthopair fuzzy information. The Yager prioritized mechanism is incorporated to reflect the relative importance of attributes and decision-makers. The effectiveness and practicality of the proposed approach are demonstrated through a numerical example and comparative analysis. Results indicate that the method provides reliable and flexible decision support for complex group decision-making problems involving uncertain information.Item Graph Topological Indices and Machine Learning for Protein Networks and Drug Property Prediction(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Ahmad Mehmood; CIIT/SP24-RMT-025/LHR; Dr. Sana Javed; LHR TP 10085Molecular property prediction for drugs is an integral part of computational chemistry in the early stages of drug discovery for efficient screening and optimization before experi- mental verification. In the present work, an exhaustive and interpretable machine learning (ML) strategy is proposed that combines SMILES-based molecular feature extraction with topological indices using molecular graph theoretical approaches for the prediction of key physicochemical properties of drugs. The dataset consisting of forty-seven pharmacolog- ically active molecules from DrugBank, with their structures characterized using sixteen topological descriptors such as the Wiener, Balaban, Harary, Randi´c, Zagreb, Schultz, and Shannon Entropy indexes, were identified using the RDKit grouping library coupled with mathematical formulations from molecular graph theory. Preprocessing steps were per- formed rigorously for missing values using imputation; removal with cutoffs using the inter-quartile range method; variance for stabilization using the Box-Cox transformation; and normalization with MinMax scaling. Four different regression models, namely Ordi- nary Least Squares (OLS), Ridge Regression, LASSO, and Elastic Net algorithms, were employed for the prediction of the MW, LogP, and HBD properties for the dataset with evaluations made in terms of R2, MAE, and Root MSE metrics; with feature interpreta- tion from SHapley Additive exPlanations (SHAP) analyses for feature interpretation. The results achieved for MW with high predictive precision with R2 ≈ 0.96 using linear mod- els effectively; moderate accuracy with R2 ≈ 0.4 for Lipophilicity; with poorer accuracy for HBD values likely due to their dependence on chemical principles underlying chem- ical groupings in chemistry. The interpretation using SHAP explained the contributions made by the Shannon entropy, Wiener, Zagreb, Schultz indices being the principal predic- ix tors in each case. The present work clearly illustrates the utility of integrating molecular chem-graph principles with interpretable ML algorithms for efficient, scalable, and insight- ful property-based predictions in early stage computational screens for drug identification strategies. Protein-protein interactions (PPIs) networks are very important to decode cellular signal- ing and mechanisms of disease. PPI also play a key to spot possible drug targets. A strong computational strategy is presented in this paper. It unites the graph theoretic approach with unsupervised machine learning. This study is used to detect biologically significant hub genes in a large scale lung cancer associated PPI network. The network retrieved from the STRING database (461 proteins, 18,704 weighted interactions). Using NetworkX, we constructed an undirected graph by extracting source and target node from the dataset. We computed 17 comprehensive topological centrality measures, confirming the networks scale free and small world properties (average clustering coefficient = 0.67, average short- est path length ≈ 1.96). The Isolation Forest algorithm identified 61 topological outliers. These outliers were related to nodes which had a very high influence. The K-means cluster- ing (that was optimized by the elbow curve) sorted the rest of the nodes into three groups. The hub genes were given ranks according to degree centrality and a composite centrality score. Among the genes, CALM3, CREB1, AKT1, MAPK1, EGFR and KRAS came out as the strongest candidates. The pathway enrichment analysis performed with KEGG and Reactome showed that the oncogenic pathways were remarkably over represented. These pathways were PI3KAkt, Ras, MAPK, and proteoglycans in cancer and EGFR signaling. Our methodology, which combines together the multiple centrality metrics, anomaly de- tection, and clustering, has indeed solved the drawbacks of traditional single metric ap- proaches. It has also provided greater sensitivity in hub detection. The hubs that were identified not only correspond to the well established lung cancer drivers but also point to potential novel biomarkers. The detected targets are thus a scalable and reproducible pipeline for systems level analysis of disease specific interactomesItem Using Cryptography and Steganography Techniques for Data Security(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Umm e Amara; CIIT/FA23-RMT-041/LHR; Dr. Tariq Javed Zia; LHR TP 10064This thesis explores a hybrid security framework that integrates cryptographic and stegano- graphic techniques to enhance data security in digital communication. The proposed model combines the RSA encryption algorithm with XOR ciphering and Least Significant Bit (LSB) image steganography. To secure the data, it is first encrypted with both RSA and XOR operations. The resulting ciphertext is then hidden in a digital image by modifying the LSB’s of the image’s pixel data. The analysis focuses on three critical metrics: the amount of data that can be hidden (embedding capacity), the resulting fidelity of the carrier image (image quality), and the processing resources required (computational complexity). Experimental results demonstrate that 1-bit and 2-bit LSB methods provide an optimal balance between security, imperceptibility, and processing efficiency, while deeper em- bedding increases data capacity at the cost of visible distortion and higher computational overhead. Evaluation metrics such as PSNR, MSE, BER, and correlation analysis con- firm the effectiveness of the hybrid approach in securing data transmission with minimal perceptual degradation. The findings highlight the potential of combining cryptography and steganography for robust, multi-layered protection of sensitive information in modern communication systems.Item On Some Solutions of Kawahara and Modified Kawahara Eqautions(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Affifa Shafique; CIIT/SP24-RMT-006/LHR; Prof Dr. Sarfraz Ahmad; LHR TP 10069The study introduces the Quantum-Inspired Real-Time Decomposition Method (Q-RDM) in the shape of a new calculation tool that can be applied to solve the complicated mathe- matical model according to the Fractional Kawahara and Modified Kawahara equations. The plasmas and shallow water behavior of waves are explained with the help of the equations and the traditional approaches are not able to explain this properly due to the high computational cost and the numerical instability. Q-RDM offers the answer to these problems through a better way to optimize the classical Adomian Decomposition Method through a clever and adaptive optimization loop that uses quantum control theory. This al- lows the algorithm to self-correct in real-time stabilizing the answer without even needing complex linearization. As our results have demonstrated, Q-RDM is not just a much faster means of obtaining a 56 fold minimum reduction in processing time compared to current algorithms but is also a much more accurate technique since it was able to deal with errors in calculations that plague the traditional algorithms.Item Physicochemical Properties of Some Drugs by Using Machine Learning(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Amman Farzeen; CIIT/SP24-RMT-023/LHR; Dr. Kashif Ali; LHR TP 10083This study explores the integration of machine learning and multi-criteria decision-making techniques for the characterization and evaluation of potential therapeutics targeting Duchenne Muscular Dystrophy . A dataset comprising the physicochemical properties of twenty anti- Duchenne muscular dystrophy drug candidates is analyzed using topological indices de- rived from molecular graphs. These indices, computed via Python-based algorithms, served as input features for machine learning models. Artificial Neural Networks and Random Forests are employed to predict key physicochemical properties based on the structural de- scriptors. The results demonstrate that topological indices are effective predictors, enabling accurate and consistent performance across both models. To prioritize the drug candidates, the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is applied as an multi-criteria decision-making tool, leveraging the model predictions to rank compounds based on multiple beneficial attributes. The proposed framework highlights the synergy be- tween graph-theoretical descriptors, machine learning, and multi-criteria decision-making, offering a robust in-silico approach for early-stage drug evaluation. This methodology pro- vides a systematic and objective strategy for ranking therapeutic candidates, particularly valuable in the context of rare diseases such as duchenne muscular dystrophy. Key words: Machine learning; Molecular Graph; QSPR Analysis; Topological indices; Ar- tificial Neural Networks; Random Forest; TOPSIS; Python Algorithm; Anti-DMD drugs.