Department of Mathematics

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    Dynamics of a Vector-Host SIR-SI Model in Dengue Transmission
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Rayan Zahid; CIIT/SP21-BSM-001/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 9877
    Dengue fever is a major health problem, especially in tropical and subtropical areas, affect- ing around 50 countries. The disease is mainly spread by Aedes mosquitoes, with Aedes aegypti being the most important carrier. Urbanization, climate change, and population movement make it spread more easily. Dengue symptoms can range from mild fever to se- vere hemorrhagic fever, which can be fatal. To control the disease, effective strategies are needed, and epidemiological models are crucial for understanding disease spread, testing interventions, and predicting outbreaks. In this project, we will create a model that describes the relationship between humans and mosquitoes using a susceptible-infected-recovered (SIR) model for people and a susceptible- infected (SI) model for mosquitoes. The goal is to understand how these two groups in- teract and use this information to improve control measures. We will calculate the basic reproduction number (Ro) using the Next Generation Matrix method to determine whether the disease will spread (Ro > 1) or fade out (Ro < 1). We will also use stability analysis with the Lyapunov function to check these results. Furthermore, we will run simulations in MATLAB to test the model and explore the effects of control measures like mosquito population control, vaccination, and quarantine. By doing sensitivity analysis, we will identify the most important factors affecting disease spread and help prioritize the best in- terventions. The project aims to provide useful insights into how dengue spreads and guide effective ways to control it.
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    Characterizing Localized Wave Structures in High-Dimensional Nonlinear PDEs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Bismah Nazir; Fa23-rmt-008; Dr. Hafiz Muhammad Afzal Siddiqui; LHR TP 9756
    This thesis investigates the extraction of solitons that arise in nonlinear dynamics, focus ing on two important models: the Stochastic Davey–Stewartson equation and the Ben jamin–Bona–Mahony (BBM) equation well known Nonlinear Evolution equation. Numer ous physical events are described by these models, which are well known for their intricacy in nonlinear wave propagation. These models are solved using the Extended Modified Aux iliary Equation Method (EMAEMM), a potent yet effective technique for resolving nonlin ear partial differential equations (NLPDEs). By simplifying the equations, the EMAEMM approach facilitates the search for precise solutions, making it a valuable tool in nonlinear analysis. Optical fibers, fluid dynamics, and plasma physics are just a few of the many applications that benefit from the soliton’s ability to maintain its shape while propagating. A variety of graphical representations, all created using Mathematica, are displayed, in cluding 2D, 3D, density linear, 1D, slice contour plotting, and stream density plots, and the stability and sensitivity of the resulting solitons are investigated. Bright, dark, kink, periodic, and optical solitons are among the many different behaviors of the reported soli tons. This thorough analysis of soliton dynamics in nonlinear systems offers insights on the stability and representation of solitons in mathematical physics