Nonlinear Dynamics and Bifurcations in Tumor–Immune Models with Therapies

AMIN, RUCHITA

Nonlinear Dynamics and Bifurcations in Tumor–Immune Models with Therapies

Files

Amin_Ruchita_PhD_2025_Thesis.pdf (3.12 MB)

Date

2025-10-22

Authors

AMIN, RUCHITA

Abstract

Cancer is one of the most serious health problems worldwide, responsible for millions of deaths each year. It develops when cells grow in an uncontrolled way and avoid detec- tion by the immune system. Key immune players such as CD4+ T cells and cytokines are essential for recognizing and controlling tumors, and many modern therapies aim to en- hance or restore their function. Immunotherapy, in particular, has become one of the most promising approaches for reducing tumor growth and improving survival. In this thesis, mathematical models are used to study how tumors interact with immune cells and how treatments influence these interactions. We begin with a model that describes tumor, CD4+ T cell, and cytokine dynamics in the absence of treatment. This helps identify the lim- its of natural immune control and shows how stability and oscillations in tumor size can occur, especially through Hopf bifurcations. The results suggest that when tumors have very low antigenicity, they can continue to grow unchecked, even if immune responses are strong. The analysis is then extended to include immunotherapy and combined treatments (polytherapy), where stability, equilibrium points, and bifurcation patterns are examined. Using normal form theory, we study the amplitude and stability of limit cycles, revealing how therapy can shift the system toward long-term tumor control or oscillatory behavior. We also explore virotherapy, a newer treatment that uses engineered viruses to infect and destroy cancer cells. A nonlinear model is developed for tumor–virus interactions, and bi- furcation analysis shows how viral therapy can produce stable oscillations that represent ongoing cycles of tumor reduction. Numerical simulations are provided throughout to il- lustrate theoretical results. Overall, this work demonstrates how mathematical oncology can be used to understand tumor–immune–therapy interactions and to provide insights for improving cancer treatment strategies.

Summary for Lay Audience

These studies use mathematical models to analyze immune system and cancer tumor inter- actions. We considered three models to study tumor growth or shrinkage under different conditions: without treatment, immune therapy, and virus therapy. Using advanced math- ematical tools, we explored when the system show steady tumor control, oscillations, or multiple possible states, and found important phenomena called bifurcations where small parameter changes cause sudden shifts towards the outcomes. In the model without treatment, we found that very low tumor antigenicity leads to cancer progression no matter how strong the killing ability or cytokine levels are, showing that proper tumor recognizing is essential. Small parameter changes can push the system from control to escape, showing the limits of natural immunity. Introducing immunotherapy, it changes the boundaries, sometimes stabilize or destabi- lize tumor control, depending on dose and timing. Near bifurcation points, small treatment changes can change tumor–immune cycles or create situations where both stable control and stable tumor growth coexist, making the outcome highly sensitive to initial conditions. Oncolytic virotherapy adds viruses that infect and destroy tumor cells. The findings show that viral replication can either help long-term tumor control or allow cancer regrowth depending on infection rate, burst size, and clearance rate. Overall, the results highlight key factors that decide whether tumors are controlled or escaped, guiding for designing safer and more effective cancer therapies.