The emergence of the novel coronavirus SARS-CoV-2 led to the COVID-19 pandemic and a global health crisis resulting in over seven million deaths and widespread morbidity. SARS-CoV-2 is a highly contagious and transmissible virus, and the clinical manifestations of COVID-19 are heterogeneous, ranging from asymptomatic cases to severe, life-threatening illness. Immunological memory from prior infection, vaccination, or both reduces severe disease on re-exposure; however, long-term protection is challenged by waning immunity and the continual emergence of immune evasive variants. A comprehensive understanding of these immunological mechanisms is essential for explaining inter-individual differences in immune protection and for guiding the development of effective, long-term immunization strategies. In this thesis, we applied mathematical modelling and computational simulations to quantitatively study immune dynamics and predicting viral dynamics and immune response outcomes across diverse populations. In the second chapter, we applied the classical target cell-limited model and validated against to clinical plasma viral RNA data from 294 hospitalised COVID-19 patients to identify the key drivers of heterogeneous viral-shedding patterns across subgroups defined by sex, disease severity, and survival. Delayed viral elimination correlated with greater severity and mortality, whereas faster clearance characterised survivors and moderately ill patients. We also identified plasma levels of the receptor for advanced glycation end products (RAGE) as a clinically accessible surrogate biomarker of impaired viral clearance and severe disease. In the third chapter, we developed a mechanistic model of humoral immunity following the three-dose mRNA vaccine primary series. Our model describes B-cell differentiation, antibody production, and memory formation, and was calibrated to longitudinal data from 33 healthcare workers and 21 older adults. Our results highlight the T follicular helper cell decay rate as a key factor underlying age-related differences in vaccine-induced antibody responses. Simulations suggest that semi-annual half-dose boosters may offer more durable protection in older adults than standard annual regimens, supporting the development of age-tailored immunization strategies. Finally, in the fourth chapter, we developed a new model of immune response to SARS-CoV-2 that describes CD8+ T-cell and B-cell-mediated memory to better understand long-term protection. Using vaccine- and infection-induced memory models, we examined how immunological profiles, the magnitude of prior responses, and exposure timing shaped innate, cellular, and humoral responses upon re-exposure. Our model showed that hybrid immunity (vaccination plus infection) outperformed either alone, producing faster, more robust responses; sequence also mattered: vaccination after infection generally elicited higher, earlier peaks and more sustained levels of adaptive cells and antibodies than breakthrough infection. Together, this thesis advances quantitative, within-host frameworks for infectious-disease dynamics, clarifying how immunological memory is regulated. This work therefore supports the establishment of personalised immunisation schedules to improve epidemics prevention, control, and vaccine development.

Mathematical models describing the relationship between the concentration of drugs or biological agents and their effect is often assumed to be monotonic. However, drugs and biological agents can often display non-monotonic effects in cases such as cellular signaling. Therefore, the models describing the effect of combining agents like those is poorly understood. The aim of this thesis is to present current methods for estimating the effect for agents that are non-monotonic and to provide a novel mathematical model to estimate the effect of combinations of non-monotonic agents. The first chapter of this thesis presents a case study in which mathematical modelling of combination of agents is particularly relevant as well as an overview of current models used in pharmacodynamics to estimate the dose-response curves and surfaces of biological agents. In the second chapter, we present a novel mathematical model that estimates non-monotonic dose-response surfaces of biological agents. Additionally, we test our model on biological data and provide new insights to guide further experiments. Our results show that our model can accurately estimate combinations of biological agents and distinguish between synergy of efficacy and synergy of potency. Furthermore, our new model can be used in a variety of contexts that go far beyond the case studied and presents a significant step forward to study non-monotonic dose-response combinations. The final chapter of this thesis presents a summary of the work presented herein.

Developing novel cancer drugs or therapies requires years of preclinical work before translation to clinical trials and ultimately the market. Unfortunately, an overwhelming majority of compounds will fail to make this transition and will show no benefit in trials. To reduce attrition along the drug development pipeline, mathematical modelling is increasingly used in preclinical work to investigate and optimize treatment scenarios, in the hope of improving the success rate of potential therapies. Mechanistic models aim to incorporate the mechanisms of actions of drugs and physiological/cellular interactions to provide a deeper understanding of the system and rationally investigate therapeutic effectiveness. This thesis focuses on the implementation of heterogeneous, mechanistic mathematical models in preclinical contexts in cancer drug development. The first chapter of this thesis provides an overview of mathematical oncology and the drug discovery pipeline by presenting different tumour growth models and the integration of therapeutic effect through pharmacokinetic/pharmacodynamic (PK/PD) models. The second chapter of this thesis discusses granulosa cell tumour (GCT) of the ovary and the development of a mathematical model to investigate the potential of a combination therapy using a chemotherapy and an immunotherapy that produces tumour necrosis factor-related apoptosis-inducing ligand (TRAIL) through an oncolytic virus (OV). The model considers tumour cells throughout the phases of the cell cycle, the infection of these cancer cells by the OV, and the innate-immune pressure from the body. It also incorporates detailed PK/PD models for TRAIL and the chemotherapeutic drug, procaspase activating compound-1 (PAC-1). This includes a mechanistic receptor binding PK model for TRAIL as well as a two-compartment PK model for PAC-1 to properly integrate the concentrations of both compounds in the combination effect function applied to the cancer cell populations. Through simulations and hypothesis testing, we determined the minimal doses and ideal dosing regimens for PAC-1 that best controlled tumour growth. We also established how to successfully eradicate the tumour under the assumption of a much higher infection rate of the OV. 6 In the third chapter, we present different approaches to include inter-individual variability into mechanistic mathematical models, each with their own benefits and challenges. We describe how population PKs (PopPK) inform on cohort averages and variability due to covariates, and how to use this heterogeneity to recover the dynamics of drug treatment in patient populations. Variability in cohorts can also be generated through algorithms ensuring that virtual patients have realistic parameters and outcomes. We also touch upon in silico trials that help to predict a range of outcomes and treatment scenarios. These in silico clinical trials are highly valuable in quantitative system pharmacology (QSP) due to their predictive nature. Lastly, we present an application of PopPK using 300 generated patients in a QSP model for mammary stem cell differentiation under treatment with estrogen (estradiol). We investigate the effect of hormone therapy on mammary cell differentiation due to its potential application in triple negative breast cancer (TNBC), as prolactin has been proposed in experimental models to induce differentiation in TNBC stem cells. Our model and results serve as proof of concept for the continued investigation into pharmacological means of inducing stem cell differentiation to reduce cancer plasticity and severity.

Modelling biological systems with mathematical models has been a challenge due to the tendency for biological data to be heavily heterogeneous with complex relationships between the variables. Mixed effects models are an increasingly popular choice as a statistical model for biological systems since it is designed for multilevel data and noisy data. The aim of this thesis is to showcase the range of usage of mixed effects modelling for different biological systems. The second chapter aims to determine the relationship between maple syrup quality rating and various quality indicator commonly obtained by producers as well as a new indicator, COLORI, and amino acid (AA) concentration. For this, we created two mixed effects models: the first is an ordinal model that directly predicts maple syrup quality rating using transmittance, COLORI and AA; the second model is a nonlinear model that predicts AA concentration using COLORI with pH as a time proxy. Our models show that AA concentration is a good predictor for maple syrup quality, and COLORI is a good predictor for AA concentration. The third chapter involves using a population pharmacokinetics (PopPK) model to estimate estradiol dynamics in a quantitative systems pharmacokinetics (QSP) model for mammary cell differentiation into myoepithelial cells in order to capture population heterogeneity among patients. Our results show that the QSP model inherently includes heterogeneity in its structure since the added PopPK estradiol portion of the model does not add large variation in the estimated virtual patients. Overall, this thesis demonstrates the application of mixed effects models in biology as a way to understand heterogeneity in biological data.