Background: Vaccines are an important tool in global disease burden reduction, with vaccine dose amount (hereafter ‘dose’) being a key decision during vaccine development. Vaccine dose selection is often conducted through empirical comparison of a small number of potential doses, which is likely to fail to find the optimal dose if none of the doses are optimal. Mathematical modelling has been suggested as a method for identifying optimal vaccine dose and has been historically important in determining optimal dose of, and informing clinical trials for, drug development. Mathematical modelling is however not commonly used in either the design of vaccine dose ranging trials nor in the selection of optimal vaccine dose based on the resulting clinical trial data. To address this gap, recently ‘Immunostimulation/lmmunodynamic’ (IS/ID) modelling has been proposed to encompass quantitative modelling for vaccine dose optimisation. Initial IS/ID work has been used to find the maximally immunogenic dose for tuberculosis and influenza vaccines, and has shown that, contrary to widespread belief, vaccine dose-efficacy response may be peaking rather than saturating. However, as the field is new, there are many gaps including: uncertainty in the prevalence of such peaking dose-response curve shape, primarily only efficacy-maximisation has been considered, the impact of incorrectly assuming a peaking/saturating dose response has not been assessed, and mathematical modelling has been performed retrospectively of clinical trials rather than informing them during the trial itself (e.g. in an adaptive trial design). Further, there has also been little research into multi-dimensional vaccine dose-optimisation, where there is a need to choose prime doses, boost doses, adjuvant doses, and/or time between doses, which may complicate the dose-response relationship through potential for synergism/antagonism. My aim for this thesis was to explore and expand the field of IS/ID and mathematical modelling for vaccine dose optimisation, addressing the gaps described above. My objectives were: (1.) To gather dose-response data through a systematic review of dose-ranging studies for a specific class of vaccine (adenoviral vector), and to find the distribution of the number of doses typically investigated in these studies. (2.) Using dose-response data from objective one and mathematical models, determine the prevalence of predicted saturating versus peaking dose- response curves. (3.) To extend IS/ID beyond efficacy-maximisation into multi-factor dose optimisation by proposing alternative utility functions and investigate the impact of the choice of utility function on the selection of ‘optimal’ dose. (4.) To evaluate the potential impact of correctly or incorrectly assuming a peaking/saturating dose-efficacy response, along with the impact of adaptive trial design, on optimal vaccine dose selection. (5.) To evaluate the use of a non-parametric dose-response model for the purpose of optimal vaccine dose selection in the adaptive trial design setting, with emphasis on multi-dimensional vaccine dose-optimisation. Methods: For objective one, a class of vaccine (adenoviral vector) was selected, and dose- response data were extracted from a systematic review of single-dose dose-ranging studies. I conducted a descriptive analysis of these studies to investigate the number of doses that were typically investigated. For objective two, representative peaking and saturating dose-response models were calibrated to the data from objective one. I assessed which of the two mathematical models best described the data through the use of Akaike Information Criterion. I determined the prevalence of dose-response data which was peaking or saturating and investigated potential covariates that may impact dose-response shape. For objective three, I calibrated dose-response models to efficacy and toxicity data from a phase I dose-ranging study of a recombinant adenovirus type-5 COVID-19 single-dose vaccine (Ad5-nCoV). Using these mathematical models, I predicted optimal dose for three potential dose selection criteria, namely i) achieving herd immunity, ii) balancing efficacy and toxicity, and iii) balancing efficacy, toxicity, and cost. For objective four, I used a simulation-based study to assess the impacts of different assumed efficacy models and trial dose selection methods on optimal dose selection and ethical trial design. Comparison was done using simulated clinical trials, using a utility function that involved both efficacy and ordinal toxicity and both peaking and saturating dose-efficacy curves across 14 dose-optimisation scenarios. For objective five, I conducted a second simulation-based study to assess a novel non-parametric dose-response model, the ‘Continuously Correlated Beta Process’ model, in identifying optimal dose. This was compared to other mathematical model- based and mathematical model-free methods of vaccine dose optimisation. The simulation study included both single-dose and multi-dimensional dose-optimisation scenarios. Results: For objective one, data from 35 studies were extracted and I found that adenoviral vector vaccine dose ranging trials were designed around selecting between a small number of doses (94% of studies investigated < six doses). For objective two, I found that the data from the available dose-ranging trials were often insufficient to provide evidence for either a peaking or aturating dose-efficacy curve, with the peaking model best describing 22% of the data, the saturating model best describing 4.7% of the data, and there being no significant difference for 73.3% of the data. Further, the species being vaccinated and the response type of interest may be more predictive of dose-response curve shape than the adenoviral serotype or route of administration of a vaccine. For objective three, I found that vaccine optimal dose depends on the utility function that is being maximised, with the optimal doses for the Ad5-nCoV vaccine being i) 1.3 x 1011, ii) 1.5 x 1011 oriii) 1.1 x 1011 viral particles.For objective four, I showed that assuming a peaking dose-efficacy curve or using weighted model averaging was typically preferable to assuming a saturating dose­ efficacy curve for the purpose of selecting optimal dose. Adaptive trial design may not typically improve dose selection relative to a ‘sufficiently explorative’ trial design but may lead to trial participants receiving more optimal doses. For objective five, I found that the non-parametric model was able to optimise dose well despite using a simpler set of assumptions than the parametric models, in particular for multi-dimensional dose-optimisation problems. Using mathematical model-based and/or adaptive design-based approaches of vaccine dose optimisation consistently selected a more optimal dose using less trial participants than using neither mathematical modelling nor adaptive design. Discussion: In this work I explored and expanded the field of IS/ID and mathematical modelling for vaccine dose optimisation. I collated and summarised adenoviral vector vaccine data that can now be used for quantitative adenoviral vectored vaccine dose optimisation analysis. There is evidence from these dose-ranging trial data to support the hypothesis that for some adenoviral vector vaccines the dose-efficacy response was peaking, so vaccine adenoviral vector developers should not assume that increasing dose always leads to more efficacious vaccine response. I have shown that the ‘optimal dose’ predicted by modelling is likely to depend on which utility function is used to define ‘optimal’, so vaccine developers should have a clear definition of optimal dose prior to conducting clinical trials. I have also shown that adaptive trial design informed by mathematical modelling can be used both to improve vaccine dose selection and benefit clinical trial participants, suggesting that vaccine developers could consider adaptive trial design.Additionally, I showed that using a weighted average of peaking and saturating models to describe the dose-efficacy response may be beneficial, suggesting that vaccine developers should consider using a weighted average of peaking and saturating models to describe dose-efficacy response. However, my results also suggested that a non-parametric model was able to optimise dose at least well as using parametric models, despite using a simpler set of assumptions, in particular for multi-dimensional dose-optimisation problems, suggesting that vaccine developers could consider using non-parametric models as an alternative. Finally, my results showed that using mathematical modelling and/or adaptive trial design reduced the number of trial participants required to find an optimal vaccine dose when compared to using neither modelling nor adaptive design, and so vaccine developers should consider using modelling and/or adaptive design in vaccine dose- finding trials to increase efficiency.I believe there is merit to continued development and validation of IS/ID methods in order to provide tools for identifying optimal vaccine dose, ultimately saving lives.