This thesis develops novel statistical methodology for estimating the incidence and the prevalence of Human Immunodeficiency Virus (HIV) using routinely collected surveillance data. The robust estimation of HIV incidence and prevalence is crucial to correctly evaluate the effectiveness of targeted public health interventions and to accurately predict the HIV- related burden imposed on healthcare services.

Bayesian CD4-based multi-state back-calculation methods are a key tool for monitoring the HIV epidemic, providing estimates of HIV incidence and diagnosis rates by disentangling their competing contribution to the observed surveillance data. Improving the effectiveness of public health interventions, requires targeting specific age-groups at high risk of infection; however, existing methods are limited in that they do not allow for such subgroups to be identified.

Therefore the methodological focus of this thesis lies in developing a rigorous statistical framework for age-dependent back-calculation in order to achieve the joint estimation of age-and-time dependent HIV incidence and diagnosis rates. Key challenges we specifically addressed include ensuring the computational feasibility of proposed methods, an issue that has previously hindered extensions of back-calculation, and achieving the joint modelling of time-and-age specific incidence. The suitability of non-parametric bivariate smoothing methods for modelling the age-and-time specific incidence has been investigated in detail within comprehensive simulation studies.

Furthermore, in order to enhance the generalisability of the proposed model, we developed back-calculation that can admit surveillance data less rich in detail; these handle surveillance data collected from an intermediate point of the epidemic, or only available on a coarse scale, and concern both age-dependent and age-independent back-calculation. The applicability of the proposed methods is illustrated using routinely collected surveillance data from England and Wales, for the HIV epidemic among men who have sex with men (MSM).