Human T-lymphotropic virus type-1 (HTLV-1) infects approximately 10-20 million people worldwide. The virus persists within hosts via de novo infection and infected cell proliferation, creating a population structure of multiple clones (infected cell populations with identical genomic proviral integration sites). The number of clones in one host is unknown, and is determined by the rate of de novo infection. Our primary objectives are: i) to estimate HTLV-1 clonal diversity; and ii) to develop a model of HTLV-1 dynamics that can estimate the relative contributions of de novo infection and mitotic replication. We use a combination of mathematical modelling, computer simulation and statistical methods to interpret experimental observation. We develop an estimator (named DivE) to estimate the number of HTLV-1 clones. DivE uses the ecological method of rarefaction, and includes novel model selection criteria. We show that DivE is more accurate than widely-used estimators from population ecology, and we demonstrated that this holds across many systems. Differences between these systems and ecological populations are investigated, and DivE is applied to patients with a range of HTLV-associated diseases. HTLV-1 clonal abundance varies by several orders of magnitude: quantifying within-host HTLV-1 dynamics requires mathematical modelling at multiple scales. Stochastic processes, important for modelling small populations, are introduced, and we explore properties and approximations of a mass-action birth-death process for biologically realistic species extinction scenarios. We combine ordinary differential equations with stochastic processes in a hybrid model and explore its consequences. The estimated HTLV-1 clonal diversity is substantially higher than previously thought, which strongly implies higher rates of de novo infection. The hybrid model captures known behaviour of HTLV-1, and can be used to infer rates of viral persistence. DivE and the hybrid model are applicable to other biological systems, in particular the study of T and B cell receptor repertoires.