When an HIV-infected mother gives birth, the virus may be passed on to the child. Fortunately, an HIV infection is not a death sentence anymore. Since international guidelines changed in 2008 to always start treatment immediately after birth, many children nowadays survive into adolescence. They, however, depend lifelong on antiretroviral drugs to control the virus. As part of the EPIICAL project, concerned with the improvement of the life of children with HIV, the research presented in this thesis has aimed to quantitatively assess the response to early initiation of treatment and to describe differences among children. To do so, we have developed mathematical models describing viral load (VL) and CD4+ T-cell dynamics for every child individually. The models were fitted to data taken from previous observational cohort studies and focuses on those children who successfully suppress HIV after initiating treatment within 6 months of age. By characterising the VL decay patterns in these children, we can mathematically express the time to viral suppression (TTS). In children without treatment complications, we can predict TTS based on the baseline VL and the percentage of CD4+ T-cells at the start of treatment. Next, we have aimed to quantify CD4+ T-cell reconstitution in response to the treatment. To this end, we have developed novel normalisation functions for T-lymphocyte counts and percentages, especially capturing the dynamic immunological changes during the first year of life. We have shown that children typically have the potential to recover their deteriorated CD4+ T-cell levels “fully” once the HIV VL is under control. Lastly, we have shifted our attention to disease progression in untreated children, who tend to progress towards AIDS fast and have high VLs. We have shown that their immature immune response most parsimoniously explains the observed small contractions in VLs. Thus, perinatally HIV-infected children rely on very early treatment initiation to survive. Our quantification of many time courses of the VL and CD4+ T-cell counts in early treated children with HIV broadens our knowledge on paediatric HIV infections and provides a first step towards informing clinicians about the timing and selection of alternative treatment strategies, hopefully paving the way for an HIV cure in the future.