Estimates of infant HIV-1-free survival and the timing of vertical transmission of HIV-1 are complicated by the fact that the infants' HIV-1 status at death may be unknown, the time of HIV-1 infection may be interval censored, and the tests used to detect HIV-1 infection in the infants may be less than 100% sensitive or specific. To address these problems we develop nonparametric and semiparametric approaches to estimating the joint distribution of HIV-1 infection time and death time. .Dysbiosis refers to compositional and functional changes of the microbiome and it can result into changes of all or any of the following microbiome characteristics: Microbiome stability, Microbiome diversity, and Microbiome resilience . Microbiome stability also referred to as microbiome resistance is the amenability of microbiota to perturbations. Microbiome diversity refers to richness of the microbiota ecosystem. Microbiome resilience is the ability of microbiota to get restored to pre-perturbation state. Dysbiosis can be driven by environmental and host related factors. However, variability of microbiota among healthy individuals of different age, geographical limits and dietary habits, limits the definition of what actually constitutes the dysbiotic state. In view of this limitation, have defined dysbiosis as a microbial community state that is not only statistically associated with disease, but also functionally contributes to etiology, diagnosis, or treatment of the disease.