The COVID-19 pandemic is a global challenge that impacted 200+ countries. India ranks in the second and third positions in terms of number of reported cases and deaths. Being a populous country with densely packed cities, SARS-CoV-2 spread exponentially. India sequenced ≈0.14% isolates from confirmed cases for pandemic surveillance and contributed ≈1.58% of complete genomes sequenced globally. This study was designed to map the circulating lineage diversity and to understand the evolution of SARS-CoV-2 in India using comparative genomics and population genetics approaches. Despite varied sequencing coverage across Indian States and Union Territories, isolates belonging to variants of concern (VoC) and variants of interest (VoI) circulated, persisted, and diversified during the first seventeen months of the pandemic. Delta and Kappa lineages emerged in India and spread globally. The phylogenetic tree shows lineage-wise monophyletic clusters of VoCs/VoIs and diversified tree topologies for non-VoC/VoI lineages designated as ‘Others’ in this study. Evolutionary dynamics analyses substantiate a lack of spatio-temporal clustering, which is indicative of multiple global and local introductions. Sites under positive selection and significant variations in spike protein corroborate with the constellation of mutations to be monitored for VoC/VoI as well as substitutions that are characteristic of functions with implications in virus–host interactions, differential glycosylation, immune evasion, and escape from neutralization.

In this paper, under the assumption that the offspring distributions of lag one and lag two are identical, we discuss the joint estimation of the offspring mean and the offspring variance of a second order branching process. The proposed estimator is shown to be finite sample optimal, if the common offspring distribution is mesokurtic and symmetric. The conditional asymptotic joint normality of the proposed estimator is established using the martingale limit theory.

We study a multiplicative controlled branching process which is subjected to an independent immigration at each generation. We discuss probabilistic properties such as generating function, mean and long run behavior of the generation sizes. Further, we consider an extension of proposed process and discuss its limiting behavior. We also propose the estimators of parameters of the process and discuss their asymptotic properties.