Multiple partition Markov model for B.1.1.7, B.1.351, B.1.617.2, and P.1 variants of SARS-CoV 2 virus

Abstract

With tools originating from Markov processes, we investigate the similarities and differences between genomic sequences in FASTA format coming from four variants of the SARS-CoV 2 virus, B.1.1.7 (UK), B.1.351 (South Africa), B.1.617.2 (India), and P.1 (Brazil). We treat the virus’ sequences as samples of finite memory Markov processes acting in We model each sequence, revealing some heterogeneity between sequences belonging to the same variant. We identified the five most representative sequences for each variant using a robust notion of classification, see Fernández et al. (Math Methods Appl Sci 43(13):7537–7549. 10.1002/mma.5705 ). Using a notion derived from a metric between processes, see García et al. (Appl Stoch Models Bus Ind 34(6):868–878. 10.1002/asmb.2346), we identify four groups, each group representing a variant. It is also detected, by this metric, global proximity between the variants B.1.351 and B.1.1.7. With the selected sequences, we assemble a multiple partition model, see Cordeiro et al. (Math Methods Appl Sci 43(13):7677–7691. 10.1002/mma.6079), revealing in which states of the state space the variants differ, concerning the mechanisms for choosing the next element in A . Through this model, we identify that the variants differ in their transition probabilities in eleven states out of a total of 256 states. For these eleven states, we reveal how the transition probabilities change from variant (group of variants) to variant (group of variants). In other words, we indicate precisely the stochastic reasons for the discrepancies.

A Metric Based on the Efficient Determination Criterion

“…[4] demonstrates that the EDC, under certain conditions on the term w n (Equation ( 9)), provides a strongly consistent estimate of the minimal partition, as defined in Definition 2. Building on the results from [5], we conjectured that using w n = ln(ln(n)) might preserve strong consistency, even though this term does not satisfy the second condition imposed by Equation (9). We confirm in Theorem 1 and Corollary 1 that strong consistency is indeed achieved using the EDC with the penalization term w n = ln(ln(n)).…”

“…Another option is to use w n = n a ln(n) for a ∈ (0, 1). Figure 1 shows penalty functions w n verifying Equation (9). We see in the figure that functions w n are positioned between the functions n and ln(ln(n)).…”

“…Remark 1. In order to guarantee the consistent estimation of the partition given by Definition 2, we note that Theorem 2 must be used for a large enough n and with weights w n following Equation (9) or w n = ln(ln(n)).…”

“…In [8], a variant of the model specified in Definition 2 is designed specifically to model the first DNA sequence (FASTA format) of SARS-CoV-2 virus, Genbank number MN908947 (accessible at: https://www.ncbi.nlm.nih.gov/nuccore/MN908947, accessed on 10 March 2024). Furthermore, partition Markov models have been instrumental in modeling the DNA of the SARS-CoV-2 virus, exposing the evolution of the various variants during the pandemic period (see, for example, [9]).…”