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Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) lineage C.37 (Lambda) has spread rapidly in Peru and other Latin American countries. However, most studies in Peru have focused on Lima, the capital city, without knowing the dynamics of the spread of the variant in other departments. Cusco, Peru, is one of the most popular departments in the country for tourists, so the introduction of new variants of SARS-CoV-2 might occur despite closure of the borders. Therefore, in this work, we analyzed the variants circulating in Cusco. The aim of this work was to better understand the distribution of SARS-CoV-2 lineages circulating in Cusco and to characterize the genomes of these strains. To this end, 46 SARS-CoV-2 genomes from vaccinated and unvaccinated patients were sequenced in the first half of 2021. The genomes were analyzed using phylogenetic and natural selection methods. Phylogenetic trees from Cusco showed dominance of the Lambda lineage over the variants of concern (VOCs), and there was no clustering of variants by district. Natural selection analysis revealed mutations, mainly in the spike protein, at positions 75, 246, 247, 707, 769, and 1020. In addition, we found that unvaccinated patients accumulated more new mutations than did vaccinated patients, and these included the F101Y mutation in ORF7a, E419A in NSP3, a deletion in S (21,618-22,501), and a deletion in ORF3a (25,437-26,122).

In this paper we study the Maki-Thompson rumor model on infinite Cayley trees. The basic version of the model is defined by assuming that a population represented by a graph is subdivided into three classes of individuals: ignorants, spreaders and stiflers. A spreader tells the rumor to any of its (nearest) ignorant neighbors at rate one. At the same rate, a spreader becomes a stifler after a contact with other (nearest neighbor) spreaders, or stiflers. In this work we study this model on infinite Cayley trees, which is formulated as a continuous-times Markov chain, and we extend our analysis to the generalization in which each spreader ceases to propagate the rumor right after being involved in a given number of stifling experiences. We study sufficient conditions under which the rumor either becomes extinct or survives with positive probability.

Evolution algebras are non-associative algebras. In this work we provide an extension of this class of algebras, in a framework of Hilbert spaces, and illustrate the applicability of our approach by discussing a connection with discrete-time Markov chains with infinite countable state space. Specifically, if we associate to each possible state of such a Markov process a generator of the Hilbert evolution algebra structure, then the whole dynamics of the process can be described through consecutive applications of the evolution operator, provided certain boundedness conditions on the transition probabilities hold.

We consider the Maki–Thompson model for the stochastic propagation of a rumour within a population. In this model the population is made up of “spreaders”, “ignorants” and “stiflers”; any spreader attempts to pass the rumour to the other individuals via pair-wise interactions and in case the other individual is an ignorant, it becomes a spreader, while in the other two cases the initiating spreader turns into a stifler. In a finite population the process will eventually reach an equilibrium situation where individuals are either stiflers or ignorants. We extend the original hypothesis of homogenously mixed population by allowing for a small-world network embedding the model, in such a way that interactions occur only between nearest-neighbours. This structure is realized starting from a k-regular ring and by inserting, in the average, c additional links in such a way that k and c are tuneable parameters for the population architecture. We prove that this system exhibits a transition between regimes of localization (where the final number of stiflers is at most logarithmic in the population size) and propagation (where the final number of stiflers grows algebraically with the population size) at a finite value of the network parameter c. A quantitative estimate for the critical value of c is obtained via extensive numerical simulations.

We study a class of growing systems of random walks on regular trees, known as frog models with geometric lifetime in the literature. With the help of results from renewal theory, we derive new bounds for their critical parameters. Our approach also improves the existing bounds for the critical parameter of a percolation model on trees known as cone percolation.

In the classical Erdős–Rényi random graph G(n, p) there are n vertices and each of the possible edges is independently present with probability p. The random graph G(n, p) is homogeneous in the sense that all vertices have the same characteristics. On the other hand, numerous real-world networks are inhomogeneous in this respect. Such an inhomogeneity of vertices may influence the connection probability between pairs of vertices. The purpose of this paper is to propose a new inhomogeneous random graph model which is obtained in a constructive way from the Erdős-Rényi random graph G(n, p). Given a configuration of n vertices arranged in N subsets of vertices (we call each subset a super-vertex), we define a random graph with N super-vertices by letting two super-vertices be connected if and only if there is at least one edge between them in G(n, p). Our main result concerns the threshold for connectedness. We also analyze the phase transition for the emergence of the giant component and the degree distribution. Even though our model begins with G(n, p), it assumes the existence of some community structure encoded in the configuration. Furthermore, under certain conditions it exhibits a power law degree distribution. Both properties are important for real-world applications.

We study two rumor processes on N , the dynamics of which are related to an SI epidemic model with long range transmission. Both models start with one spreader at site 0 and ignorants at all the other sites of N , but differ by the transmission mechanism. In one model, the spreaders transmit the information within a random distance on their right, and in the other the ignorants take the information from a spreader within a random distance on their left. We obtain the probability of survival, information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders. The key step of our proofs is to show that, in each model, the position of the spreaders on N can be related to a suitably chosen discrete renewal process.

We consider Stavskaya’s process, which is a two-state probabilistic cellular automaton defined on a one-dimensional lattice. The state of any vertex depends only on itself and on the state of its right-adjacent neighbour. This process was one of the first multicomponent systems with local interaction for which the existence of a kind of phase transition has been rigorously proved. However, the exact localisation of its critical value remains as an open problem. We provide a new lower bound for the critical value.

We propose a stochastic model describing a process of awareness, evaluation, and decision making by agents on the d-dimensional integer lattice. Each agent may be in any of the three states belonging to the set {0, 1, 2. In this model 0 stands for ignorants, 1 for aware, and 2 for adopters. Aware and adopters inform its nearest ignorant neighbors about a new product innovation at rate λ. At rate α an agent in aware state becomes an adopter due to the influence of adopters' neighbors. Finally, aware and adopters forget the information about the new product, thus becoming ignorant, at rate 1. Our purpose is to analyze the influence of the parameters on the qualitative behavior of the process. We obtain sufficient conditions under which the innovation diffusion (and adoption) either becomes extinct or propagates through the population with positive probability.

Terrestrial thermal springs are widely distributed globally, and these springs harbor a broad diversity of organisms of biotechnological interest. In Mexico, few studies exploring this kind of environment have been described. In this work, we explore the microbial community in Chignahuapan hot springs, which provides clues to understand these ecosystems’ diversity. We assessed the diversity of the microorganism communities in a hot spring environment with a metagenomic shotgun approach. Besides identifying similarities and differences with other ecosystems, we achieved a systematic comparison against 11 metagenomic samples from diverse localities. The Chignahuapan hot springs show a particular prevalence of sulfur-oxidizing bacteria from the genera Rhodococcus, Thermomonas, Thiomonas, Acinetobacter, Sulfurovum, and Bacillus, highlighting those that are different from other recovered bacterial populations in circumneutral hot springs environments around the world. The co-occurrence analysis of the bacteria and viruses in these environments revealed that within the Rhodococcus, Thiomonas, Thermonas, and Bacillus genera, the Chignahuapan samples have specific species of bacteria with a particular abundance, such as Rhodococcus erytropholis. The viruses in the circumneutral hot springs present bacteriophages within the order Caudovirales (Siphoviridae, Myoviridae, and Podoviridae), but the family of Herelleviridae was the most abundant in Chignahuapan samples. Furthermore, viral auxiliary metabolic genes were identified, many of which contribute mainly to the metabolism of cofactors and vitamins as well as carbohydrate metabolism. Nevertheless, the viruses and bacteria present in the circumneutral environments contribute to the sulfur cycle. This work represents an exhaustive characterization of a community structure in samples collected from hot springs in Mexico and opens opportunities to identify organisms of biotechnological interest.