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Viktor Bezborodov

Publications and source records attributed to Viktor Bezborodov.

At least 19 recordsLinked to original sources
arXiv2026

Household size can explain 40% of the variance in cumulative COVID-19 incidence across Europe

Household size impacts the spread of respiratory infectious diseases: Larger households tend to boost transmission by acquiring external infections more frequently and subsequently transmitting them back into the community. Furthermore, mandatory interventions primarily modulate contagion between households rather than within them. We developed an approach to quantify the role of household size in epidemics by separating within-household from out-household transmission, and found that household size explains 41% of the variability in cumulative COVID-19 incidence across 34 European countries (95% confidence interval: [15%, 46%]). The contribution of households to the overall dynamics can be quantified by a boost factor that increases with the effective household size, implying that countries with larger households require more stringent interventions to achieve the same levels of containment. This suggests that households constitute a structural (dis-)advantage that must be considered when designing and evaluating mitigation strategies.

q-bio.PE↗
arXiv2025

The maximal displacement of radially symmetric branching random walk in $\mathbb{R}^d$

We consider discrete-time branching random walks with a radially symmetric distribution. Independently of each other individuals generate offspring whose relative locations are given by a copy of a radially symmetric point process $\mathcal{L}$. The number of particles at time $t$ form a supercritical Galton-Watson process. We investigate the maximal distance to the origin of such branching random walks. Conditioned on survival, we show that, under some assumptions on $\mathcal{L}$, it grows in the same way as for branching Brownian motion or a broad class of one-dimensional branching random walks: the first term is linear in time and the second logarithmic. The constants in front of these terms are explicit and depend only on the mean measure of $\mathcal{L}$ and dimension. Our main tool in the proof is a ballot theorem with moving barrier which may be of independent interest.

math.PR↗
arXiv2025

Inter-city infections and the role of size heterogeneity in containment strategies

This study examines the effectiveness of regional lockdown strategies in mitigating pathogen spread across regional units, termed cities hereinafter. We develop simplified models to analyze infection spread across cities within a country during an epidemic wave. Isolation of a city is initiated when infection numbers within the city surpass defined thresholds. We compare two strategies: strategy (P) consists in prescribing thresholds proportionally to city sizes, while the same threshold is used for all cities under strategy (U). Given the heavy-tailed distribution of city sizes, strategy (P) may result in more secondary infections from larger cities than strategy (U). Random graph models are constructed to represent infection spread as a percolation process. In particular, we consider a model in which mobility between cities only depends on city sizes. We assess the relative efficiency of the two strategies by comparing the ratios of the number of individuals under isolation to the total number of infections by the end of the epidemic wave under strategy (P) and (U). Additionally, we derive analytical formulas for disease prevalence and basic reproduction numbers. Our models are calibrated using mobility data from France, Poland and Japan, validated through simulation. The findings indicate that mobility between cities in France and Poland is mainly determined by city sizes. However, a poor fit was observed with Japanese data, highlighting the importance to include other factors like e.g. geography for some countries in modeling. Our analysis suggest similar effectiveness for both strategies in France and Japan, while strategy (U) demonstrates distinct merits in Poland.

physics.soc-ph↗
arXiv2025

Block Graph Neural Networks for tumor heterogeneity prediction

Accurate tumor classification is essential for selecting effective treatments, but current methods have limitations. Standard tumor grading, which categorizes tumors based on cell differentiation, is not recommended as a stand-alone procedure, as some well-differentiated tumors can be malignant. Tumor heterogeneity assessment via single-cell sequencing offers profound insights but can be costly and may still require significant manual intervention. Many existing statistical machine learning methods for tumor data still require complex pre-processing of MRI and histopathological data. In this paper, we propose to build on a mathematical model that simulates tumor evolution (Ożański (2017)) and generate artificial datasets for tumor classification. Tumor heterogeneity is estimated using normalized entropy, with a threshold to classify tumors as having high or low heterogeneity. Our contributions are threefold: (1) the cut and graph generation processes from the artificial data, (2) the design of tumor features, and (3) the construction of Block Graph Neural Networks (BGNN), a Graph Neural Network-based approach to predict tumor heterogeneity. The experimental results reveal that the combination of the proposed features and models yields excellent results on artificially generated data ($89.67\%$ accuracy on the test data). In particular, in alignment with the emerging trends in AI-assisted grading and spatial transcriptomics, our results suggest that enriching traditional grading methods with birth (e.g., Ki-67 proliferation index) and death markers can improve heterogeneity prediction and enhance tumor classification.

cs.CV↗
arXiv2023

Explosion and non-explosion for the continuous-time frog model

We consider the continuous-time frog model on $\mathbb{Z}$. At time $t = 0$, there are $η(x)$ particles at $x\in \mathbb{Z}$, each of which is represented by a random variable. In particular, $(η(x))_{x \in \mathbb{Z} }$ is a collection of independent random variables with a common distribution $μ$, $μ(\mathbb{Z}_+) = 1$. The particles at the origin are active, all other ones being assumed as dormant, or sleeping. Active particles perform a simple symmetric continuous-time random walk in $\mathbb{Z} $ (that is, a random walk with $\exp(1)$-distributed jump times and jumps $-1$ and $1$, each with probability $1/2$), independently of all other particles. Sleeping particles stay still until the first arrival of an active particle to their location; upon arrival they become active and start their own simple random walks. Different sets of conditions are given ensuring explosion, respectively non-explosion, of the continuous-time frog model. Our results show in particular that if $μ$ is the distribution of $e^{Y \ln Y}$ with a non-negative random variable $Y$ satisfying $\mathbb{E} Y < \infty$, then a.s. no explosion occurs. On the other hand, if $a \in (0,1)$ and $μ$ is the distribution of $e^X$, where $\mathbb{P} \{X \geq t \} = t^{-a}$, $t \geq 1$, then explosion occurs a.s. The proof relies on a certain type of comparison to a percolation model which we call totally asymmetric discrete inhomogeneous Boolean percolation.

math.PR↗
arXiv2023

Interacting particle systems with continuous spins

We study a general class of interacting particle systems over a countable state space $V$ where on each site $x \in V$ the particle mass $η(x) \geq 0$ follows a stochastic differential equation. We construct the corresponding Markovian dynamics in terms of strong solutions to an infinite coupled system of stochastic differential equations and prove a comparison principle with respect to the initial configuration as well as the drift of the process. Using this comparison principle, we provide sufficient conditions for the existence and uniqueness of an invariant measure in the subcritical regime and prove convergence of the transition probabilities in the Wasserstein-1-distance. Finally, for sublinear drifts, we establish a linear growth theorem showing that the spatial spread is at most linear in time. Our results cover a large class of finite and infinite branching particle systems with interactions among different sites.

math.PR↗
arXiv2023

Minimal controllability time for systems with nonlinear drift under a compact convex state constraint

In this paper we estimate the minimal controllability time for a class of non-linear control systems with a bounded convex state constraint. An explicit expression is given for the controllability time if the image of the control matrix is of co-dimension one. A lower bound for the controllability time is given in the general case. The technique is based on finding a lower dimension system with the similar controllability properties as the original system. The controls corresponding to the minimal time, or time close to the minimal one, are discussed and computed analytically. The effectiveness of the proposed approach is illustrated by a few examples.

math.OC↗
arXiv2023

Linear and superlinear spread for continuous-time frog model

Consider a stochastic growth model on $\mathbb{Z} ^d$. Start with some active particle at the origin and sleeping particles elsewhere. The initial number of particles at $x \in \mathbb{Z} ^d$ is $η(x)$, where $η(x)$ are independent random variables distributed according to $μ$. Active particles perform a simple continuous-time random walk while sleeping particles stay put until the first arrival of an active particle to their location. Upon the arrival all sleeping particles at the site activate at once and start moving according to their own simple random walks. The aim of this paper is to give conditions on $μ$ under which the spread of the process is linear or faster than linear. The proofs rely on comparison to various percolation models.

math.PR↗
arXiv2022

Spatial birth-and-death processes with a finite number of particles

Spatial birth-and-death processes with time dependent rates are obtained as solutions to certain stochastic equations. The existence, uniqueness, uniqueness in law and the strong Markov property of unique solutions are proven when the integral of the birth rate over $\mathbb{R} ^ \mathrm{d}$ grows not faster than linearly with the number of particles of the system. Martingale properties of the constructed process provide a rigorous connection to the heuristic generator. We also study pathwise behavior of an aggregation model. The probability of extinction and the growth rate of the number of particles conditioning on non-extinction are estimated.

math.PR↗
arXiv2021

Non-triviality in a totally asymmetric one-dimensional Boolean percolation model on a half-line

It is well known that there are two regimes in a standard one-dimensional Boolean percolation model: either the entire space is covered a.s., or the covered volume fraction is strictly less than one. The aim of this work is to demonstrate that there is a third possibility in a Boolean model with totally asymmetric grains on a half-line: a.s. there is no unbounded component, but the covered volume fraction is one. An explicit condition is given characterizing the existence of an unbounded occupied component.

math.PR↗
arXiv2020

Lattice birth-and-death processes

Lattice birth-and-death Markov dynamics of particle systems with spins from the set of non-negative integers are constructed as unique solutions to certain stochastic equations. Pathwise uniqueness, strong existence, Markov property and joint uniqueness in law are proven, and a martingale characterization of the process is given. Sufficient conditions for the existence of an invariant distribution are formulated in terms of Lyapunov functions. We apply obtained results to discrete analogs of the Bolker--Pacala--Dieckmann--Law model and an aggregation model.

math.PR↗
arXiv2020

Spatial growth processes with long range dispersion: microscopics, mesoscopics, and discrepancy in spread rate

We consider the speed of propagation of a {continuous-time continuous-space} branching random walk with the additional restriction that the birth rate at any spatial point cannot exceed $1$. The dispersion kernel is taken to have density that decays polynomially as $|x|^{- 2α}$, $x \to \infty$. We show that if $α> 2$, then the system spreads at a linear speed, {while for $α\in (\frac 12 ,2]$ the spread is faster than linear}. We also consider the mesoscopic equation corresponding to the microscopic stochastic system. We show that in contrast to the microscopic process, the solution to the mesoscopic equation spreads exponentially fast for every $α> \frac 12$.

math.PR↗
arXiv2019

Fecundity regulation in a spatial birth-and-death process

We study a Markov birth-and-death process on a space of locally finite configurations, which describes an ecological model with a density dependent fecundity regulation mechanism. We establish existence and uniqueness of this process and analyze its properties. In particular, we show global time-space boundedness of the population density and, using a constructed Foster-Lyapunov-type function, we study return times to certain level sets of tempered configurations. We find also sufficient conditions that the degenerate invariant distribution is unique for the considered process.

math.PR↗