Spatial extent of an outbreak in animal epidemics

Abstract

Characterizing the spatial extent of epidemics at the outbreak stage is key to controlling the evolution of the disease. At the outbreak, the number of infected individuals is typically small, and therefore, fluctuations around their average are important: then, it is commonly assumed that the susceptible-infected-recovered mechanism can be described by a stochastic birth-death process of GaltonWatson type. The displacements of the infected individuals can be modeled by resorting to Brownian motion, which is applicable when long-range movements and complex network interactions can be safely neglected, like in the case of animal epidemics. In this context, the spatial extent of an epidemic can be assessed by computing the convex hull enclosing the infected individuals at a given time. We derive the exact evolution equations for the mean perimeter and the mean area of the convex hull, and we compare them with Monte Carlo simulations.branching Brownian motion | extreme value statistics M odels of epidemics traditionally consider three classes of populations-namely, the susceptible (S), the infected (I), and the recovered (R). This framework provides the basis of the so-called SIR model (1, 2), a fully connected mean-field model where the population sizes of the three species evolve with time t by the coupled nonlinear equations: dS/dt = −βIS, dI/dt = βIS − γI, and dR/dt = γI. Here, γ is the rate at which an infected individual recovers, and β denotes the rate at which it transmits the disease to a susceptible (3-5). In the simplest version of these models, the recovered cannot be infected again. These rate equations conserve the total population size I(t) + S(t) + R(t) = N; one assumes that, initially, there is only one infected individual, and the rest of the population is susceptible: I(0) = 1, S(0) = N − 1, and R(0) = 0. Of particular interest is the outbreak stage (i.e., the early times of the epidemic process), when the susceptible population is much larger than the number of infected or recovered. During this regime, for large N, the susceptible population hardly evolves and stays S(t) ∼ N; therefore, nonlinear effects can be safely neglected, and one can just monitor the evolution of the infected population alone: dI/dt ∼ (βN − γ)I(t). Thus, the ultimate fate of the epidemics depends on the key dimensionless parameter R 0 = βN/ γ, which is called the reproduction rate. If R 0 > 1, the epidemic explodes and invades a finite fraction of the population; if R 0 < 1, the epidemic goes to extinction, and in the critical case R 0 = 1 the infected population remains constant (6-8).This basic deterministic SIR has been generalized to a variety of both deterministic as well as stochastic models, and distinct advantages and shortcomings are discussed at length in refs. 9-11. Generally speaking, stochastic models are more suitable in the presence of a small number of infected individuals, when fluctuations around the average may be relevant (9, 10). During the outbreak of epidemics, the infected population is typically ...

Neutron fluctuations: The importance of being delayed

“…However, because of fluctuations on the number of individuals in the population, a non-trivial finite extinction probability exists for the whole system [3]. A super-critical regime is typically found also during the early stages of an epidemic (the so-called 'outbreak' phase), where a fast growth of the infected population is observed, until non-linear effects due to the depletion of susceptible individuals ultimately slow down the epidemic [14]. In the intermediate regime, the population stays constant on average, and the system is said to be exactly critical.…”

Universal Order and Gap Statistics of Critical Branching Brownian Motion

“…For example, the first maximum x 1 (t) ∼ vt typically increases linearly with t and its cumulative distribution satisfies a nonlinear Fisher-Kolmogorov-PetrovkyPiscounov equation [22,37] with a traveling front solution with velocity v [24,25]. The statistics of this first maximum, in the supercritical phase, also appears in numerous other applications in mathematics [38,39] and physics [20,21,34]. More recently, the statistics of the gaps between successive maxima have also been studied in the supercritical phase [20,21] and the average gap between the k-th and (k + 1)-th maximum was shown to tend to a k-dependent constant, independent of time t, at large t. The stationary probability distribution function (PDF) of the first gap was also computed numerically and an analytical argument was given to explain its exponential tail [20,21].…”

“…[34,[40][41][42]. In this Letter, we show that, in contrast to the supercritical case, the order and the gap statistics can be computed exactly for the critical case b = d. In the critical case where n(t) = 1 at all times, to make sense of the gaps between particles, it is necessary to work in the fixed particle number sector, i.e., condition the process to have exactly n(t) = n particles at time t, with their ordered positions denoted by x 1 (t) > .…”

“…1. BBM is a prototypical model of evolution, but has also been extensively used as a simple model for reaction-diffusion systems, disordered systems, nuclear reactions, cosmic ray showers, epidemic spreads amongst others [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. In one dimension, the position of the existing particles at time t constitute a set of strongly correlated variables that are naturally ordered according to their positions on the line with x 1 (t) > x 2 (t) > x 3 (t) .…”

Universality and time-scale invariance for the shape of planar Lévy processes

“…non-Markovian anomalous diffusion processes [38], random-acceleration processes [39] and constrained processes like excluded-volume or non self-intersecting processes [3,5]): will there be other universality classes or just case-by-case formulae for the average number of edges appearing on the convex hull of sample paths? Also relevant will be the study of the global convex hull of multiple dependent or independent paths [31,40].…”