Law of large numbers in an epidemic model

Abstract

113Over the last decade, a huge number of publica tions have been devoted to the construction of epi demic models on graphs (see, e.g., [1][2][3][4][5][6]) and to the study of probability properties of these models. An additional stimulus was provided by the Internet, since epidemic models play an important role in the study of the evolution of web graphs. Many similar models were described in [7]. This paper deals with a general ization of the model proposed by Machado, Mashu rian, and Matzinger [8], who considered the following situation. Let {x 1 , x 2 , …, x n } be a set of points. At the time t = 1, there is a single particle at each point. The point x 1 contains an active particle, while the other points contain inactive ones. At each time t, exactly one active particle with an index having a uniform dis tribution on the index set of active particles available at t moves to a point x ξ(t) . The random variable ξ(t) also has a uniform distribution on {1, 2, …, n}. If the parti cle moves to a point with an inactive particle, then the latter is activated and the former remains active as well. An active particle dies if it moves to a point where there is or was an active particle (if there is an active particle at this point, it survives). Specifically, if a par ticle moves to its own point, it dies. Let D n (t) denote the number of inactive particles at the time t and σ n be the time at which the last active particle dies. The fol lowing important result was derived for the random variable X n = n -D n (σ n ) in [8].Theorem 1 [8]. Let q be a unique solution of the equation 2p = -ln(1 -p) in the interval (0, 1), and let σ = . Then we have the central limit theoremWe study a more complicated model, assuming that several particles move at each time and the number ofthese particles is random. Consider the sets {x 1 (n),x 2 (n), …, x n (n)}, n ∈ ‫.ގ‬ As in the above model, the time t is discrete and we assume that there is a single particle at each point at the time t = 1. The point x 1 = x 1 (n) contains an active particle, while the other point con tain inactive ones. For each n ∈ ‫,ގ‬ the particles are indexed according to their original order (at the time 1, the ith particle is at the point x i ). At each time, an active particle moves with probability p irrespective of the other active particles. Moreover, for a moving particle, the probability of hitting any point is . If a particle moves to a point where there was (or is) an active parti cle, then the former instantly dies. If a particle moves to a point with an inactive particle, then the latter becomes active and the former survives. If several particles move to the same point with an inactive particle at the time, then all of them survive and the inactive particle is acti vated. Inactive and active particles are called alive. Assume that all the considered random variables are defined on some probability space (Ω, Ᏺ, P). Let A n (t) denote the number of active particles at the time t, while D n (t), as before, denote the number of inactive particles. The...

Information Transmission under Random Emission Constraints

“…Assuming without any loss of generality that 32) the second line following from the LDP we proved above for (r, t) satisfying 0 < r < t ∧ 1. This is enough to conclude the proof.…”

“…Each active particle starts with L lives and looses one life unit whenever it jumps on a vertex which has already been visited by the process. For another similar model with simultaneous jumps in discrete time, the number of informed servers has fluctuations of order n 3/4 [32].…”