On the Expected Total Number of Infections for Virus Spread on a Finite Network

In this paper we consider a simple virus infection spread model on a finite population of $n$ agents connected by some neighborhood structure. Given a graph $G$ on $n$ vertices, we begin with some fixed number of initial infected vertices. At each discrete time step, an infected vertex tries to infect its neighbors with probability $\beta \in (0,1)$ independently of others and then it dies out. The process continues till all infected vertices die out. We focus on obtaining proper lower bounds on the expected number of ever infected vertices. We obtain a simple lower bound, using $\mathit{\text{breadth-first}}\mathit{\text{search}}$ algorithm and show that for a large class of graphs which can be classified as the ones which locally “look like” a tree in sense of the \emph{local weak convergence}, this lower bound gives better approximation than some of the known approximations through matrix-method based upper bounds derived by Draief, Ganesh and Massoulie in 2008.