On the expected total number of infections for virus spread on a finite network.

*(English)*Zbl 1322.60206Summary: In this work we consider a simple SIR infection spread model on a finite population of \(n\) agents represented by a finite graph \(G\). Starting with a fixed set of initial infected vertices the infection spreads in discrete time steps, where each infected vertex tries to infect its neighbors with a fixed probability \(\beta\in(0,1)\), independently of others. It is assumed that each infected vertex dies out after a unit time and the process continues till all infected vertices die out. This model was first studied by M. Draief et al. [Ann. Appl. Probab. 18, No. 2, 359–378 (2008; Zbl 1137.60051)]. In this work, we find a simple lower bound on the expected number of ever infected vertices using a breath-first search algorithm and show that it asymptotically performs better for a fairly large class of graphs than the upper bounds obtained in [loc. cit.]. As a by product, we also derive the asymptotic value of the expected number of the ever infected vertices when the underlying graph is the random \(r\)-regular graph and \(\beta<\frac{1}{r-1}\).

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

60J85 | Applications of branching processes |

05C80 | Random graphs (graph-theoretic aspects) |

92D30 | Epidemiology |

##### Keywords:

percolation; finite graphs; random \(r\)-regular graphs; virus infections; breath-first search; local weak convergence; susceptible infected removed model##### References:

[1] | Aldous, D. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1-72. Springer, Berlin. · Zbl 1037.60008 |

[2] | Alon, N., Benjamini, I. and Stacey, A. (2004). Percolation on finite graphs and isoperimetric inequalities. Ann. Probab. 32 1727-1745. · Zbl 1046.05071 |

[3] | Athreya, K. B. and Ney, P. E. (2004). Branching Processes . Dover Publications, Inc., Mineola, NY. · Zbl 1070.60001 |

[4] | Bollobás, B. (2001). Random Graphs , 2nd ed. Cambridge Univ. Press, Cambridge. · Zbl 0979.05003 |

[5] | Borgs, C., Chayes, J. T., van der Hofstad, R., Slade, G. and Spencer, J. (2005). Random subgraphs of finite graphs. I. The scaling window under the triangle condition. Random Structures Algorithms 27 137-184. · Zbl 1076.05071 |

[6] | Borgs, C., Chayes, J. T., van der Hofstad, R., Slade, G. and Spencer, J. (2005). Random subgraphs of finite graphs. II. The lace expansion and the triangle condition. Ann. Probab. 33 1886-1944. · Zbl 1079.05087 |

[7] | Draief, M., Ganesh, A. and Massoulié, L. (2008). Thresholds for virus spread on networks. Ann. Appl. Probab. 18 359-378. · Zbl 1137.60051 |

[8] | Grimmett, G. (1999). Percolation , 2nd ed. Springer, Berlin. · Zbl 0926.60004 |

[9] | Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs . Wiley, New York. |

[10] | Kozma, G. and Nachmias, A. (2011). A note about critical percolation on finite graphs. J. Theoret. Probab. 24 1087-1096. · Zbl 1235.60138 |

[11] | Wästlund, J. (2012). Replica symmetry of the minimum matching. Ann. of Math. (2) 175 1061-1091. · Zbl 1262.91046 |

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