1 Lecture 16 Epidemics University of Nevada – Reno Computer Science & Engineering Department Fall 2015 CS 791 Special Topics: Network Architectures and Economics

Outline  Epidemics  Branching process  SIR model  SIS model  SIRS model  Synchronization 2

Epidemiology  The study of populations and the transfer of disease between subsets of those populations.  Models the rate at which a healthy population becomes infected and recovers 3 What if we have a network of contact between individuals?

Worm/Virus Spreading in the Internet  How to control the spread of the threat by reinforcing local rules?  E.g.: What is the maximum probability of contagion between two neighbors so that the overall network is protected against a complete invasion by the threat?  How to model?  Resembles a lot to the epidemiology 4

Model Basics and Goals 5  A contagious threat/idea  Key goals:  Will the epidemic invade the whole network?  How long will the epidemic take to invade?  Contagion  One person “infects” another  p – probability that an infectious person transmits the threat to a neighbor he/she meets  k – # of new people an infected node meets during a time slot  Reproductive Number: R 0 – expected # of new nodes getting infected at a time slot

Branching Process 6  k = 3  A 3-ary tree

Branching Process  high p 7

Branching Process  low p 8

Branching Process  If R 0 < 1, the epidemic ends in a finite amount of time  How long time?  If R 0 > 1, the epidemic runs forever with a positive probability  How large is the probability?  Birth-Death Processes from queuing theory 9

SIR Model  Susceptible, Infected, Removed Model  The process  Some in I and some in S state initially  Each infectious node stay in the I state for t I slots  During each step an I node has p probability of infecting a susceptible neighbor  After t I slots, an I node “recovers” and goes to R state 10

SIR Model Susceptible: 11

SIR Model Infected 12

SIR Model  Removed 13

SIR Model  Recovered 14

SIR Model  Recovered 15

SIR Model and Network Shape  Network shape can change dynamics significantly..  p = 2/3, k = 2  R 0 = 4/3 > 1  A highly infectious threat  But, (1/3) 4 = 1/81 probability that the epidemic will stop at each step!  With probability 1, the epidemic will stop in a finite number of steps! 16

SIR Model and Percolation  Another way to look at the SIR model  Run the probability of contagion for each link  Then see if there is a path from the root(s) of the threat to a node  Helps to figure out if a particular node is more/less likely to be infected  Percolation Theory: Also used for understanding the end-to-end connectivity patterns 17

SIS Model  Only two states: Susceptible and Infected  The Removed state is eliminated  The process  Some in I and some in S state initially  Each infectious node stays in the I state for t I slots  During each step an I node has p probability of infecting a susceptible neighbor  After t I slots, an I node goes back to S state 18

SIS Model  More general and maybe more realistic  Can model larger set of scenarios  A threat disappearing and appearing over cycles  An inbounded supply of nodes 19

SIS and SIR Models 20

SIRS Model  Three states: Susceptible, Infected, Removed  The process  Some in I and some in S state initially  Each infectious node stays in the I state for t I slots  During each step an I node has p probability of infecting a susceptible neighbor  After t I slots, an I node goes back to R state and stays there for t R slots, and then returns to the S state 21

Synchronization  Small-world can change things..  c – probability that a node has long-range links  Again, network characteristics in action.. 22

Optimal Marketing – Revisited  Which k people should we “infect” so that the spread of the “disease” is as large as possible?  We can greedily pick these k people: first, pick the single best individual x1, then the best individual paired with x1, … 23

Optimal Marketing – Revisited  The problem is in general NP-Hard, even for 0/1 transmission probabilities.  The greedy algorithm picks an initial set of k people so that the spread is at least 63% as big as the largest possible spread over all k node starting sets. 24

Lecture 16: Summary  Reproductive number w.r.t. 1  Network shape has serious effects  Synchronization effects due to small- world 25

Lecture 16: Reading  Easley & Kleinberg, Chapter 21  Chen, Gao, and Kwiat, Modeling the Spread of Active Worms, IEEE INFOCOM,