Interacting Viruses in Networks: Can Both Survive? Authors: Alex Beutel, B. Aditya Prakash, Roni Rosenfeld, and Christos Faloutsos Presented by: Zachary Stanfield March 3, 2015

Motivation Central Question: Given two partially competing products/entities, is it possible that they both survive? Current Theory: The Competitive Exclusion Principle - when two species compete under constant conditions, the more fit one will drive the less fit one to extinction. No complete theory for two competing species under partial immunity that allows/predicts coexistence.

Background and Related Work Single virus models: SIS, SIR, SIRS (S = susceptible, I = infected) Notable limitation: Most studies assume that the network is a clique (homogeneous model) Focus placed on finding the “epidemic threshold”. Under what conditions will a virus become extinct? Threshold recently discovered as the leading eigenvalue of the graph adjacency matrix for a single virus model. Two virus-models of SIS with full mutual immunity recently studied on arbitrary graphs -> Winner takes all

Goals Provide analytical proof of a competitive epidemic threshold, ϵ critical Define a novel model Solve the closed-form steady-state behavior under the SIS (flu-like) setting Show that the model applies to real world instances

Model Formulation State Diagram for a single node in the graph for partial competition

Mathematical Analysis At steady state, the derivatives are equal to zero, and three equilibrium points are found. And possibly one more where

Finding critical For κ 1, κ 2 > 0, where κ 1 = ((I 1 + I 1,2 )/N) = fraction of nodes infected with virus 1, they prove that there exist valid equilibrium points κ 1 * and κ 2 * for which the population infected by each virus is non-zero Given a fully connected graph, their SI 1|2 S model, and virus strengths σ 1 ≥σ 2 ≥1, the above equilibrium points exist if ϵ > ϵ critical

Experiments *(6-4)/(4*(6-1)) = 0.1

Case Studies Data from Google-Insights. Two examples of partial immunity – people can use both, but the use of one lessens the use of the other. Introduce Hulu and Chrome while Blockbuster and Firefox begin in steady state. Hulu and Chrome have higher “infection rates” δ

Conclusions This model setup is novel. Mathematical proof of the transition from winner takes all to coexistence Experiments and Case-studies agree with model prediction Impressions Logical approach to an unsolved problem Simulations accurately capture real data in many application areas Limited to fully connected graphs and parameters have to be chosen correctly to fit the data, but there is predictive power.