Global Dynamics of Epidemic Spread over Complex Networks Hyoung Jun Ahn and Babak Hassibi California Institute of Technology Introduction

Upper Bound of Nonlinear Model by Linear Model

Modeling epidemic spread in networks can play a key role in preventing the spread of threatening diseases. It is also helpful to understand the spread of computer viruses. We study the threshold of epidemic spread and how it is related with linear, and nonlinear models, as well as the Markov chain model.

Epidemic State 1

≤ 𝑃𝑖 𝑡 1 − δ + 1 −

1 − 𝛽𝑃𝑗 𝑡 𝑃𝑗 (𝑡)

𝑗~𝑖

𝐴 ∶ Adjacency Matrix of G,

𝑃 𝑡 = (𝑃1 𝑡 , 𝑃2 𝑡 , ⋯ , 𝑃𝑛 𝑡 ) 𝑃 𝑡 + 1 ≤ 1 − δ 𝐼 + 𝛽𝐴 𝑃(𝑡) We get a linear model by replacing inequality to equality just above.

4 5

If 𝜆𝑚𝑎𝑥 1 − δ 𝐼 + 𝛽𝐴 < 1 , the origin is a unique fixed point and it is globally stable in both linear and nonlinear model.

1001010

7

𝑗~𝑖

≤ 𝑃𝑖 𝑡 1 − δ + 𝛽

3

6

1 − 𝛽𝑃𝑗 𝑡

𝑗~𝑖

Blue(0) : Healthy Red(1) : Infected

2

𝑃𝑖 𝑡 + 1 = 𝑃𝑖 𝑡 1 − δ + (1 − 1 − δ 𝑃𝑖 𝑡 ) 1 −

Epidemic Spread Probabilistic Transition 1

What happens to the nonlinear model if the linear model is not stable?

1

2

3

2

3 4

4

5

5

6

6 7

Limit cycle (Orbit)

7

X=0111011

Epidemic spread is independent for each node. 𝑛

Pr 𝑌𝑖 𝑋 𝑖=1 3

𝛽 : infection rate of a node from one’s neighbor 𝛿 : recovery rate

Pr 𝑌1 𝑋 = 1 − (1 − 𝛽) Pr 𝑌2 𝑋 = 1 − 𝛿(1 − 𝛽) Pr 𝑌3 𝑋 = 1 − 𝛿(1 − 𝛽) Pr 𝑌4 𝑋 = 𝛿(1 − 𝛽)2 Pr 𝑌5 𝑋 = (1 − 𝛽)2 Pr 𝑌6 𝑋 = 𝛿(1 − 𝛽)2 Pr 𝑌7 𝑋 = 1 − 𝛿(1 − 𝛽)2

The probability of infection depends on the number of infected neighbors. An infected node becomes healthy when it recovers from the disease and is not infected from any of neighbors at same time.

We can make a Markov chain model with the probability as above. The transition matrix is 2𝑛 × 2𝑛 huge matrix where 𝑛 is the number of nodes in the network. X = 0000000 ⇒ no more transition All-healthy-state is an absorbing state. At the stationary distribution, all nodes are healthy with probability 1. It means that the epidemic dies out after long enough time since probability distribution of Markov chain goes to the stationary distribution as time passes. How long does it take to converge to the stationary distribution? ⇒ We study the mixing time of the Markov chain model.

Approximation of the Markov Chain Model

Epidemic dies out If 𝜆𝑚𝑎𝑥 𝐴 Epidemic spreads If 𝜆𝑚𝑎𝑥 𝐴

We focus on the marginal probability of each node being infected at time 𝑡 𝐼(𝑡) : set of infected nodes at time 𝑡, 𝑝𝑖 𝑡 = Pr[𝑖 ∈ 𝐼 𝑡 ] the marginal probability Marginal probability at time 𝑡 + 1 where 𝑚𝑖 is the number of 𝑖’s infected neighbors 𝑝𝑖 𝑡 + 1 = Pr 𝑖 ∈ 𝐼 𝑡 + 1 𝑖 ∈ 𝐼 𝑡 × 𝑝𝑖 𝑡 + Pr 𝑖 ∈ 𝐼 𝑡 + 1 𝑖 ∉ 𝐼 𝑡 × (1 − 𝑝𝑖 𝑡 ) = 1 − 𝛿 1 − 𝛽 𝑚𝑖 𝑝𝑖 𝑡 + 1 − 1 − 𝛽 𝑚𝑖 1 − 𝑝𝑖 𝑡 𝑗~𝑖

1 − 𝛽1𝐼

𝑡

𝑗

𝑝𝑖 𝑡 + 1 −

𝑗~𝑖

1 − 𝛽1𝐼

𝑡

𝑗

(1 − 𝑝𝑖 𝑡 )

Assume that each infection is independent. 𝑗~𝑖

1 − 𝛽1𝐼

𝑡

𝑗 ≈ 𝑗~𝑖

𝐄 1 − 𝛽1𝐼

𝑡

𝑗

= 𝑗~𝑖

1 − 𝛽𝑝𝑗 𝑡

Distinguish 𝑃𝑖 𝑡 , the approximated one from 𝑝𝑖 𝑡 , the exact one. We get Chakrabarti’s model (Nonlinear model) 𝑃𝑖 𝑡 + 1 = 𝑃𝑖 𝑡 1 − δ + (1 − 1 − δ 𝑃𝑖 𝑡 ) 1 −

1 − 𝛽𝑃𝑗 𝑡 𝑗~𝑖

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Chaos

𝛿 < 𝛽 𝛿 > 𝛽

Epidemic threshold

The Mixing Time of the Markov Chain Model Returning back to the Markov chain model, Formal definition of the mixing time 𝑡𝑚𝑖𝑥 𝜀 = min{𝑡: sup𝜇 𝜇𝑆 𝑡 − 𝜋 𝑇𝑉 < 𝜀} where 𝑆 is transition matrix of the Markov chain. 𝜋 is stationary distribution and 𝜇 is any initial distribution. All the nodes are healthy with probability 1 at 𝜋 Theorem. The nonlinear model at time 𝒕 gives an upper bound for the probability that the epidemic state at time 𝒕 is not a steady state with given initial condition in the true Markov chain model. If 𝝀𝒎𝒂𝒙 𝟏 − 𝜹 𝑰 + 𝜷𝑨 < 𝟏, the mixing time of the Markov chain is 𝑶(𝐥𝐨𝐠 𝒏).

2𝑛 × 2𝑛 ⇒ too large to analyze !!!

= 1−𝛿

Unique fixed point

Theorem. If 𝝀𝒎𝒂𝒙 𝟏 − 𝜹 𝑰 + 𝜷𝑨 < 𝟏 , the origin is the unique fixed point of the nonlinear model and it is globally stable. If 𝝀𝒎𝒂𝒙 𝟏 − 𝜹 𝑰 + 𝜷𝑨 > 𝟏 , the origin is an unstable fixed point and there exists a second nontrivial fixed point of the nonlinear model which is also globally stable. i.e. all but the origin (the trivial fixed point) converges to the nontrivial fixed point as time passes.

Y=1110001 Pr 𝑌 𝑋 =

Multiple fixed points

Summary Convergence to the origin, all-zero-point, means that epidemic dies out. At the stationary distribution in the Markov chain model, all nodes are healthy with probability 1. 𝜹 𝝀𝒎𝒂𝒙 (𝑨) < 𝜷 Linear Nonlinear

Stable The origin is the unique fixed point and it is globally stable.

Markov Chain Mixes fast. 𝑂(log 𝑛)

𝝀𝒎𝒂𝒙

𝜹 𝑨 > 𝜷

Unstable The origin is an unstable fixed point. There exists the second nontrivial fixed point which is globally stable. Mixes exponentially slow.

When the mixing time of the Markov chain model is exponentially long, it could take time equivalent to the age of the universe. In that case, the epidemic practically never dies out.