Epidemic Spread in Complex Networks Babak Hassibi joint work with Elizabeth Barron-Bodine, Subhonmesh Bose, Hyoung Jun Ahn and Navid Azizan-Ruhi California Institute of Technology IMA Workshop on the Analysis and Control of Network Dynamics University of Minnesota, October 22, 2015 Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 1 / 57
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Epidemic Spread in Complex Networks€¦ · Though initially proposed to understand the spread of contagious diseases, they apply to many other settings: I network security (to understand
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Epidemic Spread in Complex Networks
Babak Hassibi
joint work with Elizabeth Barron-Bodine, Subhonmesh Bose, Hyoung Jun Ahnand Navid Azizan-Ruhi
California Institute of Technology
IMA Workshop on the Analysis and Control of Network DynamicsUniversity of Minnesota, October 22, 2015
Network Model:I nodes are vertices of a graph, described by an n× n adjacency matrix A
Infection Model: (SIS: Susceptible-Infectious-Susceptible)I each node in the population transitions between two possible states,
i.e., susceptible and infected, characterized by two parameters, δ and βthat represent the recovery and infection rates, respectively
I During each time step:1 an infected node can recover with probability δ and become susceptible2 a susceptible node can become infected by each of its infected
I each node in the population transitions between two possible states,i.e., susceptible and infected, characterized by two parameters, δ and βthat represent the recovery and infection rates, respectively
I During each time step:1 an infected node can recover with probability δ and become susceptible2 a susceptible node can become infected by each of its infected
Network Model:I nodes are vertices of a graph, described by an n× n adjacency matrix A
Infection Model: (SIS: Susceptible-Infectious-Susceptible)I each node in the population transitions between two possible states,
i.e., susceptible and infected, characterized by two parameters, δ and βthat represent the recovery and infection rates, respectively
I During each time step:1 an infected node can recover with probability δ and become susceptible2 a susceptible node can become infected by each of its infected
Network Model:I nodes are vertices of a graph, described by an n× n adjacency matrix A
Infection Model: (SIS: Susceptible-Infectious-Susceptible)I each node in the population transitions between two possible states,
i.e., susceptible and infected, characterized by two parameters, δ and βthat represent the recovery and infection rates, respectively
I During each time step:1 an infected node can recover with probability δ and become susceptible2 a susceptible node can become infected by each of its infected
The state transition matrix that describes the evolution from state ξ(t) tostate ξ(t + 1) is
P(ξi (t+1) = Yi |ξ(t) = X ) =
(1− β)|Ni∩SX | if (Xi ,Yi ) = (0, 0)
1− (1− β)|Ni∩SX | if (Xi ,Yi ) = (0, 1)
δ(1− β)|Ni∩SX | if (Xi ,Yi ) = (1, 0)
1− δ(1− β)|Ni∩SX | if (Xi ,Yi ) = (1, 1)(1)
Here Ni is the neighborhood of i and SX is the support set of the vectorX , i.e., the set of nodes in X that are infected. Ni ∩ Sx is thus the set ofinfected neighboring nodes of i .
Analyzing the Markov chain (1) over arbitrary large graphs has beenvery challenging
The state transition matrix that describes the evolution from state ξ(t) tostate ξ(t + 1) is
P(ξi (t+1) = Yi |ξ(t) = X ) =
(1− β)|Ni∩SX | if (Xi ,Yi ) = (0, 0)
1− (1− β)|Ni∩SX | if (Xi ,Yi ) = (0, 1)
δ(1− β)|Ni∩SX | if (Xi ,Yi ) = (1, 0)
1− δ(1− β)|Ni∩SX | if (Xi ,Yi ) = (1, 1)(1)
Here Ni is the neighborhood of i and SX is the support set of the vectorX , i.e., the set of nodes in X that are infected. Ni ∩ Sx is thus the set ofinfected neighboring nodes of i .
Analyzing the Markov chain (1) over arbitrary large graphs has beenvery challenging
The state transition matrix that describes the evolution from state ξ(t) tostate ξ(t + 1) is
P(ξi (t+1) = Yi |ξ(t) = X ) =
(1− β)|Ni∩SX | if (Xi ,Yi ) = (0, 0)
1− (1− β)|Ni∩SX | if (Xi ,Yi ) = (0, 1)
δ(1− β)|Ni∩SX | if (Xi ,Yi ) = (1, 0)
1− δ(1− β)|Ni∩SX | if (Xi ,Yi ) = (1, 1)(1)
Here Ni is the neighborhood of i and SX is the support set of the vectorX , i.e., the set of nodes in X that are infected. Ni ∩ Sx is thus the set ofinfected neighboring nodes of i .
Analyzing the Markov chain (1) over arbitrary large graphs has beenvery challenging
Mean-Field Approximation (Chakrabarti et al, Wang et al)
To get an approximate model it is customary to propagate Pi (t), themarginal probability of node i being infected, rather than the probability ofthe entire state P(ξ(t)).
At time t, denote the set of infected nodes by I(t). We may write
Pi (t + 1) = P(i ∈ I(t + 1)|i /∈ I(t))(1− Pi (t)) +
Mean-Field Approximation (Chakrabarti et al, Wang et al)
To get an approximate model it is customary to propagate Pi (t), themarginal probability of node i being infected, rather than the probability ofthe entire state P(ξ(t)).At time t, denote the set of infected nodes by I(t). We may write
Pi (t + 1) = P(i ∈ I(t + 1)|i /∈ I(t))(1− Pi (t)) +
Mean-Field Approximation (Chakrabarti et al, Wang et al)
To get an approximate model it is customary to propagate Pi (t), themarginal probability of node i being infected, rather than the probability ofthe entire state P(ξ(t)).At time t, denote the set of infected nodes by I(t). We may write
Pi (t + 1) = P(i ∈ I(t + 1)|i /∈ I(t))(1− Pi (t)) +
However, while A may not be explicitly known, it is often the case that wenow the random graph ensemble from which A is drawn.
A fairly general random graph model, which subsumes Erdos-Renyi as aspecial case, and allows the incorporation of arbitrary degree distributionsis Gn,pn(·). Consider a degree distribution pn(·) and draw n iid samples
from it to obtain a weight vector w =[w1 w2 . . . wn
]. From this
weight vector construct a random graph with adjacency matrix
However, while A may not be explicitly known, it is often the case that wenow the random graph ensemble from which A is drawn.
A fairly general random graph model, which subsumes Erdos-Renyi as aspecial case, and allows the incorporation of arbitrary degree distributionsis Gn,pn(·).
Consider a degree distribution pn(·) and draw n iid samples
from it to obtain a weight vector w =[w1 w2 . . . wn
]. From this
weight vector construct a random graph with adjacency matrix
However, while A may not be explicitly known, it is often the case that wenow the random graph ensemble from which A is drawn.
A fairly general random graph model, which subsumes Erdos-Renyi as aspecial case, and allows the incorporation of arbitrary degree distributionsis Gn,pn(·). Consider a degree distribution pn(·)
and draw n iid samples
from it to obtain a weight vector w =[w1 w2 . . . wn
]. From this
weight vector construct a random graph with adjacency matrix
However, while A may not be explicitly known, it is often the case that wenow the random graph ensemble from which A is drawn.
A fairly general random graph model, which subsumes Erdos-Renyi as aspecial case, and allows the incorporation of arbitrary degree distributionsis Gn,pn(·). Consider a degree distribution pn(·) and draw n iid samples
from it to obtain a weight vector w =[w1 w2 . . . wn
].
From thisweight vector construct a random graph with adjacency matrix
However, while A may not be explicitly known, it is often the case that wenow the random graph ensemble from which A is drawn.
A fairly general random graph model, which subsumes Erdos-Renyi as aspecial case, and allows the incorporation of arbitrary degree distributionsis Gn,pn(·). Consider a degree distribution pn(·) and draw n iid samples
from it to obtain a weight vector w =[w1 w2 . . . wn
]. From this
weight vector construct a random graph with adjacency matrix
Consider an epidemic spread over a graph in Gn,pn(·) with parameters δ andβ. Assume that pn(·) has finite variance and that the parameters are suchthat the system matrix M is almost surely stable. Then
One can use these results to optimize the social cost of containing anepidemic by randomly vaccinating a fraction π < 1 of the nodes.Assuming the cost per vaccination is cv , we obtain
Back to the Nonlinear Mean-Field Approximation (2)
Recall the nonlinear model
Pi (t + 1) = (1− δ)Pi (t) + (1− (1− δ)Pi (t))
1−∏j∈Ni
(1− βPj(t))
.
There has been very little analysis of this. Such a highly nonlinear andcoupled nonlinear dynamical system can potentially have very complicaeddynamics which could depend on the underlying graph. Note that we mayrewrite this as
Back to the Nonlinear Mean-Field Approximation (2)
Recall the nonlinear model
Pi (t + 1) = (1− δ)Pi (t) + (1− (1− δ)Pi (t))
1−∏j∈Ni
(1− βPj(t))
.
There has been very little analysis of this.
Such a highly nonlinear andcoupled nonlinear dynamical system can potentially have very complicaeddynamics which could depend on the underlying graph. Note that we mayrewrite this as
Back to the Nonlinear Mean-Field Approximation (2)
Recall the nonlinear model
Pi (t + 1) = (1− δ)Pi (t) + (1− (1− δ)Pi (t))
1−∏j∈Ni
(1− βPj(t))
.
There has been very little analysis of this. Such a highly nonlinear andcoupled nonlinear dynamical system can potentially have very complicaeddynamics which could depend on the underlying graph.
Back to the Nonlinear Mean-Field Approximation (2)
Recall the nonlinear model
Pi (t + 1) = (1− δ)Pi (t) + (1− (1− δ)Pi (t))
1−∏j∈Ni
(1− βPj(t))
.
There has been very little analysis of this. Such a highly nonlinear andcoupled nonlinear dynamical system can potentially have very complicaeddynamics which could depend on the underlying graph. Note that we mayrewrite this as
Back to the Nonlinear Mean-Field Approximation (2)
Now define the setS = {x : Φ(x)− x ≥ 0}.
We can establish the following facts
1 S is closed under pairwise maximization
x ∈ S , x ′ ∈ S ⇒ max(x , x ′) ∈ S .
2 S has a unique maximal point, x∗
3 If (1− δ)I + βA is unstable, x∗ is a fixed point of Φ(·)4 If (1− δ)I + βA is unstable, x∗ is the only nonzero fixed point of Φ(·)5 If (1− δ)I + βA is unstable, x∗ attracts every nonzero point in the
Back to the Nonlinear Mean-Field Approximation (2)
Now define the setS = {x : Φ(x)− x ≥ 0}.
We can establish the following facts
1 S is closed under pairwise maximization
x ∈ S , x ′ ∈ S ⇒ max(x , x ′) ∈ S .
2 S has a unique maximal point, x∗
3 If (1− δ)I + βA is unstable, x∗ is a fixed point of Φ(·)4 If (1− δ)I + βA is unstable, x∗ is the only nonzero fixed point of Φ(·)5 If (1− δ)I + βA is unstable, x∗ attracts every nonzero point in the
Back to the Nonlinear Mean-Field Approximation (2)
Now define the setS = {x : Φ(x)− x ≥ 0}.
We can establish the following facts
1 S is closed under pairwise maximization
x ∈ S , x ′ ∈ S ⇒ max(x , x ′) ∈ S .
2 S has a unique maximal point, x∗
3 If (1− δ)I + βA is unstable, x∗ is a fixed point of Φ(·)4 If (1− δ)I + βA is unstable, x∗ is the only nonzero fixed point of Φ(·)5 If (1− δ)I + βA is unstable, x∗ attracts every nonzero point in the
Back to the Nonlinear Mean-Field Approximation (2)
Now define the setS = {x : Φ(x)− x ≥ 0}.
We can establish the following facts
1 S is closed under pairwise maximization
x ∈ S , x ′ ∈ S ⇒ max(x , x ′) ∈ S .
2 S has a unique maximal point, x∗
3 If (1− δ)I + βA is unstable, x∗ is a fixed point of Φ(·)4 If (1− δ)I + βA is unstable, x∗ is the only nonzero fixed point of Φ(·)5 If (1− δ)I + βA is unstable, x∗ attracts every nonzero point in the
Back to the Nonlinear Mean-Field Approximation (2)
Now define the setS = {x : Φ(x)− x ≥ 0}.
We can establish the following facts
1 S is closed under pairwise maximization
x ∈ S , x ′ ∈ S ⇒ max(x , x ′) ∈ S .
2 S has a unique maximal point, x∗
3 If (1− δ)I + βA is unstable, x∗ is a fixed point of Φ(·)4 If (1− δ)I + βA is unstable, x∗ is the only nonzero fixed point of Φ(·)5 If (1− δ)I + βA is unstable, x∗ attracts every nonzero point in the
1 If the matrix M = (1− δ)I + βA is stable, then the origin is a globallystable fixed point.
2 If the matrix M = (1− δ)I + βA is unstable, then there exists asecond unique ”non-origin” fixed point that attracts every point inthe state-space, except for the origin.
Based on extensive numerical simulations for Erdos-Renyi randomgraphs, we conjecture that the converse is also true:
I If the matrix M = (1− δ)I + βA is unstable,then for Erdos Renyirandom graphs the Markov chain has an exponential mixing time.
A Markov chain with exponential mixing time is, for all practicalpurposes, unstable: the all-healthy state will never be observed in anyreasonable amount of time.
Based on extensive numerical simulations for Erdos-Renyi randomgraphs, we conjecture that the converse is also true:
I If the matrix M = (1− δ)I + βA is unstable,then for Erdos Renyirandom graphs the Markov chain has an exponential mixing time.
A Markov chain with exponential mixing time is, for all practicalpurposes, unstable: the all-healthy state will never be observed in anyreasonable amount of time.
Based on extensive numerical simulations for Erdos-Renyi randomgraphs, we conjecture that the converse is also true:
I If the matrix M = (1− δ)I + βA is unstable,then for Erdos Renyirandom graphs the Markov chain has an exponential mixing time.
A Markov chain with exponential mixing time is, for all practicalpurposes, unstable: the all-healthy state will never be observed in anyreasonable amount of time.
Figure: The evolution of a) SIRS, b) SIV-Vaccination-Dominant, c)SIV-Infection-Dominant epidemics over an Erdos-Renyi graph with n = 2000nodes. The blue curves show fast extinction of the epidemic. The red curves showepidemic spread around the nontrivial fixed point (convergence is not observed.)
I introduced Markov chain model, mean-field approximation, linearapproximation
I analyzed the social cost of an epidemic for linear modelI full stability analysis for mean-field model (at most two fixed points)I related fast-mxing of the underlying Markov chain to stability of the
mean-field approximationI studied SIRS/SIV models
Future work:I studying the social cost of epidemic for Markov chain and mean-field
modelsI control of epidemics using social cost as a metricI tighter bounds for when the chain is fast mixingI study of more complicated epidemic models: SIS/SIRS with birth and
Studied the SIS model for epidemic spread:I introduced Markov chain model, mean-field approximation, linear
approximation
I analyzed the social cost of an epidemic for linear modelI full stability analysis for mean-field model (at most two fixed points)I related fast-mxing of the underlying Markov chain to stability of the
mean-field approximationI studied SIRS/SIV models
Future work:I studying the social cost of epidemic for Markov chain and mean-field
modelsI control of epidemics using social cost as a metricI tighter bounds for when the chain is fast mixingI study of more complicated epidemic models: SIS/SIRS with birth and
Studied the SIS model for epidemic spread:I introduced Markov chain model, mean-field approximation, linear
approximationI analyzed the social cost of an epidemic for linear model
I full stability analysis for mean-field model (at most two fixed points)I related fast-mxing of the underlying Markov chain to stability of the
mean-field approximationI studied SIRS/SIV models
Future work:I studying the social cost of epidemic for Markov chain and mean-field
modelsI control of epidemics using social cost as a metricI tighter bounds for when the chain is fast mixingI study of more complicated epidemic models: SIS/SIRS with birth and
Studied the SIS model for epidemic spread:I introduced Markov chain model, mean-field approximation, linear
approximationI analyzed the social cost of an epidemic for linear modelI full stability analysis for mean-field model (at most two fixed points)
I related fast-mxing of the underlying Markov chain to stability of themean-field approximation
I studied SIRS/SIV models
Future work:I studying the social cost of epidemic for Markov chain and mean-field
modelsI control of epidemics using social cost as a metricI tighter bounds for when the chain is fast mixingI study of more complicated epidemic models: SIS/SIRS with birth and
Studied the SIS model for epidemic spread:I introduced Markov chain model, mean-field approximation, linear
approximationI analyzed the social cost of an epidemic for linear modelI full stability analysis for mean-field model (at most two fixed points)I related fast-mxing of the underlying Markov chain to stability of the
mean-field approximation
I studied SIRS/SIV models
Future work:I studying the social cost of epidemic for Markov chain and mean-field
modelsI control of epidemics using social cost as a metricI tighter bounds for when the chain is fast mixingI study of more complicated epidemic models: SIS/SIRS with birth and
Studied the SIS model for epidemic spread:I introduced Markov chain model, mean-field approximation, linear
approximationI analyzed the social cost of an epidemic for linear modelI full stability analysis for mean-field model (at most two fixed points)I related fast-mxing of the underlying Markov chain to stability of the
mean-field approximationI studied SIRS/SIV models
Future work:I studying the social cost of epidemic for Markov chain and mean-field
modelsI control of epidemics using social cost as a metricI tighter bounds for when the chain is fast mixingI study of more complicated epidemic models: SIS/SIRS with birth and
Studied the SIS model for epidemic spread:I introduced Markov chain model, mean-field approximation, linear
approximationI analyzed the social cost of an epidemic for linear modelI full stability analysis for mean-field model (at most two fixed points)I related fast-mxing of the underlying Markov chain to stability of the
mean-field approximationI studied SIRS/SIV models
Future work:
I studying the social cost of epidemic for Markov chain and mean-fieldmodels
I control of epidemics using social cost as a metricI tighter bounds for when the chain is fast mixingI study of more complicated epidemic models: SIS/SIRS with birth and
Studied the SIS model for epidemic spread:I introduced Markov chain model, mean-field approximation, linear
approximationI analyzed the social cost of an epidemic for linear modelI full stability analysis for mean-field model (at most two fixed points)I related fast-mxing of the underlying Markov chain to stability of the
mean-field approximationI studied SIRS/SIV models
Future work:I studying the social cost of epidemic for Markov chain and mean-field
models
I control of epidemics using social cost as a metricI tighter bounds for when the chain is fast mixingI study of more complicated epidemic models: SIS/SIRS with birth and
Studied the SIS model for epidemic spread:I introduced Markov chain model, mean-field approximation, linear
approximationI analyzed the social cost of an epidemic for linear modelI full stability analysis for mean-field model (at most two fixed points)I related fast-mxing of the underlying Markov chain to stability of the
mean-field approximationI studied SIRS/SIV models
Future work:I studying the social cost of epidemic for Markov chain and mean-field
modelsI control of epidemics using social cost as a metric
I tighter bounds for when the chain is fast mixingI study of more complicated epidemic models: SIS/SIRS with birth and
Studied the SIS model for epidemic spread:I introduced Markov chain model, mean-field approximation, linear
approximationI analyzed the social cost of an epidemic for linear modelI full stability analysis for mean-field model (at most two fixed points)I related fast-mxing of the underlying Markov chain to stability of the
mean-field approximationI studied SIRS/SIV models
Future work:I studying the social cost of epidemic for Markov chain and mean-field
modelsI control of epidemics using social cost as a metricI tighter bounds for when the chain is fast mixing
I study of more complicated epidemic models: SIS/SIRS with birth anddeath, SEIS, SEIR, etc.
Studied the SIS model for epidemic spread:I introduced Markov chain model, mean-field approximation, linear
approximationI analyzed the social cost of an epidemic for linear modelI full stability analysis for mean-field model (at most two fixed points)I related fast-mxing of the underlying Markov chain to stability of the
mean-field approximationI studied SIRS/SIV models
Future work:I studying the social cost of epidemic for Markov chain and mean-field
modelsI control of epidemics using social cost as a metricI tighter bounds for when the chain is fast mixingI study of more complicated epidemic models: SIS/SIRS with birth and