Epidemic Spread in Complex Networks€¦ · Though initially proposed to understand the spread of...
Transcript of Epidemic Spread in Complex Networks€¦ · Though initially proposed to understand the spread of...
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
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 1 / 57
Outline
IntroductionI epidemic spread: SIR and SIS modelsI Markov chain model, mean-field approximation, linear approximation
Social Cost of an EpidemicI random graphsI optimizing the social cost
Global Stability Analysis of Mean-Field ApproximationI at most two fixed points
Mixing Time of Markov ChainI connection to stability of mean-field modelI SIRS model and vaccination
Extensions and Future Work
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 2 / 57
Introduction
Epidemic models have been extensively studied since the SIR(Susceptible-Infectious-Recovered) model was proposed in 1927 inKermack and McKendrick.
Though initially proposed to understand the spread of contagiousdiseases, they apply to many other settings:
I network security (to understand and limit the spread of computerviruses, e.g.)
I viral advertising (to create an epidemic to propagate interest in aproduct, e.g.)
I information propagation (to understand how quickly new ideaspropagate through a network, e.g.)
Questions of interestI existence of fixed-points, stability, transient behavior, cost of an
epidemic, how to control an epidemic, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 3 / 57
Introduction
Epidemic models have been extensively studied since the SIR(Susceptible-Infectious-Recovered) model was proposed in 1927 inKermack and McKendrick.
Though initially proposed to understand the spread of contagiousdiseases, they apply to many other settings:
I network security (to understand and limit the spread of computerviruses, e.g.)
I viral advertising (to create an epidemic to propagate interest in aproduct, e.g.)
I information propagation (to understand how quickly new ideaspropagate through a network, e.g.)
Questions of interestI existence of fixed-points, stability, transient behavior, cost of an
epidemic, how to control an epidemic, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 3 / 57
Introduction
Epidemic models have been extensively studied since the SIR(Susceptible-Infectious-Recovered) model was proposed in 1927 inKermack and McKendrick.
Though initially proposed to understand the spread of contagiousdiseases, they apply to many other settings:
I network security (to understand and limit the spread of computerviruses, e.g.)
I viral advertising (to create an epidemic to propagate interest in aproduct, e.g.)
I information propagation (to understand how quickly new ideaspropagate through a network, e.g.)
Questions of interestI existence of fixed-points, stability, transient behavior, cost of an
epidemic, how to control an epidemic, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 3 / 57
Introduction
Epidemic models have been extensively studied since the SIR(Susceptible-Infectious-Recovered) model was proposed in 1927 inKermack and McKendrick.
Though initially proposed to understand the spread of contagiousdiseases, they apply to many other settings:
I network security (to understand and limit the spread of computerviruses, e.g.)
I viral advertising (to create an epidemic to propagate interest in aproduct, e.g.)
I information propagation (to understand how quickly new ideaspropagate through a network, e.g.)
Questions of interestI existence of fixed-points, stability, transient behavior, cost of an
epidemic, how to control an epidemic, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 3 / 57
Introduction
Epidemic models have been extensively studied since the SIR(Susceptible-Infectious-Recovered) model was proposed in 1927 inKermack and McKendrick.
Though initially proposed to understand the spread of contagiousdiseases, they apply to many other settings:
I network security (to understand and limit the spread of computerviruses, e.g.)
I viral advertising (to create an epidemic to propagate interest in aproduct, e.g.)
I information propagation (to understand how quickly new ideaspropagate through a network, e.g.)
Questions of interestI existence of fixed-points, stability, transient behavior, cost of an
epidemic, how to control an epidemic, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 3 / 57
Introduction
Epidemic models have been extensively studied since the SIR(Susceptible-Infectious-Recovered) model was proposed in 1927 inKermack and McKendrick.
Though initially proposed to understand the spread of contagiousdiseases, they apply to many other settings:
I network security (to understand and limit the spread of computerviruses, e.g.)
I viral advertising (to create an epidemic to propagate interest in aproduct, e.g.)
I information propagation (to understand how quickly new ideaspropagate through a network, e.g.)
Questions of interestI existence of fixed-points, stability, transient behavior, cost of an
epidemic, how to control an epidemic, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 3 / 57
Model
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
neighbors with i.i.d. probability β
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 4 / 57
Model
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
neighbors with i.i.d. probability β
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 4 / 57
Model
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
neighbors with i.i.d. probability β
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 4 / 57
Model
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 susceptible
2 a susceptible node can become infected by each of its infectedneighbors with i.i.d. probability β
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 4 / 57
Model
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
neighbors with i.i.d. probability β
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 4 / 57
SIR Model
𝛽
𝛿
1 − 𝛽 1 − 𝛿
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 5 / 57
A Markov Chain Model
The resulting epidemic spread can be modeled as a Markov chain with 2n
states.
Let each state be described by an n-dimensional binary vectorξ(t) =
[ξ1(t) ξ2(t) . . . ξn(t)
], where ξi (t) = 0 if node i is
healthy and ξi (t) = 1 if it is infected
Given the current state ξ(t), each node i recovers or gets infectedindependently of other nodes
P(ξ(t + 1) = Y |ξ(t) = X ) =n∏
i=1
P(ξi (t + 1) = Yi |ξ(t) = X )
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 6 / 57
A Markov Chain Model
The resulting epidemic spread can be modeled as a Markov chain with 2n
states.
Let each state be described by an n-dimensional binary vectorξ(t) =
[ξ1(t) ξ2(t) . . . ξn(t)
], where ξi (t) = 0 if node i is
healthy and ξi (t) = 1 if it is infected
Given the current state ξ(t), each node i recovers or gets infectedindependently of other nodes
P(ξ(t + 1) = Y |ξ(t) = X ) =n∏
i=1
P(ξi (t + 1) = Yi |ξ(t) = X )
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 6 / 57
A Markov Chain Model
The resulting epidemic spread can be modeled as a Markov chain with 2n
states.
Let each state be described by an n-dimensional binary vectorξ(t) =
[ξ1(t) ξ2(t) . . . ξn(t)
], where ξi (t) = 0 if node i is
healthy and ξi (t) = 1 if it is infected
Given the current state ξ(t), each node i recovers or gets infectedindependently of other nodes
P(ξ(t + 1) = Y |ξ(t) = X ) =n∏
i=1
P(ξi (t + 1) = Yi |ξ(t) = X )
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 6 / 57
A Markov Chain Model
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
Most researchers have resorted to approximations
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 7 / 57
A Markov Chain Model
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
Most researchers have resorted to approximations
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 7 / 57
A Markov Chain Model
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
Most researchers have resorted to approximations
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 7 / 57
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)) +
P(i ∈ I(t + 1)|i ∈ I(t))Pi (t)
=
1−∏j∈Ni
(1− β1j∈I(t)
) (1− Pi (t)) +
1− δ∏j∈Ni
(1− β1j∈I(t)
)Pi (t)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 8 / 57
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)) +
P(i ∈ I(t + 1)|i ∈ I(t))Pi (t)
=
1−∏j∈Ni
(1− β1j∈I(t)
) (1− Pi (t)) +
1− δ∏j∈Ni
(1− β1j∈I(t)
)Pi (t)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 8 / 57
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)) +
P(i ∈ I(t + 1)|i ∈ I(t))Pi (t)
=
1−∏j∈Ni
(1− β1j∈I(t)
) (1− Pi (t)) +
1− δ∏j∈Ni
(1− β1j∈I(t)
)Pi (t)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 8 / 57
Mean-Field Approximation (Chakrabarti et al, Wang et al)
We now use the mean-field approximation:∏j∈Ni
(1− β1j∈I(t)
)≈∏j∈Ni
(1− βPj(t))
which gives us the approximate model
Pi (t + 1) =(1−
∏j∈Ni
(1− βPj(t)))
(1− Pi (t)) +(
1− δ∏
j∈Ni(1− βPj(t))
)Pi (t)
= (1− δ)Pi (t) + (1− (1− δ)Pi (t))(
1−∏
j∈Ni(1− βPj(t))
)(2)
which is an n-dimensional nonlinear dynamical system.
Analyzing this is often difficult for an arbitrary graph.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 9 / 57
Mean-Field Approximation (Chakrabarti et al, Wang et al)
We now use the mean-field approximation:∏j∈Ni
(1− β1j∈I(t)
)≈∏j∈Ni
(1− βPj(t))
which gives us the approximate model
Pi (t + 1) =(1−
∏j∈Ni
(1− βPj(t)))
(1− Pi (t)) +(
1− δ∏
j∈Ni(1− βPj(t))
)Pi (t)
= (1− δ)Pi (t) + (1− (1− δ)Pi (t))(
1−∏
j∈Ni(1− βPj(t))
)(2)
which is an n-dimensional nonlinear dynamical system.
Analyzing this is often difficult for an arbitrary graph.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 9 / 57
Mean-Field Approximation (Chakrabarti et al, Wang et al)
We now use the mean-field approximation:∏j∈Ni
(1− β1j∈I(t)
)≈∏j∈Ni
(1− βPj(t))
which gives us the approximate model
Pi (t + 1) =(1−
∏j∈Ni
(1− βPj(t)))
(1− Pi (t)) +(
1− δ∏
j∈Ni(1− βPj(t))
)Pi (t)
= (1− δ)Pi (t) + (1− (1− δ)Pi (t))(
1−∏
j∈Ni(1− βPj(t))
)(2)
which is an n-dimensional nonlinear dynamical system.
Analyzing this is often difficult for an arbitrary graph.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 9 / 57
Mean-Field Approximation (Chakrabarti et al, Wang et al)
We now use the mean-field approximation:∏j∈Ni
(1− β1j∈I(t)
)≈∏j∈Ni
(1− βPj(t))
which gives us the approximate model
Pi (t + 1) =(1−
∏j∈Ni
(1− βPj(t)))
(1− Pi (t)) +(
1− δ∏
j∈Ni(1− βPj(t))
)Pi (t)
= (1− δ)Pi (t) + (1− (1− δ)Pi (t))(
1−∏
j∈Ni(1− βPj(t))
)(2)
which is an n-dimensional nonlinear dynamical system.
Analyzing this is often difficult for an arbitrary graph.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 9 / 57
A Linear Model
Using ∏j∈Ni
(1− βPj(t)) ≥ 1− β∑j∈Ni
Pj(t),
we may write
Pi (t + 1) ≤ β∑j∈Ni
Pj(t)(1− Pi (t)) +
1− δ + δβ∑j∈Ni
Pj(t)
Pi (t)
= (1− δ)Pi (t) + β∑j∈Ni
Pj(t)− (1− δ)β∑j∈Ni
Pj(t)Pi (t)
≤ (1− δ)Pi (t) + β∑j∈Ni
Pj(t)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 10 / 57
A Linear Model
Using ∏j∈Ni
(1− βPj(t)) ≥ 1− β∑j∈Ni
Pj(t),
we may write
Pi (t + 1) ≤ β∑j∈Ni
Pj(t)(1− Pi (t)) +
1− δ + δβ∑j∈Ni
Pj(t)
Pi (t)
= (1− δ)Pi (t) + β∑j∈Ni
Pj(t)− (1− δ)β∑j∈Ni
Pj(t)Pi (t)
≤ (1− δ)Pi (t) + β∑j∈Ni
Pj(t)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 10 / 57
A Linear Model
Using ∏j∈Ni
(1− βPj(t)) ≥ 1− β∑j∈Ni
Pj(t),
we may write
Pi (t + 1) ≤ β∑j∈Ni
Pj(t)(1− Pi (t)) +
1− δ + δβ∑j∈Ni
Pj(t)
Pi (t)
= (1− δ)Pi (t) + β∑j∈Ni
Pj(t)− (1− δ)β∑j∈Ni
Pj(t)Pi (t)
≤ (1− δ)Pi (t) + β∑j∈Ni
Pj(t)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 10 / 57
A Linear Model
Using ∏j∈Ni
(1− βPj(t)) ≥ 1− β∑j∈Ni
Pj(t),
we may write
Pi (t + 1) ≤ β∑j∈Ni
Pj(t)(1− Pi (t)) +
1− δ + δβ∑j∈Ni
Pj(t)
Pi (t)
= (1− δ)Pi (t) + β∑j∈Ni
Pj(t)− (1− δ)β∑j∈Ni
Pj(t)Pi (t)
≤ (1− δ)Pi (t) + β∑j∈Ni
Pj(t)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 10 / 57
A Linear Model
The linear model
Pi (t + 1) = (1− δ)Pi (t) + β∑j∈Ni
Pj(t), (3)
is thus an upper bound on the approximate model (2).
Let P(t) be then-dimensional column vector obtained from the Pi (t). Then in matrixnotation,
P(t + 1) = ((1− δ)I + βA)︸ ︷︷ ︸=M
P(t). (4)
It is also easy to see that it is the linearization of (2) around the origin(the ”all-healthy” state)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 11 / 57
A Linear Model
The linear model
Pi (t + 1) = (1− δ)Pi (t) + β∑j∈Ni
Pj(t), (3)
is thus an upper bound on the approximate model (2). Let P(t) be then-dimensional column vector obtained from the Pi (t). Then in matrixnotation,
P(t + 1) = ((1− δ)I + βA)︸ ︷︷ ︸=M
P(t). (4)
It is also easy to see that it is the linearization of (2) around the origin(the ”all-healthy” state)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 11 / 57
A Linear Model
The linear model
Pi (t + 1) = (1− δ)Pi (t) + β∑j∈Ni
Pj(t), (3)
is thus an upper bound on the approximate model (2). Let P(t) be then-dimensional column vector obtained from the Pi (t). Then in matrixnotation,
P(t + 1) = ((1− δ)I + βA)︸ ︷︷ ︸=M
P(t). (4)
It is also easy to see that it is the linearization of (2) around the origin(the ”all-healthy” state)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 11 / 57
The Stability Condition
Theorem
The origin in
Pi (t + 1) = (1− δ)Pi (t) + (1− (1− δ)Pi (t))
1−∏j∈Ni
(1− βPj(t))
,
is globally stable if, and only if, the matrix (1− δ)I + βA is stable. If(1− δ)I + βA is unstable, then the origin in (2) is not even locally stable.
If we denote the spectral radius of A by ρ(A), then the stability conditioncan be written as
(1− δ) + βρ(A) < 1,
orβρ(A)
δ< 1.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 12 / 57
The Stability Condition
Theorem
The origin in
Pi (t + 1) = (1− δ)Pi (t) + (1− (1− δ)Pi (t))
1−∏j∈Ni
(1− βPj(t))
,
is globally stable if, and only if, the matrix (1− δ)I + βA is stable. If(1− δ)I + βA is unstable, then the origin in (2) is not even locally stable.
If we denote the spectral radius of A by ρ(A), then the stability conditioncan be written as
(1− δ) + βρ(A) < 1,
orβρ(A)
δ< 1.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 12 / 57
The Stability Condition
Theorem
The origin in
Pi (t + 1) = (1− δ)Pi (t) + (1− (1− δ)Pi (t))
1−∏j∈Ni
(1− βPj(t))
,
is globally stable if, and only if, the matrix (1− δ)I + βA is stable. If(1− δ)I + βA is unstable, then the origin in (2) is not even locally stable.
If we denote the spectral radius of A by ρ(A), then the stability conditioncan be written as
(1− δ) + βρ(A) < 1,
orβρ(A)
δ< 1.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 12 / 57
The Cost of an Epidemic
Stability is the first concern in the study of an epidemic
However, we are often interested in other metrics such as the cost ofan epidemic
I this is defined to be proportional to the total number of nodes infected,summed over the course of the epidemic, i.e.,
C = cd
∞∑t=0
1TP(t).
for the general nonlinear model this is hard to compute; however, it iseasy to upper bound using the linearized model
C = cd
∞∑t=0
1TP(t) = cd
∞∑t=0
1TMtP(0) = cd1T (I −M)−1P(0).
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 13 / 57
The Cost of an Epidemic
Stability is the first concern in the study of an epidemic
However, we are often interested in other metrics such as the cost ofan epidemic
I this is defined to be proportional to the total number of nodes infected,summed over the course of the epidemic, i.e.,
C = cd
∞∑t=0
1TP(t).
for the general nonlinear model this is hard to compute; however, it iseasy to upper bound using the linearized model
C = cd
∞∑t=0
1TP(t) = cd
∞∑t=0
1TMtP(0) = cd1T (I −M)−1P(0).
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 13 / 57
The Cost of an Epidemic
Stability is the first concern in the study of an epidemic
However, we are often interested in other metrics such as the cost ofan epidemic
I this is defined to be proportional to the total number of nodes infected,summed over the course of the epidemic, i.e.,
C = cd
∞∑t=0
1TP(t).
for the general nonlinear model this is hard to compute; however, it iseasy to upper bound using the linearized model
C = cd
∞∑t=0
1TP(t) = cd
∞∑t=0
1TMtP(0) = cd1T (I −M)−1P(0).
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 13 / 57
The Cost of an Epidemic
Stability is the first concern in the study of an epidemic
However, we are often interested in other metrics such as the cost ofan epidemic
I this is defined to be proportional to the total number of nodes infected,summed over the course of the epidemic, i.e.,
C = cd
∞∑t=0
1TP(t).
for the general nonlinear model this is hard to compute;
however, it iseasy to upper bound using the linearized model
C = cd
∞∑t=0
1TP(t) = cd
∞∑t=0
1TMtP(0) = cd1T (I −M)−1P(0).
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 13 / 57
The Cost of an Epidemic
Stability is the first concern in the study of an epidemic
However, we are often interested in other metrics such as the cost ofan epidemic
I this is defined to be proportional to the total number of nodes infected,summed over the course of the epidemic, i.e.,
C = cd
∞∑t=0
1TP(t).
for the general nonlinear model this is hard to compute; however, it iseasy to upper bound using the linearized model
C = cd
∞∑t=0
1TP(t) = cd
∞∑t=0
1TMtP(0) = cd1T (I −M)−1P(0).
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 13 / 57
The Cost of an Epidemic
It is customary to assume that the in the initial state fraction α < 1 of thenodes are randomly infected.
Thus, P(0) = α1 and
C = αcd1T (I −M)−11 = αcd1T (I − (1− δ)I − βA)−1 1
= αcd1T (δ − βA)−11. (5)
Thus, computing the cost C requires explicit knowledge of the matrixM = (1− δ)I + βA, which is usually not available since A is not explicitlyknown.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 14 / 57
The Cost of an Epidemic
It is customary to assume that the in the initial state fraction α < 1 of thenodes are randomly infected.Thus, P(0) = α1 and
C = αcd1T (I −M)−11 = αcd1T (I − (1− δ)I − βA)−1 1
= αcd1T (δ − βA)−11. (5)
Thus, computing the cost C requires explicit knowledge of the matrixM = (1− δ)I + βA, which is usually not available since A is not explicitlyknown.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 14 / 57
The Cost of an Epidemic
It is customary to assume that the in the initial state fraction α < 1 of thenodes are randomly infected.Thus, P(0) = α1 and
C = αcd1T (I −M)−11 = αcd1T (I − (1− δ)I − βA)−1 1
= αcd1T (δ − βA)−11. (5)
Thus, computing the cost C requires explicit knowledge of the matrixM = (1− δ)I + βA, which is usually not available since A is not explicitlyknown.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 14 / 57
Random Graphs
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
Aij = Aji =
{1 with probability
wiwj∑i wi
0 with probability 1− wiwj∑i wi
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 15 / 57
Random Graphs
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
Aij = Aji =
{1 with probability
wiwj∑i wi
0 with probability 1− wiwj∑i wi
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 15 / 57
Random Graphs
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
Aij = Aji =
{1 with probability
wiwj∑i wi
0 with probability 1− wiwj∑i wi
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 15 / 57
Random Graphs
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
Aij = Aji =
{1 with probability
wiwj∑i wi
0 with probability 1− wiwj∑i wi
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 15 / 57
Random Graphs
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
Aij = Aji =
{1 with probability
wiwj∑i wi
0 with probability 1− wiwj∑i wi
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 15 / 57
Erdos-Renyi Random Graphs
Erdos-Renyi random graphs are a special case with
pn(w) = δ(w − np),
as a result of which
w =[np np . . . np
]and nodes i and j are connected with probability
wiwj∑i wi
=(np)(np)
n(np)= p.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 16 / 57
Erdos-Renyi Random Graphs
Erdos-Renyi random graphs are a special case with
pn(w) = δ(w − np),
as a result of which
w =
[np np . . . np
]and nodes i and j are connected with probability
wiwj∑i wi
=(np)(np)
n(np)= p.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 16 / 57
Erdos-Renyi Random Graphs
Erdos-Renyi random graphs are a special case with
pn(w) = δ(w − np),
as a result of which
w =[np np . . . np
]
and nodes i and j are connected with probability
wiwj∑i wi
=(np)(np)
n(np)= p.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 16 / 57
Erdos-Renyi Random Graphs
Erdos-Renyi random graphs are a special case with
pn(w) = δ(w − np),
as a result of which
w =[np np . . . np
]and nodes i and j are connected with probability
wiwj∑i wi
=(np)(np)
n(np)= p.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 16 / 57
Random Graphs
Defining W = diag(w), note that we may write the adjacency matrix A as
A = W 1/2GW 1/2 − wwT
1Tw,
where G is a Wigner matrix.
The cost can now be written as
C = nαcd1
n1T
δI − βW 1/2GW 1/2︸ ︷︷ ︸X
+βwwT
1Tw
−1
1
= nαcd
(1
n1TX−11−
β1Tw
(1n1TX−1w
)2
−1Twnβ +
(1nw
TX−1w))
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 17 / 57
Random Graphs
Defining W = diag(w), note that we may write the adjacency matrix A as
A = W 1/2GW 1/2 − wwT
1Tw,
where G is a Wigner matrix.The cost can now be written as
C = nαcd1
n1T
δI − βW 1/2GW 1/2︸ ︷︷ ︸X
+βwwT
1Tw
−1
1
= nαcd
(1
n1TX−11−
β1Tw
(1n1TX−1w
)2
−1Twnβ +
(1nw
TX−1w))
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 17 / 57
Random Graphs
Defining W = diag(w), note that we may write the adjacency matrix A as
A = W 1/2GW 1/2 − wwT
1Tw,
where G is a Wigner matrix.The cost can now be written as
C = nαcd1
n1T
δI − βW 1/2GW 1/2︸ ︷︷ ︸X
+βwwT
1Tw
−1
1
= nαcd
(1
n1TX−11−
β1Tw
(1n1TX−1w
)2
−1Twnβ +
(1nw
TX−1w))
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 17 / 57
Random Graphs
With some effort, it can be shown that each of the terms
1
n1TX−11,
1
n1TX−1w , and
1
nwTX−1w
self-average.
For example,
1
n1TX−11→ E
1
ntraceX−1 = E
1
ntrace
(δI − βW 1/2GW 1/2
)−1.
The latter is related to the Stieltjes transform of the random matrixX = δI − βW 1/2GW 1/2.
These can be computed with some additional effort.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 18 / 57
Random Graphs
With some effort, it can be shown that each of the terms
1
n1TX−11,
1
n1TX−1w , and
1
nwTX−1w
self-average. For example,
1
n1TX−11→ E
1
ntraceX−1 = E
1
ntrace
(δI − βW 1/2GW 1/2
)−1.
The latter is related to the Stieltjes transform of the random matrixX = δI − βW 1/2GW 1/2.
These can be computed with some additional effort.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 18 / 57
Random Graphs
With some effort, it can be shown that each of the terms
1
n1TX−11,
1
n1TX−1w , and
1
nwTX−1w
self-average. For example,
1
n1TX−11→ E
1
ntraceX−1 = E
1
ntrace
(δI − βW 1/2GW 1/2
)−1.
The latter is related to the Stieltjes transform of the random matrixX = δI − βW 1/2GW 1/2.
These can be computed with some additional effort.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 18 / 57
Random Graphs
Theorem
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
limn→∞
1
nC =
αcdδ
(1 + κF 2 − κ2F 2
1− Ew(
1F − δκ2
)) a.s. (6)
where κ =√β
δEw and F satisfies the implicit equation
F =
∫p(w)
w−1 − κ2Fdw .
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 19 / 57
Erdos Renyi Random Graphs
For an Erdos-Renyi graph the implicit equation for F becomes
F =
∫δ(w − np)
w−1 − κ2Fdw =
11np − κ2F
.
Solving this quadratic equation and plugging into the expression for thecost yields
C =nαcdδ − βnp
.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 20 / 57
Erdos Renyi Random Graphs
For an Erdos-Renyi graph the implicit equation for F becomes
F =
∫δ(w − np)
w−1 − κ2Fdw =
11np − κ2F
.
Solving this quadratic equation and plugging into the expression for thecost yields
C =nαcdδ − βnp
.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 20 / 57
Numerical Results: Erdos Renyi n = 1000
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
10
20
30
40
50
60
70
p
Cd(n
)
InverseTheorem 4.1Theorem 4.3Simulated
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 21 / 57
Numerical Results: Exponential Weight Distributionn = 1000
5 10 15 20 250
2
4
6
8
10
µ
Cd(n
)
Inverse
Theorem 4.1
Theorem 4.3
Simulated
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 22 / 57
Numerical Results: Heavy-Tail Weight Distributionn = 1000
2.5 3 3.5 4 4.5 50
0.5
1
1.5
2
2.5
3
3.5
θ
Cd(n
)
InverseTheorem 4.1Theorem 4.3Simulated
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 23 / 57
Optimizing the Social Cost
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
social cost = nπcv + (1− π)C (n(1− π)) .
0 0.2 0.4 0.6 0.8 10.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
π
Socialco
st
Simulated
S (π)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 24 / 57
Optimizing the Social Cost
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
social cost = nπcv + (1− π)C (n(1− π)) .
0 0.2 0.4 0.6 0.8 10.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
π
Socialco
st
Simulated
S (π)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 24 / 57
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
P(t + 1) = Φ(P(t)),
where
Φi (x) = (1− δ)xi + (1− (1− δ)xi )
1−∏j∈Ni
(1− βxj)
.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 25 / 57
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
P(t + 1) = Φ(P(t)),
where
Φi (x) = (1− δ)xi + (1− (1− δ)xi )
1−∏j∈Ni
(1− βxj)
.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 25 / 57
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
P(t + 1) = Φ(P(t)),
where
Φi (x) = (1− δ)xi + (1− (1− δ)xi )
1−∏j∈Ni
(1− βxj)
.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 25 / 57
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
P(t + 1) = Φ(P(t)),
where
Φi (x) = (1− δ)xi + (1− (1− δ)xi )
1−∏j∈Ni
(1− βxj)
.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 25 / 57
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
phase space
Critical to the proof is the fact that ∂Φi (x)∂xj
≥ 0 for all i and j .
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 26 / 57
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
phase space
Critical to the proof is the fact that ∂Φi (x)∂xj
≥ 0 for all i and j .
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 26 / 57
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
phase space
Critical to the proof is the fact that ∂Φi (x)∂xj
≥ 0 for all i and j .
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 26 / 57
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
phase space
Critical to the proof is the fact that ∂Φi (x)∂xj
≥ 0 for all i and j .
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 26 / 57
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 thephase space
Critical to the proof is the fact that ∂Φi (x)∂xj
≥ 0 for all i and j .
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 26 / 57
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
phase space
Critical to the proof is the fact that ∂Φi (x)∂xj
≥ 0 for all i and j .
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 26 / 57
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
phase space
Critical to the proof is the fact that ∂Φi (x)∂xj
≥ 0 for all i and j .
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 26 / 57
Stability of the Mean-Field Approximation
We have proven the following strong result.
Theorem
Consider the model (2)
Pi (t + 1) =
(1− δ)Pi (t) + (1− (1− δ)Pi (t))(
1−∏
j∈Ni(1− βPj(t))
).
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.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 27 / 57
Back to the Markov Chain Model (1)
There has been even less analysis of the true underlying Markov chainmodel
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)
Does the Markov chain model have a unique stationary distribution?Yes.
What is it? The distribution where all states are healthy withprobability one.
Thus, the Markov chain model is globally stable to the ”all-healthy”state.
But how does this jive with the results on the instability of the models(2) and (3)?
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 28 / 57
Back to the Markov Chain Model (1)
There has been even less analysis of the true underlying Markov chainmodel
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)
Does the Markov chain model have a unique stationary distribution?
Yes.
What is it? The distribution where all states are healthy withprobability one.
Thus, the Markov chain model is globally stable to the ”all-healthy”state.
But how does this jive with the results on the instability of the models(2) and (3)?
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 28 / 57
Back to the Markov Chain Model (1)
There has been even less analysis of the true underlying Markov chainmodel
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)
Does the Markov chain model have a unique stationary distribution?Yes.
What is it? The distribution where all states are healthy withprobability one.
Thus, the Markov chain model is globally stable to the ”all-healthy”state.
But how does this jive with the results on the instability of the models(2) and (3)?
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 28 / 57
Back to the Markov Chain Model (1)
There has been even less analysis of the true underlying Markov chainmodel
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)
Does the Markov chain model have a unique stationary distribution?Yes.
What is it?
The distribution where all states are healthy withprobability one.
Thus, the Markov chain model is globally stable to the ”all-healthy”state.
But how does this jive with the results on the instability of the models(2) and (3)?
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 28 / 57
Back to the Markov Chain Model (1)
There has been even less analysis of the true underlying Markov chainmodel
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)
Does the Markov chain model have a unique stationary distribution?Yes.
What is it? The distribution where all states are healthy withprobability one.
Thus, the Markov chain model is globally stable to the ”all-healthy”state.
But how does this jive with the results on the instability of the models(2) and (3)?
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 28 / 57
Back to the Markov Chain Model (1)
There has been even less analysis of the true underlying Markov chainmodel
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)
Does the Markov chain model have a unique stationary distribution?Yes.
What is it? The distribution where all states are healthy withprobability one.
Thus, the Markov chain model is globally stable to the ”all-healthy”state.
But how does this jive with the results on the instability of the models(2) and (3)?
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 28 / 57
Back to the Markov Chain Model (1)
There has been even less analysis of the true underlying Markov chainmodel
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)
Does the Markov chain model have a unique stationary distribution?Yes.
What is it? The distribution where all states are healthy withprobability one.
Thus, the Markov chain model is globally stable to the ”all-healthy”state.
But how does this jive with the results on the instability of the models(2) and (3)?
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 28 / 57
A Linear Programming Approach
Consider the Markov chain and the problem
maxPj (t)
Pi (t + 1).
We can spell this out as
max∑
X P(ξi (t + 1) = 1|ξ(t) = X )P(ξ(t) = X )P(ξ(t) = X ) ≥ 0∑X P(ξ(t) = X ) = 1∑
Xj=1 P(ξ(t) = X ) = Pj(t)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 29 / 57
A Linear Programming Approach
Consider the Markov chain and the problem
maxPj (t)
Pi (t + 1).
We can spell this out as
max∑
X P(ξi (t + 1) = 1|ξ(t) = X )P(ξ(t) = X )P(ξ(t) = X ) ≥ 0∑X P(ξ(t) = X ) = 1∑
Xj=1 P(ξ(t) = X ) = Pj(t)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 29 / 57
A Linear Programming Approach
Using Lagrange duality, we get
maxPj (t)
Pi (t + 1) = minλ0+
∑j∈S λj−P(ξi (t+1)=1|ξS (t)=1,ξSc (t)=0)≥0
λ0 +n∑j=
λjPj(t)
It is not hard to show that the optimal solution is
λ0 = 0 , λi = 1− δ , λj∈Ni= β , λj /∈Ni
= 0
so thatmaxPj (t)
Pi (t + 1) ≤ (1− δ)Pi (t) + β∑j∈Ni
Pj(t),
But this is nothing but the linear model we have been considering!
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 30 / 57
A Linear Programming Approach
Using Lagrange duality, we get
maxPj (t)
Pi (t + 1) = minλ0+
∑j∈S λj−P(ξi (t+1)=1|ξS (t)=1,ξSc (t)=0)≥0
λ0 +n∑j=
λjPj(t)
It is not hard to show that the optimal solution is
λ0 = 0 , λi = 1− δ , λj∈Ni= β , λj /∈Ni
= 0
so thatmaxPj (t)
Pi (t + 1) ≤ (1− δ)Pi (t) + β∑j∈Ni
Pj(t),
But this is nothing but the linear model we have been considering!
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 30 / 57
A Linear Programming Approach
Using Lagrange duality, we get
maxPj (t)
Pi (t + 1) = minλ0+
∑j∈S λj−P(ξi (t+1)=1|ξS (t)=1,ξSc (t)=0)≥0
λ0 +n∑j=
λjPj(t)
It is not hard to show that the optimal solution is
λ0 = 0 , λi = 1− δ , λj∈Ni= β , λj /∈Ni
= 0
so thatmaxPj (t)
Pi (t + 1) ≤ (1− δ)Pi (t) + β∑j∈Ni
Pj(t),
But this is nothing but the linear model we have been considering!
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 30 / 57
Mixing Time of the Markov Chain Model (1)
Theorem
Consider the Markov chain model (1):
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)
The linearized system (3) provides an upper bounds on the ”true”marginal probability of a node being infected as given by the Markovchain model.
If the matrix M = (1− δ)I + βA is stable, then the Markov chain hasfast mixing to the all-healthy state (the mixing time is O(log n)).
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 31 / 57
What About the Mean-Field Approximation?
The nonlinear mean-field approximation does not generally provide anupper bound for the true Pi (t).
However, it does if we initialize the chain with the ”all-infected” state.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 32 / 57
What About the Mean-Field Approximation?
The nonlinear mean-field approximation does not generally provide anupper bound for the true Pi (t).
However, it does if we initialize the chain with the ”all-infected” state.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 32 / 57
What About the Mean-Field Approximation?
The nonlinear mean-field approximation does not generally provide anupper bound for the true Pi (t).
However, it does if we initialize the chain with the ”all-infected” state.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 32 / 57
Mixing Time of the Markov Chain Model (1)
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.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 33 / 57
Mixing Time of the Markov Chain Model (1)
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.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 33 / 57
Mixing Time of the Markov Chain Model (1)
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.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 33 / 57
βρ(A)δ = 1.009 and n = 2000
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 34 / 57
βρ(A)δ = 0.999 and n = 2000
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 35 / 57
Mixing Time of the Markov Chain Model (1)
However, in general, the condition βρ(A)δ < 1 is only sufficient for fast
mixing.
For other graphs (e.g., star, or star followed by a line) the conditionβρ(A)δ < 1 gives a lower bound on when the phase transition to slow
mixing occurs.
Tighter bounds can be obtained by looking at the n + n(n−1)2 variables
Pi (t) = P(ξi (t) = 1) and Pij(t) = P(ξi (t) = 1, ξj(t) = 1).
and attemtping to upper bound their values using the linearprogramming approach. For example, for Pi (t + 1) this yields:
Pi (t + 1) ≤ (1− δ)Pi (t) + β∑j ∼i
Pj (t)−[β −
1
di
((1− β)di − (1− diβ)
)]∑j∼i
Pij (t)
− 2(1− β)di − (1− diβ)
di (di − 1)
∑j ∼i,k∼i
Pjk (t)
where di is the degree of node i .
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 36 / 57
Mixing Time of the Markov Chain Model (1)
However, in general, the condition βρ(A)δ < 1 is only sufficient for fast
mixing.
For other graphs (e.g., star, or star followed by a line) the conditionβρ(A)δ < 1 gives a lower bound on when the phase transition to slow
mixing occurs.
Tighter bounds can be obtained by looking at the n + n(n−1)2 variables
Pi (t) = P(ξi (t) = 1) and Pij(t) = P(ξi (t) = 1, ξj(t) = 1).
and attemtping to upper bound their values using the linearprogramming approach. For example, for Pi (t + 1) this yields:
Pi (t + 1) ≤ (1− δ)Pi (t) + β∑j ∼i
Pj (t)−[β −
1
di
((1− β)di − (1− diβ)
)]∑j∼i
Pij (t)
− 2(1− β)di − (1− diβ)
di (di − 1)
∑j ∼i,k∼i
Pjk (t)
where di is the degree of node i .
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 36 / 57
Mixing Time of the Markov Chain Model (1)
However, in general, the condition βρ(A)δ < 1 is only sufficient for fast
mixing.
For other graphs (e.g., star, or star followed by a line) the conditionβρ(A)δ < 1 gives a lower bound on when the phase transition to slow
mixing occurs.
Tighter bounds can be obtained by looking at the n + n(n−1)2 variables
Pi (t) = P(ξi (t) = 1) and Pij(t) = P(ξi (t) = 1, ξj(t) = 1).
and attemtping to upper bound their values using the linearprogramming approach.
For example, for Pi (t + 1) this yields:
Pi (t + 1) ≤ (1− δ)Pi (t) + β∑j ∼i
Pj (t)−[β −
1
di
((1− β)di − (1− diβ)
)]∑j∼i
Pij (t)
− 2(1− β)di − (1− diβ)
di (di − 1)
∑j ∼i,k∼i
Pjk (t)
where di is the degree of node i .
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 36 / 57
Mixing Time of the Markov Chain Model (1)
However, in general, the condition βρ(A)δ < 1 is only sufficient for fast
mixing.
For other graphs (e.g., star, or star followed by a line) the conditionβρ(A)δ < 1 gives a lower bound on when the phase transition to slow
mixing occurs.
Tighter bounds can be obtained by looking at the n + n(n−1)2 variables
Pi (t) = P(ξi (t) = 1) and Pij(t) = P(ξi (t) = 1, ξj(t) = 1).
and attemtping to upper bound their values using the linearprogramming approach. For example, for Pi (t + 1) this yields:
Pi (t + 1) ≤ (1− δ)Pi (t) + β∑j ∼i
Pj (t)−[β −
1
di
((1− β)di − (1− diβ)
)]∑j∼i
Pij (t)
− 2(1− β)di − (1− diβ)
di (di − 1)
∑j ∼i,k∼i
Pjk (t)
where di is the degree of node i .
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 36 / 57
Star Network βρ(A)δ = 1.9 and n = 2000
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
Time Step
Num
ber
of In
fect
ed N
odes
λ = 44.7, δ = 0.9, β = 0.0382, βλ/δ = 1.9
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 37 / 57
Star Network βρ(A)δ = 2.1 and n = 2000
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
Time Step
Num
ber
of In
fect
ed N
odes
λ = 44.7, δ = 0.9, β = 0.0423, βλ/δ = 2.1
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 38 / 57
Star-Line Network βρ(A)δ = 1.9 and n = 1200
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
Time Step
Num
ber
of In
fect
ed N
odes
λ = 4.48, δ = 0.9, β = 0.382, βλ/δ = 1.9
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 39 / 57
Star-Line Network βρ(A)δ = 2.1 and n = 1200
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
Time Step
Num
ber
of In
fect
ed N
odes
λ = 4.48, δ = 0.9, β = 0.422, βλ/δ = 2.1
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 40 / 57
SIRS Model
𝛽
1 − 𝛽 1 − 𝛿 1 − 𝛾
𝛿
𝛾
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 41 / 57
SIRS Model
This is a more reasonable model, since a recovered node is notimmediately susceptible
The SIRS model has 3n states
ξi (t) = 0 if node i is susceptible, ξi (t) = 1 if it is infected, andξi (t) = 2 if it is recovered
P(ξ(t + 1) = Y |ξ(t) = X ) =n∏
i=1
P(ξi (t + 1) = Yi |ξ(t) = X )
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 42 / 57
SIRS Model
This is a more reasonable model, since a recovered node is notimmediately susceptible
The SIRS model has 3n states
ξi (t) = 0 if node i is susceptible, ξi (t) = 1 if it is infected, andξi (t) = 2 if it is recovered
P(ξ(t + 1) = Y |ξ(t) = X ) =n∏
i=1
P(ξi (t + 1) = Yi |ξ(t) = X )
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 42 / 57
SIRS Model
This is a more reasonable model, since a recovered node is notimmediately susceptible
The SIRS model has 3n states
ξi (t) = 0 if node i is susceptible, ξi (t) = 1 if it is infected, andξi (t) = 2 if it is recovered
P(ξ(t + 1) = Y |ξ(t) = X ) =n∏
i=1
P(ξi (t + 1) = Yi |ξ(t) = X )
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 42 / 57
SIRS Model
P(ξi (t + 1) = Yi |ξ(t) = X ) =
(1− β)|Ni∩I (t)|, if (Xi ,Yi ) = (0, 0)
1− (1− β)|Ni∩I (t)|, if (Xi ,Yi ) = (0, 1)
0, if (Xi ,Yi ) = (0, 2)
0, if (Xi ,Yi ) = (1, 0)
1− δ, if (Xi ,Yi ) = (1, 1)
δ, if (Xi ,Yi ) = (1, 2)
γ, if (Xi ,Yi ) = (2, 0)
0, if (Xi ,Yi ) = (2, 1)
1− γ, if (Xi ,Yi ) = (2, 2)
(7)
The “all-healthy” state is the unique stationary distribution of theMarkov chain
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 43 / 57
SIRS Model
P(ξi (t + 1) = Yi |ξ(t) = X ) =
(1− β)|Ni∩I (t)|, if (Xi ,Yi ) = (0, 0)
1− (1− β)|Ni∩I (t)|, if (Xi ,Yi ) = (0, 1)
0, if (Xi ,Yi ) = (0, 2)
0, if (Xi ,Yi ) = (1, 0)
1− δ, if (Xi ,Yi ) = (1, 1)
δ, if (Xi ,Yi ) = (1, 2)
γ, if (Xi ,Yi ) = (2, 0)
0, if (Xi ,Yi ) = (2, 1)
1− γ, if (Xi ,Yi ) = (2, 2)
(7)
The “all-healthy” state is the unique stationary distribution of theMarkov chain
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 43 / 57
SIRS Model
Mean-Field Approximation:
PR,i (t + 1) =(1− γ)PR,i (t) + δPI ,i (t), (8)
PI ,i (t + 1) =(1− δ)PI ,i (t)+(1−
∏j∈Ni
(1− βPI ,j(t)))
(1− PR,i (t)− PI ,i (t)) (9)
Linear Model: [PR(t + 1)
PI (t + 1)
]= M
[PR(t)
PI (t)
], (10)
where
M =
[(1− γ)In δIn
0n×n (1− δ)In + βA
](11)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 44 / 57
SIRS Model
Mean-Field Approximation:
PR,i (t + 1) =(1− γ)PR,i (t) + δPI ,i (t), (8)
PI ,i (t + 1) =(1− δ)PI ,i (t)+(1−
∏j∈Ni
(1− βPI ,j(t)))
(1− PR,i (t)− PI ,i (t)) (9)
Linear Model: [PR(t + 1)
PI (t + 1)
]= M
[PR(t)
PI (t)
], (10)
where
M =
[(1− γ)In δIn
0n×n (1− δ)In + βA
](11)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 44 / 57
SIRS Model
Theorem
If βρ(A)δ < 1, the Markov chain is “fast-mixing” (and the mean-field
approximation is globally stable to the all-healthy state).
Theorem
If βρ(A)δ > 1, the mean-field approximation has a second unique non-origin
fixed point.
γ does not seem to play a role.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 45 / 57
SIRS Model
Theorem
If βρ(A)δ < 1, the Markov chain is “fast-mixing” (and the mean-field
approximation is globally stable to the all-healthy state).
Theorem
If βρ(A)δ > 1, the mean-field approximation has a second unique non-origin
fixed point.
γ does not seem to play a role.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 45 / 57
SIRS Model
Theorem
If βρ(A)δ < 1, the Markov chain is “fast-mixing” (and the mean-field
approximation is globally stable to the all-healthy state).
Theorem
If βρ(A)δ > 1, the mean-field approximation has a second unique non-origin
fixed point.
γ does not seem to play a role.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 45 / 57
SIV Model
𝛽𝛽
1 − 𝛿𝛿 1 − 𝛾𝛾
𝛿𝛿
𝛾𝛾
𝜃𝜃
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 46 / 57
SIV Model
SIV: Susceptible-Infected-Vaccinated
SIRS+Vaccination: transition from S to R is also permitted now
Depending on the value of γ, this can model temporary immunization(γ 6= 0) or permanent immunization (γ = 0).
Infection-Dominant SIV vs. Vaccination-Dominant SIV
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 47 / 57
SIV Model
SIV: Susceptible-Infected-Vaccinated
SIRS+Vaccination: transition from S to R is also permitted now
Depending on the value of γ, this can model temporary immunization(γ 6= 0) or permanent immunization (γ = 0).
Infection-Dominant SIV vs. Vaccination-Dominant SIV
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 47 / 57
SIV Model
SIV: Susceptible-Infected-Vaccinated
SIRS+Vaccination: transition from S to R is also permitted now
Depending on the value of γ, this can model temporary immunization(γ 6= 0) or permanent immunization (γ = 0).
Infection-Dominant SIV vs. Vaccination-Dominant SIV
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 47 / 57
SIV Model
Infection-Dominant Model:
P {ξi (t + 1) = Yi | ξ(t) = X} =
(1− β)|Ni∩I (t)|(1− θ), if (Xi ,Yi ) = (0, 0)
1− (1− β)|Ni∩I (t)|, if (Xi ,Yi ) = (0, 1)
(1− β)|Ni∩I (t)|θ, if (Xi ,Yi ) = (0, 2)
0, if (Xi ,Yi ) = (1, 0)
1− δ, if (Xi ,Yi ) = (1, 1)
δ, if (Xi ,Yi ) = (1, 2)
γ, if (Xi ,Yi ) = (2, 0)
0, if (Xi ,Yi ) = (2, 1)
1− γ, if (Xi ,Yi ) = (2, 2)
, (12)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 48 / 57
SIV Model
Vaccination-Dominant Model:
P {ξi (t + 1) = Yi | ξ(t) = X} =
(1− β)|Ni∩I (t)|(1− θ), if (Xi ,Yi ) = (0, 0)
(1− (1− β)|Ni∩I (t)|)(1− θ), if (Xi ,Yi ) = (0, 1)
θ, if (Xi ,Yi ) = (0, 2)
0, if (Xi ,Yi ) = (1, 0)
1− δ, if (Xi ,Yi ) = (1, 1)
δ, if (Xi ,Yi ) = (1, 2)
γ, if (Xi ,Yi ) = (2, 0)
0, if (Xi ,Yi ) = (2, 1)
1− γ, if (Xi ,Yi ) = (2, 2)
(13)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 49 / 57
SIV Model
Stationary distribution of the Markov chain:
different from that of SIS/SIRS cases, in which all the nodes becamesusceptible
once there is no infected node, the Markov chain reduces to a simplerone, where the nodes are all decoupled
The stationary distribution of each single node is then P∗S = γγ+θ and
P∗R = θγ+θ (γθ 6= 1)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 50 / 57
SIV Model
Stationary distribution of the Markov chain:
different from that of SIS/SIRS cases, in which all the nodes becamesusceptible
once there is no infected node, the Markov chain reduces to a simplerone, where the nodes are all decoupled
The stationary distribution of each single node is then P∗S = γγ+θ and
P∗R = θγ+θ (γθ 6= 1)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 50 / 57
SIV Model
Stationary distribution of the Markov chain:
different from that of SIS/SIRS cases, in which all the nodes becamesusceptible
once there is no infected node, the Markov chain reduces to a simplerone, where the nodes are all decoupled
The stationary distribution of each single node is then P∗S = γγ+θ and
P∗R = θγ+θ (γθ 6= 1)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 50 / 57
SIV Model (Infection-Dominant)
Mean-Field Approximation:
PR,i (t + 1) =(1− γ)PR,i (t) + δPI ,i (t)
+∏j∈Ni
(1− βPI ,j(t))θ(1− PR,i (t)− PI ,i (t)), (14)
PI ,i (t + 1) =(1− δ)PI ,i (t)
+(
1−∏j∈Ni
(1− βPI ,j(t)))
(1− PR,i (t)− PI ,i (t)) (15)
Linear Model: [PR(t + 1)
PI (t + 1)
]=
[P∗R1n
0n
]+ M
[PR(t)− P∗R1nPI (t)− 0n
], (16)
where
M =
[(1− γ − θ)In (δ − θ)In − θP∗SβA
0n×n (1− δ)In + P∗SβA
](17)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 51 / 57
SIV Model (Infection-Dominant)
Mean-Field Approximation:
PR,i (t + 1) =(1− γ)PR,i (t) + δPI ,i (t)
+∏j∈Ni
(1− βPI ,j(t))θ(1− PR,i (t)− PI ,i (t)), (14)
PI ,i (t + 1) =(1− δ)PI ,i (t)
+(
1−∏j∈Ni
(1− βPI ,j(t)))
(1− PR,i (t)− PI ,i (t)) (15)
Linear Model: [PR(t + 1)
PI (t + 1)
]=
[P∗R1n
0n
]+ M
[PR(t)− P∗R1nPI (t)− 0n
], (16)
where
M =
[(1− γ − θ)In (δ − θ)In − θP∗SβA
0n×n (1− δ)In + P∗SβA
](17)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 51 / 57
SIV Model (Vaccination-Dominant)
Mean-Field Approximation:
PR,i (t + 1) =(1− γ)PR,i (t) + δPI ,i (t)
+ θ(1− PR,i (t)− PI ,i (t)), (18)
PI ,i (t + 1) =(1− δ)PI ,i (t) + (1− θ)
·(
1−∏j∈Ni
(1− βPI ,j(t)))
(1− PR,i (t)− PI ,i (t)) (19)
Linear Model:[PR(t + 1)
PI (t + 1)
]=
[P∗R1n
0n
]+ M
[PR(t)− P∗R1nPI (t)− 0n
], (20)
where
M =
[(1− γ − θ)In (δ − θ)In − θP∗SβA
0n×n (1− δ)In + (1− θ)P∗SβA
](21)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 52 / 57
SIV Model (Vaccination-Dominant)
Mean-Field Approximation:
PR,i (t + 1) =(1− γ)PR,i (t) + δPI ,i (t)
+ θ(1− PR,i (t)− PI ,i (t)), (18)
PI ,i (t + 1) =(1− δ)PI ,i (t) + (1− θ)
·(
1−∏j∈Ni
(1− βPI ,j(t)))
(1− PR,i (t)− PI ,i (t)) (19)
Linear Model:[PR(t + 1)
PI (t + 1)
]=
[P∗R1n
0n
]+ M
[PR(t)− P∗R1nPI (t)− 0n
], (20)
where
M =
[(1− γ − θ)In (δ − θ)In − θP∗SβA
0n×n (1− δ)In + (1− θ)P∗SβA
](21)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 52 / 57
SIV Model (Infection-Dominant)
Proposition
The main fixed point of the mean-field approximation (14, 15) is
1 locally stable, if γγ+θ
βδ ρ(A) < 1, and
2 globally stable, if βδ ρ(A) < 1 .
Theorem
If βρ(A)δ < 1, the mixing time of the Markov chain is O(log n).
Theorem
If γγ+θ
βλmax(A)δ > 1, the mean-field approximation (14, 15) has a second
unique nontrivial fixed point.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 53 / 57
SIV Model (Infection-Dominant)
Proposition
The main fixed point of the mean-field approximation (14, 15) is
1 locally stable, if γγ+θ
βδ ρ(A) < 1, and
2 globally stable, if βδ ρ(A) < 1 .
Theorem
If βρ(A)δ < 1, the mixing time of the Markov chain is O(log n).
Theorem
If γγ+θ
βλmax(A)δ > 1, the mean-field approximation (14, 15) has a second
unique nontrivial fixed point.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 53 / 57
SIV Model (Infection-Dominant)
Proposition
The main fixed point of the mean-field approximation (14, 15) is
1 locally stable, if γγ+θ
βδ ρ(A) < 1, and
2 globally stable, if βδ ρ(A) < 1 .
Theorem
If βρ(A)δ < 1, the mixing time of the Markov chain is O(log n).
Theorem
If γγ+θ
βλmax(A)δ > 1, the mean-field approximation (14, 15) has a second
unique nontrivial fixed point.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 53 / 57
SIV Model (Vaccination-Dominant)
Proposition
The main fixed point of the mean-field approximation (18, 19) is
1 locally stable, if (1− θ) γγ+θ
βδ ρ(A) < 1, and
2 globally stable, if (1− θ)βδ ρ(A) < 1 .
Theorem
If (1− θ)βρ(A)δ < 1, the mixing time of the Markov chain is O(log n).
Theorem
If (1− θ) γγ+θ
βδ ρ(A) > 1, the mean-field approximation (18, 19) has a
second unique nontrivial fixed point.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 54 / 57
SIV Model (Vaccination-Dominant)
Proposition
The main fixed point of the mean-field approximation (18, 19) is
1 locally stable, if (1− θ) γγ+θ
βδ ρ(A) < 1, and
2 globally stable, if (1− θ)βδ ρ(A) < 1 .
Theorem
If (1− θ)βρ(A)δ < 1, the mixing time of the Markov chain is O(log n).
Theorem
If (1− θ) γγ+θ
βδ ρ(A) > 1, the mean-field approximation (18, 19) has a
second unique nontrivial fixed point.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 54 / 57
SIV Model (Vaccination-Dominant)
Proposition
The main fixed point of the mean-field approximation (18, 19) is
1 locally stable, if (1− θ) γγ+θ
βδ ρ(A) < 1, and
2 globally stable, if (1− θ)βδ ρ(A) < 1 .
Theorem
If (1− θ)βρ(A)δ < 1, the mixing time of the Markov chain is O(log n).
Theorem
If (1− θ) γγ+θ
βδ ρ(A) > 1, the mean-field approximation (18, 19) has a
second unique nontrivial fixed point.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 54 / 57
Simulation Results
100
101
102
103
100
101
102
103
Time Step
Num
ber
of In
fect
ed N
odes
β‖A‖δ
= 0.99 < 1
β‖A‖δ
= 1.2 > 1
100
101
102
103
100
101
102
103
Time Step
Num
ber
of In
fect
ed N
odes
γ
γ+θ
β‖A‖δ
= 0.99 < 1
γγ+θ
β‖A‖δ
= 1.2 > 1
100
101
102
103
100
101
102
103
Time Step
Num
ber
of In
fect
ed N
odes
(1− θ) γγ+θ
β‖A‖δ
= 0.99 < 1
(1− θ) γγ+θ
β‖A‖δ
= 1.2 > 1
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.)
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 55 / 57
Comparison
MC: Fast mixing
MFA: Global stability MFA: 2nd unique fixed point
MC: Fast mixing
MFA: Global stability MFA: 2nd unique fixed point MFA: Local stability
MC: Fast mixing
MFA: Global stability MFA: 2nd unique fixed point MFA: Local stability
SIS & SIRS
SIV Infection-Dominant
SIV Vaccination-Dominant
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 56 / 57
Summary and Future Work
Studied the SIS model for epidemic spread:
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
death, SEIS, SEIR, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 57 / 57
Summary and Future Work
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
death, SEIS, SEIR, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 57 / 57
Summary and Future Work
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
death, SEIS, SEIR, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 57 / 57
Summary and Future Work
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
death, SEIS, SEIR, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 57 / 57
Summary and Future Work
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
death, SEIS, SEIR, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 57 / 57
Summary and Future Work
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
death, SEIS, SEIR, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 57 / 57
Summary and Future Work
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
death, SEIS, SEIR, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 57 / 57
Summary and Future Work
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
death, SEIS, SEIR, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 57 / 57
Summary and Future Work
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
death, SEIS, SEIR, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 57 / 57
Summary and Future Work
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.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 57 / 57
Summary and Future Work
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
death, SEIS, SEIR, etc.
Babak Hassibi (Caltech) Epidemic Spread Oct 22, 2015 57 / 57