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

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.

Introduction

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

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

SIR Model

1 − 𝛽 1 − 𝛿

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∏

P(ξi (t + 1) = Yi |ξ(t) = X )

states.

[ξ1(t) ξ2(t) . . . ξn(t)

P(ξ(t + 1) = Y |ξ(t) = X ) =n∏

P(ξi (t + 1) = Yi |ξ(t) = X )

states.

[ξ1(t) ξ2(t) . . . ξn(t)

P(ξ(t + 1) = Y |ξ(t) = X ) =n∏

P(ξi (t + 1) = Yi |ξ(t) = X )

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

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)

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)

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)

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)

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)

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−∏

which is an n-dimensional nonlinear dynamical system.

Analyzing this is often difficult for an arbitrary graph.

(1− β1j∈I(t)

)≈∏j∈Ni

(1− βPj(t))

Pi (t + 1) =(1−

∏j∈Ni

(1− βPj(t)))

(1− Pi (t)) +(

1− δ∏

)Pi (t)

= (1− δ)Pi (t) + (1− (1− δ)Pi (t))(

1−∏

(1− β1j∈I(t)

)≈∏j∈Ni

(1− βPj(t))

Pi (t + 1) =(1−

∏j∈Ni

(1− βPj(t)))

(1− Pi (t)) +(

1− δ∏

)Pi (t)

= (1− δ)Pi (t) + (1− (1− δ)Pi (t))(

1−∏

(1− β1j∈I(t)

)≈∏j∈Ni

(1− βPj(t))

Pi (t + 1) =(1−

∏j∈Ni

(1− βPj(t)))

(1− Pi (t)) +(

1− δ∏

)Pi (t)

= (1− δ)Pi (t) + (1− (1− δ)Pi (t))(

1−∏

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

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

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)) +

Pi (t)

= (1− δ)Pi (t) + β∑j∈Ni

Pj(t)Pi (t)

≤ (1− δ)Pi (t) + β∑j∈Ni

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)) +

Pi (t)

= (1− δ)Pi (t) + β∑j∈Ni

Pj(t)Pi (t)

≤ (1− δ)Pi (t) + β∑j∈Ni

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)) +

Pi (t)

= (1− δ)Pi (t) + β∑j∈Ni

Pj(t)Pi (t)

≤ (1− δ)Pi (t) + β∑j∈Ni

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)

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)

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)

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.

Theorem

The origin in

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

1−∏j∈Ni

(1− βPj(t))

(1− δ) + βρ(A) < 1,

orβρ(A)

δ< 1.

Theorem

The origin in

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

1−∏j∈Ni

(1− βPj(t))

(1− δ) + βρ(A) < 1,

orβρ(A)

δ< 1.

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).

C = cd

∞∑t=0

1TP(t).

C = cd

∞∑t=0

1TP(t) = cd

∞∑t=0

1TMtP(0) = cd1T (I −M)−1P(0).

C = cd

∞∑t=0

1TP(t).

C = cd

∞∑t=0

1TP(t) = cd

∞∑t=0

1TMtP(0) = cd1T (I −M)−1P(0).

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).

C = cd

∞∑t=0

1TP(t).

C = cd

∞∑t=0

1TP(t) = cd

∞∑t=0

1TMtP(0) = cd1T (I −M)−1P(0).

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.

It is customary to assume that the in the initial state fraction α < 1 of thenodes are randomly infected.Thus, P(0) = α1 and

= αcd1T (δ − βA)−11. (5)

It is customary to assume that the in the initial state fraction α < 1 of thenodes are randomly infected.Thus, P(0) = α1 and

= αcd1T (δ − βA)−11. (5)

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

Random Graphs

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 this

Aij = Aji =

{1 with probability

wiwj∑i wi

Random Graphs

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 this

Aij = Aji =

{1 with probability

wiwj∑i wi

Random Graphs

From thisweight vector construct a random graph with adjacency matrix

Aij = Aji =

{1 with probability

wiwj∑i wi

Random Graphs

]. From this

Aij = Aji =

{1 with probability

wiwj∑i wi

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.

pn(w) = δ(w − np),

[np np . . . np

wiwj∑i wi

=(np)(np)

n(np)= p.

pn(w) = δ(w − np),

w =[np np . . . np

and nodes i and j are connected with probability

wiwj∑i wi

=(np)(np)

n(np)= p.

pn(w) = δ(w − np),

w =[np np . . . np

wiwj∑i wi

=(np)(np)

n(np)= p.

Random Graphs

Defining W = diag(w), note that we may write the adjacency matrix A as

A = W 1/2GW 1/2 − wwT

where G is a Wigner matrix.

The cost can now be written as

C = nαcd1

δI − βW 1/2GW 1/2︸︷︷︸X

+βwwT

= nαcd

n1TX−11−

(1n1TX−1w

−1Twnβ +

TX−1w))

Random Graphs

A = W 1/2GW 1/2 − wwT

where G is a Wigner matrix.The cost can now be written as

C = nαcd1

δI − βW 1/2GW 1/2︸︷︷︸X

+βwwT

= nαcd

n1TX−11−

(1n1TX−1w

−1Twnβ +

TX−1w))

Random Graphs

A = W 1/2GW 1/2 − wwT

where G is a Wigner matrix.The cost can now be written as

C = nαcd1

δI − βW 1/2GW 1/2︸︷︷︸X

+βwwT

= nαcd

n1TX−11−

(1n1TX−1w

−1Twnβ +

TX−1w))

Random Graphs

With some effort, it can be shown that each of the terms

n1TX−11,

n1TX−1w , and

nwTX−1w

self-average.

For example,

n1TX−11→ E

ntraceX−1 = E

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.

Random Graphs

n1TX−11,

n1TX−1w , and

nwTX−1w

self-average. For example,

n1TX−11→ E

ntraceX−1 = E

ntrace

(δI − βW 1/2GW 1/2

)−1.

Random Graphs

n1TX−11,

n1TX−1w , and

nwTX−1w

self-average. For example,

n1TX−11→ E

ntraceX−1 = E

ntrace

(δI − βW 1/2GW 1/2

)−1.

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→∞

αcdδ

(1 + κF 2 − κ2F 2

1− Ew(

1F − δκ2

)) a.s. (6)

where κ =√β

δEw and F satisfies the implicit equation

∫p(w)

w−1 − κ2Fdw .

Erdos Renyi Random Graphs

For an Erdos-Renyi graph the implicit equation for F becomes

∫δ(w − np)

w−1 − κ2Fdw =

11np − κ2F

Solving this quadratic equation and plugging into the expression for thecost yields

C =nαcdδ − βnp

Erdos Renyi Random Graphs

For an Erdos-Renyi graph the implicit equation for F becomes

∫δ(w − np)

w−1 − κ2Fdw =

11np − κ2F

Solving this quadratic equation and plugging into the expression for thecost yields

C =nαcdδ − βnp

Numerical Results: Erdos Renyi n = 1000

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

InverseTheorem 4.1Theorem 4.3Simulated

Numerical Results: Exponential Weight Distributionn = 1000

5 10 15 20 250

Inverse

Theorem 4.1

Theorem 4.3

Simulated

Numerical Results: Heavy-Tail Weight Distributionn = 1000

2.5 3 3.5 4 4.5 50

InverseTheorem 4.1Theorem 4.3Simulated

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

Socialco

Simulated

S (π)

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

Socialco

Simulated

S (π)

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)),

Φi (x) = (1− δ)xi + (1− (1− δ)xi )

1−∏j∈Ni

(1− βxj)

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)),

Φi (x) = (1− δ)xi + (1− (1− δ)xi )

1−∏j∈Ni

(1− βxj)

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)),

Φi (x) = (1− δ)xi + (1− (1− δ)xi )

1−∏j∈Ni

(1− βxj)

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)),

Φi (x) = (1− δ)xi + (1− (1− δ)xi )

1−∏j∈Ni

(1− βxj)

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 .

x ∈ S , x ′ ∈ S ⇒ max(x , x ′) ∈ S .

phase space

x ∈ S , x ′ ∈ S ⇒ max(x , x ′) ∈ S .

phase space

x ∈ S , x ′ ∈ S ⇒ max(x , x ′) ∈ S .

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

x ∈ S , x ′ ∈ S ⇒ max(x , x ′) ∈ S .

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

x ∈ S , x ′ ∈ S ⇒ max(x , x ′) ∈ S .

phase space

x ∈ S , x ′ ∈ S ⇒ max(x , x ′) ∈ S .

phase space

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−∏

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.

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)?

P(ξi (t + 1) = Yi |ξ(t) = X ) =

(1− β)|Ni∩SX | if (Xi ,Yi ) = (0, 0)

1− δ(1− β)|Ni∩SX | if (Xi ,Yi ) = (1, 1)

Does the Markov chain model have a unique stationary distribution?

P(ξi (t + 1) = Yi |ξ(t) = X ) =

(1− β)|Ni∩SX | if (Xi ,Yi ) = (0, 0)

1− δ(1− β)|Ni∩SX | if (Xi ,Yi ) = (1, 1)

P(ξi (t + 1) = Yi |ξ(t) = X ) =

(1− β)|Ni∩SX | if (Xi ,Yi ) = (0, 0)

1− δ(1− β)|Ni∩SX | if (Xi ,Yi ) = (1, 1)

What is it?

The distribution where all states are healthy withprobability one.

P(ξi (t + 1) = Yi |ξ(t) = X ) =

(1− β)|Ni∩SX | if (Xi ,Yi ) = (0, 0)

1− δ(1− β)|Ni∩SX | if (Xi ,Yi ) = (1, 1)

P(ξi (t + 1) = Yi |ξ(t) = X ) =

(1− β)|Ni∩SX | if (Xi ,Yi ) = (0, 0)

1− δ(1− β)|Ni∩SX | if (Xi ,Yi ) = (1, 1)

P(ξi (t + 1) = Yi |ξ(t) = X ) =

(1− β)|Ni∩SX | if (Xi ,Yi ) = (0, 0)

1− δ(1− β)|Ni∩SX | if (Xi ,Yi ) = (1, 1)

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)

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)

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

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!

maxPj (t)

Pi (t + 1) = minλ0+

λ0 +n∑j=

λjPj(t)

λ0 = 0 , λi = 1− δ , λj∈Ni= β , λj /∈Ni

so thatmaxPj (t)

Pi (t + 1) ≤ (1− δ)Pi (t) + β∑j∈Ni

Pj(t),

maxPj (t)

Pi (t + 1) = minλ0+

λ0 +n∑j=

λjPj(t)

λ0 = 0 , λi = 1− δ , λj∈Ni= β , λj /∈Ni

so thatmaxPj (t)

Pi (t + 1) ≤ (1− δ)Pi (t) + β∑j∈Ni

Pj(t),

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)).

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.

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.

βρ(A)δ = 1.009 and n = 2000

βρ(A)δ = 0.999 and n = 2000

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− 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 .

mixing.

mixing occurs.

Pi (t + 1) ≤ (1− δ)Pi (t) + β∑j ∼i

Pj (t)−[β −

((1− β)di − (1− diβ)

)]∑j∼i

Pij (t)

− 2(1− β)di − (1− diβ)

di (di − 1)

∑j ∼i,k∼i

Pjk (t)

mixing.

mixing occurs.

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− β)di − (1− diβ)

)]∑j∼i

Pij (t)

− 2(1− β)di − (1− diβ)

di (di − 1)

∑j ∼i,k∼i

Pjk (t)

mixing.

mixing occurs.

Pi (t + 1) ≤ (1− δ)Pi (t) + β∑j ∼i

Pj (t)−[β −

((1− β)di − (1− diβ)

)]∑j∼i

Pij (t)

− 2(1− β)di − (1− diβ)

di (di − 1)

∑j ∼i,k∼i

Pjk (t)

Star Network βρ(A)δ = 1.9 and n = 2000

0 100 200 300 400 500 600 700 800 900 10000

Time Step

λ = 44.7, δ = 0.9, β = 0.0382, βλ/δ = 1.9

Star Network βρ(A)δ = 2.1 and n = 2000

0 100 200 300 400 500 600 700 800 900 10000

Time Step

λ = 44.7, δ = 0.9, β = 0.0423, βλ/δ = 2.1

Star-Line Network βρ(A)δ = 1.9 and n = 1200

0 100 200 300 400 500 600 700 800 900 10000

Time Step

λ = 4.48, δ = 0.9, β = 0.382, βλ/δ = 1.9

Star-Line Network βρ(A)δ = 2.1 and n = 1200

0 100 200 300 400 500 600 700 800 900 10000

Time Step

λ = 4.48, δ = 0.9, β = 0.422, βλ/δ = 2.1

SIRS Model

1 − 𝛽 1 − 𝛿 1 − 𝛾

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∏

P(ξi (t + 1) = Yi |ξ(t) = X )

SIRS Model

P(ξ(t + 1) = Y |ξ(t) = X ) =n∏

P(ξi (t + 1) = Yi |ξ(t) = X )

SIRS Model

P(ξ(t + 1) = Y |ξ(t) = X ) =n∏

P(ξi (t + 1) = Yi |ξ(t) = X )

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)

The “all-healthy” state is the unique stationary distribution of theMarkov chain

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)

The “all-healthy” state is the unique stationary distribution of theMarkov chain

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)

[PR(t)

PI (t)

], (10)

[(1− γ)In δIn

0n×n (1− δ)In + βA

SIRS Model

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)

PI (t + 1)

[PR(t)

PI (t)

], (10)

[(1− γ)In δIn

0n×n (1− δ)In + βA

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.

SIRS Model

Theorem

fixed point.

SIRS Model

Theorem

fixed point.

SIV Model

𝛽𝛽

1 − 𝛿𝛿 1 − 𝛾𝛾

𝛿𝛿

𝛾𝛾

𝜃𝜃

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

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)

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)

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)

SIV Model

P∗R = θγ+θ (γθ 6= 1)

SIV Model

P∗R = θγ+θ (γθ 6= 1)

SIV Model (Infection-Dominant)

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)

PI (t + 1)

[P∗R1n

[PR(t)− P∗R1nPI (t)− 0n

], (16)

[(1− γ − θ)In (δ − θ)In − θP∗SβA

0n×n (1− δ)In + P∗SβA

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)

PI (t + 1)

[P∗R1n

], (16)

0n×n (1− δ)In + P∗SβA

SIV Model (Vaccination-Dominant)

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

], (20)

0n×n (1− δ)In + (1− θ)P∗SβA

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

], (20)

0n×n (1− δ)In + (1− θ)P∗SβA

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.

Proposition

βδ ρ(A) < 1, and

Theorem

If γγ+θ

Proposition

βδ ρ(A) < 1, and

Theorem

If γγ+θ

Proposition

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.

Proposition

βδ ρ(A) < 1, and

Theorem

Proposition

βδ ρ(A) < 1, and

Theorem

Simulation Results

Time Step

β‖A‖δ

= 0.99 < 1

β‖A‖δ

= 1.2 > 1

Time Step

β‖A‖δ

= 0.99 < 1

γγ+θ

β‖A‖δ

= 1.2 > 1

Time Step

(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.)

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

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.

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

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

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

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-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

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

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

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.

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