modeling propagation processes on networks by using...

MODELING PROPAGATION PROCESSES ON

NETWORKS BY USING DIFFERENTIAL EQUATIONS

Peter L. Simon

Department of Applied Analysis and Computational Mathematics,Institute of Mathematics, Eötvös Loránd University Budapest, and

Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences,Hungary

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SIS EPIDEMIC ON A NETWORK

A graph with N nodes is given

The nodes can be susceptible (S) or infected (I)

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Transitions:

S → I, rate: kτ , k is the number of I neighbours.

I → S, rate: γ

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Transitions:S → I, rate: kτ , k is the number of I neighbours.I → S, rate: γ

I

S

SI

recovery

γinfection

τ

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Transitions:


I → S, rate: γ

AIM: Derive a simple system of differential equations yielding theexpected number of infected nodes [I](t).

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Transitions:


I → S, rate: γ

AIM: Derive a simple system of differential equations yielding theexpected number of infected nodes [I](t).

Known models:

Master equation

Mean-field equation

Pairwise model

Compact pairwise model

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GENERAL MATHEMATICAL MODEL


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The nodes can be in the states {a1, a2, . . . am}.

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The state space of the graph has mN elements

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The state space of the graph has mN elements

The transitions between different states can be described by aPoisson process

Probability of a transition from state ai to state aj in a time interval oflength ∆t is:

1 − exp(−λij∆t).

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NETWORK PROCESSES

SIS epidemic

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NETWORK PROCESSES

SIS epidemic

States of the nodes: {S, I}.

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NETWORK PROCESSES

SIS epidemic

States of the nodes: {S, I}.

Transitions and their rates

S → I, λ = kτ , k is the number of I neighbours.

I → S, λ = γ

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NETWORK PROCESSES

SIR epidemic

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NETWORK PROCESSES

SIR epidemic

States of the nodes: {S, I,R}.

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NETWORK PROCESSES

SIR epidemic

States of the nodes: {S, I,R}.


S → I, λ = kτ , k is the number of I neighbours.

I → R, λ = γ

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NETWORK PROCESSES

Rumour spreading

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NETWORK PROCESSES

Rumour spreading

States of the nodes: {X ,Y ,Z} (ignorant, spreader, stifler).

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NETWORK PROCESSES

Rumour spreading

States of the nodes: {X ,Y ,Z} (ignorant, spreader, stifler).


X → Y , λ = kτ , k is the number of Y neighbours.

Y → Z , λ = γ + jp, j is the number of Y and Z neighbours.

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NETWORK PROCESSES

Propagation of activity in neuronal networks

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NETWORK PROCESSES


States of the nodes: {E+,E−, I+, I−} (active and inactive excitatory

neurons, active and inactive inhibitory neurons).

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NETWORK PROCESSES


States of the nodes: {E+,E−, I+, I−} (active and inactive excitatory

neurons, active and inactive inhibitory neurons).


E+ → E−

, λ = α.

E−→ E+, λ = tanh(iwE − jwI + hE), i, j is the number of E+ and

I+ neighbours.

I+ → I−

, λ = α.

I−→ I+, λ = tanh(iwE − jwI + hI), i, j is the number of E+ and I+

neighbours.

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AIM OF THE RESEARCH

Derive differential equations for different processes and for differenttypes of graphs.

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AIM OF THE RESEARCH


Frequently used random graphs:

Erdos-Rényi

Configuration model (Bollobás)

Small-world (Watts-Strogatz)

Graphs with scale free degree distribution (Barabási-Albert)

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AIM OF THE RESEARCH


Frequently used random graphs:

Erdos-Rényi

Configuration model (Bollobás)

Small-world (Watts-Strogatz)

Graphs with scale free degree distribution (Barabási-Albert)

Examples for network processes:

Epidemic propagation

Rumour spreading

Propagation of neuronal activity

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MARKOV CHAIN FOR SIS EPIDEMIC



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Transitions:


I → S, rate: γ

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State space for a triangle

I I

I

S I

I

I S

I

I I

S

S S

I

S I

S

I S

S

S S

S

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State space for a triangle

I I

I

S I

I

I S

I

I I

S

S S

I

S I

S

I S

S

S S

S

Infection: SIS → SII, IIS

Recovery: SIS → SSS

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SIS EPIDEMIC

Master equations

XSSS = γ(XSSI + XSIS + XISS),

XSSI = γ(XSII + XISI)− (2τ + γ)XSSI ,

XSIS = γ(XSII + XIIS)− (2τ + γ)XSIS ,

XISS = γ(XISI + XIIS)− (2τ + γ)XISS ,

XSII = γXIII + τ(XSSI + XSIS)− 2(τ + γ)XSII ,

XISI = γXIII + τ(XSSI + XISS)− 2(τ + γ)XISI ,

XIIS = γXIII + τ(XSIS + XISS)− 2(τ + γ)XIIS ,

XIII = −3γXIII + 2τ(XSII + XISI + XIIS),

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SIS EPIDEMIC

Master equations









2N equations for a graph with N nodes

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SIS EPIDEMIC

Master equations









The size of the system can be reduced by using the automorphismsof the graph:

Simon, P.L., Taylor, M., Kiss., I.Z., Exact epidemic models on graphs using graph-automorphism

driven lumping, J. Math. Biol., 62 (2011).

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MEAN-FIELD APPROXIMATION FOR SIS EPIDEMIC

Exact equation: ˙[I] = τ [SI] − γ[I]

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[SI](t): expected number of SI edges

This differential equation holds for any graphSimon, P.L., Taylor, M., Kiss., I.Z., Exact epidemic models on graphs using graph-automorphism

driven lumping, J. Math. Biol. 62 (2011), 479-508.

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Approximation [SI] ≈ n [I]N [S], where the average degree is n

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Approximating differential equation for [I]

I = τnN

I(N − I)− γI.

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I = τnN

I(N − I)− γI.

This is the well-known compartmental model, which does not giveaccurate result for networks.Reason: the approximation assumes random distribution of infectednodes.

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I = τnN

I(N − I)− γI.

This is the well-known compartmental model, which does not giveaccurate result for networks.Reason: the approximation assumes random distribution of infectednodes.

Better idea: derive a differential equation for [SI], this leaded to thepairwise model.Keeling, M.J., The effects of local spatial structure on epidemiological invasions, Proc. R. Soc.

Lond. B 266 (1999), 859-867.

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PAIRWISE APPROXIMATION

Keep the exact equation ˙[I] = τ [SI] − γ[I]

and derive a differential equation for [SI].

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Exact differential equations:

˙[I] = τ [SI]− γ[I],˙[SI] = γ([II]− [SI]) + τ([SSI] − [ISI]− [SI]),˙[II] = −2γ[II] + 2τ([ISI] + [SI]),˙[SS] = 2γ[SI]− 2τ [SSI].

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Approximation:

[ABC] ≈n − 1

n[AB][BC]

[B], n average degree

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Approximation:

[ABC] ≈n − 1

n[AB][BC]


M. Taylor, P. L. Simon, D. M. Green, T. House, I. Z. Kiss, From Markovian to pairwise epidemic

models and the performance of moment closure approximations, J. Math. Biol. 64 (2012),

1021-1042.12 / 23

COMPARISON OF ODE MODELS TO SIMULATION

Regular random graph with N = 1000 nodes, average degree n = 20,γ = 1, critical value of τ from compartmental model: τcr = γ/n

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0 10 20 30 400

0.01

0.02

0.03

0.04

0.05

τ = 0.9τcr

0 10 20 30 400

0.01

0.02

0.03

0.04

0.05

τ = τcr

0 10 20 30 400.03

0.04

0.05

0.06

0.07

0.08

τ = 1.1τcr

prev

alen

ce

t0 10 20 30 40

0.05

0.1

0.15

0.2

0.25

0.3

τ = 1.5τcr

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0 10 20 30 400

0.01

0.02

0.03

0.04

0.05

τ = 0.9τcr

0 10 20 30 400

0.01

0.02

0.03

0.04

0.05

τ = τcr

0 10 20 30 400.03

0.04

0.05

0.06

0.07

0.08

τ = 1.1τcr

prev

alen

ce

t0 10 20 30 40

0.05

0.1

0.15

0.2

0.25

0.3

τ = 1.5τcr

Mean-field: dashed, Pairwise: continuousSimulation (average of 200 runs): grey thick curve

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0 10 20 30 400

0.01

0.02

0.03

0.04

0.05

τ = 0.9τcr

0 10 20 30 400

0.01

0.02

0.03

0.04

0.05

τ = τcr

0 10 20 30 400.03

0.04

0.05

0.06

0.07

0.08

τ = 1.1τcr

prev

alen

ce

t0 10 20 30 40

0.05

0.1

0.15

0.2

0.25

0.3

τ = 1.5τcr

τ = τcr ⇔ basic reproduction number R0 = 1.

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Bimodal random graph with N = 1000 nodes, average degreen = 20, γ = 1, τ = 2τcr = 2γ/nN/2 nodes have degree d1, N/2 nodes have degree d2.

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0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5d

1=18, d

2=22

t

prev

alen

ce

0 5 100

0.1

0.2

0.3

0.4

0.5

d1=5, d

2=35

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0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5d

1=18, d

2=22

t

prev

alen

ce

0 5 100

0.1

0.2

0.3

0.4

0.5

d1=5, d

2=35

Mean-field: dashed, Pairwise: continuousSimulation (average of 200 runs): grey thick curve

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0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5d

1=18, d

2=22

t

prev

alen

ce

0 5 100

0.1

0.2

0.3

0.4

0.5

d1=5, d

2=35

Reason of inaccuracy: in the closure [ABC] ≈ n−1n

[AB][BC][B] it is

assumed that each node has the same degree n.

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COMPACT PAIRWISE APPROXIMATION

There are Nk nodes with degree dk for k = 1, 2, . . . ,K .

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[ASI] =K∑

k=1

[ASk I], [ASk I] ≈dk − 1

dk

[ASk ][Sk I][Sk ]

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[ASI] =K∑

k=1


dk

[ASk ][Sk I][Sk ]

[Sk ]: expected number of susceptible nodes of degree dk ,[Sk I]: expected number of edges connecting an infected node to asusceptible node of degree dk

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[ASI] =K∑

k=1


dk

[ASk ][Sk I][Sk ]

Differential equations are needed for the new unknowns.

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[ASI] =K∑

k=1


dk

[ASk ][Sk I][Sk ]

˙[Sk ] = γ[Ik ]− τ [Sk I], k = 1, 2, . . . ,K .

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[ASI] =K∑

k=1


dk

[ASk ][Sk I][Sk ]

˙[Sk ] = γ[Ik ]− τ [Sk I], k = 1, 2, . . . ,K .

[Sk A] ≈ [SA]dk [Sk ]

∑Kl=1 dl [Sl ]

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[ASI] =K∑

k=1


dk

[ASk ][Sk I][Sk ]

˙[Sk ] = γ[Ik ]− τ [Sk I], k = 1, 2, . . . ,K .

[Sk A] ≈ [SA]dk [Sk ]

∑Kl=1 dl [Sl ]

[ASk I] ≈[AS][SI]dk (dk − 1)[Sk ]

S21

⇒ [ASI] ≈ [AS][SI]S2 − S1

S21

S1 =N∑

k=1dk [Sk ], S2 =

K∑

k=1d2

k [Sk ].

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COMPACT PAIRWISE MODEL

˙[Sk ]c = γ[Ik ]c − τdk [Sk ]c[SI]cSs

,

˙[SI]c = γ([II]c − [SI]c) + τ([SS]c − [SI]c)[SI]cP − τ [SI]c ,˙[SS]c = 2γ[SI]c − 2τ [SS]c [SI]cP,

˙[II]c = 2τ [SI]c − 2γ[II]c + 2τ [SI]2cP,

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,



with Ss =∑K

k=1 dk [Sk ]c and P = 1S2

s

K∑

k=1(dk − 1)dk [Sk ]c .

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,



Compact pairwise model: K + 3 equations

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,




More complex and accurate models:

Pre-compact pairwise model: 5K equations

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,




More complex and accurate models:

Pre-compact pairwise model: 5K equations

Heterogeneous pairwise model: 2K 2 + K equations

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Bimodal random graph with N = 1000 nodes, average degreen1 = 20, γ = 1, τ = 3γn1/n2, ni =

∑

d ik pk

N/2 nodes have degree d1 = 5, N/2 nodes have degree d2 = 35.

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∑

d ik pk


0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

I/N

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∑

d ik pk


0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

I/N

Pairwise: dashed, Compact pairwise: continuous black,Heterogeneous pairwise: continuous red,Simulation (average of 200 runs): grey thick curve

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ANALYSIS OF THE MEAN-FIELD MODEL


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[SI](t): expected number of SI edges

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I = τnN

I(N − I)− γI.

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I = τnN

I(N − I)− γI.

Transcritical bifurcation at γ = nτ : trivial steady state loses its stabilityand a stable endemic steady state appears.

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I = τnN

I(N − I)− γI.


Global behaviour for γ > nτ : all solutions converge to thedisease-free steady state Idf = 0.

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I = τnN

I(N − I)− γI.


Global behaviour for γ > nτ : all solutions converge to thedisease-free steady state Idf = 0.

Global behaviour for γ < nτ : all solutions converge to the endemicsteady state Ie = N(1 − γ

nτ ).

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ANALYSIS OF THE PAIRWISE MODEL


˙[I] = τ [SI]− γ[I],˙[S] = γ[I]− τ [SI],˙[SI] = γ([II]− [SI]) + τ([SSI] − [ISI]− [SI]),˙[II] = −2γ[II] + 2τ([ISI] + [SI]),˙[SS] = 2γ[SI]− 2τ [SSI].

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Approximation:

[ABC] ≈n − 1

n[AB][BC]


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Conservation relations:

[I] + [S] = N, 2[SI] + [II] + [SS] = nN, [SI] + [SS] = n[S]

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Conservation relations:

[I] + [S] = N, 2[SI] + [II] + [SS] = nN, [SI] + [SS] = n[S]

Reduced system

˙[S] = γN − (γ + nτ)[S] + τ [SS],

˙[SS] = 2(n[S]− [SS])

(

γ − τ(n − 1)[SS]

n[S]

)

.

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Reduced system

˙[S] = γN − (γ + nτ)[S] + τ [SS],

˙[SS] = 2(n[S]− [SS])

(

γ − τ(n − 1)[SS]

n[S]

)

.

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Reduced system

˙[S] = γN − (γ + nτ)[S] + τ [SS],

˙[SS] = 2(n[S]− [SS])

(

γ − τ(n − 1)[SS]

n[S]

)

.

Steady states and their stability

If τ(n − 1) < γ, then there is no endemic steady state and thedisease-free steady state is asymptotically stable.

If τ(n − 1) > γ, then the endemic steady state is asymptoticallystable and the disease-free steady state is unstable.

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Reduced system

˙[S] = γN − (γ + nτ)[S] + τ [SS],

˙[SS] = 2(n[S]− [SS])

(

γ − τ(n − 1)[SS]

n[S]

)

.

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Reduced system

˙[S] = γN − (γ + nτ)[S] + τ [SS],

˙[SS] = 2(n[S]− [SS])

(

γ − τ(n − 1)[SS]

n[S]

)

.

Phase plane analysis

(N,nN)

[S]s

[SS

] s

(N,nN)

[S]s

[SS

] s

Direction field: τ(n − 1) < γ (left panel), τ(n − 1) > γ (right panel).

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SUMMARY

Processes

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SUMMARY

Processes

Network types

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SUMMARY

Processes

Network types

Mathematical model: Markov chain with 2N states, masterequations

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SUMMARY

Processes

Network types


Mean-field approximation

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SUMMARY

Processes

Network types



Pairwise approximation

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SUMMARY

Processes

Network types




Compact pairwise approximation

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SUMMARY

Processes

Network types




Compact pairwise approximation

Kiss., I.Z., Miller, J.C., Simon, P.L., Mathematics of Epidemics onNetworks, Springer, Interdisciplinary Applied Mathematics (2017).

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Köszönöm a figyelmet!

Kiss., I.Z., Miller, J.C., Simon, P.L., Mathematics of Epidemics on Networks, Springer,

Interdisciplinary Applied Mathematics (2017).

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