Graph theory and epidemicsBerkeley Math Circle

15 April, 2020

About me

Mark PenneyPerimeter Institute /UWaterlooVisitor at MSRI

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About me

From Newfoundland, CanadaIts nice enough

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About me

From Newfoundland, CanadaIts nice enough... Except in the winter

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Mathematics of infectious disease

I don’t study infectious diseasesI Topology (knots, surfaces,...) applied to theoretical physics(particle physics, quantum computing)

Wanted to understand how we model epidemicsI Personal reasons: Knowledge is powerI Maybe contribute to COVID-19 research

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COVID-19: Flattening the curve

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Fundamental number: Reproductive rate

R0 = average number of newinfectionsPhase transition at R0 = 1:I R0 < 1: disease dies o�I R0 > 1: epidemic outbreak

Determines the trajectory of outbreakFlattening the curve⇔ reducing R0

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How do we determine R0?

Depends on assumptions of modelSimplest model: R0 = µ · TI µ = average number of contactsI T = probability of spread

To reduce R0:I Reduce µ: social distancingI Reduce T: Handwashing, masks

What are the assumptions of this model?Main assumption: ‘Homogeneous mixing’. People interact witheach other at random.Does not take into account social structure!

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Contact networks

People are not like molecules of a gas!More realistic: Understand the network or graph describingcontacts

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Contact networks

Recall: GraphsA graph is a collection of vertices joined by edges

Coronaviruses spread by closeproximity interactions (CPIs)Build contact network for group ofpeople:I Vertices: People in groupI Edges: connect people having CPIs

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Examples of contact networks

What sort of social structure leads to the following contactnetworks?

5 people all sharing CPIsI Household, group of friends, small sports team

6 people having CPI with 1 otherI Doctor visiting patients, tutor

7 people in 2 clustersI The households of 2 friends

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Degrees of contact networks

Recall: DegreeThe degree of a vertex in the number of edges connected to it

All vertices have degree 4I Every family member is in contact with theremaining 4

Central vertex has degree 6, remainder degree 1I Doctor as 6 contacts, patients have 1

Within each household, each family member is incontact with the remainder. Friends have higherdegree

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Degree distribution

Takeaway messageSocial structures lead to varying degrees in contact networkHelpful to plot number of vertices having each degree

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Degree distribution

De�nitionThe degree distribution of a graph is

d 7→ pd =# vertices of degree d

# vertices

Probability that randomly chosen vertex has degree dObtained from previous graphs by dividing by # vertices

ExerciseCompute the average degree of our 3 examples. Can you �nd asimple, general formula?

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Average degree

Recall: Average of probability distributionGiven probability distribution pd, the average is:

µ = 〈d〉 = 0 · p0 + 1 · p1 + 2 · p2 + . . . =∑dd · pd

The average degrees are:‘Household’ network: µ = 4‘Doctor’ network: µ = 1 · 67 + 6 · 17 =

127 = 1.7...

‘Friends’ network: µ = 2 · 27 + 3 · 47 + 4 · 17 =207 = 2.9...

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Average degree formula

PropositionThe average degree of a graph is

µ =2 ·# edges# vertices

Proof

µ =

∑d d · # vertices of degree d

# vertices∑dd · # vertices of degree k =

∑vertices v

# edges meeting v

This sum counts each edge twice

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Regular graphs and homogeneous mixing

Regular graphA graph is called d-regular if each vertex has degree d.

Homogeneous mixingAssumption lead to R0 = µ · T. Assumes that contact network isd-regular with d ≈ µ

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Constructing regular graphs

Can you make a 3-regular graph with 5 vertices?7?No! Must have 3 = 2·# edges

# vertices

Must d ·# vertices is even!

QuestionCan you make a 4-regular graph with 6 vertices?

Arrange vertices in circle

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Constructing regular graphs

Can you make a 3-regular graph with 5 vertices?7?No! Must have 3 = 2·# edges

# vertices

Must d ·# vertices is even!

QuestionCan you make a 4-regular graph with 6 vertices?

Arrange vertices in circleJoin each vertex with 2 nearest on each side

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Constructing regular graphs

Can you make a 3-regular graph with 5 vertices?7?No! Must have 3 = 2·# edges

# vertices

Must d ·# vertices is even!

QuestionCan you make a 4-regular graph with 6 vertices?

Arrange vertices in circleJoin each vertex with 2 nearest on each side

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Constructing regular graphs

QuestionCan you make a 5-regular graph with 8 vertices?

Arrange vertices in circleJoin each vertex with 2 nearest on each side

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Constructing regular graphs

QuestionCan you make a 5-regular graph with 8 vertices?

Arrange vertices in circleJoin each vertex with 2 nearest on each sideJoin opposite vertices

ObservationCan use these constructions to make d-regular graph with Vvertices, assuming d · V is even and V is large enough

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Realistic degree distributions

Simplest model assumes regular contact networkSaw that even simple contact networks are far from regular

QuestionWhat are the degree distributions of realistic contact networks?

Number of daily CPIs in UShigh schoolBlue = students, pink =teacher, green = sta�What other statistical ideasmight be useful here?

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Standard deviation

Standard deviation σ measure of ‘width’ of distribution

σ2 = (0−µ)2p0+(1−µ)2p1+(2−µ)2p2+ · · · =∑d(d−µ)2pd

σ2 = Average of (distance from µ)2

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Computing standard deviation

Formulaσ2 = (0− µ)2p0 + (1− µ)2p1 + (2− µ)2p2 + · · · =

∑d(d− µ)2pd

µ = 4I σ2 = (4− 4)2 55 = 0

µ = 127

I σ2 =(1− 12

7

)267 +

(6− 12

7

)217 =

15049 = 3.06...

µ = 207

I σ2 =(2− 20

7

)227 +

(3− 20

7

)247 +

(4− 20

7

)217 =

2049 = 0.408...

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Reproductive rate v2

Better model for R0Correction factor accounting for width

R0 = Tµ(1+ σ2

µ2

)

‘Household’ network: σ2

µ2 =042 = 0

I For regular graphs σ = 0 so simple = better model

‘Doctor’ network: σ2

µ2 =(15049

)2 ( 712)2

= 625196 = 3.19...

I Better model gives R0 4 times larger!

‘Friends’ network: σ2

µ2 =(2049

)2 ( 720)2

= 149 = 0.02...

I Better model is only slightly larger than simple model

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Reproductive rate v2

σ2

µ2 = 0.118Better model for R0 isalmost 12% higher

Takeaway messages

Since σ2

µ2 ≥ 0, simple R0 ≤ better R0Wider degree distribution⇒ faster spread of disease

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How popular are your friends?

How to make a random graph

1. Create vertices with chosendegrees

2. Connect the edges at random

TheoremOn average, a contact has µ

(1+ σ2

µ2

)contacts

⇒ since ≥ µ, your friends have more friends than you!

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Proof

Question:What is the probability that an edge meets a vertex of degree d?

Answer:

#edges meeting degree d2E =

d ·#vertices of degree d2E

=

(d

2E/V

)(#vertices of degree d

V

)=d · pdµ

The average degree of a vertex meeting a random edge is∑dd ·(d · pdµ

)=1µ

∑dd2 · pd

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Proof

Observe that,

σ2 =∑d(d− µ)2pd =

∑dd2 · pd − 2µ

∑dd · pd + µ2

∑dpd

=∑dd2 · pd − µ2

Therefore,

µ

(1+ σ2

µ2

)=µ2 + σ2

µ=1µ

∑dd2 · pd

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Final words

Reproductive rate R0 determinesspread of diseaseGood model: R0 = T

[µ(1+ σ2

µ2

)]I Second factor is average degreeof a neighbor

To �atten the curve need to alsoreduce ‘width’ of contact networkI e.g., ‘super-spreaders’, largegatherings, schools

Thank you very much for your attention!

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