Who Do You Know?A Simulation Study of Infectious Disease

Control Through Contact Tracing

Benjamin Armbruster and Margaret L. Brandeau

Stanford University

Contact Tracing

Health care provider’s perspective:1. Infected person found (index case)2. Treated3. Asked for list of contacts4. Contacts found and tested5. If contact infected go to step 1.

• Standard practice for Tuberculosis (TB)• Common for HIV and other STDs

– called partner notification

1

54

susceptible

infected

? being tested

2

76

3

node 2 infected

Contact Tracing

5

1

4

susceptible

infected

? being tested

2

76

3

node 2 infects nodes 1,4,5

Contact Tracing

5

1

4

susceptible

infected

? being tested

2

76

3

node 4 gets tested (maybe has symptoms)

?

Contact Tracing

5

1

4

susceptible

infected

? being tested

2

76

3

node 4• tests positive, gets treated• becomes a contact tracing index case• names nodes 1,2,3,6,7 as contacts

nodes 1,2,3,6,7 scheduled to be tested

?

??

??

Contact Tracing

5

1

4

susceptible

infected

? being tested

2

76

3

node 3 tests negative

??

??

Contact Tracing

5

1

4

susceptible

infected

? being tested

2

76

3

node 6 tests negative

?

??

Contact Tracing

5

1

4

susceptible

infected

? being tested

2

76

3

?

??

node 1• tests positive, gets treated• becomes a contact tracing index case• names nodes 2,4 as contacts

node 4 already testedtesting node 2 gets higher priority as named by both nodes 1,4

Contact Tracing

2

5

1

4

susceptible

infected

? being tested

76

3

?

?

node 2• tests positive, gets treated• becomes a contact tracing index case• names nodes 1,4,5 as contacts

nodes 1,4 already testednode 5 scheduled to be tested

Contact Tracing

2

5

1

4

susceptible

infected

? being tested

76

3

?node 5• tests positive, gets treated• becomes a contact tracing index case• names node 2 as a contact

node 2 already tested

Contact Tracing

2

5

1

4

susceptible

infected

? being tested

76

3

node 7 tests negative

Contact Tracing

Outline

• Details of the simulation• What is the optimal contact tracing policy?• Does spending more money help?

• Conclusions• Limitations and future work

Simulation Dynamics

t1

90 dayst2

30 dayst3

90 dayst4,t

55 days

η 1/9000 new cases/day/person

di = number of infected neighbors of node i

IT

RST

S

S

S

I

S

SST

S

S

I

I

R

S

Network

A small-word graph with n nodes:

1. Create nodes 0,...,n-1

2. Connect node i to i±1,2 (mod n)

3. Add link (i,j) with probability 1/n

n=500 people

Simulation Details

Each prevalence data point is the• Average over many (400-1600) runs• Error bars show 95% confidence interval

• Steady-state prevalence in a run:1.Infect a random node2.Simulate for 5 years3.Average the prevalence

at the end of day 181, day 182, ...

Contact Tracing Policy

• Budget B

Currently tracing1. Tutankhamen2. Frederic Chopin3. Eleanor Roosevelt…B empty slot

5

1

4

2

7

6

3

?

?

?

? ?

9

8

10

5

9

8

Contact Tracing Policy

• Budget B• New index case: choose

k contacts to trace

1

4

2

7

6

3

10

10

5

9

8

Contact Tracing Policy

• Budget B• New index case: choose

k contacts to trace

1

4

2

7

6

3

Once we have time1. Niels Abel2. node 23. node 34. node 75. …

Contact Tracing Policy

• Budget B• New index case: choose k

contacts to trace• If we have resources, then

trace from top of list

Currently tracing1. Tutankhamen2. Frederic Chopin3. Eleanor Roosevelt…B empty slot

Once we have time1. Niels Abel2. node 23. node 34. node 75. …

Contact Tracing Policy

• Budget B• New index case: choose k

contacts to trace• If we have resources, then

trace from top of list

Currently tracing1. Tutankhamen2. Frederic Chopin3. Eleanor Roosevelt…B Niels Abel

Once we have time1. node 22. node 33. node 74. …

Nothing Random Most Named List Length

0.01

0.015

0.02

0.025

0.03

0.035

prev

alen

ce

k=5B=8

Which Policy is Best?

0 2 4 6 80

0.005

0.01

0.015

0.02

0.025

0.03

prev

alen

ce

k

How Many Contacts to Trace per Index Case?

Increasing the Budget

0 2 4 6 8 100.01

0.015

0.02

0.025

0.03

0.035

prev

alen

ce

budget

0 10 20 30 40 50 60

0

5

10

15

budget [$1000/year]

pers

ons

infe

cted

on

aver

age

What is the Best Level of Contact Tracing?

α=$10,000 n = 500

Choosing α

Resource allocation problem– total budget Btotal

– budget bi for program is.t. b1+ · · · +bm ≤ Btotal

– benefit fi(bi)

max f1(b1)+ · · · +fm(bm)

s.t. b1+ · · · +bm ≤ Btotal

Cost Effectiveness

Resource allocation problem– total budget Btotal

– budget bi for program is.t. b1+ · · · +bm ≤ Btotal

– benefit fi(bi)

– cost effectiveness 1 / fi′(bi)

min maxi 1 / fi′(bi)

s.t. b1+ · · · +bm ≤ Btotal

α=

0 10 20 30 40 50 600

5

10

15

budget [$1000/year]

pers

ons

infe

cted

on

aver

age

Budget Allocation

1. Choose α large2. For each program

– find bi s.t. α = 1 / fi′(bi)– if doesn't exist, then set bi =0

3. Calculate money spent b1+ · · · +bm

4. If >Btotal, then 1. decrease α a bit2. go to step 2

5. Else at optimal allocation

Conclusions

• First detailed model of contact tracing

• Found a better prioritization of contacts

• Diminishing returns to scale

• Cost-effectiveness should play a role when choosing a budget

Limitations / Future Work• Dynamics are missing

– a latent or asymptomatic stage– the male-female distinction– variety of exogenous infections

• Network stylized:

– pairs, pair-formation needed for HIV / STDs• Policy gives no priority to vulnerable contacts• Genotype information• Dynamic control