1. Epidemiology

2. Epidemiology Infectious disease spread: flu, tuberculosis
Biological terrorism
How to contain epidemics?

3. Epidemiology Many potential interventions:
Immunization/vaccination
Public awareness campaigns
Closing facilities
Tracing contacts of infected people
Limited resources: how to target most effectively?

4. Models Describe how disease spreads through the population
Lead to different optimization problems later

5. SIS models Most basic class of models, widely studied
Population can be in 2 studies:
Susceptible
Infected

6. SIS models Most basic class of models, widely studied
Population can be in 2 studies:
Susceptible
Infected
𝑑𝐼 𝑑𝑡 =𝑆 𝛽 𝐼 𝐼+𝑆 −𝜈𝐼

7. SIS models Most basic class of models, widely studied
Population can be in 2 studies:
Susceptible
Infected
𝑑𝐼 𝑑𝑡 =𝑆 𝛽 𝐼 𝐼+𝑆 −𝜈𝐼
Rate of contact between agents

8. SIS models Most basic class of models, widely studied
Population can be in 2 studies:
Susceptible
Infected
𝑑𝐼 𝑑𝑡 =𝑆 𝛽 𝐼 𝐼+𝑆 −𝜈𝐼
Rate of contact between agents
Fraction infected

9. SIS models Most basic class of models, widely studied
Population can be in 2 studies:
Susceptible
Infected
𝑑𝐼 𝑑𝑡 =𝑆 𝛽 𝐼 𝐼+𝑆 −𝜈𝐼
Rate of contact between agents
Clearance rate
Fraction infected

10. SIS models Most basic class of models, widely studied
Population can be in 2 studies:
Susceptible
Infected
𝑑𝐼 𝑑𝑡 =𝑆 𝛽 𝐼 𝐼+𝑆 −𝜈𝐼
Rate of contact between agents
Clearance rate
Fraction infected
Sometimes add a third “recovered” state when cured agents are immune

11. Models Most models build on the basic SIS framework
Aim to capture more realistic population structure
Usually discrete time
Sometimes makes analytical characterizations of model behavior harder
But much nicer algorithmically

12. Graph-based models Population of agents who comprise nodes of a graph
Each node can be susceptible or infected
Disease spreads between neighbors
Often similar to ICM – infect a neighbor with probability 𝑝 𝑢,𝑣

13. Graph-based models Usual assumption: graph/population is static
Good approximation for “fast” diseases like flu

14. Graph-based models Usual assumption: graph/population is static
Good approximation for “fast” diseases like flu

15. Graph-based models Algorithmic problems:
Immunize 𝑘 nodes (can never become infected)
Cure 𝑘 nodes (return to susceptible state)
Remove edges

16. Graph-based models Usual goal: drive process below critical threshold
Graph models often exhibit phase transition
Sufficiently high infection rate: (almost) entire population gets infected
Sufficiently low infection rate: disease becomes extinct

17. Graph-based models
Formally: long-run behavior depends spectrum of the adjacency matrix
Let 𝜌(𝐺) be the spectral radius of 𝐺 (largest eigenvalue)
Theorem [Ganesh et al]: There is a threshold 𝜏 (depending on model parameters) such that if 𝜌 𝐺 <𝜏, the disease becomes extinct in 𝑂( log 𝑛 ) steps.

18. Graph-based models
Formally: long-run behavior depends spectrum of the adjacency matrix
Let 𝜌(𝐺) be the spectral radius of 𝐺 (largest eigenvalue)
Theorem [Ganesh et al. 2005]: There is a threshold 𝜏 (depending on model parameters) such that if 𝜌 𝐺 <𝜏, the disease becomes extinct in 𝑂( log 𝑛 ) steps.

19. Graph-based algorithms
Common optimization problems (immunizing nodes, removing edges) are NP-hard [Tong et al 2012]

20. Graph-based algorithms
Common optimization problems (immunizing nodes, removing edges) are NP-hard [Tong et al 2012]
Proposed approaches often define a surrogate function to greedily optimize
Rank edges by product of the eigenvector entries of their endpoints [Tong et al 2012]
Greedily remove edges to minimize # of closed walks [Saha et al 2015]
Some approximation guarantees for these methods

21. Generalizing the model
Real-world epidemics don’t neatly fit the graph model
Many agents, interacting in different kinds of ways, with population changing over time
Here: two ways of expanding it
Jointly consider node immunizations and facility closures
More realistic population-based models

22. Joint individual/facility decisions
Introduced in [Deng, Shen, Vorobeychik 2013]
Set of individuals I and facilities F
Bipartite graph describing each individual’s probability of visiting each facility
Infected agents can infect susceptibles at the same facility

23. Joint individual/facility decisions
Each individual starts out infected w.p. ℎ 𝑖
Visits location 𝑗 w.p. 𝑝 𝑖𝑗
Probability location 𝑗 has infection: 1 − 𝑖∈𝐼 𝑝 𝑖𝑗 ℎ 𝑖

24. Interventions
Vaccinate individual: reduces probability they contract disease if exposed
Close facility: no one can get infected there
Also consider compensatory model: agents choose a different place to visit

25. Interventions
Vaccinate individual: reduces probability they contract disease if exposed
Close facility: no one can get infected there
Also consider compensatory model: agents choose a different place to visit

26. Optimization 𝑥 𝑗 ∈ 0,1 : Close facility j?
𝑧 𝑖 ∈{0,1}: Vaccinate individual i?
Vaccine: contract w.p. 𝑟 𝑉 , otherwise 𝑟 𝑁𝑉

27. Optimization 𝑥 𝑗 ∈ 0,1 : Close facility j?
𝑧 𝑖 ∈{0,1}: Vaccinate individual i?
Vaccine: contract w.p. 𝑟 𝑉 , otherwise 𝑟 𝑁𝑉
min 𝑗∈𝐹 Pr⁡[𝑗 infected] 1 − 𝑥 𝑗 𝑖∈𝐼 𝑝 𝑖𝑗 1 − ℎ 𝑖 (𝑧 𝑖 𝑟 𝑣 + 1− 𝑧 𝑖 𝑟 𝑁𝑉 )

28. Optimization 𝑥 𝑗 ∈ 0,1 : Close facility j?
𝑧 𝑖 ∈{0,1}: Vaccinate individual i?
Vaccine: contract w.p. 𝑟 𝑉 , otherwise 𝑟 𝑁𝑉
min 𝑗∈𝐹 Pr⁡[𝑗 infected] 1 − 𝑥 𝑗 𝑖∈𝐼 𝑝 𝑖𝑗 1 − ℎ 𝑖 (𝑧 𝑖 𝑟 𝑣 + 1− 𝑧 𝑖 𝑟 𝑁𝑉 )
Probability facility j is infected

29. Optimization 𝑥 𝑗 ∈ 0,1 : Close facility j?
𝑧 𝑖 ∈{0,1}: Vaccinate individual i?
Vaccine: contract w.p. 𝑟 𝑉 , otherwise 𝑟 𝑁𝑉
min 𝑗∈𝐹 Pr⁡[𝑗 infected] 1 − 𝑥 𝑗 𝑖∈𝐼 𝑝 𝑖𝑗 1 − ℎ 𝑖 (𝑧 𝑖 𝑟 𝑣 + 1− 𝑧 𝑖 𝑟 𝑁𝑉 )
Probability facility j is infected
Probability individual i visits j, was not infected to start with, and becomes newly infected

30. Optimization Linearize the 𝑥 𝑗 𝑧 𝑖 terms by adding auxiliary variables
Mixed integer linear program!
Also: more efficient greedy heuristic and exact dynamic programming algorithm

31. Population-based models
Motivation: reconcile epidemiological and CS modeling approaches
Epidemiology [White et al 2005, Chan et al 2011, Dowdy et al 2012]: realistic models, often complex, hard to optimize
Computer science: more abstract, but well-characterized analytically and amenable to optimization
Aim: a model which includes “realistic” population dynamics but admits principled optimization approach
Here: MCF-SIS model [W, Suen, Tambe 2018]

32. MCF-SIS

33. Segmented population
Age 0-15
Age 45-60
Age 30-45
Age 15-30
Age 60+

34. Birth, death, aging Birth, death, aging Age 0-15 Age 45-60 Age 30-45

35. Disease spread
Age 0-15
Age 45-60
Age 30-45
Age 15-30
Age 60+

36. Optimization problem Policymaker gets to control the cure rate 𝜈
Pre-campaign: 𝜈=𝐿
Policymaker conducts campaign, targeting selected groups
Post-campaign: any feasible 𝜈
𝜈−𝐿 1 ≤𝐾 (total budget of 𝐾)
𝐿 𝑖 ≤𝜈 𝑖 ≤ 𝑈 𝑖

37. Optimization problem
Let 𝐹 𝜈 denote total infected agents summed over time t = 1…T
min 𝜈 𝐹(𝜈)
𝜈−𝐿 1 ≤𝐾
𝐿 𝑖 ≤𝜈 𝑖 ≤ 𝑈 𝑖

38. Challenges Can’t just target the groups with the most infected agents
Maybe demographics cause more between-group spread

39. Challenges Can’t just target the groups with the most infected agents
Maybe demographics cause more between-group spread
Also can’t just look at contact patterns
Demographics shape future population
If you want to cure age 30, may need to start targeting at age 27

40. Challenges Can’t just target the groups with the most infected agents
Maybe demographics cause more between-group spread
Also can’t just look at contact patterns
Demographics shape future population
If you want to cure age 30, may need to start targeting at age 27
Many parameters will not be known in practice
Initial prevalence 𝐼 0
Contact pattern 𝛽
We’ll come back to this

41. What to do? Notice: treatment resource have diminishing returns
If increasing 𝜈 𝑖 averts some infections, those can’t be averted by increasing 𝜈 𝑗

42. Submodularity
Normally a property of set functions with diminishing returns
Greedy algorithm etc.
Not as well known: submodularity for continuous functions
𝑓 𝐵∪ 𝑣 −𝑓 𝐵 ≤𝑓(𝐴∪ {𝑣}) −𝑓(𝐴) ∀ 𝐴⊆𝐵
𝜕 2 𝐹 𝜕 𝑥 𝑖 𝑥 𝑗 ≤ 𝑖, 𝑗=1…𝑛

43. Theorem: Minimizing infection in the MCF-SIS model is equivalent to a continuous submodular maximization problem

44. DOMO algorithm Frank-Wolfe approach [Bian et al. 2017]
Serious of iterations 1…R
Maintain a feasible solution at each iteration
At each iteration, take a small step towards feasible point furthest in direction of gradient
𝜈 0 =𝐿
For k = 1…R:
𝑦 𝑘 = arg max 𝑦, 𝛻𝐹( 𝜈 𝑘−1 )
𝜈 𝑘 = 𝜈 𝑘− 𝑅 𝑦 𝑘

45. Theorem: DOMO produces a 1 − 1 e − 𝜖 approximate solution using 𝑂 𝐾 𝑇 2 𝜖 iterations. Each iteration can be implemented in time 𝑂 𝑇 𝑛 𝜔 , where 𝜔 is the matrix multiplication constant.
Proof: General analysis of Frank-Wolfe for continuous submodular functions [Bian et al 2017] + domain-specific bound for convergence rate.

46. Stochastic problem min 𝜈 𝐸 𝜉∼𝐷 𝐹(𝜈, 𝜉)
Many of those parameters won’t actually be known exactly
Let Ξ be an uncertainty set for their joint values, with distribution 𝐷
min 𝜈 𝐸 𝜉∼𝐷 𝐹(𝜈, 𝜉)
No closed form access to objective or gradient – previous algorithm doesn’t work anymore!

47. Stochastic problem Provide a stochastic extension to DOMO
Key idea: only need access to gradient
Replace exact gradient with stochastic approximation:
Only need sample access! Arbitrary distributions OK
𝛻 = 𝑖=1 𝑟 𝛻𝐹(𝜈, 𝜉 𝑖 ) , 𝜉 1 … 𝜉 𝑟 ∼𝐷

48. Theorem: In the stochastic setting, DOMO provides a 1 − 1 𝑒 −𝜖 approximation using the same number of iterations and 𝑂 𝐾 2 𝑇 2 𝜖 gradient samples per iteration.
Generalizes to any smooth, continuous submodular function!

49. Evaluation: TB in India
Model parameters fit from variety of data sources (Indian government reports, U.N., epidemiological literature…)
But still substantial uncertainty
Contact patterns 𝛽 not known
Initial infected prevalence 𝐼 0 very uncertain
Many patients do not report to approved treatment facilities

50. Side note: disease data sources
Often available at aggregate level
India RNTCP/WHO: estimates of TB incidence by year, typically with some age segmentation
US reportable diseases (e.g. flu): usually available by week, segmented by either state or by age but not both
In-between space: some amount of data is available, but not detailed enough to directly fit complicated models

51. Evaluation: TB in India
For 𝐼 0 : assume a Gaussian distribution within confidence intervals
For 𝛽: find matrix minimizing MSE between MCF-SIS predictions and observations

52. Baselines Compare to an array of baseline approaches
degree: spend budget on groups with highest degree in 𝛽
eigen: highest eigenvector centrality in 𝛽
prevalence: allocate to groups with most infected agents
equal: split budget equally
SQ: split budget proportional to 𝜈 produced by status quo policies

53. Improvement in person-years of TB

54. Future work
Collaboration with IIT Guwahati and state government of Assam, India
Use machine learning to predict risk of default from treatment
Optimize interventions to increase treatment completion rate