1. Why are Epidemics so Unpredictable? Duncan Watts, Roby Muhamad, Daniel Medina, Peter Dodds Columbia University

2. An Obvious Question Whenever a novel outbreak of infectious disease is announced (SARS, Avian Influenza, Ebola, etc.), one of the most pressing questions is: “How big will it get”? One could also ask an analogous question for existing epidemics (HIV, TB, Malaria) Amazingly, mathematical epidemiology currently has no way to answer these questions

3. Mathematical Epidemiology Started with Daniel Bernoulli’s analysis of smallpox epidemic (1760’s) Has been developed extensively since late 1920’s (Kermack and McKendrick) Now hundreds of models deal with many variations of human, animal, and plant diseases Incredible diversity of models, which can be extremely complex, but most are variants of the original

4. The Standard (SIR) Model (1) Individuals cycle between three states: Susceptible; Infected; and Removed (2) Mixing is uniformly random “Mass Action” assumption   r

5. Basic Reproduction Number R 0 Mass Action assumption means that epidemics depend only total fraction of infectives (I) and susceptibles (S) Condition for an epidemic is simple: R 0 > 1 R 0 is the “Basic Reproduction Number” Average number of new infectives generated by a single infected individual in a susceptible population R 0 =  S depends on Infectiousness of disease (  Recovery rate (  Density of susceptibles near outbreak (S) Preventing an epidemic thus becomes equivalent to keeping R 0 < 1

6. Standard models imply outbreaks are “bi-modal” When R 0 < 1 Epidemics never occur When R 0 > 1, Only one of two outcomes possible: Outbreak fails to achieve “epidemic” status (left peak) Outbreak becomes a full-fledged epidemic, infecting a significant fraction of the entire population (right peak)

7. Epidemic size should therefore be predictable Together, R 0 and N (population size) completely Determine the expected number of cases.

8. Also, epidemics should “peak” only once

9. Result: Epidemics should follow classic “logistic” curve

10. Differ dramatically in size 1918-19 “Spanish Flu” ~ 500,000 deaths in US (20- 80 Million world-wide) 1957-58 “Asian Flu” ~ 70,000 deaths in US 1968-69 “Hong Kong Flu” ~ 34,000 deaths in US 2003 SARS Epidemic ~ 800 deaths world-wide All these diseases have about same R 0 ! How different in size can epidemics of seemingly similar diseases be? Unfortunately, historical data on large epidemics is hard to collect. Thus true size distributions are unknown Real Epidemics, However…

11. Seem to be “multi-modal” Size distribution of epidemics for (A) measles and (B) pertussis (whooping cough) in Iceland, 1888-1990

12. Real Epidemics also“Resurgent” Global Daily Case Load for 2003 SARS Epidemic: Epidemic had several peaks, interspersed with lulls

13. Result is unpredictability Multi-modal size distributions imply that any given outbreak of the same disease can have dramatically different outcomes Resurgence implies that even epidemics which seem to be burning out can regenerate themselves by invading new populations

14. What makes epidemics unpredictable? Key insight from the literature on social networks: populations exhibit structure What kind of structure? Inhomogeneous population distribution Transportation and infrastructure networks Social, Organizational, and Sexual Networks Result is that Uniform mixing occurs only in small, relatively confined contexts (where standard model applies) Large epidemics are not single events: they are concatenations of many, small epidemics

15. Influenza Pandemic, 1957

16. How do network models help? Last 20 years has seen rapid growth in “network epidemiology” In principal, tremendously appealing Problem is that in a SARS-like epidemic, many kinds of networks can potentially matter Social, organizational, infrastructural, transport Result is both empirically and analytically intractable What to include and what to exclude? How to estimate parameters? How to balance realism with complexity?

17. Compromise between realism and complexity Assume mass-action assumption holds (approximately) in “local contexts” (schools, hospitals, apartment buildings, villages, etc.) Then incorporate main insight of population structure: Local contexts are embedded in a series of successively larger contexts (neighborhoods, cities, counties, states, regions, countries, continents…) Global populations are “nested” Result is a “multiscale metapopulation model” Individuals can “escape” current local context and move to another (with probability p) Once escaped, move to another local context, chosen from contexts at characteristic length 

18. Multiscale metapopulation model Incorporates essence of population structure while remaining simple

19. Previous metapopulation models Metapopulation models hardly new idea Some very detailed models worked out using airline network data (Longini), and even detailed maps of individual cities (Halloran et al.; Eubanks et al.) These models all restricted to two scales (local and global) Multiscale models have been advocated previously (Bailey; Cliff and Haggett; Ferguson et al.) but not formally specified Technically, extension to multiple scales is trivial, but consequential nonetheless

20. Difference with network models Critical feature of biological disease (in contrast with information spread) is that individuals must be physically co-located Thus i cannot infect j 1 and j 2 sequentially unless j 1 and j 2 are also co-located. Very different from “small world” and other network models in which individuals can sustain multiple “long-range” contacts simultaneously Subtle difference, but turns out to be critical Network models also generate bimodal distributions

21. What does epidemic size depend on? Average epidemic size vs. P 0 (expected number of infectives “escaping” a local context) When P 0 > 1: Average epidemic size vs.  (“typical” distance traveled) Mobility (R 0 = 3) Range

22. What do these results tell us? Still need R 0 > 1 as necessary condition Everything has to start locally, and locally, mass- action models might be OK But also need some non-local mobility P 0 > 1 and R 0 > 1 are sufficient for “non-local” epidemic (on average -- remember stochastic) Average size of non-local epidemic then depends on transport range (  Average size appears more sensitive to range (  than to volume (P 0 ) of transport Simply restricting range of travel may be an effective intervention strategy E.g. issuing travel advisories

23. When non-local epidemics do occur: Multiscale populations generate “flat” distributions of epidemic sizes Very different outcomes possible for same R 0 Very similar distributions for very different R 0

24. What’s up with R 0 ? R 0 is supposed to be the one number that accounts for “almost everything” It is a necessary condition for an epidemic However, as long as R 0 > 1, the value of R 0 tells us very little about size or duration in a multi-scale world Similar R 0 can lead to very different outcomes Very different R 0 correspond to similar distributions of outcomes Result is many-to-many mapping between R 0 and epidemic size

25. Are we just computing it wrong? If our model were deterministic, we could compute R 0 in terms of the eigenvalues of the inter-group mixing matrix (i.e. in terms of p and  ) However, we would still get just one value In fact, can estimate its value directly Some variance, but very small. The ambiguity is real Method of computation doesn’t eliminate “many to many” mapping problem

26. Where does the variance come from? Problem is that model is not deterministic Stochasticity is well known to cause problems when epidemic is small, even for mass-action models Here, combination of multiple scales and individual transport across them, means that stochasticity continues to matter throughout course of epidemic Thus a few individuals (“rare events”) can have huge impact on size and duration

27. Same disease can have very different trajectories Resurgence driven by “rare events”

28. Importance of Network Thinking Large populations exhibit network structure Social, sexual, infrastructure, transportation Large epidemics need to be understood as many small epidemics linked by networks But taken too literally, network models are a losing proposition Complexity is virtually unlimited Empirical estimation impossible Modeling impossible Need some compromise between tractability and realism

29. Multi-scale metapopulation models Incorporating “multi-scale” structure of the world in epidemic models can explain multi-modality and resurgence of epidemics Knowledge of disease itself (R 0 ) doesn’t help predict size or duration of epidemic Reason is that “rare events” (e.g. one person getting on a plane) can have big consequences Population structure itself can be used as control measure (e.g. travel advisories) See Watts et al. PNAS, 102(32), 11157- 11162 for details