1. Disease signatures – a simple combinatorial-type exploitation of them for our own evil purposes
Prof. Nina H. Fefferman
Visiting DIMACS from :
Tufts Univ. School of Medicine, Dept. Public Health and Family Medicine

2. Plan for today: Looking very quickly at traditional SIR models
Communication problems
Tweaking parameter definitions
Using these definitions to clear up communication
Building disease signatures
Decomposing reported disease into component signature curves
Checking this method against reality
Where this method can take us from here…

3. A quick look at SIR models
I(t) = number of infected
S(t) = number of susceptibles
R(t) = number of recovered
in the population at time t       
And if we want spatial spread :                                              
                  
          
Keep R, but I(t, x, y) and S(t, x, y) become functions of position (x, y), and a is replaced by an expression involving two other constants related to the rate at which the infection diffuses through space
Go ask HHS or NIH or CDC for a and b for the next flu season so our models can predict it.
Good luck.
Pictures of equations stolen from :

4. Leads us to : Communication Problems
Parameters/Variables used by epidemiologists are warm and fuzzy and not rigorously defined
So modelers made up their own (you just saw them) – these aren’t things doctors/public health people can really measure  we can’t get accurate parameter values
Example: MANY people are worried about outbreaks
There is no good definition of what constitutes an outbreak
BIG problem (mostly just ignored)
Modelers use the concept of R0 – the reproductive number of disease (in the differential equation model, it’s the ratio of S to a/b)
It’s when the average number of new infections caused by contact with a current infection is greater than 1

5. How can we say this mathematically?
Communication Problems cont.
R0 gives us a rigorous definition of something good, but not of what we really need ‘outbreak’ to mean
Really, if we think about it, public health people want ‘outbreak’ to refer to “times when we need to pay attention to disease spread for some reason”
How can we say this mathematically?

6. What can public health people/ doctors measure (at least sometimes)?
Communication Problems cont.
What can public health people/ doctors measure (at least sometimes)?
Infectivity : Probability of becoming infectious after becoming exposed
Attack rate : Probability of developing disease after becoming exposed
Pathogenicity : Probability of developing disease after becoming infected
Virulence : Probability of dying after becoming ill
Immunogenicity : Attack rate for re-exposure

7. So : Tweaking Parameter Definitions
E(X,T)= Probability of exposure in population X at time T
I = Probability of infection from exposure
ST = Probability that infection at time 0 leads to manifestation of symptoms at time T (a distribution function which does not need to sum to one if not all of the infected develop symptoms)
Really, these are all functions of time, but my journal referees got upset with functions, so most are now subscripts
CT = Probability that infection takes T days to become contagion
MT = Probability that the time from the onset of symptoms to death from the disease is T days
NT = Size of the population possibly exposed to infection on day T (this will be our disease signature curve)
IT = Probability of infection from current exposure, given previous infection T days ago

8. With those we can build :
Clearing up communication
With those we can build :
Pathogenicity : The probability of developing disease after becoming infected
= ST , for n the maximum recovery time n
T=0 
Virulence : The probability of dying after becoming ill = MT , for n the maximum recovery time n
T=0 
Infectivity : The probability of becoming infectious after becoming exposed
= I* CT , for n the end of the window for the disease n
T=0 

9. Clearing up communication cont.
And :
Attack rate : The probability of developing disease after becoming exposed
= I * ST , for n the end of the window for disease expression n
T=0 
But now we notice that, from our original list, Immunogenicity is not a truly meaningful idea, so we define instead:
PsuedoImmunogenicity : Probability of infection from current exposure, given previous infection T days ago = IT
We won’t be using all of these today, but they’re still useful to have if you ever need to talk to health people

10. Now both the math and health people have the same picture!
Clearing up communication cont.
Uses a slightly different notation
Now both the math and health people have the same picture!
But this is only one town
The SIR models could handle spatial spread with PDEs…

11. With multiple locations and central reporting :
Clearing up communication cont.
With multiple locations and central reporting : ?

12. Let’s call these narrower things subpopulations
Clearing up communication cont.
Notice : different occurrences don’t have to be separated only spatially or temporally
Can be different demographic populations, or anything that allows narrower, more accurate estimations of exposure or susceptibility
Let’s call these narrower things subpopulations

13. So, using our definitions and our flow chart:
Building Disease Signatures
So, using our definitions and our flow chart:
For a given subpopulation, we can compute a ‘disease signature curve’ representing the number of cases predicted over time from a single instance of exposure
Notice : these signature curves depend on subpopulation-specific etiology, including the shape of the distribution for some parameters – not just averages

14. Decomposing curves into signatures
So, if we have a total reported disease curve, we can iteratively define
(Notice populations exposed on different days are disjoint sets due to the definitions)
Now we can think of a single reported curve CT as the composition of these curves

15. Yielding the total disease incidence curve:
Decomposing curves into signatures cont.
Since we are interested in exploiting the heterogeneity of etiological response within a diverse population, we can specify these curves by subpopulation Y:
Yielding the total disease incidence curve:

16. Decomposing curves into signatures cont.
And we can even exploit immune memory by further dividing subpopulations into classes of those with similar immune protection from previous infection
With IT = Probability of infection given previous infection T days ago
And T* = the last day of most recent prior infection
Giving us

17. How many different combinations of coins can make $1.50…
Decomposing curves into signatures cont.
Important because public health people may trust it
Now we can use high school math to find combinations of signature curves that make up the total reported cases curve!
How many different combinations of coins can make $1.50…
10¢ 5¢
25¢
Coins
Sub-Populations
Similarly, we can ask how many combinations of ‘signature curves’ can go into a ‘Total Reported Cases’ curve:

18. Now let’s come back to the idea of an outbreak:
Decomposing curves into signatures cont.
Now let’s come back to the idea of an outbreak:
Remember, we wanted ‘outbreak’ to mean “times when we need to pay attention to disease spread for some reason”
Suppose that the only combination of disease signature curves was to have EVERY subpopulation just beginning to show symptoms from a disease – that means that soon many many more people will be sick – we should probably pay attention to that OR
Maybe the only combination of signature curves indicates that only one location has been exposed – we might want to use that to find out what the source of exposure was, or quarantine the area
No matter how we choose to define it (will be arbitrary), this method can tell us WHY we should care now

19. There was an actual “outbreak” in MA in 1995
Decomposing curves into signatures cont.
Let’s take a look at an example of how this can work
To begin with, let’s look at something very simple :
Giardiasis – a waterborne infection causing diarrheal disease in humans with extremely low levels of secondary transmission (makes life simpler)
There was an actual “outbreak” in MA in 1995

20. Decomposing curves into signatures cont.
Reported incidence for MA (all of it)
HIPPA requires aggregation of data released to public and to most researchers without special access

21. Decomposing curves into signatures cont.

22. To use this method, we need some measured parameter values
Decomposing curves into signatures cont.
To use this method, we need some measured parameter values
I’m cheating a little because I’m assuming values for I, but we could in theory measure this

23. We know that most of the reporting came from 3 urban centers:
Decomposing curves into signatures cont.
We know that most of the reporting came from 3 urban centers:

24. Then we can decompose by demographic subgroup for each town:
Decomposing curves into signatures cont.
Then we can decompose by demographic subgroup for each town:

25. It gets MUCH more complicated…
Decomposing curves into signatures cont.
That was a really simple disease without any secondary transmission
So what happens if there is secondary spread?
It gets MUCH more complicated…
First of all, the probability of exposure in each subpopulation can start to depend on the levels of infection in each other subpopulation
Now we start getting into the social network stuff

26. Social Networks : Oy vey
An aside
Social Networks : Oy vey
Since this is a talk and not a course, I can’t leave this as an exercise to the reader, but I can use the ‘we only have a little over an hour’ excuse to hand-wave some of the modeling details on this – I’m going to talk about the concepts
If you are interested in the details, well, that’s why I’m going to be around for the year

27. Again, rather than using mass averages, let’s still keep the idea of a disease signature
So exposure isn’t a simple underlying rate - it’s based on contacting an infected individual
We can think of individuals in each subgroup as having certain probabilities of interacting with others, possibly in other subgroups
(People in the room who think of social interactions as edges in a graph, this is almost the same - it’s like weighted edges in a complete graph)
Also, membership in particular subgroups can changes over time (e.g. children becoming adults)
(In this case, both vertex states and edge weights can be thought of as vertex-state dependent progressions)

28. Checking Reality
This all gets complicated enough that it’s nerve wracking not to check model outcomes against some form of reality
Need to :
measure all model parameters
create disease outbreaks
check predicted spread against what actually happens
(I tried to get Thus Spake Zarathustra to play now, but I couldn’t make it work)
My beautiful termites

29. The particular details: Temporary Immunity
Checking Reality cont.
On Thursday, at the DIMACS Mixer, I’ll be talking to you about ‘Why Termites’
For now, just go with it
The particular details:
Temporary Immunity
Zootermopsis angusticollis
Allogroomed off
Spores land on termite
Burrow through cuticle
Not a termite
Metarhizium anisopliae
Death

30. So we built some CA simulation models
Checking Reality cont.
So we built some CA simulation models
Including age-based differences in :
direction of wandering through nest
interaction rates
exposure rates
susceptibility to infection from exposure
mortality from infection
efficacy/duration of induced immunity (via social vaccination)
As the model ran, individuals aged and behaved accordingly

31. We’re even getting some interesting new directions
Checking Reality cont.
And…
Thank god, all the work so far has shown that the models predict spread accurately
Whew!
We’re even getting some interesting new directions

32. Now that we know the model can work, we can work backwards
Regardless of why specific outputs happen
Now that we know the model can work, we can work backwards
Fit model outcome to observed data and look at which sets of parameter values and behavioral mixing rates produce them
This might provide an odd way of understanding human social networks – especially since they can so dramatically affect model output
Maybe this last part is a pipe-dream.
Who knows, but it’s so crazy it just might work…

33. Thanks for asking me to speak to you I hope you’ve had fun
Some of what I’ve talked about has been accomplished in collaboration with Elena Naumova, James Traniello and Rebeca Rosengaus
My thanks to the NIH for funding support for this research