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Can societies be both safe
and efficient?
Different Scales of BioDefense:
Social interactions are key to transmission of infectious disease
Oh dear.
Germs
Societal structure and social organization shape social interactions
Family
Work
Schools
Social Gathering
s
Public Transportatio
n
Hospitals
Most of these are controlled at a societal level
Family
Work
Schools
Social Gathering
s
Public Transportatio
n
Hospitals
But even saying “societal” may be too broad
We’ve actually got a variety of scales:
• individual
• neighborhood
• company
• local
• national
• international
Each scale probably leads to a different robustness goal
So, could there be ways to structure societies to maximize robustness to
disease?
What could the ‘maximal robustness’ goals be?
1. Minimizing the number of infections
2. Minimizing the number of deaths
Or maybe we’re more concerned about societal effects
3. Minimizing the economic costs
4. Minimizing the effect on population growth
5. Minimizing crowding in hospitals
6. Minimizing the compromise of societal infrastructure (keeping a minimum number of people in crucial positions at all times)
Pipe Dream #1:
To build a single model of infectious disease epidemiology that incorporates measures of
each of these effects and, weighting each goal according to our policies/needs, tells us how to re-structure social interactions in a minimally intrusive way that still doesn’t interfere with a functioning society
Ideas welcome
Each of these goals leads us to a different question & (for now) a different modelToday we’ll focus on a model that can be
interpreted to examine both3. Minimizing the economic costs
&6. Minimizing the compromise of societal infrastructure
In previous talks, we’ve discussed a few experiments that focused on
4. Minimizing the effect on population growth
&5. Minimizing crowding in hospitals
If you would like to refresh your memory on those, please talk to me later
Starting on the largest scale:
We got to this point by thinking about social interactions guiding exposure risks, but let’s pull back for a bit and think only about primary exposure
This should let us focus on the efficiency question and then we can add back the layers of complexity for individual secondary exposureWe talked briefly about this work when it was in it’s planning stage
To answer questions about economic and infrastructure efficiency, we need a way to represent costs and benefits and disease
risk
To start with, let’s look at the simplest trade-off system
Yes folks, that’s right…
It’s another termite talk!
Once again, social insects provide all of the crucial facets of social organization without most of the
incredible complexities of humans
• They need to complete a variety of tasks, as a society
• Each task has different associated primary exposure risks
Some Bees
Some Wasps
Ants
Termites
So adorable and so useful!
Age of worker
Amount of ‘work’ in each task completed
in each unit of time
Is the task currently a
limiting factor for
the colony?Disease
risk associated with task
completion
4 Basic elements of concern:
How do they all relate?
In social insects, there are four basic theories for task allocation decisions:
1) Defined permanently by physiological caste
2) Determined by age
3) Repertoire increases with age
4) Completely random
So which does better under what assumptions of pathogen risk?
And can we predict a social organization by what we know about the different pathogen risks of different insects?
Examples of what I mean:
1. We know that some ants are really good at combating pathogens by glandular secretions – Their social organization should be willing to ‘compromise safety’
for greater efficiency since they can handle the risks individually
2. Termites are (comparatively) quite bad at combating pathogen risks – So we would expect that they should sacrifice colony
performance in favor of greater safety
3. Honey bees are differentially susceptible to pathogens based on age – So we might expect an age-specific exploitation of labor
So what do we do:
First we make a basic assumption: that disease risk is a substantial and independent selective pressure, operating
on a population-wide level, during the evolutionary history of social insects
This is probably not a bad assumption, but it doesn’t hurt to
keep in mind that it might not be true
Model formulation – (discrete)
Three basic counterbalancing parameters:
1. Mortality risks for each task Mt
2. Rate of energy production for each task Bt
3. The cost of switching to task t from some other task (either to learn how, or else to get
to where the action is), St
We simulate the following via a stochastic state-dependent Markov process of
successive checks of randomly generated values against threshold values
Notice that we actually can write this in closed form – we don’t need to simulate
anything stochastically to get meaningful results
HOWEVER – part of what we want to see is the range and distribution of the
outcome when we incorporate stochasticity into the process
We have individuals I and tasks (t) in iteration (x), so we write It,x
In each iteration of the Markov process, each individual It,x contributes to some Pt,x the size of the population working on their task (t) in iteration (x) EXCEPT
1) The individual doesn’t contribute if they are dead
2) The individual doesn’t contribute if they are in the ‘learning phase’
They’re in the learning phase if they’ve switched into their
current task (t) for less than St iterations
In each iteration, for each individual in Pt,x there is a probability Mt of dying from task related pathogen exposure and once you die, that’s it, you stay dead
To run the model, for every x, we generate an independent random value [0,1] for each individual in Pt,x and use Mt as a threshold – above survives, below dies
Individuals also die if they exceed a maximum life span (iteration based)
We also replenish the population periodically: every 30 iterations, we add 30 new individuals
This mimics the oviposition patterns of termites, we’d change it for other social insect species
Then for each iteration (x), the total amount of work produced is
And the total for all the iterations is just
t
xttPB ,
x t
xttPB ,
Now we just need to define the different task allocation strategies as transition probabilities
Prob(It,x Ij[T\t],x+1)
1) Defined permanently by physiological caste
When born, individuals are assigned at random into a permanent task
So Prob(It,1)=1/|T| for each t and is then constant over all x
2) Determined by age
We assign individuals into |T| age classes and for age class a, we deterministically assign the individual into task t=a
3) Repertoire increases with age
Individuals in each age class a choose at random from among the first a tasks
4) Completely random
Individuals change tasks when they change age classes, but switch into any other task
Transition from one age class into another is defined to happen every (life span/|T|) iterations
So what were our strategies again?
Now we can examine how these strategies do in the face of different relationships among the
parameters:Suppose that we choose some combination of the
following:
Increasing linearly Bt=ρ1t, Decreasing linearly Bt= ρ1(|T|-t),
Even Bt=½ ρ1|T|
Increasing linearly St= ρ2t, Decreasing linearly St=ρ2|T|-t,
Even St=½ ρ2|T|
Increasing linearly Mt=2 ρ3t, Decreasing linearly Mt=ρ32|T|-2t,
Even Mt= ρ3|T|
ρ is some proportionality constant (in the examples shown, it’s just 1)
So what sorts of results do we see?
Total work
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
3,000,000
3,500,000
bd,
md,
sd
bd,
md,
se
bd,
md,
si
bd,
me,
sd
bd,
me
sebd
, m
e, s
ibd
, m
i, sd
bd,
mi,
sebd
, m
i, si
be,
md,
sd
be,
md,
se
be,
md,
si
be,
me,
sd
be,
me,
se
be,
me,
si
be,
mi,
sdbe
, m
i, se
be,
mi,
sibi
, m
d, s
dbi
, m
d, s
ebi
, m
d, s
ibi
, m
e, s
dbi
, m
e, s
ebi
, m
e, s
ibi
, m
i, sd
bi,
mi,
sebi
, m
i, si
randomrepdiscretedetermined
These are averages from 1000 runs each
randomrep.
castes
age based
But what can this help us to say about social structure and pathogen
exposure risks?
This becomes a matter of prior knowledge –
What relationships between the parameters do we know we can expect?
How can we structure society based on that knowledge?
This last graph was “complete knowledge”, but what if we don’t know anything about the risks or benefits or switching costs of
each tasks?Total work
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
Random Rep Discrete Determined
stdev
average
random rep age based castes
What if we only know one thing?Random Total
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
bd be bi
stdev
average
Random total
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
md me mi
stdev
average
Random total
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
sd se si
stdev
average
These graphs are from the Random strategy
Random total s
Random total b Random total m
Rep total b
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
1,600,000
1,800,000
2,000,000
bd be bi
stdev
average
Rep total m
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
1,600,000
1,800,000
2,000,000
md me mi
stdev
average
Rep total S
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
1,600,000
1,800,000
2,000,000
sd se si
stdev
average
These graphs are from the
Repertoire strategy
Rep total s
Rep total mRep total b
Discrete total b
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
bd be bi
stdev
average
Discrete total m
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
md me mi
stdev
average
Discrete total s
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
sd se si
stdev
average
These graphs are from the
age based strategy
Age based total s
Age based total b Age based total m
Determined total b
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
bd be bi
stdev
average
Determined total m
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
md me mi
stdev
average
Determined total s
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
sd se si
stdev
average
These graphs are from the castes
strategy
Castes total s
Castes total b Castes total m
Random total pairs
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
bd m
dbd
me
bd m
ibd
sd
bd s
ebd
si
md
bem
d bi
md
sdm
d se
md
sisd
be
sd b
isd
me
sd m
ibe
me
be m
ibe
se
be s
im
e bi
me
sem
e si
se b
ise
mi
bi m
ibi
si
mi s
i
stdev
average
Random total pairs
Rep total
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
3,000,000
3,500,000bd
md
bd m
ebd
mi
bd s
dbd
se
bd s
im
d be
md
bim
d sd
md
sem
d si
sd b
esd
bi
sd m
esd
mi
be m
ebe
mi
be s
ebe
si
me
bim
e se
me
sise
bi
se m
ibi
mi
bi s
im
i si
stdev
average
Rep total pairs
Discrete total pairs
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
3,000,000
3,500,000
bd m
dbd
me
bd m
ibd
sd
bd s
ebd
si
md
bem
d bi
md
sdm
d se
md
sisd
be
sd b
isd
me
sd m
ibe
me
be m
ibe
se
be s
im
e bi
me
sem
e si
se b
ise
mi
bi m
ibi
si
mi s
i
stdev
average
Age-based total pairs
Determined total pairs
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
bd m
dbd
me
bd m
ibd
sd
bd s
ebd
si
md
bem
d bi
md
sdm
d se
md
sisd
be
sd b
isd
me
sd m
ibe
me
be m
ibe
se
be s
im
e bi
me
sem
e si
se b
ise
mi
bi m
ibi
si
mi s
i
stdev
average
Castes total pairs
But, alas, this is not the whole picture
Sometimes we need specific tasks more than usual, or more than any other…
how do we hedge our bets to make sure that we can always have enough
workers to devote to those when we need them?
This could be thought of as a buffer zone for each task against that task becoming “rate
limiting”
Maintaining this buffer zone might be at odds with maximizing efficiency, even under the
same pathogen exposure risks
For every given chunk of time, we choose one of the tasks to be “the most pressing” task of
the moment (i)
We don’t ask any individuals to switch which task they perform, we just measure only how much work is
produced in the “most pressing task”So instead, for each iteration (x), the total amount of most pressing work produced is
And for all iterations is
xiiPB ,
x
xiiPB ,
The most pressing task changes every 100 iterations and is selected at random from T
Called for work
4.400
4.600
4.800
5.000
5.200
5.400
5.600bd
, m
d, s
dbd
, m
d, s
ebd
, m
d, s
ibd
, m
e, s
dbd
, m
e se
bd,
me,
si
bd,
mi,
sdbd
, m
i, se
bd,
mi,
sibe
, m
d, s
dbe
, m
d, s
ebe
, m
d, s
ibe
, m
e, s
dbe
, m
e, s
ebe
, m
e, s
ibe
, m
i, sd
be,
mi,
sebe
, m
i, si
bi,
md,
sd
bi,
md,
se
bi,
md,
si
bi,
me,
sd
bi,
me,
se
bi,
me,
si
bi,
mi,
sdbi
, m
i, se
bi,
mi,
si
random
rep
discrete
determined
And from this we get:
castes
agerep.
random
Called for
4.80000
4.85000
4.90000
4.95000
5.00000
5.05000
5.10000
5.15000
5.20000
5.25000
5.30000
Random Rep Discrete Determined
stdev
average
MPW work
random rep age based castes
Total work
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
Random Rep Discrete Determined
stdev
average
random rep age based castes
Random cfw s
4.85
4.90
4.95
5.00
5.05
5.10
sd se si
stdev
average
Random mpw s
Random cfw b
4.85
4.90
4.95
5.00
5.05
5.10
bd be bi
stdev
average
Random mpw b
Random cfw m
4.85
4.90
4.95
5.00
5.05
5.10
md me mi
stdev
average
Random mpw m
Random total
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
sd se si
stdev
average
Random total s
Random Total
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
bd be bi
stdev
average
Random total b
Random total
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
md me mi
stdev
average
Random total m
Rep cfw s
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
sd se si
stdev
average
Rep mpw s
Rep cfw b
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
bd be bi
stdev
average
Rep mpw b
Rep cfw m
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
md me mi
stdev
average
Rep mpw m
Rep total S
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
1,600,000
1,800,000
2,000,000
sd se si
stdev
average
Rep total s
Rep total m
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
1,600,000
1,800,000
2,000,000
md me mi
stdev
average
Rep total m
Rep total b
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
1,600,000
1,800,000
2,000,000
bd be bi
stdev
average
Rep total b
Discrete cfw s
4.80
4.90
5.00
5.10
5.20
5.30
5.40
sd se si
stdev
average
Age based mpw s
Discrete cfw b
4.80
4.90
5.00
5.10
5.20
5.30
5.40
bd be bi
stdev
average
Age based mpw b
Discrete cfw m
4.80
4.90
5.00
5.10
5.20
5.30
5.40
md me mi
stdev
average
Age based mpw m
Discrete total s
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
sd se si
stdev
average
Age based total s
Discrete total b
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
bd be bi
stdev
average
Age based total b
Discrete total m
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
md me mi
stdev
average
Age based total m
Determined cfw s
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
sd se si
stdev
average
Castes mpw s
Determined cfw b
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
bd be bi
stdev
average
Castes mpw b
Determined cfw m
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
md me mi
stdev
average
Castes mpw m
Determined total s
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
sd se si
stdev
average
Castes total s
Determined total b
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
bd be bi
stdev
average
Castes total b
Determined total m
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
md me mi
stdev
average
Castes total m
So we have a few cases where making the colony the most efficient, even under
the same parameter scenarios should lead us to a different choice than if we
were trying to make sure that our buffer against being unable to complete the
most important tasks of the moment is sufficiently large
And we compare each of these with the mortality costs by looking at the size of
the population left alive
Population at End
0
50
100
150
200
250
300
350
Random Rep build Discrete Determined
stdev
average
Population Surviving
Called for
4.80000
4.85000
4.90000
4.95000
5.00000
5.05000
5.10000
5.15000
5.20000
5.25000
5.30000
Random Rep Discrete Determined
stdev
average
MPW work
random rep age based castes
Total work
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
Random Rep Discrete Determined
stdev
average
random rep age based castes
Population at End
0
50
100
150
200
250
300
350
Random Rep build Discrete Determined
stdev
average
Population Surviving
Okay, these
didn’t all fit so well
random rep age based castes
Random cfw s
4.85
4.90
4.95
5.00
5.05
5.10
sd se si
stdev
average
Random mpw s
Random cfw b
4.85
4.90
4.95
5.00
5.05
5.10
bd be bi
stdev
average
Random mpw b
Random cfw m
4.85
4.90
4.95
5.00
5.05
5.10
md me mi
stdev
average
Random mpw m
Random total
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
sd se si
stdev
average
Random total s
Random Total
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
bd be bi
stdev
average
Random total b
Random total
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
md me mi
stdev
average
Random total m
Random Pop b
0
20
40
60
80
100
120
140
bd be bi
stdev
average
Random pop m
0
20
40
60
80
100
120
140
md me mi
stdev
average
Random pop s
0
20
40
60
80
100
120
140
sd se si
stdev
average
Rep cfw s
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
sd se si
stdev
average
Rep mpw s
Rep cfw b
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
bd be bi
stdev
average
Rep mpw b
Rep cfw m
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
md me mi
stdev
average
Rep mpw m
Rep total S
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
1,600,000
1,800,000
2,000,000
sd se si
stdev
average
Rep total s
Rep total m
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
1,600,000
1,800,000
2,000,000
md me mi
stdev
average
Rep total m
Rep total b
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
1,600,000
1,800,000
2,000,000
bd be bi
stdev
average
Rep total bRep pop b
0
50
100
150
200
250
300
350
400
bd be bi
stdev
average
Rep pop m
0
50
100
150
200
250
300
350
400
md me mi
stdev
average
Rep pop s
0
50
100
150
200
250
300
350
400
sd se si
stdev
average
Discrete cfw s
4.80
4.90
5.00
5.10
5.20
5.30
5.40
sd se si
stdev
average
Age based mpw s
Discrete cfw b
4.80
4.90
5.00
5.10
5.20
5.30
5.40
bd be bi
stdev
average
Age based mpw b
Discrete cfw m
4.80
4.90
5.00
5.10
5.20
5.30
5.40
md me mi
stdev
average
Age based mpw m
Discrete total s
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
sd se si
stdev
average
Age based total s
Discrete total b
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
bd be bi
stdev
average
Age based total b
Discrete total m
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
md me mi
stdev
average
Age based total m
Age based pop b
0
10
20
30
40
50
60
70
80
90
100
bd be bi
stdev
average
Age based pop m
0
10
20
30
40
50
60
70
80
90
100
md me mi
stdev
average
Age based pop s
0
10
20
30
40
50
60
70
80
90
100
sd se si
stdev
average
Determined cfw s
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
sd se si
stdev
average
Castes mpw s
Determined cfw b
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
bd be bi
stdev
average
Castes mpw b
Determined cfw m
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
md me mi
stdev
average
Castes mpw m
Determined total s
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
sd se si
stdev
average
Castes total s
Determined total b
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
bd be bi
stdev
average
Castes total b
Determined total m
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
md me mi
stdev
average
Castes total m
Castes pop m
0
2
4
6
8
10
12
14
16
md me mi
stdev
average
Castes pop s
0
2
4
6
8
10
12
14
16
sd se si
stdev
average
Castes pop b
0
2
4
6
8
10
12
14
16
bd be bi
stdev
average
This research is ongoing, so I haven’t finished all the ‘interpreting of results’ yet, however, clearly we have a few points of trade-off
A society as a whole needs to balance {survival against efficiency against ‘buffering’} in incredibly complex ways, but this allows a first step into examining those trade-offs
As a next step, to more accurately reflect social interaction governing disease dynamics, even at this scale, it’s time to introduce a new variable Dt to represent the density of infected individuals performing each task and make Mt dependent on Dt…
At least that’s the plan
This work is ongoing and is in collaboration with Sam Beshers at University of Illinois at Urbana-Champaign
I’m also now working on shifting
the parameter structure a little to
reflect human societies with
Ramanan Laxminarayan
(thanks to DIMACS!)
Thanks very much!