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

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e, s

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mi,

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md,

sd

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be,

md,

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me,

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si

be,

mi,

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mi,

sibi

, m

d, s

dbi

, m

d, s

ebi

, m

d, s

ibi

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dbi

, m

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ebi

, m

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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

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400,000

600,000

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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

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bd be bi

stdev

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md me mi

stdev

average

Random total

0

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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

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bd be bi

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md me mi

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average

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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

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bd be bi

stdev

average

Discrete total m

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md me mi

stdev

average

Discrete total s

0

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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

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Determined total m

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Determined total s

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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

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1,200,000

bd m

dbd

me

bd m

ibd

sd

bd s

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si

md

bem

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md

sdm

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sd b

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me

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me

be m

ibe

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im

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me

sem

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se b

ise

mi

bi m

ibi

si

mi s

i

stdev

average

Random total pairs

Rep total

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3,000,000

3,500,000bd

md

bd m

ebd

mi

bd s

dbd

se

bd s

im

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md

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sem

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mi

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me

bim

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bi

se m

ibi

mi

bi s

im

i si

stdev

average

Rep total pairs

Discrete total pairs

0

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1,000,000

1,500,000

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3,500,000

bd m

dbd

me

bd m

ibd

sd

bd s

ebd

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sdm

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me

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im

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me

sem

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se b

ise

mi

bi m

ibi

si

mi s

i

stdev

average

Age-based total pairs

Determined total pairs

0

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bd m

dbd

me

bd m

ibd

sd

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ebd

si

md

bem

d bi

md

sdm

d se

md

sisd

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sd b

isd

me

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ibe

me

be m

ibe

se

be s

im

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me

sem

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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

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, m

d, s

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bd,

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, 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

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Random Rep Discrete Determined

stdev

average

random rep age based castes

Random cfw s

4.85

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5.00

5.05

5.10

sd se si

stdev

average

Random mpw s

Random cfw b

4.85

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5.10

bd be bi

stdev

average

Random mpw b

Random cfw m

4.85

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average

Random mpw m

Random total

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Random total s

Random Total

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Random total b

Random total

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Random total m

Rep cfw s

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Rep mpw s

Rep cfw b

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Rep mpw b

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Discrete cfw s

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Age based mpw s

Discrete cfw b

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Age based mpw b

Discrete cfw m

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Determined cfw s

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Castes mpw b

Determined cfw m

4.80

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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

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Population Surviving

Called for

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Population at End

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Population Surviving

Okay, these

didn’t all fit so well

random rep age based castes

Random cfw s

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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!