Preparing Societal Infrastructure Against
Disease-Related Workforce Depletion
Nina H. Fefferman, Ph.D.
InForMID Tufts Univ.
DIMACS Rutgers Univ.
feferman@math.princeton.edu
Disease can affect a large percentage of a
population
This can be All at once
Over time
Such diseases pose not only direct threats,
but indirect threats to the public
health of a community
What do I mean?
Direct threats:
Well people
Sick people
Pathogens of all sorts
Nothing terribly surprising about
this
Indirect threats:
Well people
Some of the sick
people have
crucial jobs and
they can’t go to
work
Sick people
Well People who
are harmed by a
lack of provision
of infrastructure
Basic idea behind this research :
Can we train or allocate our work force according to some
algorithm in order to minimize these sorts of problems?
Due to time constraints, I’m going to show the ideas, not the equations – if anyone
wants the mathematical details, please just ask me after the talk!
What elements of the system do we want to incorporate?
Different tasks that need to be accomplished
Maybe each task has its own
1) rate of production
2) time to be trained
3) minimum number of workers needed
to accomplish anything
Let’s assume for today’s talk that risk of contracting disease, and the
subsequent risk of death from disease is uniform, regardless of task –
This may not be true if there is occupational exposure, or differential
availability of medical treatment based on employment
Another strong and unlikely-to-be-correct assumption for today:
We will deal with all absence from work as
“mortality” (permanent absence from the
workforce once absent once for any reason) –
Depending on the specific disease in question,
this would definitely want to be changed to
reflect “duration of symptoms causing
absence from work” and “what is the
probability of death from infection”
In addition to disease risk, we include an “additional risk
of mortality” as a function of how many tasks have fewer
than the minimum number of workers needed to
accomplish them
This represents the indirect harm caused by the breakdown in
infrastructure support – for the models you’ll see here, it will be
kept small (an order of magnitude less) relative to the direct
disease risk – this again would change once we had a specific
problem/society to model
Given all of this, we can then simulate a
population, with new workers being recruited
into the system, staying in or learning and
progressing through new tasks over time
according to a variety of different strategies
We’ll start with four different allocation strategies
(Suggested by the
most efficient
working
organizations of
the natural world –
social insects!)
1. Defined permanently : only trained for
(Determined)
one thing
2. Allocated by seniority : progress
(Discrete)
through different
tasks over time
3. Repertoire increases with seniority :
(Repertoire)
build knowledge the
longer you work
4. Completely random : just for comparison,
(Random)
everyone switches at
random
So within the model, we are concerned with :
1. Direct disease/mortality risk (constant in all tasks t), DRiskt
2. Indirect mortality risk, IRiskt
3. Rate of production for each task, Bt
4. Cost of switching to task t from some other
task, St
5. Minimum number of individuals in task t in
order to be successful, Mt
For today, we’ll run this with t=20, Bt=t, St=t, and
t DRiskt=0.01, IRiskt=0.001*(# tasks that have failed)
And we’ll look at two scenarios of Mt :
1) Mt=21-t
and
2) t Mt=5
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
(and I do in the paper) – 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 step 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
 In each iteration, for each living individual in Pt,x there is an
associated probability (IRiskt + DRiskt) of dying
(independent for each individual)
 Individuals also die (deterministically) if they exceed a (iteration based)
maximum life span (500 time steps – arbitrarily chosen)
2) The individual doesn’t contribute during the ‘learning phase’
 They’re in the learning phase if they’ve switched into their
current task (t) for less than St iterations
We also replenish the population :
Add 10 new individuals every time step
(arbitrary)
Then for each iteration (x), the total amount of work produced is
Bt Pt , x for each t
We also keep track of how much of the population is “left
alive”, since there is a potential conflict
between
“work production” and population
survival
So, given all this, what are our results?
Deterministic Strategy – Different scenarios
Deterministic – Const. Exposure
– ↓ Minimum #s
Deterministic – Const. Exposure
– Const. Minimum #s
Constant
Exposure
Deterministic – Seas. Exposure
– ↓ Minimum #s
Seasonal
Exposure
Deterministic – Seas. Exposure
– Const. Minimum #s
Discrete Strategy – Different scenarios
Discrete – Const. Exposure
– ↓ Minimum #s
Discrete – Const. Exposure
– Const. Minimum #s
Discrete – Seas. Exposure
– ↓ Minimum #s
Discrete – Seas. Exposure
– Const. Minimum #s
Constant
Exposure
Seasonal
Exposure
Repertoire Strategy – Different scenarios
Repertoire – Const. Exposure
– ↓ Minimum #s
Repertoire
Repertoire – Const. Exposure
– Const. Minimum #s
Constant
Exposure
Repertoire – Seas. Exposure
– ↓ Minimum #s
Repertoire
Seasonal
Exposure
Repertoire
Repertoire – Seas. Exposure
– Const. Minimum #s
Random Strategy – Different scenarios
Random – Const. Exposure
– ↓ Minimum #s
Random – Const. Exposure
– Const. Minimum #s
Random – Seas. Exposure
– ↓ Minimum #s
Random – Seas. Exposure
– Const. Minimum #s
Constant
Exposure
Seasonal
Exposure
So what if we compare within the same scenario, across strategies:
Let’s compare across strategies for Constant Exposure, Constant M
Deterministic – Const. Exposure
– Const. Minimum #s
Repertoire
Repertoire – Const. Exposure
– Const. Minimum #s
Discrete – Const. Exposure
– Const. Minimum #s
Random – Const. Exposure
– Const. Minimum #s
What about for Seasonal Exposure, Constant M
Deterministic – Seas. Exposure
– Const. Minimum #s
Repertoire – Seas. Exposure
– Const. Minimum #s
Discrete – Seas. Exposure
– Const. Minimum #s
Random – Seas. Exposure
– Const. Minimum #s
And for Constant Exposure, Decreasing M
Deterministic – Const. Exposure
– ↓ Minimum #s
Repertoire – Const. Exposure
– ↓ Minimum #s
Discrete – Const. Exposure
– ↓ Minimum #s
Random – Const. Exposure
– ↓ Minimum #s
And for Seasonal Exposure, Decreasing M
Deterministic – ↓ Exposure
– ↓ Minimum #s
Repertoire – ↓ Exposure
– ↓ Minimum #s
Discrete – ↓ Exposure
– ↓ Minimum #s
Random – ↓ Exposure
– ↓ Minimum #s
Those are the results from the work produced
What about the number left living?
But just to check, did the indirect mortality actually make a difference?
Not really – if
the strategy is
Deterministic
Without Infrastructure
Compounded Mortality
These figures are all
taken only from the
scenarios of constant
disease and even
minimum numbers
required
Deterministic – Const. Exposure
– Const. Minimum #s
It makes a
huge
difference if
the strategy is
Discrete
Discrete – Const. Exposure
– Const. Minimum #s
Without Infrastructure
Compounded Mortality
It makes a
huge
difference if
the strategy is
Repertoire
Without Infrastructure
Compounded Mortality
Repertoire – Const. Exposure
– Const. Minimum #s
Not Really – if
the strategy is
Random
Without Infrastructure
Compounded Mortality
Random – Const. Exposure
– Const. Minimum #s
Take home messages:
These studies are by no means the “answer” to anything,
but they are a good way to start examining these sorts
of questions
The more specific a disease and population and
infrastructure we want to examine, the more
appropriately we can tailor the simulations
It’s unlikely that these sorts of models will provide “easy”
answers – but it IS likely that they could provide public
policy makers with “likely disease-related repercussions”
of societal organization policies
Any Questions?!
My thanks to
The organizers for inviting me
The NSF for funding to DIMACS, where I have been
happily visiting for the past year
InForMID for additional support
All of you for your time and interest
Please feel free to contact me
with further questions later!