Nina H. Fefferman, Ph.D.

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Preparing Societal Infrastructure Against
Disease-Related Workforce Depletion
Nina H. Fefferman, Ph.D. InForMID Tufts
Univ. DIMACS Rutgers Univ. feferman_at_math.princeton
.edu
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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?
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Direct threats
Well people
Sick people
Pathogens of all sorts
Nothing terribly surprising about this
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Indirect threats
Well people
Some of the sick people have crucial jobs and
they cant go to work
Sick people
Well People who are harmed by a lack of provision
of infrastructure
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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, Im 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
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Lets assume for todays 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
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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 youll 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
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Well start with four different allocation
strategies

1. Defined permanently only trained for one
   thing
2. Allocated by seniority progress
   through different tasks over time
3. Repertoire increases with seniority build
   knowledge the longer you work
4. Completely random just for comparison,
   everyone switches at random

(Suggested by the most efficient working
organizations of the natural world social
insects!)
(Determined)
(Discrete)
(Repertoire)
(Random)
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• So within the model, we are concerned with
• Direct disease/mortality risk (constant in all
  tasks t), DRiskt
• Indirect mortality risk, IRiskt
• Rate of production for each task, Bt
• Cost of switching to task t from some other
  task, St
• Minimum number of individuals in task t in
  order to be successful, Mt

For today, well run this with t20, Btt, Stt,
and ?t DRiskt0.01, IRiskt0.001( tasks that
have failed) And well look at two scenarios of
Mt 1) Mt21-t and 2) ?t Mt5
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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 dont 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
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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 doesnt
contribute if they are dead
2) The individual doesnt contribute during the
learning phase

• 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)

• Theyre in the learning phase if theyve
  switched into their current task (t) for less
  than St iterations

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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 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?
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Deterministic Strategy Different scenarios
Deterministic Const. Exposure ? Minimum s
Deterministic Const. Exposure Const. Minimum
s
Constant Exposure
Deterministic Seas. Exposure ? Minimum s
Deterministic Seas. Exposure Const. Minimum
s
Seasonal Exposure
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Discrete Strategy Different scenarios
Discrete Const. Exposure ? Minimum s
Discrete Const. Exposure Const. Minimum s
Constant Exposure
Discrete Seas. Exposure ? Minimum s
Discrete Seas. Exposure Const. Minimum s
Seasonal Exposure
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Repertoire Strategy Different scenarios
Repertoire Const. Exposure ? Minimum s
Repertoire Const. Exposure Const. Minimum s
Constant Exposure
Repertoire Seas. Exposure ? Minimum s
Repertoire Seas. Exposure Const. Minimum s
Seasonal Exposure
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Random Strategy Different scenarios
Random Const. Exposure ? Minimum s
Random Const. Exposure Const. Minimum s
Constant Exposure
Random Seas. Exposure ? Minimum s
Random Seas. Exposure Const. Minimum s
Seasonal Exposure
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So what if we compare within the same scenario,
across strategies
Lets compare across strategies for Constant
Exposure, Constant M
Discrete Const. Exposure Const. Minimum s
Deterministic Const. Exposure Const. Minimum
s
Repertoire Const. Exposure Const. Minimum s
Random Const. Exposure Const. Minimum s
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What about for Seasonal Exposure, Constant M
Discrete Seas. Exposure Const. Minimum s
Deterministic Seas. Exposure Const. Minimum
s
Random Seas. Exposure Const. Minimum s
Repertoire Seas. Exposure Const. Minimum s
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And for Constant Exposure, Decreasing M
Deterministic Const. Exposure ? Minimum s
Discrete Const. Exposure ? Minimum s
Repertoire Const. Exposure ? Minimum s
Random Const. Exposure ? Minimum s
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And for Seasonal Exposure, Decreasing M
Deterministic ? Exposure ? Minimum s
Discrete ? Exposure ? Minimum s
Repertoire ? Exposure ? Minimum s
Random ? Exposure ? Minimum s
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Those are the results from the work produced What
about the number left living?
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But just to check, did the indirect mortality
actually make a difference?
Not really if the strategy is Deterministic
These figures are all taken only from the
scenarios of constant disease and even minimum
numbers required
Deterministic Const. Exposure Const. Minimum
s
Without Infrastructure Compounded Mortality
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It makes a huge difference if the strategy is
Discrete
Discrete Const. Exposure Const. Minimum s
Without Infrastructure Compounded Mortality
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It makes a huge difference if the strategy is
Repertoire
Without Infrastructure Compounded Mortality
Repertoire Const. Exposure Const. Minimum s
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Not Really if the strategy is Random
Random Const. Exposure Const. Minimum s
Without Infrastructure Compounded Mortality
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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 Its 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
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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!