Nina H' Fefferman, Ph'D'

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Does Securing Infrastructure Against
Workforce-Depletion Depend on Whether the Risk is
Environmental or Infectious?
Nina H. Fefferman, Ph.D. DIMACS Rutgers
Univ. feferman_at_math.princeton.edu
This talk is on the continuation of the work I
presented at our last meeting, but Ill basically
proceed as though no one was there/remembers
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Disease can affect a large percentage of a
population 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 maintain a minimum
efficiency?
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!
Elements of the system Different tasks that
need to be accomplished Maybe each task
has its own 1) rate of production (depends
on having a minimum of workers on each
task) 2) time to be trained to perform the
task 3) minimum number of workers needed
to accomplish anything
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An 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/contaminant 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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Based on this framework, we can ask whether or
not infectious disease and environmental (or at
least non-coworker mediated infectious disease)
lead to different successes of task allocation
methods?
We can 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
allocation strategies We measure success by
amount of work produced (in each task and
overall) and the survival of population (also in
each task and overall) (Today Ill just show the
total measures for the whole population, even
though we measure everything in each task)
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Well examine four different allocation
strategies

• Defined permanently only trained for one
  thing
• Allocated by seniority progress
  through different tasks over time
• Repertoire increases with seniority build
  knowledge the longer you work
• 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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Model formulation (discrete)

• Three basic counterbalancing parameters
• Disease/Mortality risks for each task Mt (this
  will change over time for the infectious disease,
  based on how many other coworkers are already
  sick)
• Rate of production for each task Bt
• 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

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

• In each iteration, for each living individual in
  Pt,x there is an associated probability Mt of
  dying (independent for each individual)
• Individuals also die (deterministically) if they
  exceed a (iteration based) maximum life span

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We also replenish the population periodically
every 30 iterations, we add 30 new
individuals This is arbitrary and can be changed,
but think of it as a new class year graduating,
or a new hiring cycle, or however else the
workforce is recruited
Then for each iteration (x), the total amount of
work produced is
And the total for all the iterations is just
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
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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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Now we can examine different relationships among
the parameters

• Suppose that we take all combinations of the
  following
• Increasing Decreasing Constant
• Bt ?1t Bt ?1(T-t) Bt ?1T
• St ?2t St ?2(T-t) St ?2T
• Mt ?3t Mt ?3(T-t) Mt ?3T
• ? is some proportionality constant
• (in the examples shown, its just 1)
• Also in the examples shown the minimum
  number of individuals for each task is held
  constant for all t

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So do we actually see differences in the produced
amount of work?
So even as the relationships among the parameters
vary, we do see drastic differences in the amount
of work produced
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How about Survival?
We also see differences in the survival
probability of the population as the
relationships among the parameters vary
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So the full story as the relationships among the
parameter values vary looks like
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But notice In the examples you just saw, the
mortality cost in each task was independent of
the number of individuals in that task already
affected This is much more like an environmental
exposure risk What if we wanted to look at
infectious disease risks? Then the risk of
mortality in each task would depend on the number
of sick workers already performing that task
Mt c ß( Infectedt) where ß is the
probability of becoming infected from contact
with a sick coworker and c is any constant level
of primary exposure
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For simplicity now, lets not let the other
parameters vary in relation to each other lets
just look at
Bt ?1t Increasing St ?2t Increasing Mt
c ß( Infectedt) Constant primary
secondary And again a constant minimum number for
each task
And we will compare this with the narrower range
of non-infectious scenarios by then keeping
everything the same, but changing Mt back to just
the constant primary exposure
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So do we still actually see differences in the
produced amount of work without infectious
spread, but with the narrower range?
Non-infectious Exposure
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And when we introduce infectious spread, we still
see differences among the allocation strategies
Infectious Exposure
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And in direct comparison? Non-infectious vs
Infectious Mortality Risk?
Total work Produced

• Always better to have environmental disease
• Makes sense
• BUT the difference in outcome is drastically
  different!

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How about differences for overall survival?
Non-infectious Exposure
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So we also difference in survival
Infectious Exposure
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And again - Direct comparison?
Population Left Alive
Again, better to have only environmental exposure
(makes sense again) But again, differences in
delta between strategies
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So, are the differences seen across strategies
from environmental to infectious exposure the
same for both survival and work?
No!
Work comparisons
Survival comparisons
Larger delta
Smaller delta
Smaller delta
Larger delta
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As a weird potential extension, can this tell us
anything about how we can affect the economics of
the system with vaccination?
Yes! If we do not get to plan which allocation
method to use, we can use vaccination to create
our own Mt landscape to try and (at least) manage
within each task to keep the mortality the same,
but minimize the cost to work produced this
defines a dynamically shifting equilibrium point
for each disease state for the system
Work comparisons
Survival comparisons
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Take home messages There are important
differences strategies for task allocation and
they do depend on the type of health risk Last
talk I showed some results about the differences
between outcomes for these strategies when there
are seasonal vs constant environmental risks
those also showed drastic differences in the
efficacy of the four strategies 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 The more we look at the
problem, the better the information to the
decision makers can be
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Any Questions?! My thanks to The
Organizers!!!! DIMACS SACEMA AIMS The
NSF All of you for your time and interest
Especially for sticking around to the bitter end
to listen to me!
Please feel free to contact me with further
questions