Does Securing Infrastructure Against
Workforce-Depletion Depend on
Whether the Risk is Environmental or
Infectious?
Nina H. Fefferman, Ph.D.
DIMACS Rutgers Univ.
feferman@math.princeton.edu
This talk is on the continuation of the work I presented at
our last meeting, but I’ll basically proceed as though no one
was there/remembers
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?
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 maintain a minimum efficiency?
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!
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
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”
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 I’ll just show the “total” measures for the
whole population, even though we measure
everything in each task)
We’ll examine 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
Model formulation – (discrete)
Three basic counterbalancing parameters:
1. 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)
2. Rate of 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 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 Mt of dying (independent for each individual)
 Individuals also die (deterministically) if they exceed a (iteration based) maximum life span
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
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
B P
t
t,x
t
And the total for all the iterations is just
 B P
t t,x
x
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
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
Now we can examine different relationships among the
parameters:
Suppose that we take all combinations of the following:
Increasing
Decreasing
Constant
Bt = ρ1t
Bt = ρ1(|T|-t)
Bt = ρ1|T|
St = ρ2t
St = ρ2(|T|-t)
St = ρ2|T|
Mt = ρ3t
Mt = ρ3(|T|-t)
Mt = ρ3|T|
ρ is some proportionality constant
(in the examples shown, it’s just 1)
Also in the examples shown the minimum
number of individuals for each task is held constant for
all t
So do we actually see differences in the
produced amount of work?
Range for Total Work as Relationship Among
Parameters Varies in Non-Infectious Disease
1.0×10 7
1.0×10 6
1.0×10 5
Allocation Method
re
pe
rt
oi
re
ra
nd
om
di
sc
re
te
in
is
tic
1.0×10 4
de
te
rm
Amount of Work Produced
So even as the
relationships among
the parameters vary,
we do see drastic
differences in the
amount of work
produced
How about Survival?
100
Allocation Method
re
pe
rt
oi
re
ra
nd
om
e
di
sc
re
t
in
is
t
ic
0
de
te
rm
Number Left Alive
Range for Survival as Relationship Among
We also see
Parameters Varies in Non-Infectious Disease
differences in the
400
survival probability
of the population as
300
the relationships
among the
200
parameters vary
So the full story as the relationships among
the parameter values vary looks like:
If you want to be safest on average,
via both metrics, Repertoire wins!
1.0×10 7
Number Left Alive
6
1.0×10 5
300
200
100
Allocation Method
re
pe
rt
oi
re
ra
nd
om
e
di
sc
re
t
in
is
t
re
pe
rt
oi
re
ra
nd
om
in
is
tic
di
sc
re
te
Allocation Method
ic
0
1.0×10 4
de
te
rm
1.0×10
400
de
te
rm
Amount of Work Produced
Range for Total Work as Relationship Among Range for Survival as Relationship Among
Parameters Varies in Non-Infectious Disease Parameters Varies in Non-Infectious Disease
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 + β(# Infectedt)
where β is the probability of becoming
infected from contact with a sick
coworker and c is any constant
level of primary exposure
For simplicity now, let’s not let the other parameters vary in
relation to each other – let’s just look at :
Bt = ρ1t
Increasing
St = ρ2t
Increasing
Mt = c + β(# 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
6.3×10 7
5.3×10 7
1.9×10 7
1.8×10 7
1.7×10 7
R
ep
er
to
ire
an
do
m
R
is
cr
et
e
D
et
er
m
in
ed
1.6×10 7
D
Noninfectious
Exposure
Amount of Work Produced
So do we still actually see differences in the
produced amount of work without infectious
spread, but with the narrower range?
Allocation Method
And when we introduce infectious spread, we still
see differences among the allocation strategies
4.4×10 6
1.1×10 5
R
ep
er
to
ire
an
do
m
R
D
is
cr
et
e
et
er
m
in
ed
1.0×10 4
D
Infectious
Exposure
Amount of Work Produced
4.8×10 6
Allocation Method
And in direct comparison?
Non-infectious vs Infectious Mortality Risk?
Total work Produced
7.0×10 7
5.0×10 7
2.0×10 7
1.8×10 7
1.5×10 7
5.0×10 6
2.5×10 6
nf
R
an
do
m
-E
-I
R
an
do
m
-E
nv
D
is
cr
et
e
-I
nf
is
cr
et
e
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nv
R
ep
er
to
ire
-I
nf
R
ep
er
to
ire
-E
nv
D
et
er
m
in
ed
-E
nv
-I
nf
0
in
ed
- BUT – the
difference in
outcome is
drastically
different!
5.5×10 7
D
et
er
m
- Makes sense
6.0×10 7
Work Produced
Always better
to have
environmental
disease
6.5×10 7
Allocation Methods and Exposure Type
How about differences for overall survival?
Noninfectious
Exposure
480
R
ep
er
to
ire
an
do
m
R
is
cr
et
e
D
et
er
m
in
ed
430
D
Number Left Alive
530
Allocation Method
So we also difference in survival
300
Infectious
Exposure
200
150
100
50
R
ep
er
to
ire
an
do
m
R
is
cr
et
e
D
et
er
m
in
ed
0
D
Number Left Alive
250
Allocation Method
Number Left Alive
500
400
300
200
100
nv
-E
ep
er
to
ire
-I
nf
-E
an
do
m
R
ep
er
to
ire
nv
nf
-I
an
do
m
R
-E
nv
is
cr
et
e
-I
nf
is
cr
et
e
-E
nv
D
D
R
R
D
et
er
m
in
ed
in
ed
-I
nf
0
et
er
m
But again,
differences
in delta
between
strategies
600
D
Again, better
to have only
environment
al exposure
(makes sense
again)
And again - DirectPopulation
comparison?
Left Alive
So, are the differences seen across strategies from
environmental to infectious exposure the same for
both survival and work?
No!
Survival comparisons
Smaller
delta
600
6.5×10 7
300
Allocation Methods and Exposure Type
nv
-E
-I
nf
ep
er
to
ire
R
R
-E
an
do
m
ep
er
to
ire
nv
nf
-I
R
an
do
m
R
-I
nf
is
cr
et
e
D
-E
in
ed
et
er
m
D
nv
-I
nf
nv
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ep
er
to
ire
R
nv
ep
er
to
ire
R
an
do
m
-E
-I
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an
do
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R
-I
nf
is
cr
et
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is
cr
et
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nv
D
-E
D
et
er
m
in
ed
in
ed
-I
nf
0
nf
0
-E
nv
100
-I
nf
2.5×10 6
-E
nv
200
in
ed
1.5×10 7
5.0×10 6
400
et
er
m
1.8×10 7
Larger
delta
500
D
5.0×10 7
2.0×10 7
D
et
er
m
Work Produced
5.5×10 7
Smaller
delta
Number Left Alive
Larger
delta
6.0×10 7
is
cr
et
e
7.0×10
7
D
Work comparisons
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
600
7.0×10 7
6.5×10 7
500
6.0×10 7
Allocation Methods and Exposure Type
nv
-E
-I
nf
ep
er
to
ire
-E
ep
er
to
ire
nv
nf
-I
an
do
m
an
do
m
R
R
R
Survival comparisons
R
-I
nf
is
cr
et
e
D
is
cr
et
e
in
ed
-E
nv
-I
nf
D
et
er
m
in
ed
-E
nv
ep
er
to
ire
R
-E
nv
ep
er
to
ire
R
-I
an
do
m
R
D
is
cr
et
e
D
in
ed
et
er
m
R
an
do
m
nv
is
cr
et
e
-E
in
ed
D
D
et
er
m
Work comparisons
-I
nf
0
nf
0
-E
nv
100
-I
nf
2.5×10 6
-E
nv
200
et
er
m
1.5×10 7
5.0×10 6
300
D
1.8×10 7
400
D
Number Left Alive
5.0×10 7
2.0×10 7
-I
nf
Work Produced
5.5×10
7
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
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
The more we look at the problem, the better the
information to the decision makers can be
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