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Nina H. Fefferman, Ph.D.
Rutgers [email protected]
du
Balancing Workforce Productivity Against Disease Risks for Environmental and
Infectious Epidemics
Direct threats:Well
peopleSick people
Nothing terribly surprising about this
Pathogens of all sorts
Workers Being Productive
Sick workers
Sick Workers have a choice:
Lack of Productivity AND Sick
People
Stay home (don’t be productive)
Go to work and maybe infect coworkers
Basic idea behind this research :
Can we train or allocate our work force according to some algorithm in order to
maintain a minimum efficiency?
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
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”
An assumption for today:
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
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)
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
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 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
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
t
xttPB ,
x t
xttPB ,
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 – 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 AmongParameters Varies in Non-Infectious Disease
1.0×104
1.0×105
1.0×106
1.0×107
Allocation Method
Am
ou
nt
of
Wo
rk
Pro
du
ced
So even as the relationships among the parameters
vary, we do see drastic
differences in the amount of work produced
How about Survival?
Range for Survival as Relationship AmongParameters Varies in Non-Infectious Disease
0
100
200
300
400
Allocation Method
Nu
mb
er L
eft
Ali
ve
We also see differences in the survival
probability of the population
as the relationships among the parameters
vary
So the full story as the relationships among the
parameter values vary looks like:
Range for Survival as Relationship AmongParameters Varies in Non-Infectious Disease
deter
min
istic
discr
ete
random
reper
toire
0
100
200
300
400
Allocation Method
Nu
mb
er L
eft
Ali
ve
Range for Total Work as Relationship AmongParameters Varies in Non-Infectious Disease
1.0×104
1.0×105
1.0×106
1.0×107
Allocation Method
Am
ou
nt
of
Wo
rk
Pro
du
ced
If you want to be safest on average, via both metrics,
Repertoire wins!
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 taskMt = 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 taskAnd 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
So do we still actually see differences in the produced amount of work
without infectious spread, but with the narrower range?
Deter
min
ed
Discr
ete
Random
Reper
toire
1.6×107
1.7×107
1.8×107
1.9×1075.3×107
6.3×107
Allocation Method
Am
ou
nt
of
Wo
rk
Pro
du
ced
Non-infectio
us Exposur
e
Det
erm
ined
Discr
ete
Random
Reper
toire
1.0×104
1.1×105
4.4×106
4.8×106
Allocation Method
Am
ou
nt
of
Wo
rk P
rod
uce
dInfectio
us Exposu
re
And when we introduce infectious spread, we still see differences among
the allocation strategies
Deter
min
ed -
Inf
Deter
min
ed -
Env
Discr
ete
- Inf
Discr
ete
- Env
Random
- In
f
Random
- Env
Reper
toire
- In
f
Reper
toire
- Env
0
2.5×106
5.0×1061.5×107
1.8×107
2.0×1075.0×107
5.5×107
6.0×107
6.5×107
7.0×107
Allocation Methods and Exposure Type
Wo
rk P
rod
uce
d
And in direct comparison?
Non-infectious vs Infectious Mortality Risk?
Total work ProducedAlways
better to have
environmental
disease
- Makes sense
- BUT – the
difference in outcome
is drastically different!
How about differences for overall survival?
Deter
min
ed
Discr
ete
Random
Reper
toire
430
480
530
Allocation Method
Nu
mb
er
Lef
t A
live
Non-infectio
us Exposur
e
So we also difference in survival
Deter
min
ed
Discr
ete
Random
Reper
toire
0
50
100
150
200
250
300
Allocation Method
Nu
mb
er
Lef
t A
live
Infectious
Exposure
Det
erm
ined
- In
f
Deter
min
ed -
Env
Discr
ete
- Inf
Discr
ete
- Env
Random
- In
f
Random
- Env
Reper
toire
- In
f
Reper
toire
- Env
0
100
200
300
400
500
600
Nu
mb
er
Lef
t A
live
Population Left AliveAgain,
better to have only environm
ental exposure (makes sense again)
But again,
differences in delta between
strategies
And again - Direct comparison?
Det
erm
ined
- In
f
Deter
min
ed -
Env
Discr
ete
- Inf
Discr
ete
- Env
Random
- In
f
Random
- Env
Reper
toire
- In
f
Reper
toire
- Env
0
2.5×106
5.0×1061.5×107
1.8×107
2.0×1075.0×107
5.5×107
6.0×107
6.5×107
7.0×107
Allocation Methods and Exposure Type
Wo
rk P
rod
uce
d
Deter
min
ed -
Inf
Deter
min
ed -
Env
Discr
ete
- Inf
Discr
ete
- Env
Random
- In
f
Random
- Env
Reper
toire
- In
f
Reper
toire
- Env
0
100
200
300
400
500
600
Nu
mb
er
Lef
t A
live
Work comparisons Survival comparisons
So, are the differences seen across strategies from environmental to
infectious exposure the same for both survival and work?
Smaller delta
Larger delta
Larger delta
Smaller delta
No!
Take home messages:
YES! There are conflicts between productivity and disease risks, and the change depending on type of
diseaseIt’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