Nina H. Fefferman, Ph.D.

Rutgers Univ.fefferman@aesop.rutgers.e

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

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Env

Discr

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- Inf

Discr

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- Env

Random

- In

f

Random

- Env

Reper

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- In

f

Reper

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- 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

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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

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Discr

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Random

Reper

toire

430

480

530

Allocation Method

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live

Non-infectio

us Exposur

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So we also difference in survival

Deter

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Reper

toire

0

50

100

150

200

250

300

Allocation Method

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Infectious

Exposure

Det

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- In

f

Deter

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Env

Discr

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- Inf

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- In

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f

Reper

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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

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Env

Discr

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- Inf

Discr

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Random

- In

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Reper

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f

Reper

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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

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Env

Discr

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Reper

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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