1. Nina H. Fefferman, Ph.D. DIMACS Rutgers Univ. feferman@math.princeton.edu Does Securing Infrastructure Against Workforce-Depletion Depend on Whether the Risk is Environmental or Infectious? 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

2. 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?

3. Direct threats: Well peopleSick people Nothing terribly surprising about this Pathogens of all sorts

4. Indirect threats: Well people Sick people Some of the sick people have crucial jobs and they can’t go to work Well People who are harmed by a lack of provision of infrastructure

5. 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

6. 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:

7. 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)

8. 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)

9. Model formulation – (discrete) Three basic counterbalancing parameters: 1. Disease/Mortality risks for each task M t (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 B t 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), S t

10. We have individuals I and tasks (t) in iteration (x), so we write I t,x In each step of the Markov process, each individual I t,x contributes to some P t,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 S t iterations  In each iteration, for each living individual in P t,x there is an associated probability M t of dying (independent for each individual)  Individuals also die (deterministically) if they exceed a (iteration based) maximum life span

11. 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

12. 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

13. Now we can examine different relationships among the parameters: Suppose that we take all combinations of the following: IncreasingDecreasing Constant B t = ρ 1 t B t = ρ 1 ( |T|-t) B t = ρ 1 |T| S t = ρ 2 t S t = ρ 2 ( |T|-t)S t = ρ 2 |T| M t = ρ 3 t M t = ρ 3 ( |T|-t)M t = ρ 3 |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

14. 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

15. How about Survival? We also see differences in the survival probability of the population as the relationships among the parameters vary

16. 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!

17. 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 M t = c + β(# Infected t ) where β is the probability of becoming infected from contact with a sick coworker and c is any constant level of primary exposure

18. For simplicity now, let’s not let the other parameters vary in relation to each other – let’s just look at : B t = ρ 1 t Increasing S t = ρ 2 t Increasing M t = c + β(# Infected t )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 M t back to just the constant primary exposure

19. So do we still actually see differences in the produced amount of work without infectious spread, but with the narrower range? Non- infectious Exposure

20. Infectious Exposure And when we introduce infectious spread, we still see differences among the allocation strategies

21. 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!

22. How about differences for overall survival? Non- infectious Exposure

23. So we also difference in survival Infectious Exposure

24. Population Left Alive Again, better to have only environment al exposure (makes sense again) But again, differences in delta between strategies And again - Direct comparison?

25. Work comparisonsSurvival comparisons So, are the differences seen across strategies from environmental to infectious exposure the same for both survival and work? Smaller delta Larger delta Smaller delta No!

26. Work comparisonsSurvival 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 M t 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

27. 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

28. 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