Nina H. Fefferman, Ph.D. InForMID Tufts Univ. DIMACS Rutgers Univ. feferman@math.princeton

1. Preparing Societal Infrastructure Against Disease-Related Workforce Depletion Nina H. Fefferman, Ph.D. InForMID Tufts Univ. DIMACS Rutgers Univ. feferman@math.princeton.edu

2. Disease can affect a large percentage of a population This can be - All at once Over time 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 people Sick people Pathogens of all sorts Nothing terribly surprising about this

4. 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

5. Basic idea behind this research : Can we train or allocate our work force according to some algorithm in order to minimize these sorts of problems? 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! What elements of the system do we want to incorporate? Different tasks that need to be accomplished Maybe each task has its own 1) rate of production 2) time to be trained 3) minimum number of workers needed to accomplish anything

6. Let’s assume for today’s talk that risk of contracting disease, and the subsequent risk of death from disease is uniform, regardless of task – This may not be true if there is occupational exposure, or differential availability of medical treatment based on employment Another strong and unlikely-to-be-correct 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 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”

7. In addition to disease risk, we include an “additional risk of mortality” as a function of how many tasks have fewer than the minimum number of workers needed to accomplish them This represents the indirect harm caused by the breakdown in infrastructure support – for the models you’ll see here, it will be kept small (an order of magnitude less) relative to the direct disease risk – this again would change once we had a specific problem/society to model Given all of this, we can then 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 strategies

8. We’ll start with 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)

9. So within the model, we are concerned with : • Direct disease/mortality risk (constant in all tasks t), DRiskt • Indirect mortality risk, IRiskt • Rate of production for each task, Bt • Cost of switching to task t from some other task, St • Minimum number of individuals in task t in order to be successful, Mt For today, we’ll run this with t=20, Bt=t, St=t, and tDRiskt=0.01, IRiskt=0.001*(# tasks that have failed) And we’ll look at two scenarios of Mt : 1) Mt=21-t and 2) tMt=5

10. We simulate the following via a stochastic state-dependent Markov process of successive checks of randomly generated values against threshold values • 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

11. We have individuals I and tasks (t) in iteration (x), so we writeIt,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 during the ‘learning phase’ • In each iteration, for each living individual in Pt,x there is an associated probability (IRiskt + DRiskt) of dying (independent for each individual) • Individuals also die (deterministically) if they exceed a (iteration based)maximum life span (500 time steps – arbitrarily chosen) • They’re in the learning phase if they’ve switched into their current task (t) for less than St iterations

12. We also replenish the population : Add 10 new individuals every time step(arbitrary) Then for each iteration (x), the total amount of work produced is for each 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 So, given all this, what are our results?

13. Deterministic Strategy – Different scenarios Deterministic – Const. Exposure – ↓ Minimum #s Deterministic – Const. Exposure – Const. Minimum #s Constant Exposure Deterministic – Seas. Exposure – ↓ Minimum #s Deterministic – Seas. Exposure – Const. Minimum #s Seasonal Exposure

14. Discrete Strategy – Different scenarios Discrete – Const. Exposure – ↓ Minimum #s Discrete – Const. Exposure – Const. Minimum #s Constant Exposure Discrete – Seas. Exposure – ↓ Minimum #s Discrete – Seas. Exposure – Const. Minimum #s Seasonal Exposure

15. Repertoire Repertoire Repertoire Repertoire Strategy – Different scenarios Repertoire – Const. Exposure – ↓ Minimum #s Repertoire – Const. Exposure – Const. Minimum #s Constant Exposure Repertoire – Seas. Exposure – ↓ Minimum #s Repertoire – Seas. Exposure – Const. Minimum #s Seasonal Exposure

16. Random Strategy – Different scenarios Random – Const. Exposure – ↓ Minimum #s Random – Const. Exposure – Const. Minimum #s Constant Exposure Random – Seas. Exposure – ↓ Minimum #s Random – Seas. Exposure – Const. Minimum #s Seasonal Exposure

17. Repertoire So what if we compare within the same scenario, across strategies: Let’s compare across strategies for Constant Exposure, Constant M Discrete – Const. Exposure – Const. Minimum #s Deterministic – Const. Exposure – Const. Minimum #s Repertoire – Const. Exposure – Const. Minimum #s Random – Const. Exposure – Const. Minimum #s

18. What about for Seasonal Exposure, Constant M Discrete – Seas. Exposure – Const. Minimum #s Deterministic – Seas. Exposure – Const. Minimum #s Random – Seas. Exposure – Const. Minimum #s Repertoire – Seas. Exposure – Const. Minimum #s

19. And for Constant Exposure, Decreasing M Deterministic – Const. Exposure – ↓ Minimum #s Discrete – Const. Exposure – ↓ Minimum #s Repertoire – Const. Exposure – ↓ Minimum #s Random – Const. Exposure – ↓ Minimum #s

20. And for Seasonal Exposure, Decreasing M Deterministic – ↓ Exposure – ↓ Minimum #s Discrete – ↓ Exposure – ↓ Minimum #s Repertoire – ↓ Exposure – ↓ Minimum #s Random – ↓ Exposure – ↓ Minimum #s

21. Those are the results from the work produced What about the number left living?

22. But just to check, did the indirect mortality actually make a difference? Not really – if the strategy is Deterministic These figures are all taken only from the scenarios of constant disease and even minimum numbers required Without Infrastructure Compounded Mortality Deterministic – Const. Exposure – Const. Minimum #s

23. It makes a huge difference if the strategy is Discrete Discrete – Const. Exposure – Const. Minimum #s Without Infrastructure Compounded Mortality

24. It makes a huge difference if the strategy is Repertoire Without Infrastructure Compounded Mortality Repertoire – Const. Exposure – Const. Minimum #s

25. Not Really – if the strategy is Random Without Infrastructure Compounded Mortality Random – Const. Exposure – Const. Minimum #s

26. Take home messages: • These studies are by no means the “answer” to anything, but they are a good way to start examining these sorts of questions • The more specific a disease and population and infrastructure we want to examine, the more appropriately we can tailor the simulations • 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

27. Any Questions?! My thanks to The organizers for inviting me The NSF for funding to DIMACS, where I have been happily visiting for the past year InForMID for additional support All of you for your time and interest Please feel free to contact me with further questions later!