Sketches of Applications in Temporal Paths

Disease Transmission

An early goal that manifested from the efforts undertaken in exploring temporal sheaves is the tracking of disease transmission. Specifically, the timing by which individuals infect one another.

Note: The origins of this particular sketch is indebted to Nate Osgood, Nelson Niu, Juxin Liu, and Priyaa Varshinee Srinivasan! 😄

Zigzag Category with Temporal Sets

Consider the following Zigzag category defined for three time points (which I refer to as , , and ):

At each of these time points, there exist an associated temporal set and inclusion morphisms that we can declare explicitly as follows:

Scenario: Flu Transmission

For this scenario, we are interested in the transmission of flu. Elaborating on what the temporal sets "mean" in this scenario, we say that the temporal sets contain people who were in some form of contact with one another. Moreover, each person has an associated "state" of infection which is the following:

• "S" - person susceptible to infection

• "I" - person infected

• "R" - person recovered

Before continuing, let's describe the story of this scenario:

• At , Jacob and Sanna were in contact.

  • Sanna is infected

  • Jacob is susceptible

• At , Jacob, Sanna, and Tino were in contact.

  • Sanna is infected

  • Jacob is infected

  • Tino is susceptible

    • Tino appeared in this temporal set at this time point meaning Tino came into contact with Sanna and Jacob.

• At , Sanna and Tino were in contact.

  • Sanna is recovered

  • Tino is infected

  • Jacob left this temporal set at this time point meaning Jacob was no longer in contact with Sanna and Tino.

We can draw this depiction as follows:

Here, we have another cospan below the temporal sets that represent the notion of "state" associated to each element of the temporal set (i.e. each person has a state of infection).

Note: Categorically, I do not know what to call this structure just yet or even how to exactly define it. I fear I may be getting ahead of the mathematics of the ZigZag category because I effectively want something like  .

Analyses on the Temporal Sets

We can summarize the information present in the attributed combinatorial data on each temporal set as the following table:

Person
Jacob	S	I	-
Sanna	I	I	R
Tino	-	S	I

This data can be derived from the temporal sets at each time point. Taking this forward, we can think of deriving path graphs that represent the path of infection for each person as follows:

Here, the red nodes represent if that person was infected at that time point and the black nodes represent if they were not (I am treating "S" and "R" as the same for simplification purposes). However, the question that arises from this information is, how can we determine the lineage of infection for Tino? Or put another way, "where did the infection that Tino contracted arise from?" Thinking about this question leads to consideration of gluing operations.

At this stage, the actual mathematical machinery is limited, but the thing I am trying to say here is that:

1. Looking back at this table, we know when the earliest infected person arose (Sanna).

2. We know that at , Jacob was infected and Tino was in contact with Sanna and Jacob.

3. We know that while Sanna maintained contact with Tino, Sanna recovered while Tino became infected.

If we want to order the flu transmission, taking these facts, we could know that Sanna was initially infected. While Tino came into contact with Jacob and Sanna at the same time, Jacob wasn't infected initially. A potential path graph that orders these temporal occurrences in disease spread is as follows:

Where This Can Be Applied

Some areas on where to expand this to are to investigate questions such as:

• How long it takes from exposure to infection

• Understanding how infectious the disease is within the infection time period

• Principled approach to record information during outbreak investigation

And there are several more. Moreover, what has been absent discussion is the fact of conducting additional analysis on the categorical structure scaffolding the temporal sets themselves. Conducting actual investigations on hom-sets, for example, could yield completely novel new ways to investigate the temporal dynamics of a disease spread through the lens of a "compositional public health" approach where public health methods are grounded in categorical machinery.

Treatment Patterns

Treatment pattern analysis is an emerging concept across health informatics (particularly in the context of retrospective and observational health settings involving patients) that looks at a patient's continuum of care. For a given patient population, the goal is to find the treatment pathway consisting of selected "treatments of interest" (i.e. event cohorts). These treatments of interest (or events) can be drug prescribing, other therapies, procedures, or measurements. The treatment pathway are such sequences of events over time. [2]

Note: This idea is less developed; I wanted to draft the idea as a sketch so I could later return to it.

Scenario: Event Relationships in an Emergency Operation

A patient encounter with a care provider (such as a hospital, outpatient clinic, walk-in clinic, etc.) presents a panoply of data around just that one patient visit. A way to temporally view this is as follows:

Each of these Zigzag categories of only two time points represents one patient across three different perspectives being:

• Visit Occurrence (Red): where persons engage with the healthcare system for a duration of time.

• Condition Occurrence (Green): the presence of a disease or medical condition stated as a diagnosis, a sign, or a symptom, which is either observed by a Provider or reported by the patient.

• Drug Exposure (Blue): exposure to a drug ingested or otherwise introduced into the body.

To further explain this example, the zigzags record a sequence of events for one individual patient. Each zigzag is equipped with temporal sets associated to what is being tracked with the red zigzag contains visit information, the green zigzag contains condition diagnosis information, and the blue zigzag contains prescription information. The black arrow does not mean anything but to provide a sense of depth to the figure as you can imagine the three zigzags occuring in parallel

As an example, this individual patient could be someone who has a medical emergency requiring surgery. A flow could be:

• At :

  • The patient visits the emergency department

  • The patient is diagnosed with a condition requiring immediate surgical attention

  • The patient is then prescribed anti-inflammatory medications and painkillers for recovery

• At :

  • The patient is moved into recovery

  • The patient is re-assessed and receives another diagnosis

  • The patient no longer needs painkillers and is moved to maintenance medication

• This continues until the patient is discharged

Implicit in this example is a preorder that can exist between the elements of the zigzags being something like:

While the zigzags here are denoted as having the same number of discrete time points, this approach does not mean in reality, they occur at the exact same time. Instead, the zigzag structure endows the relationships across and within the zigzag to have temporally the "same" sequence but the timings can be related across one another agnostic of the actual timing of the event.

Note: The definitions provided above come from within observational health research settings and reflect general perspectives across health informatics [3].

Where This Can Be Applied

Some general ideas about where this could be applied are:

• Overprescription of opioids affecting vulnerable populations

• Principled understanding of prescription cascades in geriatric populations

• End-to-end understanding care pathways for a given disease

The current way to address these questions and concerns are often laborious and ad hoc methods. Moreover, given the lack of structure with much clinical data, the zigzag structure provides one of the most advantageous and generalized approaches to recording temporal data.

1. A. F. Markus, K. M. Verhamme, J. A. Kors and P. R. Rijnbeek. TreatmentPatterns: an R package to facilitate the standardized development and analysis of treatment patterns across disease domains. Computer Methods and Programs in Biomedicine 225, 107081 (2022).

2. OHDSI Community. Observational Medical Outcomes Partnership Common Data Model (OMOP CDM), https://github.com/OHDSI/CommonDataModel (2023). Version 5.4.