Notes on Complex Systems, Part 2

Megan Molteni at Wired has a great story about mathematical modeling and epidemiology ("The Mathematics of Predicting the Course of the Coronavirus," Wired, March 30, 2020). One key passage:

As with simulations of Earth’s changing climate or what happens when a nuclear bomb detonates in a city, the goal here is to make an informed prediction—within a range of uncertainty—about the future. When data is sparse, which happens when a virus crosses over into humans for the first time, models can vary widely in terms of assumptions, uncertainties, and conclusions. But governors and task force leads still tout their models from behind podiums, increasingly famous modeling labs release regular reports into the content mills of the press and social media, and policymakers still use models to make decisions. In the case of Covid-19, responding to those models may yet be the difference between global death tolls in the thousands or the millions. Models are imperfect, but they’re better than flying blind—if you use them right.

The basic math of a computational model is the kind of thing that seems obvious after someone explains it. Epidemiologists break up a population into “compartments,” a sorting-hat approach to what kind of imaginary people they’re studying. A basic version is an SIR model, with three teams: susceptible to infection, infected, and recovered or removed (which is to say, either alive and immune, or dead). Some models also drop in an E—SEIR—for people who are “exposed” but not yet infected. Then the modelers make decisions about the rules of the game, based on what they think about how the disease spreads. Those are variables like how many people one infected person infects before being taken off the board by recovery or death, how long it takes one infected person to infect another (also known as the interval generation time), which demographic groups recover or die, and at what rate. Assign a best-guess number to those and more, turn a few virtual cranks, and let it run.

Their discussion of the SIR model is key. As the link in the story above lays out, the acronym is derived from the three key components that structure the model: The susceptible fraction of the population; the infected fraction; and the recovered fraction of the population. But one of the challenges of this model seems to be that it takes geography for granted. Necessarily, it assumes that a given population exists within a given container. There are good reasons to do this, of course: It simplifies the model and limits the numbers of variables we need to worry about.

So this took me to a recent review article about modeling infection disease (Dongmei Chen, "Modeling the Spread of Infectious Diseases: A Review," in Analyzing and Modeling Spatial and Temporal Dynamics of Infectious Diseases [ResearchGate link]). Chen lays out several different kinds of modeling [all new to me, so this is useful]:

• Mathematical models [like the SIR model above]: This approach begins from a mathematical model to describe how diseases spread and propagate. 
• Statistical models: This approach differs in that it begins from data and then seeks to fit an equation to that data.
• Gravity models: Here, the underlying assumption seems to be that interactions (both social and natural) conform to (or at least echo) the basic premise of Newton's Law of Gravity - namely, that the attraction between two objects is proportional to their mass (bigger objects are more strongly attracted to each other) and inversely proportional to the square of the distance between them [and I'm sure somebody has written about Tobler's Law in relation to this?]
• Network-based models: This differs in that what matters seems to be less the nature of the objects (their mass and distance) and more the number and type of connections between various nodes. [Echoes of Latour.]

As Chen continues, increasing computing power has made it possible to run simulations that mobilize the strengths of various models, in turn helping us better understand how diseases move not simply within populations but also spatially. This, in turn, opens up a way to model disease spread and propagation not based on populations [with their implicit geographies] but based on individuals [agent-based modeling]. Crucially, however, she notes that this is not a predictive model - which is one of the challenges that Molteni's article raises (namely, the way that models and predictions come to be confused in practice):

The underlying framework of this model may be applicable for modeling the transmission of many epidemic diseases, and offers researchers an opportunity to better understand the conditions under which an epidemic may occur. In agent-based simulation practices, parameter tuning for individual heterogeneity is a significant challenge due to the complicity of individual behaviors and/or data insufficiency (Chao et al. 2010). (Chen, "Modeling the Spread of Infectious Diseases," p. 33)

[Now a cardinal in the magnolia; signs of spring]

Chen's cautionary note about the limits of agent-based models in helping us predict echoes something I noted the other day in Andy Dobson's interview.