Cite as:

Alexander F. Siegenfeld, Pratyush K. Kollepara, and Yaneer Bar-Yam, Modeling complex systems: A case study of compartmental models in epidemiology, Complexity (October 29, 2022)

Abstract

Compartmental epidemic models have been widely used for predicting the course of epidemics, from estimating the basic reproduction number to guiding intervention policies. Studies commonly acknowledge these models’ assumptions but less often justify their validity in the specific context in which they are being used. Our purpose is not to argue for specific alternatives or modifications to compartmental models, but rather to show how assumptions can constrain model outcomes to a narrow portion of the wide landscape of potential epidemic behaviors. This concrete examination of well-known models also serves to illustrate general principles of modeling that can be applied in other contexts

Introduction

Compartmental models such as the SIR model have been widely used to study infectious disease outbreaks [1–5]. These models have informed policy makers of the risks of inaction and have been used to analyze various policy responses. The limitations of the assumptions of compartmental models are wellknown [6–8]; we intend to explore which assumptions are appropriate in which contexts and when and why the models do or do not succeed. No model accurately captures all the details of the system that it represents, but some models are nonetheless accurate because certain large-scale behaviors of systems do not depend on all these details [9]. (For example, modeling material phase transitions generally does not require including the quantum mechanical details of individual atoms.) The key to good modeling is understanding which details matter and which do not. Paradoxically, failing to recognize that a model can be accurate in spite of certain unrealistic assumptions can lead to models in which all assumptions are excused: the impossibility of getting all the details right may discourage a careful analysis of which assumptions are appropriate in which contexts. During a pandemic, it is crucial that models complement decision-making. In an attempt to obtain better predictions, it may be tempting to include more details and fine-tune the model assumptions. However, focusing on irrelevant assumptions and details while losing sight of the large scale behavior is counterproductive [10]. Which details are relevant depends on the question at hand; the inclusion or exclusion of details in a model must be justified depending on the modeling objectives. Compartmental models tend to include some details (e.g. disease stages) while not including others (e.g. stochasticity and heterogeneity) that, in many cases, have a far larger effect on forecasting the epidemic trajectory, estimating the final epidemic size, and analyzing the impact of interventions. In this work, we examine some common assumptions of compartmental models—such as the distribution of generation intervals, homogeneity in population characteristics and connectivity, and the use of continuous variables—in order to determine their relevance for various model outcomes. Our purpose is not to argue for specific alternatives to compartmental models or for specific modifications but rather to illustrate how the assumptions of these models affect their results.

FIG. 1. Schematic representation of the impact of various modeling choices/assumptions. The left column lists various details that can be incorporated into a compartmental model (boxes with dashed borders indicate modeling choices that are analyzed in section III), and the right column lists typical potential impacts on the model output. The three panels classify the system details by ‘scale’, with the largest scale details typically having the most impact on model output, and the smallest scale details typically having the least impact, although the impact of any given assumption ultimately depends on precisely for what purpose the model is being used. For instance, an SIRS model may not be needed if only the initial growth of the epidemic is being modeled. Furthermore, various assumptions can compound non-linearly to affect the model output. For instance, policy interventions such as travel restrictions, which both rely on and affect heterogeneity in geographical connectivity, can play a decisive role in determining whether or not a stable elimination is achieved [11]. Of course, the actual effect of any assumption depends on its precise mathematical implementation, as well as the presence or absence of other assumptions within the model, and so this figure should be considered as a rough schematic rather than as a definitive guide.

FIG. 1. Schematic representation of the impact of various modeling choices/assumptions. The left column lists various details that can be incorporated into a compartmental model (boxes with dashed borders indicate modeling choices that are analyzed in section III), and the right column lists typical potential impacts on the model output. The three panels classify the system details by ‘scale’, with the largest scale details typically having the most impact on model output, and the smallest scale details typically having the least impact, although the impact of any given assumption ultimately depends on precisely for what purpose the model is being used. For instance, an SIRS model may not be needed if only the initial growth of the epidemic is being modeled. Furthermore, various assumptions can compound non-linearly to affect the model output. For instance, policy interventions such as travel restrictions, which both rely on and affect heterogeneity in geographical connectivity, can play a decisive role in determining whether or not a stable elimination is achieved [11]. Of course, the actual effect of any assumption depends on its precise mathematical implementation, as well as the presence or absence of other assumptions within the model, and so this figure should be considered as a rough schematic rather than as a definitive guide.

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