Abstract
We model the Western Roman Empire from 500 BCE to 500 CE, aiming to understand the interdependent dynamics of army size, conquered territory and the production and debasement of coins within the empire. The relationships are represented through feedback relationships and modelled mathematically via a dynamical system, specified as a set of ordinary differential equations. We analyze the stability of a subsystem and determine that it is neutrally stable. Based on this, we find that to prevent decline, the optimal policy was to stop debasement and reduce the army size and territory during the rule of Marcus Aurelius. Given the nature of the stability of the system and the kind of policies necessary to prevent decline, we argue that a high degree of centralized control was necessary, in line with basic tenets of structural-demographic theory. This article was updated on 01/09/2020 to correct an error in equation (3.5). Page numbers were updated on 01/05/2021.
Highlights
Mathematical and numerical modelling has increasingly been passing through disciplinary boundaries, with quantitative models in the social sciences becoming more common
The maximum time horizon for our model is from 500 BCE to 500 CE, and we aim to understand the interdependent dynamics of the size of the army, the territorial expanse and the production and debasement of coins within the empire
For the subsystem of ordinary differential equations (3.1)–(3.3), we find the following approximation for the solution up to t ≤ td: x(t) = A[1 + sin(w[t + Δ])]
Summary
Mathematical and numerical modelling has increasingly been passing through disciplinary boundaries, with quantitative models in the social sciences becoming more common. Cliodynamics is one such recent development (Turchin 2008, 2011), wherein the integration of quantitative methods and historical knowledge brings new insights into human behavior and social institutions. Our research focus is to quantify feedback relationships between a set of variables in the many factors at play in the decline of the Western Roman Empire. There are centuries’ worth of explanations for the decline of the Western Roman Empire. An early starting point for the study of the Roman Empire and its decline is the work of Gibbon (1776). At least 210 reasons and theories have been put forth for the fall of the Roman Empire (Storey and Storey 2017), a list of which was originally compiled by Demandt
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