The limitations of discrete-time approaches to continuous-time contagion dynamics

TL;DR

This paper critically examines the limitations of discrete-time methods in modeling continuous-time contagion processes, highlighting potential biases and inaccuracies in parameter estimation and epidemic thresholds.

Contribution

It demonstrates how time discretization restricts parameter accuracy, compares simulation schemes, and advocates for event-based methods like Gillespie for better fidelity.

Findings

Discrete-time approaches can bias parameter estimates.

Synchronous updating schemes may distort results.

Event-based simulations accurately replicate continuous-time dynamics.

Abstract

Continuous-time Markov process models of contagions are widely studied, not least because of their utility in predicting the evolution of real-world contagions and in formulating control measures. It is often the case, however, that discrete-time approaches are employed to analyze such models or to simulate them numerically. In such cases, time is discretized into uniform steps and transition rates between states are replaced by transition probabilities. In this paper, we illustrate potential limitations to this approach. We show how discretizing time leads to a restriction on the values of the model parameters that can accurately be studied. We examine numerical simulation schemes employed in the literature, showing how synchronous-type updating schemes can bias discrete-time formalisms when compared against continuous-time formalisms. Event-based simulations, such as the Gillespie algorithm, are proposed as optimal simulation schemes both in terms of replicating the continuous-time process and computational speed. Finally, we show how discretizing time can affect the value of the epidemic threshold for large values of the infection rate and the recovery rate, even if the ratio between the former and the latter is small.