  Infectious diseases are notorious for their complex dynamics, which make it
difficult to fit models to test hypotheses. Methods based on state-space
reconstruction have been proposed to infer causal interactions in noisy,
nonlinear dynamical systems. These "model-free" methods are collectively known
as convergent cross-mapping (CCM). Although CCM has theoretical support,
natural systems routinely violate its assumptions. To identify the practical
limits of causal inference under CCM, we simulated the dynamics of two pathogen
strains with varying interaction strengths. The original method of CCM is
extremely sensitive to periodic fluctuations, inferring interactions between
independent strains that oscillate with similar frequencies. This sensitivity
vanishes with alternative criteria for inferring causality. However, CCM
remains sensitive to high levels of process noise and changes to the
deterministic attractor. This sensitivity is problematic because it remains
challenging to gauge noise and dynamical changes in natural systems, including
the quality of reconstructed attractors that underlie cross-mapping. We
illustrate these challenges by analyzing time series of reportable childhood
infections in New York City and Chicago during the pre-vaccine era. We comment
on the statistical and conceptual challenges that currently limit the use of
state-space reconstruction in causal inference.
