Strong information delay as a driver of epidemic waves: Mathematical modeling for drug trends and epidemic bio-preparedness.

Avoidance behaviors can have a substantial influence on both the spread of an infectious disease and on the long-term dynamics of substance use disorder. However, this behavior is typically reactive based on delayed information related to personal risk as well as both the length of exposure to and retention of that information. This suggests that in epidemic models, feedback delays should be strong - that is, not exponentially distributed (Erlang-1) but instead peaking at a point strictly in the past (Erlang-2 or greater). However, almost all studies of infectivity feedback delays are exponential with qualitatively different results than are seen with strong delays. To address this gap, we analyze two compartmental models for infectious disease epidemiology. Our results demonstrate that sustained oscillations in the total number of active cases may appear even as early as the first two years of an outbreak, suggesting that human behavior may be an explanatory factor for periodic fluctuations evident in recent pandemic time-series data (e.g. COVID-19). We then extend our analysis to a study of the role of information feedback on substance use disorder epidemiology, with a focus on both the transient and asymptotic dynamics of drug waves. We show that under certain conditions, oscillations in substance use disorder can become sustained and that models without strong feedback delays can fail to produce important qualitative transient behavior in substance use incidence rates. To our knowledge, this work represents the first mathematical model exhibiting oscillations with non-contact (linear) pathways to substance use disorder.