History-dependent SEIR models: equivalent formulations, fractional foundation, and growth and decay principles
- 1Harbin Institute of Technology
Abstract
Non-exponential residence times give rise to history-dependent or memory effects in epidemic dynamics, yet their mathematical formulations remain fragmented. Here we develop a unified framework for a history-dependent SEIR model with general latent- and infectious-period distributions. We first establish the equivalence between the stage-age and renewal formulations and show that both admit a unified transition-flow representation. We then introduce the concept of a transition-flow multiplier, which provides a direct link between residence-time distributions and transition flows. When the multiplier is a weighted sum of shifted power functions, the model admits an exact representation in terms of Riemann-Liouville or tempered Riemann-Liouville fractional operators, providing a first-principles foundation for fractional epidemic models. Finally, using Euler-Lotka equations and stochastic-order arguments, we characterize how the latent- and infectious-period distributions shape epidemic growth and decay. A longer latent period slows both early growth and late decay, while, at fixed means, greater variability in the latent period accelerates early growth but slows late decay, whereas greater variability in the infectious period slows both.
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BibTeX
@misc{liu2026historydependent,
title = {{History-dependent SEIR models: equivalent formulations, fractional foundation, and growth and decay principles}},
author = {S. Liu and J. Wu and Y. Lou and C. Jia},
howpublished = {medRxiv},
year = {2026},
month = sep,
doi = {10.64898/2026.09.30.26364380},
url = {https://www.medrxiv.org/content/10.64898/2026.09.30.26364380v1},
note = {CC BY},
}
RIS
TY - UNPB
TI - History-dependent SEIR models: equivalent formulations, fractional foundation, and growth and decay principles
AU - S. Liu
AU - J. Wu
AU - Y. Lou
AU - C. Jia
PY - 2026
DA - 2026/09/30
PB - medRxiv
DO - 10.64898/2026.09.30.26364380
UR - https://www.medrxiv.org/content/10.64898/2026.09.30.26364380v1
AB - Non-exponential residence times give rise to history-dependent or memory effects in epidemic dynamics, yet their mathematical formulations remain fragmented. Here we develop a unified framework for a history-dependent SEIR model with general latent- and infectious-period distributions. We first establish the equivalence between the stage-age and renewal formulations and show that both admit a unified transition-flow representation. We then introduce the concept of a transition-flow multiplier, which provides a direct link between residence-time distributions and transition flows. When the multiplier is a weighted sum of shifted power functions, the model admits an exact representation in terms of Riemann-Liouville or tempered Riemann-Liouville fractional operators, providing a first-principles foundation for fractional epidemic models. Finally, using Euler-Lotka equations and stochastic-order arguments, we characterize how the latent- and infectious-period distributions shape epidemic growth and decay. A longer latent period slows both early growth and late decay, while, at fixed means, greater variability in the latent period accelerates early growth but slows late decay, whereas greater variability in the infectious period slows both.
ER -