# Reinfection induced multistability in an epidemic model

Pierre-Alexandre Bliman, Marcel Fang

► To cite this version:

Pierre-Alexandre Bliman, Marcel Fang. Reinfection induced multistability in an epidemic model. Biomath 2025 - international conference on Mathematical Methods and Models in Biosciences, Jun 2025, Sofia, Bulgaria. hal-05172573

HAL Id: hal-05172573

https://hal.science/hal-05172573v1

Submitted on 21 Jul 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Reinfection induced multistability in an
epidemic model

Pierre-Alexandre Bliman¹, Marcel Fang¹

¹Sorbonne Université, Université Paris Cité, CNRS, Inria, Laboratoire Jacques-Louis Lions, LJLL, EPC MUSCLEES, F-75005 Paris, France

pierre-alexandre.bliman@inria.fr, marcel.fang@inria.fr

We consider in this presentation the effects induced on the transmission of a
disease with imperfect immunity, by differences between primary and subsequent
infections (i.e. reinfections), due e.g. to differential susceptibility or differential
infectiousness. To this end, an 8-dimensional *two-stage* SEIRS reinfection model
that extends the classical SEIRS model [1] is proposed, in which the parameters
characteristic of the disease dynamics are different between primary infections
and secondary infections. The number of steady states of this model is assessed,
depending upon the parameter values. It is shown that the reinfection induced
heterogeneity may cause up to two endemic equilibria when the basic reproduc-
tion number of the model is less than one, and up to three endemic equilibria
when it is greater than one. This suggests that, according to the parameter
values, the model may present backward bifurcation, a feature quite important
from the point of view of disease control. Simulations confirm this situation.
The persistence of the disease is also examined. Finally, we turn our attention to
the study of several models with partial immunity, which can be considered as
particular cases of the previous one. Based on a geometric approach developed
by Li and Muldowney [3], it is shown that every trajectory of these simpler
models converges asymptotically to an equilibrium, which depends upon the
initial condition in case of multistability. More details may be found in [2].

Keywords: Compartmental models in epidemiology; reinfection models; multistability; backward bifurcation; persistence; Li and Muldowney theory

References

[1] O.N. Björnstad, K. Shea, M. Krzywinski, N. Altman, The SEIRS model for infectious disease dynamics, *Nature Methods*, **17** (6), 2020.

[2] M. Fang, *Mathematical modelling, observation and identification of epidemiological models with reinfection*, PhD thesis, Sorbonne Université, 2024.

[3] M.Y. Li, J.S. Muldowney, A geometric approach to global-stability problems. *SIAM Journal on Mathematical Analysis*, **27** (7), 1996.