A DETERMINISTIC COMPARTMENTAL MODEL
FOR FLAVOBACTERIOSIS CONTROL IN RAINBOW
TROUT FARMING: FROM EPIDEMIOLOGICAL
INSIGHTS TO MANAGEMENT AND GENETIC
SELECTION STRATEGIES

Auguste Caen, Florence Phocas

► To cite this version:

Auguste Caen, Florence Phocas. A DETERMINISTIC COMPARTMENTAL MODEL FOR FLAVOBACTERIOSIS CONTROL IN RAINBOW TROUT FARMING: FROM EPIDEMIOLOGICAL INSIGHTS TO MANAGEMENT AND GENETIC SELECTION STRATEGIES. Aquaculture Europe 2025, European Aquaculture Society (EAS), Sep 2025, Valence (Espagne), Spain. hal-05246160

HAL Id: hal-05246160

https://hal.inrae.fr/hal-05246160v1

Submitted on 9 Sep 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
A DETERMINISTIC COMPARTMENTAL MODEL FOR FLAVOBACTERIOSIS CONTROL IN RAINBOW TROUT FARMING: FROM EPIDEMIOLOGICAL INSIGHTS TO MANAGEMENT AND GENETIC SELECTION STRATEGIES

Auguste Caen* [1], Florence Phocas [1]

[1]: Université Paris-Saclay, INRAE, AgroParisTech, GABI, Jouy-en-Josas, France

E-mail: auguste.caen@inrae.fr

Statement of the Problem

Flavobacteriosis, caused by Flavobacterium psychrophilum, is a major threat to rainbow trout (Oncorhynchus mykiss) aquaculture, with outbreaks causing up to 70% mortality [1]. Current control strategies are limited, and understanding the epidemiological drivers—such as pathogen excretion, transmission efficiency, and environmental factors—is crucial for effective intervention. While existing studies focus on resistance through challenge experiments, few have developed comprehensive epidemiological models. Agent-based approaches, like that of Brenden et al. [2], provide simulations but lack analytical insights. To address this, we propose a deterministic compartmental model that enables both numerical and analytical examination of flavobacteriosis dynamics. This model offers practical guidance for managing aquaculture parameters during outbreaks and identifies key targets for genetic selection, advancing disease control strategies.

Modeling Approach

Our model follows a SIR-type framework [3]. It represents fish populations through compartmental states: susceptible (S), exposed but non-excreting (E), infected (I) and excreting (at rate β<sub>I(htarrowP)</sub>, recovered and immune (R), and dE) but still excreting (at rate β<sub>D→P</sub>). The pathogen concentration in the system (P) is reduced by water outflow at rate γ. Epidemiological parameters β govern transitions between states, while the mortality removal frequency μ and the water exchange γ are aquaculture-specific factors. The full system is formalized in Equation (1) and illustrated in Figure 1.

$$
\begin{align}
\dot{S} &= -\beta_{S \to E} PS & \dot{R} &= \beta_{I \to R} I \nonumber \\
\dot{E} &= \beta_{S \to E} PS - \beta_{E \to I} E & \dot{D} &= \beta_{I \to D} - \mu D \tag{1} \nonumber \\
\dot{I} &= \beta_{E \to I} E - (\beta_{I \to R} + \beta_{I \to D}) I & \dot{P} &= \beta_{I \to P} I + \beta_{D \to P} D - \gamma P \nonumber
\end{align}
$$

<figure><img src="image_2.png" /><figcaption>Figure 1: Schematic representation of the epidemiological model. Each compartment corresponds to a state variable, and each arrow indicates a transition flow.</figcaption></figure>

Epidemic Intensity and Control

We analytically demonstrate that epidemics always extinguish ($P \to 0$), with a subset of fish remaining uninfected. We pinpoint both farm management strategies – such as increasing the water outflow $\gamma$ – and critical epidemiological traits – such as recovery rate ($\beta_{I \to R}$) and pathogen excretion rate ($\beta_{I \to P}$) – as targets for genetic selection, aiming to
optimize disease resilience in farmed populations. However, the mortality rate ($\beta_{I \to D}$) requires careful consideration: it should be maximized if $\frac{\beta_{I \to P}}{\beta_{I \to R}} > \frac{\beta_{D \to P}}{\mu}$ (prioritizing rapid removal of infected fish) or minimized otherwise (reducing overall mortality). We introduce an epidemic intensity indicator, $Q = E + I + aD + bC$, where $a > 0$ and $b > 0$ are positive scalars dependent on model parameters. $Q$ equals zero if and only if the epidemic is extinct. Our analysis reveals that $Q$ exhibits only two possible behaviors: it either **rises and then declines to zero** (indicating an epidemic that takes hold before fading) or **declines immediately to zero** (indicating an epidemic that fails to establish itself). Through analytical study, we derive a critical threshold for epidemic growth: the epidemic spreads if and only if

$$ \zeta = \frac{\gamma\mu}{\beta_{S \to E} \beta_{D \to C} \beta_{I \to D} + \mu\beta_{I \to C}} < 1 $$

This result provides a clear target for intervention: parameters must be adjusted until $\zeta \ge 1$ to contain the epidemic.

## Model Calibration and Simulation

Calibrating this model is challenging due to the lack of comprehensive monitoring data from aquaculture systems. While existing infection experiments in the literature only allow analysis of specific epidemiological fluxes, we combined multiple datasets to calibrate the model and generate realistic simulations. Figure 2 illustrates two scenarios, corresponding to $\zeta > 1$ (epidemic growth) and $\zeta < 1$ (epidemic extinction).

<figure><img src="image_3.png" /><figcaption>Figure 2: For each simulation, the upper panel shows the proportion of fish in each compartment (susceptible, exposed, infected, recovered, and dead) within the aquaculture system, while the lower panel depicts the epidemic intensity indicator $Q$. Panel (a) corresponds to $\zeta > 1$, where $Q$ declines immediately, indicating an epidemic that fails to establish. Panel (b) corresponds to $\zeta < 1$, where $Q$ initially increases before decaying to zero, reflecting an epidemic that takes hold before subsiding.</figcaption></figure>

## References

[1] A Nematollahi, Annemie Decostere, Frank Pasmans, and Freddy Haesebrouck. Flavobacterium psychrophilum infections in salmonid fish. *Journal of fish diseases*, 26(10):563–574, 2003.

[2] Travis O Brenden, Lori N Ivan, and Thomas P Loch. Reducing flavobacterium psychrophilum transmission risk via hatchery-rearing practices: an individual-based modeling evaluation. *Aquaculture*, 563:738868, 2023.

[3] William Ogilvy Kermack and Anderson G McKendrick. A contribution to the mathematical theory of epidemics. *Proceedings of the royal society of London. Series A, Containing papers of a mathematical and physical character*, 115(772):700–721, 1927.