Using mathematical modeling to study the dynamics of Legionnaires' disease and consider management options

Mark Z. Wang; Christina J. Edholm; Lihong Zhao; Mark Z. Wang; Christina J. Edholm; Lihong Zhao

doi:10.3934/mbe.2025045

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 5: 1226-1242. doi: 10.3934/mbe.2025045

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Using mathematical modeling to study the dynamics of Legionnaires' disease and consider management options

1.
Department of Mathematics, Pitzer College, 1050 N Mills Ave, Claremont, CA 91711, USA
2.
Department of Mathematics, Scripps College, 1030 N Columbia Ave, Claremont, CA 91711, USA
3.
Department of Mathematics, Kennesaw State University, 1100 South Marietta Pkwy, Marietta, GA 30060, USA

Academic Editor: Maia Martcheva

Received: 31 December 2024 Revised: 06 March 2025 Accepted: 18 March 2025 Published: 18 April 2025

Legionnaires' disease (LD) is a largely understudied and underreported pneumonic environmentally transmitted disease caused by the bacteria Legionella. It primarily occurs in places with poorly maintained artificial sources of water. There is currently a lack of mathematical models on the dynamics of LD. In this paper, we formulate a novel ordinary differential equation-based susceptible-exposed-infected-recovered (SEIR) model for LD. One issue with LD is the difficulty in its detection, as the majority of countries around the world lack the proper surveillance and diagnosis methods. Thus, there is not much publicly available data or literature on LD. We use parameter estimation for our model with one of the few outbreaks with time series data from Murcia, Spain in 2001. Furthermore, we apply a global sensitivity analysis to understand the contributions of parameters to our model output. To consider managing LD outbreaks, we explore implementing sanitizing individual sources of water by constructing an optimal control problem. Using our fitted model and the optimal control problem, we analyze how different parameters and controls might help manage LD outbreaks in the future.
- Legionnaires' disease,
- optimal control,
- ordinary differential equation,
- mathematical model,
- mathematical biology,
- epidemiological modeling
Citation: Mark Z. Wang, Christina J. Edholm, Lihong Zhao. Using mathematical modeling to study the dynamics of Legionnaires' disease and consider management options[J]. Mathematical Biosciences and Engineering, 2025, 22(5): 1226-1242. doi: 10.3934/mbe.2025045

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Abstract

Legionnaires' disease (LD) is a largely understudied and underreported pneumonic environmentally transmitted disease caused by the bacteria Legionella. It primarily occurs in places with poorly maintained artificial sources of water. There is currently a lack of mathematical models on the dynamics of LD. In this paper, we formulate a novel ordinary differential equation-based susceptible-exposed-infected-recovered (SEIR) model for LD. One issue with LD is the difficulty in its detection, as the majority of countries around the world lack the proper surveillance and diagnosis methods. Thus, there is not much publicly available data or literature on LD. We use parameter estimation for our model with one of the few outbreaks with time series data from Murcia, Spain in 2001. Furthermore, we apply a global sensitivity analysis to understand the contributions of parameters to our model output. To consider managing LD outbreaks, we explore implementing sanitizing individual sources of water by constructing an optimal control problem. Using our fitted model and the optimal control problem, we analyze how different parameters and controls might help manage LD outbreaks in the future.

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