Extensions of mean-field approximations for environmentally-transmitted pathogen networks

Kale Davies; Suzanne Lenhart; Judy Day; Alun L. Lloyd; Cristina Lanzas; Kale Davies; Suzanne Lenhart; Judy Day; Alun L. Lloyd; Cristina Lanzas

doi:10.3934/mbe.2023075

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 2: 1637-1673. doi: 10.3934/mbe.2023075

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Research article

Extensions of mean-field approximations for environmentally-transmitted pathogen networks

1.
Department of Mathematics, University of Chicago, Chicago, IL, USA
2.
Department of Population Health and Pathobiology, North Carolina State University, Raleigh, NC, USA
3.
Department of Mathematics, University of Tennessee, Knoxville, TN, USA
4.
Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN, USA
5.
Biomathematics Graduate Program and Department of Mathematics, North Carolina State University, Raleigh, NC, USA

Academic Editor: Yang Kuang

Received: 12 August 2022 Revised: 26 September 2022 Accepted: 08 October 2022 Published: 04 November 2022

Many pathogens spread via environmental transmission, without requiring host-to-host direct contact. While models for environmental transmission exist, many are simply constructed intuitively with structures analogous to standard models for direct transmission. As model insights are generally sensitive to the underlying model assumptions, it is important that we are able understand the details and consequences of these assumptions. We construct a simple network model for an environmentally-transmitted pathogen and rigorously derive systems of ordinary differential equations (ODEs) based on different assumptions. We explore two key assumptions, namely homogeneity and independence, and demonstrate that relaxing these assumptions can lead to more accurate ODE approximations. We compare these ODE models to a stochastic implementation of the network model over a variety of parameters and network structures, demonstrating that with fewer restrictive assumptions we are able to achieve higher accuracy in our approximations and highlighting more precisely the errors produced by each assumption. We show that less restrictive assumptions lead to more complicated systems of ODEs and the potential for unstable solutions. Due to the rigour of our derivation, we are able to identify the reason behind these errors and propose potential resolutions.
- environmental transmission,
- moment closure approximations,
- mean-field models,
- network models,
- individual-based models,
- infectious disease
Citation: Kale Davies, Suzanne Lenhart, Judy Day, Alun L. Lloyd, Cristina Lanzas. Extensions of mean-field approximations for environmentally-transmitted pathogen networks[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 1637-1673. doi: 10.3934/mbe.2023075

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Abstract

Many pathogens spread via environmental transmission, without requiring host-to-host direct contact. While models for environmental transmission exist, many are simply constructed intuitively with structures analogous to standard models for direct transmission. As model insights are generally sensitive to the underlying model assumptions, it is important that we are able understand the details and consequences of these assumptions. We construct a simple network model for an environmentally-transmitted pathogen and rigorously derive systems of ordinary differential equations (ODEs) based on different assumptions. We explore two key assumptions, namely homogeneity and independence, and demonstrate that relaxing these assumptions can lead to more accurate ODE approximations. We compare these ODE models to a stochastic implementation of the network model over a variety of parameters and network structures, demonstrating that with fewer restrictive assumptions we are able to achieve higher accuracy in our approximations and highlighting more precisely the errors produced by each assumption. We show that less restrictive assumptions lead to more complicated systems of ODEs and the potential for unstable solutions. Due to the rigour of our derivation, we are able to identify the reason behind these errors and propose potential resolutions.

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