Chapter 21 : Mathematical Modeling Tools to Study Preharvest Food Safety

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Preharvest food safety is a complex system because pathogen transmission and dissemination within a farm environment are determined by multiple interrelated factors, including ecological, evolutionary, environmental, and management drivers that act on different scales of time, space, and organizational complexity. The nonlinear dynamics of pathogen transmission and the complexity of the systems involved pose challenges in understanding key determinants of preharvest food safety and in identifying critical points and designing effective mitigation strategies. Mathematical modeling provides tools to explicitly represent the variability, interconnectedness, and complexity of such systems ( ). In biology, mathematical models have contributed to numerous insights and theoretical advances, as well as to changes in public policy, health practice, and management ( ). Mathematical biology has, indeed, become one of the most prominent interdisciplinary areas of research, but the use of mathematical models in preharvest food safety is recent ( ). Results from modeling research in preharvest food safety have been published mostly since the 2000s. Most of these publications describe the development of epidemiological models to represent foodborne pathogen transmission in animal farming, following the longstanding tradition of applying mathematical modeling in epidemiology ( ).

Citation: Lanzas C, Chen S. 2018. Mathematical Modeling Tools to Study Preharvest Food Safety, p 383-400. In Thakur S, Kniel K (ed), Preharvest Food Safety. ASM Press, Washington, DC. doi: 10.1128/microbiolspec.PFS-0001-2013
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Figure 1

The modeling cycle.

Citation: Lanzas C, Chen S. 2018. Mathematical Modeling Tools to Study Preharvest Food Safety, p 383-400. In Thakur S, Kniel K (ed), Preharvest Food Safety. ASM Press, Washington, DC. doi: 10.1128/microbiolspec.PFS-0001-2013
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Figure 2

Time series (days) of infection prevalence simulated by a stochastic susceptible-infectious model. Ten realizations of the model are presented.

Citation: Lanzas C, Chen S. 2018. Mathematical Modeling Tools to Study Preharvest Food Safety, p 383-400. In Thakur S, Kniel K (ed), Preharvest Food Safety. ASM Press, Washington, DC. doi: 10.1128/microbiolspec.PFS-0001-2013
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Figure 3

Examples of flow charts for epidemiological models commonly used for modeling foodborne pathogens in the animal host. –poultry ( ). –dairy cattle ( ). –swine ( ).

Citation: Lanzas C, Chen S. 2018. Mathematical Modeling Tools to Study Preharvest Food Safety, p 383-400. In Thakur S, Kniel K (ed), Preharvest Food Safety. ASM Press, Washington, DC. doi: 10.1128/microbiolspec.PFS-0001-2013
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Figure 4

An example of a deterministic compartmental infectious disease model. The flowchart represents an SIRS model. Compartments S, I, and R track the number of individuals who are susceptible, infectious, and recovered, respectively. The triangles represent change in the number of individuals in the compartments. Models that contain both demographic (e.g., births and deaths) and infection flows are called endemic models. Compartmental models are often described mathematically as ordinary differential equations. The ordinary differential equations here describe the inflows and outflows by which the number of individuals in each epidemiological state (S, I, and R) changes over time. The total population, N is equal to S + I + R; υ = birth and death rate; γ = recovery rate of infectious individuals; β = transmission coefficient (transmission is frequency dependent); and = immunity loss rate. The inflow and outflow of a compartment is reflected in the equation. For instance, the S compartment has four arrows associated with it: two inflows and two outflows. The two inflows are birth and loss of immunity from recovered compartment (hence the + sign in the first equation); the two outflows are natural death and infection to infectious compartment (hence the – sign in the first equation). The basic reproduction number of this model.

Citation: Lanzas C, Chen S. 2018. Mathematical Modeling Tools to Study Preharvest Food Safety, p 383-400. In Thakur S, Kniel K (ed), Preharvest Food Safety. ASM Press, Washington, DC. doi: 10.1128/microbiolspec.PFS-0001-2013
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Figure 5

Network models are usually represented as graphs. A graph contains a set of nodes (individuals) and edges (contacts) that are associated with these nodes. Shown is an example with five interacting calves. The calves with contact with each other have a line (edge) between them to indicate their mutual relationship . The graph can be translated to an adjacency matrix, which contains as many rows and columns as there are nodes . The elements of the matrix record information about the edges between each pair of nodes, with 1 indicating a contact and 0 meaning no contact.

Citation: Lanzas C, Chen S. 2018. Mathematical Modeling Tools to Study Preharvest Food Safety, p 383-400. In Thakur S, Kniel K (ed), Preharvest Food Safety. ASM Press, Washington, DC. doi: 10.1128/microbiolspec.PFS-0001-2013
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