Compartmental Models for Predicting the Level Listerial Contamination in a Damaged Silage Bale. L. A. Kelly and G. Gettinby, Department of Statistics and Modelling Science, University of Strathclyde, Glasgow, UK; and G. Gibson, BioSS, University of Edinburgh, Edinburgh, UK
INTRODUCTION
Listeriosis is a disease of both humans and animals which is caused by the bacterium Listeria monocytogenes, the pathogenic species of the genus Listeria. The disease occurs both sporadically and in epidemic form (Ryser and Marth, 1991) and although it is considered rare, it often has fatal consequences.
In domestic animals, listeriosis is most frequently observed in ruminants and in particular sheep. During the late 1970s and throughout the 1980s an increasing incidence of ovine listeriosis was observed and during this period, peak occurrence was in late winter and early spring. The increase in incidence and seasonality of ovine listeriosis have recently been attributed to big-bale silage feeding (Wilesmith and Gitter, 1986). This type of fodder, which became popular during the 1980s, is fed during periods of winter housing and when grass is limited.
Silage bales are produced by the controlled anaerobic. i.e. oxygen-free, fermentation of high moisture content crops such as grass (McDonald, 1981). To achieve anaerobic conditions, the silage is wrapped in plastic covering. However, this covering is easily damaged in the farm environment and as a result, oxygen may permeate the material. The presence of oxygen breaks down the fermentation process, initiates aerobic deterioration and renders the silage suitable for the growth of L. monocytogenes. Under these conditions, any dormant organisms which have survived initial fermentation rapidly begin to grow, multiply and spread through the bale. As a result, when the bale is opened, the fodder may present a risk to feeding animals.
Mould is a characteristic of decomposed silage and current advice to farmers focuses on the removal of mouldy material from the bale. However, as Fenlon (1986) points out, totally aerobic silage may not harbour L. monocytogenes as a result of competition from more vigorous aerobic organisms. A more structured decision making process for feeding silage is thus required.
Here we present a mathematical model which has been developed to introduce such structure into the decision making process. The model is a compartmental model which predicts the level of listerial contamination of a silage bale at time points following damage. From these predictions, the risk of feeding at different times can be quantified and key parameters in the process can be identified.
THE COMPARTMENTAL MODELLING FRAMEWORK
Compartmental models are used to describe the evolution of a population of individuals over time. Individuals are partitioned into various compartments and the evolutionary process results from transitions from one compartment to another. Perhaps the most well known example of compartmental models are the epidemic models for human and animal diseases (see e.g. Bailey, 1975). Here susceptibles, infectives and removals are used to predict quantities such as the size and duration of the epidemic.
The compartmental framework discussed here is based on the following biological hypothesis. We consider the bale as n distinct sites and at any time, each of these sites is either populated or not populated with L. monocytogenes. In a populated site, organisms may be dormant or active, where activity refers to growth and multiplication. Hence a three state process results and these states are dormant, active and unpopulated. At time t = 0, all populated sites are dormant and at this time damage occurs to the bale. Following damage, activity is initiated. Dormant organisms may become active, spread to other unpopulated sites and then either die or return to the dormant state. At the same time, dormant organisms may die, thus rendering the site they occupy to the unpopulated state. This process continues throughout the storage of the bale and at opening, a proportion of the bale will be populated with L. monocytogenes.
Linear, non-linear, deterministic, stochastic, spatial and non-spatial transitions have been explored to determine the proportion of the silage bale which is contaminated at any time following damage. As the models become more complex, more realism is incorporated and in turn the solution becomes more difficult. The model presented here is non-linear, due to interactions between active and unpopulated sites, and non-spatial, because sites are not considered in relation to their location within the bale.
A NON LINEAR MODEL FOR THE RISK OF SILAGE FEEDING
We make the following definitions
Xd(t) - number of dormant sites at time t
Xa(t) - number of active sites at time t
Xu(t) - number of unpopulated sites at time t
Pc(t) - proportion of silage contaminated at time t
The system is represented by the compartmental diagram in
figure 1. Here nodes represent the various compartments or states
and arcs from these nodes define the state transitions which can
occur. The transitions are defined by the transition parameters 
These
parameters depend on the site size and hence on the value of n.
The transitions in figure 1 lead to the deterministic system of
differential equations (1) for the population variables Xd(t),
Xa(t) and Xu (t).
 |
(1) |
 |
Figure 1: Compartmental representation of the non linear model.
In this model, it is assumed that spreading between active and unpopulated sites is homogeneous, hence any active site can populate any unpopulated site. The non-linear nature of the model means that, in this case, the system cannot be solved exactly. Numerical solutions can however be obtained with ease.
The process can also be modelled stochastically, by considering the probabilities of transitions occurring in small time intervals. As in the deterministic case, the non linear nature of the process renders the stochastic model intractable. As an alternative to analytical investigation, the process can be simulated on a computer to calculate the means and variances of the population variables. By generating such quantities, an insight into the random variation which will be inherent in nature is obtained.
RESULTS
Figure 2 shows the numerical solution for Pc(t),
obtained using the numerical integration package SOLVER ©STAMS,
together with one stochastic realistion, obtained using a Pascal
program, for the parameter set 
= 0.1, 
= 0.001
and n = 1000. At the start, the number of dormant sites is
500, i.e. Xd(0) = 500, the number of
active sites is zero, i.e. Xa(0) = 0,
and the number of unpopulated sites is 500, i.e. Xu(0)
= 500. This parameter set is for demonstration purposes only
and does not reflect the true biological situation.
 |
Figure 2: Numerical solution and one stochastic realistion for Pc(t).
Consider the numerical solution. The proportion of silage contaminated, Pc(t), decreases slightly then increases exponentially to an equilibrium value of 0.85. This equilibrium results when the flows into and out of the three states, dormant, active and unpopulated, are zero. Equilibrium is reached in approximately 11 days since damage and thus if the bale is damaged 11 or more days prior to opening, 85% of the material will pose a risk to feeding animals. The proportion Pc(t) can also be thought of as the probability that the bale is contaminated, i.e. the probability that the bale poses a risk.
The stochastic realistion also decreases and then increases, fluctuating around the deterministic numerical solution. We would expect similar fluctuations from other stochastic realisations.
DISCUSSION
The model presented here is one in a series of compartmental models which have been developed to obtain estimates for the proportion of a silage bale which is contaminated in a time period following damage. More realistic and complex models include a spatial element and also incorporate the notion of site suitability for bacterial growth and multiplication. In a spatial sense, spreading is assumed to be nearest neighbour and the results generated from these models show the variation in contamination levels with depth into the silage from the site of damage. To incorporate site suitability for bacterial growth, results from a mathematical model for the aerobic deterioration of silage (Ruxton and Gibson, in press) have been used.
The models are a first attempt at providing a quantitative measure for the risk of silage feeding. This measurement can be enhanced by considering the digestion and disease processes for feeding animals. It is vital for a structured decision process for feeding and in turn for highlighting strategies for disease control disease control.
REFERENCES
Bailey, N. T. J. (1975) The mathematical theory of infectious diseases. Charles Griffin and Co. Ltd, High Wycombe.
Fenlon, D. R. (1986) Growth of naturally occurring Listeria spp. in silage: a comparative study of laboratory and farm ensiled grass. Grass and Forage Science, 41, 375-378.
McDonald, P. (1981) The biochemistry of silage. John Wiley and Sons Ltd., Chichester.
Ruxton, G. D. and Gibson, G. J. (in press) A mathematical model for the aerobic deterioration of big-bale silage and it's implications for the growth of Listeria monocytogenes.
Ryser, E. T. and Marth, E. H. (1991) Listeria, listeriosis and food safety. Dekker Inc.. New York.
Wilesmith, J. W. and Gitter, M. (1986) Epidemiology of ovine
listeriosis in Great Britain. Veterinary Record, 119,
467-470.