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**|**Appl Environ Microbiol**|**v.76(1); 2010 January**|**PMC2798664

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- Abstract
- Continuous growth models.
- Inactivation models.
- Population dynamics models of growth and/or inactivation.
- METHODS
- RESULTS AND DISCUSSION
- REFERENCES

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Appl Environ Microbiol. 2010 January; 76(1): 230–242.

Published online 2009 November 13. doi: 10.1128/AEM.01527-09

PMCID: PMC2798664

Department of Mathematics and Statistics,^{1} Department of Food Science, University of Massachusetts, Amherst, Massachusetts 01003,^{2} Instituto de Tecnología, Facultad de Ingeniería y Ciencias Exactas, Universidad Argentina de la Empresa, Buenos Aires, Argentina^{3}

Received 2009 June 29; Accepted 2009 November 4.

Copyright © 2010, American Society for Microbiology

This article has been cited by other articles in PMC.

After a short time interval of length δ*t* during microbial growth, an individual cell can be found to be divided with probability *P _{d}*(

Most traditional microbial isothermal growth models are based on the assumption that the growth curve has three regions: a lag time, an exponential growth phase, and a stationary phase during which the number of cells remains unchanged (19, 20). The corresponding mathematical models are generally of two types: empirical analytical expressions and rate equations. Among the first type, the most widely known is the Gompertz model (19), one form of which is

(1)

where *N*(*t*) is the number of cells in a given volume or mass at time *t* and *A*, *B*, *C*, and μ are adjustable, temperature-dependent growth parameters.

The second type of isothermal growth model consists of various modified versions of the continuous logistic equation, Verhulst's model (31):

(2)

where *r* is a temperature-dependent rate constant and *N*_{asympt}, commonly but erroneously dubbed *N*_{max}, is the asymptotic growth level, representing the carrying capacity of the habitat. The model is based on the assumption that the instantaneous growth rate, *dN*(*t*)/*dt*, of an organism introduced into a closed habitat is proportional to the current population size, *N*(*t*), and the fraction of resources still unutilized, 1 − *N*(*t*)/*N*_{asympt}.

According to equation 2, when *t* is ~0 and *N*(*0*) is *N*_{asympt}, *dN*(*t*)/*dt* is ~*rN*(*t*), i.e., the initial growth is approximately exponential. Therefore, Verhulst's model in its original form is unsuitable for growth curves that exhibit a long lag time, unless the initial growth rate is extremely low. A growth model widely used in food microbiology is a modified version of the logistic equation proposed by Baranyi and Roberts (5):

(3)

where *dq*(*t*)/*dt* equals μ_{max}*q*(*t*), *q*(*t*) being the concentration of limiting substrate and μ_{max} being the maximum specific growth rate, and *m* is a constant frequently assumed to have the value of 1.

The term *q*(*t*)/[1 *+ q*(*t*)], which is the same as *q*_{0}exp(μ_{max}*t*)/[1*+ q*_{0}exp(μ_{max}*t*)], is supposed to account for the physiological state of the population at the time of its introduction into a new habitat. Therefore, the Baranyi-Roberts model implies that there is a universal relationship between the lag time and the maximum growth rate (μ_{max}), a highly questionable proposition (9, 22, 23). A lag time (*t*_{lag}) can, of course, be incorporated into any model by replacing the original term *N*(*t*) by *N*_{lag}(*t*), equal to *N*([*t* − *t*_{lag}]^{+}), where (*t* − *t*_{lag})^{+} is 0 for *t* of <*t*_{lag} and is (*t* − *t*_{lag}) for *t* of ≥*t*_{lag}. Then *N*_{lag}(*t*) equals *N*(0) for *t* of <*t*_{lag}, after which *N*_{lag}(*t*) develops according to the original model, with the time origin shifted to *t*_{lag}, for *t* of ≥*t*_{lag}. Other modifications of the logistic equation, such as that by Fujikawa et al. (15), were intended only to improve the logistic model's fit to experimental growth data. Consequently, the authors made no attempt to assign a mechanistic meaning to the added parameters.

None of the above-mentioned models and their variants can describe situations where cell mortality plays a significant role. This is not a serious handicap in modeling food spoilage, for example, where *N*(*t*) is defined as the net number of viable organisms. Thus, even if some cells die out during the exponential phase, their departure will hardly have any practical consequences. This is because most untreated or contaminated foods become inedible or unsafe to eat long before the stationary phase is reached and, therefore, what might happen when the bacterial population starts to die off is of little interest. The situation may be different, however, when pathogens rather than spoilage organisms are involved. In such a case, the pathogen's ability merely to survive in the food may determine the associated health risk.

Both types of traditional models can describe only the sigmoid part of the growth curve. If extended for long periods, they would imply the existence of a stationary phase that lasts indefinitely. This would be a totally unrealistic scenario because a crowded and polluted habitat, depleted of its material resources, cannot support a large population indefinitely. Consequently, mortality must take over at a certain point and reduce the population (19, 20). Under less than favorable conditions, mortality can become predominant even before a stationary phase is reached, resulting in a clearly discernible growth peak (9, 23). The onset of mortality can be incorporated into continuous isothermal growth models in different ways (30). In the first group of models, this can be achieved by multiplying the expression for the population size by a decay term having a characteristic time constant (11, 26), but this is an ad hoc approach to the problem of modeling mortality.

Kinetic inactivation models have been developed primarily to account for or predict microbial survival patterns when the organism is exposed to lethal agents such as heat, extreme cold, chemical preservatives and disinfectants, and ultrahigh hydrostatic pressure, etc., alone or in combination. The literature on the subject is rich and describes several types of models. The most widely known and used primary model is based on the assumption of first-order mortality kinetics (16, 17). With few exceptions, microbial survival and growth have been treated as separate phenomena and described by different kinetic models. There have been attempts to treat bacterial inactivation patterns as if they were inverted analogs of sigmoid growth curves. We consider the merit of this approach questionable because at the cellular level, growth, division, and inactivation are processes governed by very different molecular or biophysical mechanisms, which frequently have very different time scales, too.

A unique approach to modeling bacterial growth and/or inactivation has been described by Taub et al., Doona et al., and Ross et al. (12, 28, 29). Their publications also provide an extensive literature survey of previous works on the subject. These researchers, affiliated with the U.S. Army Natick Soldier Research, Development and Engineering Center, produced a dynamics model of growth and death, which they termed a quasichemical model. Their basic assumption is that the increase (or decrease) in a microbial population is regulated by several rates: the physical growth rates of the individual cells prior to division, the rate of division once cells reach the appropriate size, and the rate of cell death via different mechanistic pathways, as shown schematically in Fig. Fig.1.1. Since each rate has it own characteristic temperature dependence, the model allows for scenarios in which inactivation commences before there is any measurable growth. In other words, heat inactivation is just a special case in which the rate of cell death dominates over the rates of the other processes and consequently determines the treatment's time scale. The quasichemical model captures the essence of the growth-mortality duality, which plays a decisive role in the fate of microbial populations exposed to favorable, unfavorable, or lethal conditions. The above-described approach is not limited to temperature. The promoting or inhibitory factor can be the pH, the sugar concentration, high hydrostatic pressure, the presence of a chemical antimicrobial, or oxygen tension, etc. With its parameters adjusted, the quasichemical model can also be used to quantify the relative influences of the biological processes that control a population's growth or inactivation and how the population is affected by external conditions (12, 28, 29).

Because the mechanisms that operate at the cellular level are rarely if ever known in sufficient detail, description of their dynamics may require a probabilistic approach. In contrast with traditional kinetic models, a probabilistic population model develops from the starting point that crucial information about the underlying processes is or may be missing. This is because the molecular events within the cells that determine the history of a microbial population, including its interaction with the habitat, are or can be too numerous and/or complex to be monitored and accounted for individually. The objectives of the present work are to explore the merits of the probabilistic approach to modeling microbial populations based on cell division and mortality, to develop a prototype of a general microbial growth and inactivation model, and to assess its potential applications.

Consider a population of bacterial cells suspended in a medium of finite volume under conditions that favor growth, at least for some time. For the sake of simplicity, we will assume that an individual cell can be either viable or dead, deliberately ignoring the possibility that it can be injured and then recover or die at a later time. Microbial injury has been amply discussed in the literature (7, 8, 14). Its quantitative implications for food preservation processes have been recently addressed elsewhere (10) and will not concern us here. Similarly, when dealing with mortality, we will ignore the possibility of adaptation at sublethal temperatures or in the presence of a low chemical agent concentration, for example.

In principle, one can continuously monitor each cell, determine its size, assess its physiological state, and establish whether it is about to divide or die. Studies of this kind have actually been performed (1-4, 27), but there is little doubt that monitoring individual cells on a routine basis is impractical. The same can be said about counting the growing and dividing cells over short time intervals, on the order of minutes. Although a video camera and image-processing techniques can be used for the purpose, this approach would also be judged unfeasible under most circumstances. Without knowing the states of individual cells, all we can say is that, within a given time interval, every living cell has a certain probability that it will divide, a probability that it will expire, and a probability that it will remain alive but undivided, as shown in Fig. Fig.2.2. Of course, these three probabilities must add up to 1, and in general they will be time dependent, as will be explained below. The development of a group of cells is shown schematically in Fig. Fig.3.3. We will give a rigorous model for this situation in the next section, but for now we describe a simplified model that can be grasped more intuitively. We make the following two assumptions for this model. These can be checked by statistical techniques and are implicit in all the usual deterministic kinetic models.

Cellular events from which the probabilistic model of microbial growth and mortality has been derived.

(i) The fate of an individual cell can be treated mathematically as if it were independent of those of other cells.

(ii) The behavior of a cell alive at time *t* can be described mathematically as if it were independent of the past behavior of the whole population.

In other words, any actual interaction, current or previous, will be manifested in the division and mortality probability functions *P _{d}*(

Consider a sequence of time points, 0 ≤ *t*_{0} < *t*_{1} < *t*_{2} < …, which for simplicity we assume to be equally spaced, so that (*t _{i}*

There is plenty of evidence that cells introduced into a new habitat, even a resource-rich habitat, may undergo a period of adjustment before they start to divide, hence the lag phase. Inevitably, substantial cell growth and division will deplete the nutrients and other resources in the habitat and will alter its chemical composition through the release of metabolites. Consequently, the individual transition rates *P _{d}*(

Let *N*(0) = *N*_{0} be a given initial cell count, and let *N*(*t _{i}*) denote the population count (the number of live cells) at time

From this model, it is easy to calculate the expected population size at each time *t _{i}*. Given that

(4)

where *k* is the number of steps.

Let us examine two extreme cases of the model, namely, those in which there are (essentially) (i) no deaths and (ii) no divisions. If during a given experimental period, *P _{d}*(

(5)

Likewise, if *P _{m}*(

(6)

It can be shown analytically that, when Δ*t* approaches 0, the expected value of the population size becomes, in the limit,

(7)

where *s* is a dummy variable.

In the two extreme cases described above, this gives exponential growth and decay, respectively, and the slope varies with time. The expected population size can itself provide a continuous, deterministic model. We discuss this in more detail below.

The models specified in equations 4 to 7 are fully deterministic. Given the values of the parameters, they allow for no uncertainty in the development of the microbial population, and they describe the population evolution as a smooth curve. Such a description may be appropriate for a large population of small units, such as microbes, and it may be possible in that situation to fit data with a continuous model. Such a description, however, will generally be unrealistic for small populations of cells. First of all, the number *N*(*t*) of live cells will not be a smooth function of time but rather a step function that changes values at the time of a cell division [*N*(*t*) → *N*(*t*) + 1] or death [*N*(*t*) → *N*(*t*) − 1], and as noted above, the probability of more than one division or death in a very short time interval will be negligible. Figure Figure33 illustrates that certain cells' lineages can proliferate while those of others remain stationary or die out. Also, a lineage that is relatively prolific at the beginning can become extinct for some reason while one that initially struggles may become invigorated after a few inactive generations. One can think of many other possible scenarios in which the trend for certain progeny changes or even reverses direction. Thus, regardless of the details, the emergence of a smooth growth curve for a very small microbial population is very unlikely. The same can be said about the inactivation of a small number of cells. Differences in resistance and/or ability to adapt or recover from injury among individual cells may be expressed in their deaths at irregular times. To account for the actual pattern of growth and mortality in small populations, let us monitor the fates of individual cells. Although the transition rates of division and mortality are fixed smooth functions, the fates of individual cells can vary dramatically. As illustrated in Fig. 3, a live cell at time zero can be found dead, still alive (but not divided), or already divided at time Δ*t*.

The full stochastic model starts from the same assumptions, i and ii, given above but is developed in continuous time from the outset. To emphasize this, we write δ*t* rather than Δ*t* in what follows. By analogy to the simplified model, an individual cell alive at time *t* may divide with a probability of approximately *P _{d}*(

The stochastic model is now characterized by saying that *N*(*t*) is a continuous-time, nonhomogeneous Markov chain, taking values in the nonnegative integers, with allowable transitions *n* → (*n* + 1) and *n* → (*n* − 1) (except that no transitions away from *n* = 0 are allowed). Such a Markov chain, whose only possible transitions are jumps with sizes of ±1, is called a birth and death process, and so we shall refer to this as the birth and death (BD) model. In this context, of course, birth corresponds to cell division. To complete the model, it suffices to specify the rates for each possible transition, and these are given by analogy to the simplified model, i.e., the transition rate at time *t* for the transition *n* → (*n* + 1) is *nP _{d}*(

According to our model, *N*(*t*) starts at *N*_{0} when *t* = 0, remains there for a random period of time, and then makes a transition either to *N*_{0} + 1 or *N*_{0} − 1, corresponding to a cell division or a death. At later times, *N*(*t*) stays at its current value *n* for a random period of time and then makes the transition *n* → (*n* + 1) or *n* → (*n* − 1) (the latter cannot occur if *n* = 0) and continues on in the same fashion. No other transitions are possible.

The complete probabilistic description of a Markov chain typically involves solving an infinite system of differential equations (the Kolmogorov equations) for the transition probabilities of the chain. It is usually not feasible to solve these equations explicitly even in the special case of birth and death processes. Nonetheless, it is possible to obtain useful information about *N*(*t*) directly from the division and death rates. For example, the expected population size at time *t* and the probability that a microbial population will become extinct by time *t* are important quantities to know in the context of food safety; these can be determined explicitly (21). Let

(8)

where *g*(*t*) is the probability function limit.

The expected population size at time *t* in the BD model is the same as the expected population size as Δ*t* approaches 0 in the simplified model, as follows:

(9)

The probability of extinction at time *t*, given an initial population of size *N*_{0}, is

(10)

The integral in this formula is generally difficult to evaluate in closed form but can be calculated numerically. Note that the probability of eventual extinction is obtained by letting *t* approach ∞ in the formula; in particular, Pr(population eventually extinct) equals 1 if and only if the integral (with upper limit *t* = ∞) is infinite. Observe also that the probability of extinction will be >0 except for a pure birth process, in which case *P _{m}*(

In addition, using a standard result in probability theory (Markov's inequality), we can at least give a bound on the probability that the population size exceeds a given threshold at time *t*. Let *M* be a given threshold; then Markov's inequality yields

(11)

Whether this bound is useful or not depends on the relative sizes of *N*_{0} and *M* and on the particulars of the rate functions *P _{d}*(

The discrete equation 4 can be converted into a continuous function by making the time intervals infinitesimally small, i.e., Δ*t* → 0, and the number of steps infinitely large in the simplified model. In this case, the expected population size is given by the formula

(12)

where *P _{d}*(

The integral in equation 12 can be integrated symbolically or numerically with a program like Mathematica (Wolfram Research, Champaign, IL), the program used in this work. For convenience, we will examine only the case in which the transition rates *P _{d}*(

(13)

(14)

Consequently, the general version of the model has six parameters, *p _{d}*,

(15)

and then the function *f*(*t*) *= N*_{0} exp[*g*(*t*)] becomes

(16)

Notice that the model's six parameters can be reduced to five or even four for special cases. Examples are cases in which *P _{d}* equals

The temperature effects on the fate of a microbial population are well known. Refrigeration in most cases slows down the growth or even decimates the cells. Elevating the temperature will have the opposite effect until it is raised to well above that optimal for growth, when it will cause mortality. Extreme cold can have a similar effect, albeit usually not at the same intensity and rate. In the lethal high-temperature regime, raising the temperature accelerates the inactivation dramatically. But there may be situations where premature cooling may allow the survivors to resume growth. A similar situation can also be observed in the case of disinfection with a volatile chemical agent, for example, where the treatment's survivors resume growth. The opposite may also occur. Raw foods are sometimes kept at a temperature that allows for microbial growth, but the population so created is destroyed by adequate cooking. Such growth/inactivation and inactivation/growth scenarios entail influences of temperature on the division and mortality rates and the manner in which these rates change with time. In the model outlined above, this implies that the coefficients of *P _{d}*(

We give two simulation algorithms, one for tracking the development of individual cells and another for simulation at the population level, described in terms of the simplified model, of which the BD model is a continuous-time limit. We assume that the time interval over which the simulation is to take place is from *t* = 0 to *t = t _{f}*, 0 <

(a) Choose the initial number, *N*_{0}, of live cells at time zero and the “time horizon,” *t _{f}*.

(b) Choose Δ*t* small enough so that [*P _{m}*(

(c) At any time *t* = *t _{i}* =

(d) For a particular cell, if *R _{n}* ≤

(e) Let *N*(*t _{i + 1}*) be the new number of living cells.

(f) Repeat steps c to e with *i +* 1 instead of *i* until the desired number of steps is reached.

Notice that since a new random number is drawn for each live cell and at each step, the growth/mortality curves generated with this algorithm will be irreproducible, unless the random number generator “seed” is fixed. After even a small number of iterations, the number of live cells can be very large, so the calculation can become overly burdensome very quickly. This imposes a practical limit on the number of initial cells that can be monitored and the number of steps in the growth/mortality curve construction. The following algorithm avoids this problem to an extent but can have numerical problems when the population size is large.

(a′) Choose the initial number, *N*_{0}, of live cells at time zero and at the time horizon, *t _{f}*.

(b′) Choose Δ*t* so that [*P _{m}*(

(c′) At any time *t* = *t _{i}* =

(d′) If *R _{n}* ≤

(e′) Repeat steps c′ to d′ with *i* + 1 instead of *i* until the desired number of steps is reached.

The effect of the choice δ*t* on the curves produced by the individual-level algorithm is illustrated in Fig. Fig.6.6. As δ*t* becomes smaller, the generated growth curve becomes smoother and gets progressively closer to the continuous curve produced by the LP model based on the same underlying transition rates. As noted above, for computational reasons, the numbers generated by the stochastic (simplified) version of the model were too small to approach the LP-limit model. In some cases, the continuous LP model may not be a good approximation of the simulated model because a significant fraction of the cells can occasionally avoid mortality and exhibit extensive proliferation, resulting in a population size that will surpass that predicted by the LP model. Also, as shown above, except in the case of a “pure birth” process, there is a nonzero probability that the population becomes extinct, resulting in a growth curve that does not resemble the one predicted by the deterministic LP model.

Simulated growth curves generated with the discrete (stochastic) version of the model with the same underlying division and mortality probability functions but different δt's. Notice the curves' increased smoothness as δt decreases and **...**

Examples of life histories of individual cells and their assemblies produced with the algorithm are shown in Fig. Fig.7,7, ,8,8, and and9.9. The figures show, from left to right, model-generated growth curves of three types: those with a noticeable lag, those with no lag, and those with a lag and peak growth followed by mortality. The top graphs are the growth curves, and the ones at the bottom are three-dimensional displays of the life histories of the individual cells (Fig. (Fig.7)7) or groups of 10 and 100 cells (Fig. (Fig.88 and and9,9, respectively). As in Fig. Fig.4,4, the different growth patterns were created by adjusting the underlying transition rates. Figures Figures77 to to99 demonstrate that when the initial number of cells is very small, the fluctuations in the population size during its initial growth stage can be substantial. But if the population is initially large (or has reached a large size through successive divisions [Fig. [Fig.6]),6]), its growth curve becomes progressively smoother. This result should not come as a surprise. When many cells divide or die according to the same rule, the averaging effect suppresses the fluctuations, which could be detected had the fate of each individual cell been monitored. This is the same effect that gives a smooth appearance to the growth curves of large populations of the kind reported in the literature and is a manifestation of the large population limit alluded to earlier.

The stochastic BD model provides a single conceptual framework for comparing the distinct growth and inactivation patterns of small and large microbial populations. It is an excellent tool for simulations, especially for a small initial number of cells and a limited number of generations. For larger population sizes, the LP model provides a biologically motivated, continuous, deterministic model.

If the functions *P _{d}*(

To demonstrate how the model parameters can be estimated from growth or inactivation data, we assume that the isothermal growth/inactivation of a hypothetical organism follows the LP model, equation 12, with transition (probability) rates *P _{d}*(

As pointed out earlier, the continuous LP model differs from the BD model by a Gaussian stochastic process. In general, this error process will be correlated, though it can be expected that the correlations will be short range and hence will die out for widely separated times. Thus, it is reasonable in simulations to use Gaussian random (i.e., uncorrelated) noise for the error process. In analyzing real data, correlated errors will mean that parameter estimates are less efficient (in the technical statistical sense), but this should have little effect in practice.

An example of a simulated realistic-looking sigmoid growth curve is shown in Fig. Fig.10.10. The data were generated by the program with a small amount of random noise equivalent to 5% of the smooth *N*(*t*) level. Since at the plateau region (stationary phase) according to our model, *P _{d}*(

Parameters for growth data generation and retrieved parameters using equation 16 as a model with equations 13 and 14 as the probability rate functions

Published sigmoid growth data on *Escherichia coli* O157:H7, *Salmonella* spp., and *Yersinia enterocolitica* (18) (see ComBase record numbers B124_39 and B002_146 at http://combase.arserrc.gov/), together with the fit of equation 16 as the model, are shown in Fig. Fig.12.12. Also shown in the figure are the corresponding probability rate functions *P _{d}*(

Figure Figure1313 shows published growth/mortality data on *Lactococcus lactis*, *Y. enterocolitica*, and *Salmonella* spp. (13) (see ComBase record numbers B092_4 and B002_92 at http://combase.arserrc.gov/), together with the fit of equation 16 as a model. The corresponding transition rates *P _{d}*(

Admittedly, we have deliberately chosen data that are sufficiently dense and have a relatively small visible scatter. The method may not have yielded meaningful results had a data set with a smaller number of points and/or larger scatter been used. At this point, the functions *P _{d}*(

Most microbial kinetic models for foods have been developed in order to quantify and predict growth and inactivation patterns at the population level. Only a few modelers, notably Baranyi (2, 4), Brul et al. (6), Pin and Baranyi (27), and the Natick group (Taub et al. [29], Doona et al. [12], and Ross et al. [28]), have tried to link observed growth and inactivation kinetics to processes or events at the cellular level directly. The model described in this work is not intended to replace existing kinetic models of either kind. The main reason for its introduction is to provide a general complementary tool to interpret population evolution patterns of cell division and mortality. The approach has already been successfully implemented in the interpretation of pure inactivation patterns, such as those encountered in heat treatments at lethal temperatures. Since, with few exceptions, growth and division do not occur during such treatments, the survival curve is by definition a manifestation of the mortality events' temporal distribution, akin to (but not the same as) *P _{m}*(

The starting point of the probabilistic approach is the admission either that we do not know what happens at the cellular level or that our knowledge is insufficient to derive a plausible mechanistic model. Nevertheless, this work shows that it is possible to translate the probabilities of important events in the life cycle of individual cells into commonly observed growth and inactivation patterns and transitions between them. The work also shows that it is possible to estimate an organism's division and mortality transition rates from experimental data, at least if they are specified in parametric form. The quality of the estimates will depend on the complexity of the model and the amount and scatter of the data. Techniques to monitor the states of individual cells, through image processing, for example, already exist and may eventually be used to validate and improve the modeling approach. For now, experimental evidence can be used to test any specified structure of the transition rates *P _{d}*(

Although much of the discussion in this article focuses on temperature as a promoter of growth or cause of inactivation, the general concept is just as applicable to other factors that make a habitat and conditions favorable or hostile to microbial growth or survival. Possible examples are chemical disinfectants and preservatives, biological antimicrobials, water activity, pH, oxygen tension, and CO_{2} pressure, etc.

The stochastic BD model may be particularly useful to simulate and investigate the growth and mortality patterns of very small microbial populations, for which most if not all the traditional deterministic continuous models may not be suitable. Microbial infection or accidental contamination of a food need not always start with massive invasion by the pathogen or spoilage organism. The former, especially, can start with the ingestion of or contact with a small number of cells. According to the described model, this means that some groups of cells will die off while others may proliferate to various degrees. It also means that both the initial inoculum size and the vigor of the pathogen cells, expressed as *P _{d}*(

This is a contribution of the Massachusetts Agricultural Experiment Station at Amherst.

We thank Murray Eisenberg of the mathematics and statistics department at the University of Massachusetts, Amherst, for his most valuable help in the development of the Mathematica program for curve generation with the deterministic model and the retrieval of its parameters.

^{}Published ahead of print on 13 November 2009.

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