# Probabilistic Model for Microbial Mortality

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The semi-logarithmic survival curves of microbial cells and spores exposed to a lethal agent have a variety of shapes. A stochastic model based on the inactivation probability of each individual cell or spore can reproduce these shapes. If the underlying probability is constant, the survival curve is log-linear (“first-order kinetics”) and if it rises or falls, the semi-logarithmic survival curve has downward or upward concavity, respectively. If the underlying probability rate function is sigmoid, the semi-logarithmic survival curve can exhibit residual survival or a prominent flat shoulder. This Demonstration also shows that as the initial microbial population size grows, the corresponding survival curve becomes smoother and more deterministic.

Contributed by: Mark D. Normand and Micha Peleg (January 2014)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: survival curve of a small population

Snapshot 2: survival curve of a large population with upward concavity

Snapshot 3: survival curve of a large population with downward concavity

Snapshot 4: survival curve of a large population showing tailing and residual survival

Snapshot 5: survival curve of a large population with a prominent flat shoulder

Consider a live microbial cell or viable bacterial endospore exposed to a lethal agent, be it heat, chemical disinfectant, ultra-high hydrostatic agent, radiation, etc. Assume that the cell or spore can be in one of two states: alive/viable or dead/inactivated. In other words, growth, injury, and adaptation do not occur on the relevant time scale. If a microbe is alive at time , , then after time step , it has a mortality or inactivation probability , where is the probability rate function at time 1, and a probability to remain alive or viable (see the diagram). After a second time step, , which for simplicity remains of the same duration, the mortality probability is and that of survival is . Similarly, for the third time increment, the survival probability is , and so on. Assigning each time step a pertinent unit time, that is, , the survival probability after the time step is [1, 2].

To create a discrete survival curve of a group of cells or spores, start by generating a random number , , and check whether . If so, the cell or spore is considered inactivated and the calculation stops. If not, then generate another and check if , in which case the cell or spore is considered inactivated and the process ends for that particular spore. If not, the process is repeated with new generated and until the cell or spore dies out. The same is repeated for all spores and the tally is recorded. The discrete survival curve is the sum of the number of living cells or viable spores at 0, 1, 2, … time units.

This Demonstration lets you select the model with a setter bar: constant, linearly rising or falling, or sigmoid of two types. Using sliders you can select the seed for the random number generation, the number of points to be generated, the initial number of living cells or viable spores, and the parameters of the probability rate function . The program then plots the probability rate function and below it the corresponding survival curve on linear coordinates and as a semi-logarithmic relationship.

The objective of this Demonstration is not to match any particular micro-organism’s or spore’s survival pattern, but to provide visualization of the relationship between the stochastic model parameters and the survival curve’s shape, for either small or large populations. Consequently, not all allowed parameter combinations have a realistic microbial counterpart. As the number of spores increases, the curve created by the stochastic model becomes smoother and more deterministic. Also, when the mortality probability rate function is sigmoid of type I, the "tailing" survival curve can show residual survival. When is sigmoid of type II, the survival curve can have a prominent flat shoulder.

References

[1] J. Horowitz, M. D. Normand, M. G. Corradini, and M. Peleg, "A Probabilistic Model of Growth, Division and Mortality of Microbial Cells," *Applied & Environmental Microbiology*, 76, 2010 pp. 230–242.

[2] M. G. Corradini, M. D. Normand, and M. Peleg, "A Stochastic and Deterministic Model of Microbial Heat Inactivation," *Journal of Food Science,* 75, 2010 pp. R59–R70.

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