Author: Jade Mitchell and Sushil B. Tamrakar

General overview

Inactivation of vegetative microorganisms, there are several types of survival curves that described the inactivation rates and patterns(Xiong, Xie et al. 1999)[1]. The most commonly used mathematical linear model is first order exponential model. However, in many cases linear model alone cannot describe the prevailing pattern. There are several nonlinear models described by various investigators to fit the data (Coroller, Leguerinel et al. 2006)[2]. Seven different models described in various studies have been shown in Table 1.(Peleg and Cole 1998[3]; Juneja, Eblen et al. 2001[4]; Valdramidis, Bernaerts et al. 2005[5]; Juneja, Huang et al. 2006[6]).

The data from each treatment was fitted to a best-fit curve using an unpublished mathematical model fitting tool in Microsoft® Excel (Microsoft® Inc., Redmond, Washington) by Patrick Gurian at Drexel University and modified by Sushil Tamrakar (Michigan State University). The tool can be used to model bacterial survival in culture-dependent or culture-independent methods independent of the organism and the environmental conditions. The smallest absolute value of the Bayesian information criterion (BIC) was the criteria to choose the best fit model.

Table 1 Persistent models and equations

S.N.

Model

Equation

Curve Properties

Reference

1

First order exponential decay model

Ln (Nt/N0) = -kt

Linear, negative slope

Crane and Moore, 1986

2

Biphasic exponential decay model

for 0≤t<x: Ln (Nt/N0) = -k1*t

for t≥x: Ln (Nt/N0) = -k1*t+k2*(t-x)

Linear, negative slope, slope changes at t=x

Carret et al. 1991

3

General logistic model

Ln (Nt/N0) = ln(2/(1+e(-kt)))

Nonlinear, concave

Gonzalez, 1995

4

Exponential damped model

Ln (Nt/N0) = -kt*e(-st)

Nonlinear, concave

Cavalli-Sforza et al. 1983

5

Gompertz model

Ln (Nt/N0) = ln[C*e [-e^(-b*(ln(t)-a))]}

Nonlinear, concave

Gompertz, 1825

Gil et al. 2011[7]

6

Two-stage model Juneja and Marks (1)

Ln (Nt/N0) = -Ln(1-(1-e(-kt)^m)))

Nonlinear, concave

Juneja et al. 2006

7

Log-logistic model Juneja and Marks (2)

Ln (Nt/N0) = -Ln(1+e (a+b*ln (t)))

Convex or concave

Juneja et al. 2003

8

Gompertz-3 (gz3)

Ln (Nt/N0) =(k1*exp(-exp((-k2*exp(1)*(k3-t)/k1)+1) ) )

k1, k2, k3 > 0
nonlinear concave

Gil et al. 2011[7]

9

Gompertz-Makeham (gzm)

Ln (Nt/N0)=(-k3*t -k1/k2*(exp^((k2*t)).-1))

k1, k2, k3 > 0 
nonlinear concave

Jodra, 2009[8]

10

Weibull (wb)

Ln (Nt/N0)=-{(t/k1)}^k2

k1= treatment time for first decimal reduction 
k2=shape parameter 
concave or convex

Coroller et al., 2006[2]; Mafart et al., 2002[9]

11

Gamma (gam)

Ln (Nt/N0)=(t^(k2-1) ) exp^((-t/k2)).

Nonlinear concave

[1],[2]

12

Sigmoid-B (sB)

Ln (Nt/N0)=-(k1*t^k3)/(k2+t^k3 )

Nonlinear concave or convex

Peleg 2006[10]

13

Logistic-Fermi Combination

N(t)/N_0 =1/(1+exp(k1*(t-k2)))

Nonlinear, concave

Peleg 2006[10]

14

Log-normal

N(t)/N_0)=[1-Φ{(ln(t)-µ)/б}]

Concave, nonlinear

Aragao 2007.

15

Biphasic(3 parameters)

if (min(t)<k3)
N(t)/N_0 = exp(-k1*t[t<k3])
if (max(t)>=k3)
N(t)/N_0 = exp(-k1*t[t>=k3]+k2*(t[t>=k3]-k3))

Nonlinear, concave

Kamau et al. (1990)

16

Double exponential

N(t)/N_0 =a exp(-k1t) + (1-a) exp(-k2t)

Nonlinear, concave

Peleg 2006[10]

17

Sigmoid type A

log(N(t)/N_0 )=(a_1 t)/([1+a_2 t][a_3-t])

Nonlinear, S curve

Peleg 2006[10]

 

The data from fomite recovery experiments in Dr. Charles Gerba’s lab were fit to the different persistent models. The best fit models are shown in Table 2 and Figure 1 .

Table 2 Best-fit models

Organism

Fomite

Time(hrs)

Best fit model

B. anthracis

Polyester

0,24,672,2190

Biphasic exponential

B. anthracis

Steel

0,24,672,2190

Biphasic exponential

B. anthracis

Laminar

0,24,672,2190

Juneja & Mark(2)

 

Persistence_model_example.png

Figure 1 Best fit models

 

Persistence excel tool

Any persistent data could be analyzed by using attached excel spreadsheet. Read the instruction manual first and then use the excel tool accordingly.

Spreadsheet Instruction

Persistence Model wiki


References

  1. A mathematical model for bacterial inactivation, Xiong, R, Xie G, Edmondson AE, and Sheard MA , International journal of food microbiology, Volume 46, p.45–55, (1999)
  2. General model, based on two mixed Weibull distributions of bacterial resistance, for describing various shapes of inactivation curves, Coroller, Louis, Leguérinel Ivan, Mettler Eric, Savy Nicolas, and Mafart Pierre , Applied and Environmental Microbiology, Volume 72, p.6493–6502, (2006)
  3. Reinterpretation of microbial survival curves, Peleg, Micha, and Cole Martin B. , Critical Reviews in Food Science, Volume 38, p.353–380, (1998)
  4. Modeling non-linear survival curves to calculate thermal inactivation of Salmonella in poultry of different fat levels, Juneja, VK, Eblen BS, and Marks HM , International journal of food microbiology, Volume 70, p.37–51, (2001)
  5. An alternative approach to non-log-linear thermal microbial inactivation: modelling the number of log cycles reduction with respect to temperature, Valdramidis, Vasilis Panagiotis, Bernaerts Kristel, Van Impe Jan Frans, and Geeraerd Annemie Helena , Food Technology and Biotechnology, Volume 43, p.321–327, (2005)
  6. Predictive model for growth of Clostridium perfringens in cooked cured pork, Juneja, Vijay K., Huang Lihan, and Thippareddi Harshvardhan H. , International journal of food microbiology, Volume 110, p.85–92, (2006)
  7. On the use of the Gompertz model to predict microbial thermal inactivation under isothermal and non-isothermal conditions, Gil, Maria M., Miller Fátima A., Brandao Teresa R. S., and Silva Cristina L. M. , Food Engineering Reviews, Volume 3, p.17–25, (2011)
  8. A closed-form expression for the quantile function of the Gompertz–Makeham distribution, Jodrá, Pedro , Mathematics and Computers in Simulation, Volume 79, p.3069–3075, (2009)
  9. On calculating sterility in thermal preservation methods: application of the Weibull frequency distribution model, Mafart, Pierre, Couvert Olivier, Gaillard Stéphane, and Leguérinel Ivan , International journal of food microbiology, Volume 72, p.107–113, (2002)
  10. Advanced quantitative microbiology for foods and biosystems: models for predicting growth and inactivation, Peleg, Micha , (2006)