# Persistence Models

## Persistence Models

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). 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). Seven different models described in various studies have been shown in Table 1 .(Peleg and Cole 1998; Juneja, Eblen et al. 2001; Valdramidis, Bernaerts et al. 2005; Juneja, Huang et al. 2006).

The data from each treatment was fitted to a best fit curve using an unpublished mathematical model fitting tool in Microsoft® Excel (Microsoft® Inc., Redmond, Washington) by Patrick Gurian at Drexel University and modified by Sushil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

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
9 Gompertz-Makeham (gzm) Ln (Nt/N0)=(-k3*t -k1/k2*(exp^((k2*t)).-1)) k1, k2, k3 > 0
nonlinear concave
Jodra, 2009
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 Mafart et al., 2002
11 Gamma (gam) Ln (Nt/N0)=(t^(k2-1) ) exp^((-t/k2)). Nonlinear concave ,
12 Sigmoid-B (sB) Ln (Nt/N0)=-(k1*t^k3)/(k2+t^k3 ) Nonlinear concave or convex Peleg 2006
13 Logistic-Fermi Combination N(t)/N_0 =1/(1+exp(k1*(t-k2))) Nonlinear, concave Peleg 2006
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
17 Sigmoid type A log(N(t)/N_0 )=(a_1 t)/([1+a_2 t][a_3-t]) Nonlinear, S curve Peleg 2006

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) 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.

### Reference

Coroller, L., I. Leguerinel, et al. (2006). "General Model, Based on Two Mixed Weibull Distributions of Bacterial Resistance, for Describing Various Shapes of Inactivation Curves." Applied and Environmental Microbiology 72(10): 6493-6502.

Juneja, V. K., B. S. Eblen, et al. (2001). "Modeling non-linear survival curves to calculate thermal inactivation of Salmonella in poultry of different fat levels." International Journal of Food Microbiology 70(1–2): 37-51.

Juneja, V. K., L. Huang, et al. (2006). "Predictive model for growth of Clostridium perfringens in cooked cured pork." International Journal of Food Microbiology 110(1): 85-92.

Peleg, M. and M. B. Cole (1998). "Reinterpretation of Microbial Survival Curves." Critical Reviews in Food Science and Nutrition 38(5): 353-380.

Valdramidis, V. P., K. Bernaerts, et al. (2005). "An alternative approach to non-log-linear thermal microbial inactivation: Modelling the number of log cycles reduction with respect to temperature." Food Technology and Biotechnology 43(4): 321-327.

Xiong, R., G. Xie, et al. (1999). "A mathematical model for bacterial inactivation." International Journal of Food Microbiology 46(1): 45-55.

Gil, Maria M., Fátima a. Miller, Teresa R. S. Brandão, and Cristina L. M. Silva. 2011. “On the Use of the Gompertz Model to Predict Microbial Thermal Inactivation Under

Isothermal and Non-Isothermal Conditions.” Food Engineering Reviews 3 (1): 17–25. doi:10.1007/s12393-010-9032-2.

Jodrá, P. 2009. “A Closed-Form Expression for the Quantile Function of the Gompertz–Makeham Distribution.” Mathematics and Computers in Simulation 79 (10): 3069–75.

doi:10.1016/j.matcom.2009.02.002.

Coroller, L, I Leguerinel, E Mettler, N Savy, and P Mafart. 2006. “General Model, Based on Two Mixed Weibull Distributions of Bacterial Resistance, for Describing Various Shapes

of Inactivation Curves.” Applied and Environmental Microbiology 72 (10): 6493–6502. doi:10.1128/AEM.00876-06.

Mafart, P., O. Couvert, S. Gaillard, and I. Leguerinel. 2002. “On Calculating Sterility in Thermal Preservation Methods: Application of the Weibull Frequency Distribution Model.”

International Journal of Food Microbiology 72 (1-2): 107–13. doi:10.1016/S0168-1605(01)00624-9.

M. Peleg, Advanced quantitative microbiology for foods and biosystems, CRC Press, Boca Raton, FL, (2006)

Peleg 2006, Advanced Quantitative Microbiology for Foods and Biosystems, CRC Press

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1,

chapter 14. Wiley, New York.