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The data from each treatment was fitted to a best-fit curve using an unpublished mathematical model fitting tool in Microsoft® Excel (Microsoft® Inc., Redmond, Washington) by Patrick Gurian at Drexel University and modified by Sushil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




Curve Properties



First order exponential decay model

Ln (Nt/N0) = -kt

Linear, negative slope

Crane and Moore, 1986


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


General logistic model

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

Nonlinear, concave

Gonzalez, 1995


Exponential damped model

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

Nonlinear, concave

Cavalli-Sforza et al. 1983


Gompertz model

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

Nonlinear, concave

Gompertz, 1825

Gil et al. 2011[7]


Two-stage model Juneja and Marks (1)

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

Nonlinear, concave

Juneja et al. 2006


Log-logistic model Juneja and Marks (2)

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

Convex or concave

Juneja et al. 2003


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]


Gompertz-Makeham (gzm)

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

k1, k2, k3 > 0 
nonlinear concave

Jodra, 2009[8]


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]


Gamma (gam)

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

Nonlinear concave



Sigmoid-B (sB)

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

Nonlinear concave or convex

Peleg 2006[10]


Logistic-Fermi Combination

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

Nonlinear, concave

Peleg 2006[10]




Concave, nonlinear

Aragao 2007.


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)


Double exponential

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

Nonlinear, concave

Peleg 2006[10]


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




Best fit model

B. anthracis



Biphasic exponential

B. anthracis



Biphasic exponential

B. anthracis



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.

Spreadsheet Instruction

Persistence Model wiki


  1. Xiong, R., Xie G., Edmondson AE., & Sheard MA. (1999).  A mathematical model for bacterial inactivation. International journal of food microbiology. 46, 45–55.
  2. Coroller, L., Leguérinel I., Mettler E., Savy N., & Mafart P. (2006).  General model, based on two mixed Weibull distributions of bacterial resistance, for describing various shapes of inactivation curves. Applied and Environmental Microbiology. 72, 6493–6502.
  3. Peleg, M., & Cole M. B. (1998).  Reinterpretation of microbial survival curves. Critical Reviews in Food Science. 38, 353–380.
  4. Juneja, VK., Eblen BS., & Marks HM. (2001).  Modeling non-linear survival curves to calculate thermal inactivation of Salmonella in poultry of different fat levels. International journal of food microbiology. 70, 37–51.
  5. Valdramidis, V. Panagiotis, Bernaerts K., Van Impe J. Frans, & Geeraerd A. Helena (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, 321–327.
  6. Juneja, V. K., Huang L., & Thippareddi H. H. (2006).  Predictive model for growth of Clostridium perfringens in cooked cured pork. International journal of food microbiology. 110, 85–92.
  7. Gil, M. M., Miller F. A., Brandao T. R. S., & Silva C. L. M. (2011).  On the use of the Gompertz model to predict microbial thermal inactivation under isothermal and non-isothermal conditions. Food Engineering Reviews. 3, 17–25.
  8. Jodrá, P. (2009).  A closed-form expression for the quantile function of the Gompertz–Makeham distribution. Mathematics and Computers in Simulation. 79, 3069–3075.
  9. Mafart, P., Couvert O., Gaillard S., & Leguérinel I. (2002).  On calculating sterility in thermal preservation methods: application of the Weibull frequency distribution model. International journal of food microbiology. 72, 107–113.
  10. Peleg, M. (2006).  Advanced quantitative microbiology for foods and biosystems: models for predicting growth and inactivation.