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)[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The data from each treatment was fitted to a best-fit curve using an unpublished mathematical model fitting tool in Microsoft® Excel (Microsoft® Inc., Redmond, Washington) by Patrick Gurian at Drexel University and modified by Sushil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) = -k1t for t≥x: Ln (Nt/N0) = -k1t+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[Ce [-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) =(k1exp(-exp((-k2exp(1)*(k3-t)/k1)+1) ) )	k1, k2, k3 > 0 nonlinear concave	Gil et al. 2011[7]
9	Gompertz-Makeham (gzm)	Ln (Nt/N0)=(-k3t -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(-k1t[t<k3]) if (max(t)>=k3) N(t)/N_0 = exp(-k1t[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)