competitionmodels {nlsMicrobio}

Competition models for simultaneous growth of two bacterial flora

Description

Formulas of primary growth models used in predictive microbiology to model the simultaneous growth of two competitive bacterial flora assuming a Jameson effect

Usage

  jameson_buchanan
  jameson_baranyi
  jameson_without_lag

Details

These models describe the simultaneous evolution of the decimal logarithm of the microbial counts of two flora (LOG10N) as a function of the time (t) and of the flora (flora) coded as 1 for counts of flora 1 and 2 for counts of flora 2. These three models assume independent lag and growth parameters for flora 1 and 2, except for the saturation which is supposed to be governed by the Jameson effect and modelled by a common parameter (tmax) which represents the time at which both flora stop to multiply. Modelling the simultaneous saturation by this way enables the model to be fitted by nls, as an analytical form of the model is available.

jameson_buchanan is based on the model of Buchanan et al. (1997) for lag phase modelling and is characterized by seven parameters (LOG10N0_1, mumax_1, lag_1, LOG10N0_2, mumax_2, lag_2 and the common saturation time tmax). This model was described and used in Vimont et al. (2006).

jameson_baranyi is based on the model of Baranyi and Roberts (1994) for lag phase modelling and is characterized by seven parameters (LOG10N0_1, mumax_1, lag_1, LOG10N0_2, mumax_2, lag_2 and the common saturation time tmax)

jameson_without_lag is based on the exponential model without lag phase and is thus characterized by five parameters (LOG10N0_1, mumax_1, LOG10N0_2, mumax_2 and the common saturation time tmax)

Value

A formula

Author(s)

Florent Baty, Marie-Laure Delignette-Muller

References

Baranyi J and Roberts, TA (1994) A dynamic approach to predicting bacterial growth in food, International Journal of Food Microbiology, 23, 277-294.

Buchanan RL, Whiting RC, Damert WC (1997) When is simple good enough: a comparison of the Gompertz, Baranyi, and three-phase linear models for fitting bacterial growth curves. Food Microbiology, 14, 313-326.

Vimont A, Vernozy-Rozand C, Montet MP, Lazizzera C, Bavai C and Delignette-Muller ML (2006) Modeling and predicting the simultaneous growth of Escherichia coli O157:H7 and ground beef background microflora in various enrichment protocols. Applied and Environmental Microbiology 72, 261-268.

Examples

options(digits = 3)

### Example 1: fit of model jameson_buchanan  
data(competition1)
nls1 <- nls(jameson_buchanan, competition1, 
            list(lag_1 = 2, mumax_1 = 1, LOG10N0_1 = 1, tmax = 12,
                 lag_2 = 2, mumax_2 = 1, LOG10N0_2 = 4))

overview(nls1)

# Plot of theoretical curves with data
twocolors <- c("red","blue")
npoints <- 100
seq.t <- seq(0,max(competition1$t),length.out=npoints)
prednls1.1 <- predict(nls1,data.frame(t=seq.t,flora=rep(1,npoints)))
prednls1.2 <- predict(nls1,data.frame(t=seq.t,flora=rep(2,npoints)))
plot(competition1$t,competition1$LOG10N,col=twocolors[competition1$flora],xlab="t",ylab="LOG10N")
lines(seq.t,prednls1.1,col=twocolors[1])
lines(seq.t,prednls1.2,col=twocolors[2])


### Example 2 : fit of model jameson_baranyi  
data(competition1)
nls2 <- nls(jameson_baranyi, competition1, 
            list(lag_1 = 2, mumax_1 = 1, LOG10N0_1 = 1, tmax = 12,
                 lag_2 = 2, mumax_2 = 1, LOG10N0_2 = 4))

overview(nls2)
plotfit(nls2)

# Plot of theoretical curves with data
twocolors <- c("red","blue")
npoints <- 100
seq.t <- seq(0,max(competition1$t),length.out=npoints)
prednls2.1 <- predict(nls2,data.frame(t=seq.t,flora=rep(1,npoints)))
prednls2.2 <- predict(nls2,data.frame(t=seq.t,flora=rep(2,npoints)))
plot(competition1$t,competition1$LOG10N,col=twocolors[competition1$flora],xlab="t",ylab="LOG10N")
lines(seq.t,prednls2.1,col=twocolors[1])
lines(seq.t,prednls2.2,col=twocolors[2])


### Example 3: fit of model jameson_without_lag  
data(competition2)
nls3 <- nls(jameson_without_lag, competition2, 
            list(mumax_1 = 1, LOG10N0_1 = 1, tmax = 12,
                 mumax_2 = 1, LOG10N0_2 = 4))

overview(nls3)
plotfit(nls3)

# Plot of theoretical curves with data
twocolors <- c("red","blue")
npoints <- 100
seq.t <- seq(0,max(competition2$t),length.out=npoints)
prednls3.1 <- predict(nls3,data.frame(t=seq.t,flora=rep(1,npoints)))
prednls3.2 <- predict(nls3,data.frame(t=seq.t,flora=rep(2,npoints)))
plot(competition2$t,competition2$LOG10N,col=twocolors[competition2$flora],xlab="t",ylab="LOG10N")
lines(seq.t,prednls3.1,col=twocolors[1])
lines(seq.t,prednls3.2,col=twocolors[2])