Basic usage
Dataset
To use epifitter data at least two variables are needed, one representing the time when each of the multiple assessments of disease intensity were performed during the course of the epidemics and the other representing a disease intensity variable normalized to proportion (e.g. incidence, severity, prevalence). For the case of designed experiments with replicates, a third variable is needed (eg. rep).
Let’s produce a fake DPC dataset for one epidemics measured at replicated plots. The simulated data resemble a polyciclic epidemics of sigmoid shape. We can do that using the epifitter sim_logistic() function of epifitter (more about ?sim_logistic here).
dpcL <- sim_logistic(
N = 100, # duration of the epidemics in days
y0 = 0.01, # disease intensity at time zero
dt = 10, # interval between assessments
r = 0.1, # apparent infection rate
alpha = 0.2, # level of noise
n = 7 # number of replicates
)Let’s give a look at the simulated dataset.
## replicates time y random_y
## 1 1 0 0.01000000 0.01000000
## 2 1 10 0.02672668 0.02515774
## 3 1 20 0.06946279 0.07537630
## 4 1 30 0.16868385 0.18406412
## 5 1 40 0.35549349 0.45273476
## 6 1 50 0.59989486 0.54863092The dpc_L object generated using sim_logistic() is a dataframe with four columns. The y variable is a vector for the disease intensity as proportion (0 < y < 1). To facilitate visualization, let’s make a plot using the ggplot function of the ggplot2 package.
ggplot(
dpcL,
aes(time, y,
group = replicates
)
) +
geom_point(aes(time, random_y), shape = 1) + # plot the replicate values
geom_point(color = "steelblue", size = 2) +
geom_line(color = "steelblue") +
labs(
title = "Simulated 'complete' epidemics of sigmoid shape",
subtitle = "Produced using sim_logistic()"
)+
theme_minimal_hgrid()
Linear regression using fit_lin()
The fit_lin() requires at least the time and y arguments. In the example, we will call the random_y which represents the replicates. A quick way to call these variables attached to the dataframe is shown below.
fit_lin() outputs a list object which contains several elements. Three elements of the list are shown by default: stats of model fit, Infection rate and Initial Inoculum
## Results of fitting population models
##
## Stats:
## CCC r_squared RSE
## Logistic 0.9984 0.9968 0.1828
## Gompertz 0.9778 0.9565 0.4921
## Monomolecular 0.9361 0.8799 0.6609
## Exponential 0.9120 0.8382 0.6346
##
## Infection rate:
## Estimate Std.error Lower Upper
## Logistic 0.10089946 0.0006587818 0.09958710 0.09958710
## Gompertz 0.07205896 0.0017734637 0.06852604 0.06852604
## Monomolecular 0.05582706 0.0023816721 0.05108252 0.05108252
## Exponential 0.04507241 0.0022868136 0.04051684 0.04051684
##
## Initial inoculum:
## Estimate Linearized lin.SE Lower Upper
## Logistic 9.765421e-03 -4.619094 0.03897406 9.042513e-03 0.0105455072
## Gompertz 2.663203e-05 -2.354551 0.10491953 2.301461e-06 0.0001941929
## Monomolecular -1.817800e+00 -1.035956 0.14090162 -2.730890e+00 -1.1281774602
## Exponential 2.778837e-02 -3.583138 0.13528972 2.122345e-02 0.0363839750Model fit stats
The Stats element of the list shows how each of the four models predicted the observations based on three measures:
- Lin’s concordance correlation coefficient
CCC(Lin 2000), a measure of agreement that takes both bias and precision into account - Coefficient of determination
r_squared(R2), a measure of precision - Residual standard deviation
RSEfor each model.
The four models are sorted from the high to the low CCC. As expected because the sim_logistic function was used to create the synthetic epidemic data, the the Logistic model was superior to the others.
Model coefficients
The estimates, and respective standard error and upper and lower 95% confidence interval, for the two coefficients of interest are shown in the Infection rate and Initial inoculum elements. For the latter, both the back-transformed (estimate) and the linearized estimate are shown.
Global stats
The element f_lin$stats_all provides a wide format dataframe with all the stats for each model.
## # A tibble: 4 x 14
## best_model model r r_se r_ci_lwr r_ci_upr v0 v0_se r_squared RSE
## <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 Logi~ 0.101 6.59e-4 0.0996 0.102 -4.62 0.0390 0.997 0.183
## 2 2 Gomp~ 0.0721 1.77e-3 0.0685 0.0756 -2.35 0.105 0.957 0.492
## 3 3 Mono~ 0.0558 2.38e-3 0.0511 0.0606 -1.04 0.141 0.880 0.661
## 4 4 Expo~ 0.0451 2.29e-3 0.0405 0.0496 -3.58 0.135 0.838 0.635
## # ... with 4 more variables: CCC <dbl>, y0 <dbl>, y0_ci_lwr <dbl>,
## # y0_ci_upr <dbl>Model predictions
The predicted values are stored as a dataframe in the data element called using the same $ operator as above. Both the observed and (y) and the back-transformed predictions (predicted) are shown for each model. The linearized value and the residual are also shown.
## time y model linearized predicted residual
## 1 0 0.01000000 Exponential -4.6051702 0.02778837 -0.017788367
## 2 10 0.02515774 Exponential -3.6825897 0.04361240 -0.018454662
## 3 20 0.07537630 Exponential -2.5852624 0.06844740 0.006928893
## 4 30 0.18406412 Exponential -1.6924711 0.10742465 0.076639473
## 5 40 0.45273476 Exponential -0.7924488 0.16859741 0.284137343
## 6 50 0.54863092 Exponential -0.6003293 0.26460490 0.284026018Plot of predictions
The plot_fit() produces, by default, a panel of plots depicting the observed and predicted values by all fitted models. The arguments pont_size and line_size that control for the size of the dots for the observation and the size of the fitted line, respectively.
Publication-ready plots
The plots are ggplot2 objects which can be easily customized by adding new layers that override plot paramaters as shown below. The argument models allows to select the models(s) to be shown on the plot. The next plot was customized for the logistic model.
# Customized plots
plot_fit(f_lin,
point_size = 2,
line_size = 1,
models = "Logistic")+
theme_minimal_hgrid(font_size =18) +
scale_x_continuous(limits = c(0,100))+
scale_color_grey()+
theme(legend.position = "none")+
labs(
x = "Time",
y = "Proportion disease "
)
Non-linear regression
Two-parameters
The fit_nlin() function uses the Levenberg-Marquardt algorithm for least-squares estimation of nonlinear parameters. In addition to time and disease intensity, starting values for y0 and r should be given in the starting_par argument. The output format and interpretation is analogous to the fit_lin().
NOTE: If you encounter error messages saying “matrix at initial parameter estimates”, try to modify the starting values for the parameters to solve the problem.
f_nlin <- fit_nlin(
time = dpcL$time,
y = dpcL$random_y,
starting_par = list(y0 = 0.01, r = 0.03)
)
f_nlin## Results of fitting population models
##
## Stats:
## CCC r_squared RSE
## Logistic 0.9984 0.9967 0.0233
## Gompertz 0.9963 0.9938 0.0348
## Monomolecular 0.9129 0.8642 0.1563
## Exponential 0.8856 0.8227 0.1768
##
## Infection rate:
## Estimate Std.error Lower Upper
## Logistic 0.10161300 0.001943993 0.09774036 0.09774036
## Gompertz 0.07137414 0.001983705 0.06742240 0.06742240
## Monomolecular 0.02280140 0.001519789 0.01977382 0.01977382
## Exponential 0.01901604 0.001403773 0.01621958 0.01621958
##
## Initial inoculum:
## Estimate Std.error Lower Upper
## Logistic 9.253368e-03 8.435481e-04 7.572934e-03 1.093380e-02
## Gompertz 5.336470e-08 7.922394e-08 -1.044575e-07 2.111869e-07
## Monomolecular -1.933421e-01 4.680411e-02 -2.865807e-01 -1.001036e-01
## Exponential 1.814865e-01 2.117435e-02 1.393050e-01 2.236680e-01We can check the results using plot_fit.
Estimating K (maximum disease)
In many epidemics the last measure (final time) of a DPC does not reach the maximum intensity and, for this reason, estimation of maximum asymptote (carrying capacity K) may be necessary. The fin_lin2() provides an estimation of K in addition to the estimates provided by fit_lin().
Before demonstrating the function, we can transform our simulated data by creating another variable with y_random2 with maximum about 0.8 (80%). Simplest way is to multiply the y_random by 0.8.
Then we run the fit_nlin2() for the new dataset.
f_nlin2 <- fit_nlin2(
time = dpcL2$time,
y = dpcL2$random_y,
starting_par = list(y0 = 0.01, r = 0.2, K = 0.6)
)
f_nlin2## Results of fitting population models
##
## Stats:
## CCC r_squared RSE
## Logistic 0.9984 0.9967 0.0187
## Gompertz 0.9567 0.9592 0.0906
## Monomolecular 0.9455 0.9036 0.1028
##
## Infection rate:
## Estimate Std.error Lower Upper
## Logistic 0.10134943 0.002345119 0.09667668 0.09667668
## Gompertz 0.10563603 0.016849067 0.07206353 0.07206353
## Monomolecular 0.01628568 0.003124448 0.01006008 0.01006008
##
## Initial inoculum:
## Estimate Std.error Lower Upper
## Logistic 7.475098e-03 7.728694e-04 5.935122e-03 9.015074e-03
## Gompertz 8.193757e-19 2.077227e-17 -4.057027e-17 4.220902e-17
## Monomolecular -1.467192e-01 3.277790e-02 -2.120305e-01 -8.140777e-02
##
## Maximum disease intensity:
## Estimate Std.error Lower Upper
## Logistic 0.8009402 0.004589545 0.7917953 0.8100851
## Gompertz 0.6496748 0.018588483 0.6126365 0.6867132
## Monomolecular 1.0000000 0.108410391 0.7839876 1.2160124
NOTE: The exponential model is not included because it doesn’t have a maximum asymptote. The estimated value of
Kis the expected 0.8.
Fit models to multiple DPCs
Most commonly, there are more than one epidemics to analyse either from observational or experimental studies. When the goal is to fit a common model to all curves, the fit_multi() function is in hand. Each DPC needs an unique identified to further combined in a single data frame.
Data
Let’s use the sim_ family of functions to create three epidemics and store the data in a single data.frame. The Gompertz model was used to simulate these data. Note that we allowed to the y0 and r parameter to differ the DPCs. We should combine the three DPCs using the bind_rows() function and name the identifier (.id), automatically created as a character vector, for each epidemics as ‘DPC’.
epi1 <- sim_gompertz(N = 60, y0 = 0.001, dt = 5, r = 0.1, alpha = 0.4, n = 4)
epi2 <- sim_gompertz(N = 60, y0 = 0.001, dt = 5, r = 0.12, alpha = 0.4, n = 4)
epi3 <- sim_gompertz(N = 60, y0 = 0.003, dt = 5, r = 0.14, alpha = 0.4, n = 4)
multi_epidemic <- bind_rows(epi1,
epi2,
epi3,
.id = "DPC"
)
head(multi_epidemic)## DPC replicates time y random_y
## 1 1 1 0 0.00100000 0.001247501
## 2 1 1 5 0.01515505 0.016201410
## 3 1 1 10 0.07878459 0.075826792
## 4 1 1 15 0.21411521 0.243432343
## 5 1 1 20 0.39266393 0.565635541
## 6 1 1 25 0.56723412 0.628033892We can visualize the three DPCs in a same plot
p_multi <- ggplot(multi_epidemic,
aes(time, y, shape = DPC, group = DPC))+
geom_point(size =2)+
geom_line()+
theme_minimal_grid(font_size =18) +
labs(
x = "Time",
y = "Proportion disease "
)
p_multi
Or use facet_wrap() for ploting them separately.
Using fit_multi()
fit_multi() requires at least four arguments: time, disease intensity (as proportion), data and the curve identifier (strata_cols). The latter argument accepts one or more strata include as c("strata1",strata2"). In the example below, the stratum name is DPC, the name of the variable.
By default, the linear regression is fitted to data but adding another argument nlin = T, the non linear regressions is fitted instead.
multi_fit <- fit_multi(
time_col = "time",
intensity_col = "random_y",
data = multi_epidemic,
strata_cols = "DPC"
)All parameters of the list can be returned using the $ operator as below.
## DPC best_model model r r_se r_ci_lwr r_ci_upr
## 1 1 1 Gompertz 0.09572633 0.001979168 0.09175105 0.09970160
## 2 1 2 Monomolecular 0.06823320 0.002355758 0.06350152 0.07296488
## 3 1 3 Logistic 0.15277164 0.007573605 0.13755961 0.16798368
## 4 1 4 Exponential 0.08453844 0.008959621 0.06654252 0.10253437
## 5 2 1 Gompertz 0.11790289 0.002611386 0.11265776 0.12314801
## 6 2 2 Monomolecular 0.09132240 0.003306789 0.08468051 0.09796428
## v0 v0_se r_squared RSE CCC y0
## 1 -1.8173198 0.06997417 0.9790739 0.2670044 0.9894263 0.002122122
## 2 -0.5244078 0.08328862 0.9437531 0.3178091 0.9710628 -0.689458021
## 3 -4.3845657 0.26776737 0.8905652 1.0217352 0.9421153 0.012314758
## 4 -3.8601579 0.31677043 0.6403616 1.2087189 0.7807566 0.021064673
## 5 -1.9233287 0.09232645 0.9760592 0.3522953 0.9878845 0.001066150
## 6 -0.7039848 0.11691265 0.9384751 0.4461102 0.9682612 -1.021793215
## y0_ci_lwr y0_ci_upr
## 1 0.0008382445 0.004756387
## 2 -0.9971035452 -0.429204015
## 3 0.0072290504 0.020902978
## 4 0.0111489489 0.039799307
## 5 0.0002643816 0.003395483
## 6 -1.5569389520 -0.598648963Similarly, all data can be returned.
## DPC time y model linearized predicted residual
## 1 1 0 0.001247501 Exponential -6.6866131 0.02106467 -0.01981717
## 2 1 5 0.016201410 Exponential -4.1226570 0.03214605 -0.01594464
## 3 1 10 0.075826792 Exponential -2.5793036 0.04905695 0.02676984
## 4 1 15 0.243432343 Exponential -1.4129162 0.07486407 0.16856827
## 5 1 20 0.565635541 Exponential -0.5698053 0.11424740 0.45138814
## 6 1 25 0.628033892 Exponential -0.4651611 0.17434890 0.45368499If nonlinear regression is preferred, the nlim argument should be set to TRUE
multi_fit2 <- fit_multi(
time_col = "time",
intensity_col = "random_y",
data = multi_epidemic,
strata_cols = "DPC",
nlin = TRUE)## Warning in log(y0/1): NaNs produzidos
## Warning in log(y0/1): NaNs produzidos
## Warning in log(y0/1): NaNs produzidos
## Warning in log(y0/1): NaNs produzidos
## Warning in log(y0/1): NaNs produzidos## DPC model y0 y0_se r r_se df
## 1 1 Gompertz 0.0033772019 0.0018998339 0.09307649 0.004261185 50
## 2 1 Logistic 0.0395546960 0.0068531958 0.13442430 0.007156387 50
## 3 1 Monomolecular -0.1693302536 0.0423231768 0.04299224 0.002562883 50
## 4 1 Exponential 0.2358851284 0.0289837915 0.02691654 0.002557450 50
## 5 2 Gompertz 0.0009378605 0.0008363522 0.11605239 0.006308678 50
## 6 2 Logistic 0.0300135210 0.0063937038 0.16745156 0.010052896 50
## CCC r_squared RSE y0_ci_lwr y0_ci_upr r_ci_lwr
## 1 0.9897452 0.9799215 0.05345897 -0.0004387268 0.007193131 0.08451765
## 2 0.9873662 0.9764403 0.05924580 0.0257896471 0.053319745 0.12005028
## 3 0.9530529 0.9220550 0.10959824 -0.2543388559 -0.084321651 0.03784454
## 4 0.8649326 0.7898904 0.17702116 0.1776694699 0.294100787 0.02177975
## 5 0.9890663 0.9784282 0.05724415 -0.0007420024 0.002617723 0.10338104
## 6 0.9877757 0.9767931 0.06032242 0.0171713890 0.042855653 0.14725972
## r_ci_upr best_model
## 1 0.10163534 1
## 2 0.14879833 2
## 3 0.04813994 3
## 4 0.03205333 4
## 5 0.12872374 1
## 6 0.18764339 2Want to estimate K?
If you want to estimate K, set nlin = TRUE and estimate_K = TRUE.
NOTE: If you do not set both arguments
TRUE,Kwill not be estimated, becausenlindefaut isFALSE. Also remember that when estimating K, we don’t fit the Exponential model.
multi_fit_K <- fit_multi(
time_col = "time",
intensity_col = "random_y",
data = multi_epidemic,
strata_cols = "DPC",
nlin = T,
estimate_K = T
)## Warning in log(y0/K): NaNs produzidos
## Warning in log(y0/K): NaNs produzidos
## Warning in log(y0/K): NaNs produzidos
## Warning in log(y0/K): NaNs produzidos
## Warning in log(y0/K): NaNs produzidos## DPC model y0 y0_se r r_se K
## 1 1 Gompertz 0.0033772082 0.0023903828 0.09307648 0.006699356 1
## 2 1 Logistic 0.0395544163 0.0078484829 0.13442456 0.009150690 1
## 3 1 Monomolecular -0.1693296785 0.0467920019 0.04299219 0.006647116 1
## 4 2 Gompertz 0.0009378604 0.0009884541 0.11605239 0.008582625 1
## 5 2 Logistic 0.0300133238 0.0070056255 0.16745171 0.011756973 1
## 6 2 Monomolecular -0.1737968105 0.0519807351 0.05002160 0.007438802 1
## K_se df CCC r_squared RSE y0_ci_lwr y0_ci_upr
## 1 0.02076629 49 0.9897452 0.9799216 0.05400171 -0.001426446 0.008180862
## 2 0.01871708 49 0.9873662 0.9764402 0.05984729 0.023782299 0.055326533
## 3 0.06530513 49 0.9530529 0.9220552 0.11071095 -0.263361727 -0.075297630
## 4 0.01692887 49 0.9890663 0.9784282 0.05782532 -0.001048513 0.002924233
## 5 0.01522100 49 0.9877757 0.9767930 0.06093484 0.015934992 0.044091655
## 6 0.05641312 49 0.9475323 0.9140828 0.12063913 -0.278256009 -0.069337612
## r_ci_lwr r_ci_upr K_ci_lwr K_ci_upr best_model
## 1 0.07961362 0.10653934 0.9582686 1.041731 1
## 2 0.11603556 0.15281356 0.9623866 1.037613 2
## 3 0.02963431 0.05635006 0.8687644 1.131236 3
## 4 0.09880496 0.13329982 0.9659802 1.034020 1
## 5 0.14382519 0.19107823 0.9694123 1.030588 2
## 6 0.03507277 0.06497043 0.8866336 1.113366 3Graphical outputs
Use ggplot2 to produce elegant data visualizations of models curves and the estimated parameters.
DPCs and fitted curves
The original data and the predicted values by each model are stored in multi_fit$Data. A nice plot can be produced as follows:
multi_fit$Data %>%
ggplot(aes(time, predicted, color = DPC)) +
geom_point(aes(time, y), color = "gray") +
geom_line(size = 1) +
facet_grid(DPC ~ model, scales = "free_y") +
theme_minimal_hgrid()+
coord_cartesian(ylim = c(0, 1))
Using the dplyr function filter only the model of interest can be chosen for plotting.
multi_fit$Data %>%
filter(model == "Gompertz") %>%
ggplot(aes(time, predicted, color = DPC)) +
geom_point(aes(time, y),
color = "gray",
size = 2
) +
geom_line(size = 1.2) +
theme_minimal_hgrid() +
labs(
x = "Time",
y = "Disease Intensity"
)
Apparent infection rate
The multi_fit$Parameters element is where all stats and parameters as stored. Let’s plot the estimates of the apparent infection rate.
multi_fit$Parameters %>%
filter(model == "Gompertz") %>%
ggplot(aes(DPC, r)) +
geom_point(size = 3) +
geom_errorbar(aes(ymin = r_ci_lwr, ymax = r_ci_upr),
width = 0,
size = 1
) +
labs(
x = "Time",
y = "Apparent infection rate"
) +
theme_minimal_hgrid()