Simulating disease progress curves

Introduction

In epiffiter, opposite to the fit_ functions (estimate parameters from fitting models to the data), the sim_ family of functions allows to produce the DPC data given a set of parameters for a specific model. Currently, the same four population dynamic models that are fitted to the data can be simulated.

The functions use the ode() function of the devolve package (Soetaert,Petzoldt & Setzer 2010) to solve the differential equation form of the e epidemiological models.

Hands On

Packages

First, we need to load the packages we’ll need for this tutorial.

library(epifitter)
library(magrittr)
library(ggplot2)
library(cowplot)

The basics of the simulation

The sim_ functions, regardless of the model, require the same set of six arguments. By default, at least two arguments are required (the others have default values)

r: apparent infection rate
n: number of replicates

When n is greater than one, replicated epidemics (e.g. replicated treatments) are produced and a level of noise (experimental error) should be set in the alpha argument. These two arguments combined set will generate random_y values, which will vary randomly across the defined number of replicates.

The other arguments are:

N: epidemic duration in time units
dt: time (fixed) in units between two assessments
y0: initial inoculum
alpha: noise parameters for the replicates

Exponential

Let’s simulate a curve resembling the exponential growth.

exp_model <- sim_exponential(
  N = 100,    # total time units 
  y0 = 0.01,  # initial inoculum
  dt = 10,    #  interval between assessments in time units
  r = 0.045,  #  apparent infection rate
  alpha = 0.2,# level of noise
  n = 7       # number of replicates
)
head(exp_model)

##   replicates time          y   random_y
## 1          1    0 0.01000000 0.01034651
## 2          1   10 0.01568425 0.02079218
## 3          1   20 0.02459905 0.02433959
## 4          1   30 0.03858028 0.04131958
## 5          1   40 0.06050749 0.06037350
## 6          1   50 0.09489670 0.12089660

A data.frame object is produced with four columns:

replicates: the curve with the respective ID number
time: the assessment time
y: the simulated proportion of disease intensity
random_y: randomly simulated proportion disease intensity based on the noise

Use the ggplot2 package to build impressive graphics!

exp_plot = exp_model %>%
  ggplot(aes(time, y)) +
  geom_jitter(aes(time, random_y), size = 3,color = "gray", width = .1) +
  geom_line(size = 1) +
  theme_minimal_hgrid() +
  ylim(0,1)+
  labs(
    title = "Exponential",
    y = "Disease intensity",
    x = "Time"
  )
exp_plot

## Warning: Removed 1 row containing missing values or values outside the scale range
## (`geom_point()`).

Monomolecular

The logic is exactly the same here.

mono_model <- sim_monomolecular(
  N = 100,
  y0 = 0.01,
  dt = 5,
  r = 0.05,
  alpha = 0.2,
  n = 7
)
head(mono_model)

##   replicates time         y  random_y
## 1          1    0 0.0100000 0.0100000
## 2          1    5 0.2289861 0.2581656
## 3          1   10 0.3995322 0.4545163
## 4          1   15 0.5323535 0.5407130
## 5          1   20 0.6357949 0.6835060
## 6          1   25 0.7163551 0.7235810

mono_plot = mono_model %>%
  ggplot(aes(time, y)) +
  geom_jitter(aes(time, random_y), size = 3, color = "gray", width = .1) +
  geom_line(size = 1) +
  theme_minimal_hgrid() +
  labs(
    title = "Monomolecular",
    y = "Disease intensity",
    x = "Time"
  )
mono_plot

The Logistic model

logist_model <- sim_logistic(
  N = 100,
  y0 = 0.01,
  dt = 5,
  r = 0.1,
  alpha = 0.2,
  n = 7
)
head(logist_model)

##   replicates time          y   random_y
## 1          1    0 0.01000000 0.01000000
## 2          1    5 0.01638216 0.01065720
## 3          1   10 0.02672677 0.02521582
## 4          1   15 0.04331509 0.05050620
## 5          1   20 0.06946352 0.06467859
## 6          1   25 0.10958806 0.09057823

logist_plot = logist_model %>%
  ggplot(aes(time, y)) +
  geom_jitter(aes(time, random_y), size = 3,color = "gray", width = .1) +
  geom_line(size = 1) +
  theme_minimal_hgrid() +
  labs(
    title = "Logistic",
    y = "Disease intensity",
    x = "Time"
  )
logist_plot

Gompertz

gomp_model <- sim_gompertz(
  N = 100,
  y0 = 0.01,
  dt = 5,
  r = 0.07,
  alpha = 0.2,
  n = 7
)
head(gomp_model)

##   replicates time          y   random_y
## 1          1    0 0.01000000 0.01000000
## 2          1    5 0.03896283 0.02438837
## 3          1   10 0.10158896 0.11084215
## 4          1   15 0.19958740 0.21342554
## 5          1   20 0.32122825 0.24382870
## 6          1   25 0.44922018 0.44300414

gomp_plot = gomp_model %>%
  ggplot(aes(time, y)) +
  geom_jitter(aes(time, random_y), size = 3,color = "gray", width = .1) +
  geom_line(size = 1) +
  theme_minimal_hgrid() +
  labs(
    title = "Gompertz",
    y = "Disease intensity",
    x = "Time"
  )
gomp_plot

Combo

Use the function plot_grid() from the cowplot package to gather all plots into a grid

plot_grid(exp_plot,
          mono_plot,
          logist_plot,
          gomp_plot)

## Warning: Removed 1 row containing missing values or values outside the scale range
## (`geom_point()`).

References

Karline Soetaert, Thomas Petzoldt, R. Woodrow Setzer (2010). Solving Differential Equations in R: Package deSolve. Journal of Statistical Software, 33(9), 1–25. DOI: 10.18637/jss.v033.i09