SIR model with deSolve & ggplot2

This is my first post ever and in 2017!  Since I am recent graduate and currently un-employed, my hope is to upload some interesting material on using R on a weekly basis. Making this an informative and motivational blog to share my interests and mini-projects in R. Now on to this post’s material…

Last year in my infectious disease epidemiology course, I first learned about a basic compartmental disease transmission model. I thought it was pretty awesome to predict and model how an infectious agent could affect a population. Purely out of interest I wanted to develop some working code in R to make a function that would plot the model based on specified parameters.

The basic deterministic model is composed of three compartments that represent different categories of individuals within a population; the susceptible, infected, and recovered – hence the SIR Model. The relationship of these three groups is described by a set of differential equations first derived by Kermack and McKendrick. The SIR model details the transmission of infection through the contact of susceptible individuals with an infected host. If you are interested in learning more on this model, there is an online module.

The SIR Model

dS/dt = -βSI

dI/dt = βSI – γI

dR/dt = γI

β = cp

S – proportion of susceptible individuals in total population
I – proportion of infected individuals in total population
R – proportion of recovered individuals in total population
β – transmission parameter (rate of infection for susceptible-infected contact)
c – number of contacts each host has per unit time (contact rate)
p – probability of transmission of infection per contact (transmissibility)
γ – recovery parameter (rate of infected transitioning to recovered)

This model is very basic and has important assumptions. The first being the population is closed and fixed, in other words – no one it added into the susceptible group (no births), all individuals who transition from being infected to recovered are permanently resistant to infection and there are no deaths. Second, the population is homogenous (all individuals are the same) and only differ by their disease state. Third, infection and that individual’s “infectiveness” or ability to infect susceptible individuals, occurs simultaneously.

Now on to using R!

I used the deSolve package which was developed to solve the initial condition values of differential equations in R. The code to setup the SIR model was adapted from the MATLAB code from Modeling Infectious Diseases in Humans and Animals and an online demo in R.

To begin, I setup my R function to be dependent on the input of three parameters: time period (in days), and the values of β and γ. These are denoted by t,b,g within the function.

The initial conditions are set to have the proportion of the populationg being in the Susceptible group at >99.9% (1-1E-6 to be exact), the Infected group to be close to 0 (1E-6) and no one in the Recovered group. The SIR model is going to be plotted at 0.5 days for higher resolution.

SIR.model <- function(t, b, g){
require(deSolve)
init <- c(S=1-1e-6,I=1e-6,R=0)
parameters <- c(bet=b,gamm=g)
time <- seq(0,t,by=t/(2*length(1:t)))

Next we setup the differential equation (from above) so that we can run the ode function from the deSolve package correctly.

eqn <- function(time,state,parameters){
with(as.list(c(state,parameters)),{
	dS <- -bet*S*I
	dI <- bet*S*I-gamm*I
	dR <- gamm*I
	return(list(c(dS,dI,dR)))})}

Then we run the ode function based on the parameters we set above and save coerce the output as a data frame class.

out<-ode(y=init,times=time,eqn,parms=parameters)
out.df<-as.data.frame(out)

Now that we’ve solved the differential equation, the next task is to plot it. To do this I am going to be using the elegant visualization package ggplot2 (version 2.1.0).

First I’m going to set a theme for the plot, by modifying one of the built-in ggplot2 themes, theme_bw. I referred to ggplot2 documentation found here.

require(ggplot2)
mytheme4 <- theme_bw() +
theme(text=element_text(colour="black")) +
theme(panel.grid = element_line(colour = "white")) +
theme(panel.background = element_rect(fill = "#B2B2B2"))
theme_set(mytheme4)

Here I set the title for the plot as SIR Model: Basic. For the subtitle it will show the values of β and γ set when first running the function. The use of bquote was based reading some material on R expressions, trial and error, with some scouring of Stack Overflow (which is an incredible resource FYI).

title <- bquote("SIR Model: Basic")
subtit <- bquote(list(beta==.(parameters[1]),~gamma==.(parameters[2])))

Now I describe the plot, essentially I am selecting the data frame from the ode function. Most of the code for theme and legend is just for aesthetics.

res<-ggplot(out.df,aes(x=time))+
ggtitle(bquote(atop(bold(.(title)),atop(bold(.(subtit))))))+
geom_line(aes(y=S,colour="Susceptible"))+
geom_line(aes(y=I,colour="Infected"))+
geom_line(aes(y=R,colour="Recovered"))+
ylab(label="Proportion")+
xlab(label="Time (days)")+
theme(legend.justification=c(1,0), legend.position=c(1,0.5))+
theme(legend.title=element_text(size=12,face="bold"),
legend.background = element_rect(fill='#FFFFFF',
size=0.5,linetype="solid"),
legend.text=element_text(size=10),
legend.key=element_rect(colour="#FFFFFF",
fill='#C2C2C2',
size=0.25,
linetype="solid"))+
scale_colour_manual("Compartments",
breaks=c("Susceptible","Infected","Recovered"),
values=c("blue","red","darkgreen"))

So now the function runs, it prints the plot and write a .png image file to the file directory with the specified parameters in the filename. The ggsave function accomplishes this with ease.

print(res)
ggsave(plot=res,
filename=paste0("SIRplot_","time",t,"beta",b,"gamma",g,".png"),
width=8,height=6,dpi=180)
getwd()}

The fully annotated code can be found in my GitHub repository SIR-interact. You can download the code and tinker around with it.

So when we run the function while setting the period to 70 days, β = 1.4 and γ = 0.3 SIR.model(70,1.4,0.3) we write a file with the name “SIRplot_time70beta1.4gamma0.3.png” and get output plot looking like…

SIRplot_time70beta1.4gamma0.3.png

The values of β and γ chosen for this example were completely arbitrary. It would be meaningful to parametrize the model with values taken from real data to understand how an infectious agent would act in a population of interest. β is difficult to measure directly as it is a function of the contact rate (c) and transmissibility (p), you typically require data to estimate this parameter. On the other hand γ-1 or d, the duration of infection can be determined independent of data as it is dependent on the pathophysiology of the agent and host.

A realistic example?

b00528_h1n1_flu_blue_med

H1N1 influenza virus. Center for Disease Control and Prevention, 2010

 

Let’s incorporate some information on the 2009 H1N1 pandemic influenza virus to create a working example. Based on epidemiological data from remote and isolated communities in Canada, we’ll use the mean estimate of R0 or the basic reproduction number. I found this data through this extensive systematic review on R0 for pandemic and seasonal influenza.

Ro is the number of secondary cases produced by a single infected case over their infectious period.

R0 = cpd

Based on this equation, we can calculate β from the estimate of R0 and information on the infectious period of the H1N1 virus. For our example, we’ll simplify the infectious period by making it equivalent to the duration of infection. (This simplification is important to note because when someone is infected by the flu virus you aren’t able to infect others immediately. There is a 2-4 day incubation period before you become infectious and start shedding the virus).

β = R0/d

From the data of the community in Northern Manitoba R0 = 2.26 and the CDC’s estimate of the infectious period d = 7 days, we plug and play to get β = 0.32 as our estimate. I’ve set the initial population conditions to have 10 infected individuals in a total population of 5000. We’ll run the R function over a 70 day period.

>SIR.model(70,0.32,1/7)

sirplot_time70beta0-32gamma0-142857142857143

I’ll probably need to change the legend position so it doesn’t cover the recovered plot.

Anyways this is it for now… I’ll be expanding this SIR model to include other compartments (exposed, deceased), account for population demography (births/deaths) and infection control (vaccinations). I hope this post was able to demonstrate R’s usefulness in this application of infectious disease transmission modeling!

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