This R package provides an interface to define compartmental infectious disease models, typeset the model in LaTeX, simulate the model numerically by solving the ODE system, ot simulate the model stochastically using the Gillespie method. If you are simply interested in simulating an ODE model, this package also provides an interface to define an ODE model with ease. This package supports two types of model, generic ODE models, and a compartmental models.
The R6 class to define a generic ODE model is the Model. The equations may be directly defined in the new method, or individually specified using the compartment method. See more details below. The parameters of the model are automatically extracted from the equations. These parameters may also be defined as substitutions, i.e., as an expression.
The model may then be passed to a TexFormatter object to typeset as LaTeX equations, or be passed to an ODE object to numerically solve the ODE.
To create a model, we can use the Model$new() method with either no argument (which createa an empty model) and then use the compartment method to define each class, or directly pass the equations to the new method.
An equation can be expressed as a formula. For example, S ~ - beta*S*I defines a compartment S with the equation
[ \frac{dS}{dt} = \beta S I ]
The following code defines a SIR model with standard incidence.
m = Model$new(
S ~ -beta*S*I/N,
I ~ beta*S*I/N - gamma*I,
R ~ gamma*I
)
print(m)
It shows that there are three classes, S, I and R, and it uses three parameters, beta, N and gamma.
Here N is a free parameter, but it should be the total population. We can define a parameter as a substitution, using the where method, using an argument with the name N and the expression S+I+R as the value, i.e.,
m$where(N=S+I+R)
print(m)
A model can be typeset as LaTeX equations using an object of the TexFormatter class. A TexFormatter obect has a typeset.equation method that which accepts a list of R expressions. For example, the equations
field of a Model object contains two of such lists, the equations component contains the differential equations, and the where component contains the substitutions.
tex = TexFormatter$new()
tex$typeset.equation(m$equations$equations)
tex$typeset.equation(m$equations$where)
A TexFormatter object also has a typeset method which typesets an R expression as LaTeX.
tex$typeset(quote(beta*S*I))
A TexFormatter object automatically recognize greek letters trig functions and (\infty). We may define new symbols using the set.symbols method, which takes a named list with the names as the R names and the values latex commands. For example, the population size N in the SIR model above is not a compartment. To make it clear, we will use the corresponding latex mathcal symbol.
tex$set.symbols(list(N="\\mathcal{N}"))
tex$typeset.equation(m$equations$equations)
To numerically simulate the ODE model, we need an object of class ODE. Note that we need the deSolve package installed in order to use the class. The constructor of the ODE class takes the Model object to simulate. The actual simulation is done using the simulate class, which takes a numeric vector of time points to simulate, the initial conditions, and the parameter values. Additional arguments are passed unchanged to the underlying deSolve::ode function. The simulate method returns a data.frame which first column is the time, and the following columns are the state values. The following example simulates the SIR model defined above.
sim = ODE$new(m)
x = sim$simulate(
t=0:100,
y0=c(S=1000, I=10, R=0),
parms=c(beta=0.4, gamma=0.2))
head(x)
We may also select a subset of state variables to simulate, using the vars argument of the simulate method. It takes a vector of characters giving the variable names. We may also ask it to return the value of a substitution. The following example returns both I and N of the SIR model so that we can verify that the total population N is indeed a constant.
sim$simulate(
t=0:100,
y0=c(S=1000, I=10, R=0),
parms=c(beta=0.4, gamma=0.2),
vars=c("I", "N")
)
The R6 class for defining a compartmental model is Compartmental, which is a subclass of Model. The main advantage of using the Compartmental class over the Model class is that we can specify state transitions, i.e., we describe the flowchart instead of directly specifying the equations. We indeed provide a facility to generate the flowchart as the LaTeX code for a tikzpicture. In addition, the state transitions correspond to random events. We can thus run stochastic simulations using the Gillespie method.
The constructor of the Compartmental class takes the names of compartments. Alternatively the class names may also be specified using the compartment method, which takes one compartment name. To create the SIR model, we may start with the following statement
m = Compartmental$new(S, I, R)
The transitions are specified by the transition method using a formula
that has the form from -> to ~ rate where from and to are the compartment names, and rate is an expression for the rate of the transition. Either from or to may be NULL, specifying that new individuals is from outside the system (e.g., births) or the individuals leave the system (e.g., deaths). A transition has a name, which can be used to further modify it. It can be specified by the name argument. Alternatively, if the name is not specified, then a name is automatically generated, and is returned by the transition method. Substitutions may be passed to the transition method as named arguments. For example, the transitions for an SIR model with births and deaths may be specified as
# infection, where the population size N is defined as a substitution
m$transition(S->I ~ beta*S*I/N, N=S+I+R)
# recovery
m$transition(I->R ~ gamma*I)
# births
m$transition(NULL->S ~ lambda)
# deaths
death.S = m$transition(S->NULL ~ mu*S)
m$transition(I->NULL ~ mu*I)
m$transition(R->NULL ~ mu*R)
print(m)
The rate may be specified as a per capita rate, either by specifying percapita=TRUE in the argument, or specify the rate as percapita(rate). The following statements are equivalent.
m$transition(S->NULL ~ mu*S, name=death.S)
m$transition(S->NULL ~ percapita(mu), name=death.S)
m$transition(S->NULL ~ mu, percapita=TRUE, name=death.S)
Because the transitions of a compartmental model may correspond to random events that change the system state, we may use the Gillespie method to simulate the model. To do so, we use an object of the RGillespie R6 class, which is an R implementation. Like the ODE simulate, the constructor takes a compartmental model (note that it does not work with the Model class). To simulate, we use the simulate method, like the ODE simulation.
sim = RGillespie$new(m)
x = sim$simulate(
t=0:100,
y0=c(S=1000, I=10, R=0),
parms=c(beta=0.4, gamma=0.2, lambda=100, mu=0.1))
print(head(x))
The flowchart for a compartmental model can be generated using the flowchart function. Its first argument is the model, the second compartment is a TexFormatter object to format the compartments and rates. Following them is a set of named arguments, one for each compartment, and optionally one for each transsition.
A compartment annotation is the tikz "\node" command. It can also be specified by the Node function, the most important argument is the first one at, specifying the the position (a length 2 vector giving the tikz node coordinate). The next argument is the shape (default to "rectangle"), then a boolean rounded.corner specifying whether to use rounded corner (default to TRUE), then an R color name fill specifying the filling color for the node (default to NULL).
An arc annotation is the tikz "\path" command. It can also be specified using the "Arc" function. The first argument label is the text specifying the label on the arc (in the square bracket following the corresponding "\node" command). This can be generated using the Label function which taks two argument, the "position" (a real value btween 0 and 1, 0 being left most and 1 being right most) and the "direction" which is one of "above", "below", "left", "right". The second argument of Arc, namely "edge", is the text in the square bracket following the tikz "\path" command. For example, it could be "bend right=45". The following example generates a flowchart for the SIR model. If the transition has no origin ("from" state is NULL) or destination ("to" state is NULL), then the origin or destination of the arc must be specified using the named "from" and "to" arguments, giving the coordinates.
chart = flowchart(
m, tex,
S=Node(at=c(0,0)),
I=Node(at=c(2,0)),
R=Node(at=c(4,0)),
`->S`=Arc(from=c(-2,0)),
`S->`=Arc(to=c(0,-1), label=Label(direction="left")),
`I->`=Arc(to=c(2,-1), label=Label(direction="left")),
`R->`=Arc(to=c(4,-1), label=Label(direction="left")))
cat(chart, "\n")
A PDF flowchart may generated using the tikz.pdf function, which first argument is the PDF file name, then followed by the output of the flowchart function. A set of LaTeX preamble commands may be given in the named "preambles" argument (for example, defining some LaTeX commands used by the model). The font size may be given in the named "fontsize" argument (default to 12), in LaTeX points. The following example generates an "SIR.pdf" file for the flowchart.
tikz.pdf("SIR.pdf", chart, fontsize=9)