Introduction

This tutorial provides a mathematical and theoretical background for deterministic, compartmental models (DCMs), with instructions on how to run the built-in models designed for learning in EpiModel. For information on how to extend these models to simulate novel epidemiological processes, see the related tutorial New DCMs with EpiModel.

Deterministic compartmental models solve differential equations representing analytic epidemic systems in continuous time. The models are deterministic because their solutions are fixed mathematical functions of the input parameters and initial conditions, with no stochastic variability in the disease and demographic transition processes. The models are compartmental because they divide the population into groups representing discrete disease states (e.g., susceptible and infected), and further on demographic, biological, and behavioral traits that influence disease transmission. In contrast to the stochastic models presented below, individuals within the population are not discretely represented.

A Basic SI Model

Starting with a basic Susceptible-Infected (SI) disease model in which there is random mixing within the population, the size of each compartment over time is represented by the equations:

\[ \begin{align} \frac{dS}{dt} &= -\lambda S \notag \\[4pt] \frac{dI}{dt} &= \lambda S \end{align} \]

where \(\lambda\) is the force of infection and represents \(\frac{\beta c I}{N}\). \(\beta\) is the probability of transmission per contact, \(c\) is the rate of contact per person per unit time, \(I\) is the number infected at time \(t\) and and \(N\) is the population size at time \(t\) (we drop the \(t\) subscript as it is implicit throughout).

Because ``contact’’ has been defined several ways in the modeling literature, we use the word act to represent the action, such as face-to-face discussion or sexual intercourse, by which disease may be transmitted. The force of infection is multiplied by the current state sizes of the \(S\) to solve the differential equation yielding the rate of change for the compartment.

To simulate a deterministic model in EpiModel, use the dcm function. Prior to running the model, it is necessary to parameterize it. Model parameters, initial conditions, and controls settings are input into three helper functions that organize the model framework. In param.dcm, the epidemic model parameters are entered. The inf.prob argument sets the transmission probability per act, and act.rate sets the acts per person per unit time. The init.dcm function collects the initial conditions for the model, and since this is an SI model it is necessary to specify the initial number susceptible and infected at \(t_1\). The control.dcm finally collects other structural model controls like the model type and number of time steps for the simulation.

param <- param.dcm(inf.prob = 0.2, act.rate = 0.25)
init <- init.dcm(s.num = 500, i.num = 1)
control <- control.dcm(type = "SI", nsteps = 500)

The model parameters, initial conditions, and controls are then entered as inputs into the dcm function to run the model and save the output in our object mod.

mod <- dcm(param, init, control)

Options for analyzing the results are consistent across all the model classes and types. Printing the model object provides basic information on model input and output, including model parameters and data variables for plotting and analysis.

mod
EpiModel Simulation
=======================
Model class: dcm

Simulation Summary
-----------------------
Model type: SI
No. runs: 1
No. time steps: 500
No. groups: 1

Model Parameters
-----------------------
inf.prob = 0.2
act.rate = 0.25

Model Output
-----------------------
Variables: s.num i.num si.flow num

The output shows that two compartments and one flow are available. In EpiModel, regardless of the model class, compartments are discrete disease states and flows are the transitions between states; when demographic processes are introduced, flows represent transitions in and out of the population. The i.num compartment is the size of the infected population at each of the solved time steps. The endogenous disease flow names represent the starting and ending state: si.flow is the number of people moving from \(S\) to \(I\) at each time step. In epidemiological terms, i.num and si.flow are disease prevalence and incidence.

To plot the results of the model we use the generic plot function, which by default plots all the model compartment state sizes over time.

plot(mod)