Basic SIR Model Documentation

Model Overview

This model is a compartmental SIR (susceptible-infected-recovered) model. Compartmental means that we place individuals into distinct compartments, according to some characteristics. We then only track the total number of individuals in each of these compartments. In the simplest model, the only characteristic we track is a person’s infection status. We allow for 3 different stages/compartments:

• S - uninfected and susceptible individuals.
• I - infected and infectious individuals (note that these terms are often used interchangeably, but technically we are talking about someone who is infected and is infectious, i.e. can infect others).
• R - recovered/removed individuals. Those are individuals that do not further participate, either because they are now immune or because they died.

The SIR model is very basic. It could be extended by introducing further compartments. For instance, we could stratify according to gender, which would give us 2 sets of SIR compartments, one for males and one for females. Some of these extensions are implemented in other apps.

In addition to specifying the compartments of a model, we need to specify the processes/mechanisms determining the changes for each compartment. Broadly speaking, there are processes that increase the number of individuals in a given compartment/stage, and processes that lead to a reduction. Those processes are sometimes called inflows and outflows.

For our system, we specify only 2 processes/flows:

• A susceptible individual (S) can become infected by an infectious individual (I) at some rate (for which we use the parameter b). This leads to the susceptible individual leaving the S compartment and entering the I compartment.
  
• An infected individual dies or recovers and enters the recovered/removed (R) compartment at some. This is described by the parameter g in our model.

As with the compartments, we could extend the model and allow other processes to occur. For instance, we could allow for natural births and deaths, waning immunity, deaths due to disease, etc. Some of that will be included in other apps.

Model Representation

For compartmental models (and also often other types of models), it is useful to show a graphical schematic representation of the compartments and processes included in the model. For compartmental models, such a diagram/figure is usually called a flow diagram. Such a diagram consists of a box for each compartment, and arrows pointing in and out of boxes to describe flows and interactions. For the simple SIR model, the flow diagram looks as follows:

Flow diagram for simple SIR model.

Model Implementation I

To allow us to simulate this model, we need to implement it on the computer. For that purpose, it is often useful to write the model as mathematical equations (this is not strictly needed, some computer simulation models are never formulated as mathematical models). A very common way (but not the only one) to implement compartmental models such as the simple SIR model is a set of ordinary differential equations. Each compartment/variable gets an equation. The right side of each equation specifies the processes going on in the system and how they change the numbers in each compartment via inflows and outflows. For the model described above, the equations look like this:

\[ \begin{aligned} \dot S & = -bSI \\ \dot I & = bSI - gI \\ \dot R & = gI \end{aligned} \]

Note: If you don’t see equations but instead gibberish, try opening the app with a different browser. I have found that occasionally, on some computers/browsers, the math is not shown properly.

Model Implementation II

Continuous time models implemented as ordinary differential equations are the most common types of models. However, other implementations of the above model are possible. One alternative formulation is a discrete-time deterministic equivalent to the ODE model. For such an implementation, the equations are:

\[ \begin{aligned} S_{t+dt} & = S_t + dt * \left( - b I_t S_t \right) \\ I_{t+dt} & = I_t + dt * \left( b I_t S_t - g I_t \right) \\ R_{t+dt} & = R_t + dt * \left( g I_t \right) \end{aligned} \]

In words, the number of susceptible/infected/recovered at a time step dt in the future is given by the number at the current time, t, plus/minus the various inflow and outflow processes. The latter need to be multiplied by the time step, since less of these events can happen if the time step is smaller. As the time-step gets small, this discrete-time model approximates the continuous-time model above. In fact, when we implement a continuous-time model on a computer, the underlying simulator runs a “smart” version of a discrete-time model and makes sure the steps taken are so small that the numerical simulation is a good approximation of the continuous-time model. If you want to learn more about that, you can check out the ‘deSolve’ R package documentation, which we use to run our simulations.

Some general notes

• Throughout DSAIDE, I will use ID to abbreviate infectious disease
• You will see both the term host(s) and individual(s) used interchangeably. While we most often think of human hosts, the hosts can be any animal (or plants or bacteria infected by phages or…).
• In general, the entities that change in our model (here the number of individuals in compartments S, I and R) are called variables: They are variable and change during the simulation.
• In contrast, the quantities that are usually fixed for a given scenario/simulation are called parameters. For this model, those are the infection rate b and the recovery rate g.
• Sometimes, parameters can vary during the simulation, but most often they do not.
• If you want to study a specific ID, you choose parameters such that they match the specific disease you want to study. For now, just play around with the model without trying to relate it to some specific ID.
• There are no fixed rules concerning the naming of variables and parameters. Compartments (e.g. SIR) tend to be labeled very similarly by different researchers, while parameter labels are much more variable. Always check carefully for a given paper/model what the meaning of each parameter is.
• Some people prefer diagrams, others equations. In my opinion, it is best to show both.

A few general notes

• Some of the tasks below (and in future apps) are fairly open-ended. The idea is that these tasks give you something to get started, but you should feel free to explore the simulations any way you want. Play with them, query them, go through iterations of thinking what you expect, observing it, and if discrepancies occur, figure out why. Essentially, the best way to use these apps is to do your own science/research with them.

• You might find for some parameter settings that numbers in specific compartments (e.g. number of infected) drop below 1 (and possibly rebound later). This of course is biologically not reasonable. Numbers less than 1 are an artifact of the underlying differential equation model. All ODE models (and the discrete time model we consider here) have that problem. A ‘hacky’ solution is to monitor the simulation and if a quantity drops below 1, set it to 0 or stop the simulation. Some apps use this approach. A cleaner solution is to treat all variables as discrete units and allow them to only change in integer steps (in a probabilistic manner). This approach will be discussed in the apps under the “Stochastic models” section. For most apps, neither approach is taken, so you’ll see numbers less than 1. That’s ok for the purpose we use the apps here. Just be careful if you use these kinds of models in your research you might need to pay attention to this issue.

• While the list of tasks for each app always states what time units you should run the model in, this and most other simulations/apps in DSAIDE do not have natural time units (unless specifically stated). You could, therefore, assume that your model runs in units of days or weeks/months/years, based on what’s most suitable for the disease you want to study. You have to make sure that all your parameters are in the right time units. Always make sure to check if a given simulation can handle different time units or assumes specific ones. Again, for the tasks the time units are specified.

• The app lets you choose the maximum simuation time. Some apps also let you set the starting time and time step. Starting time generally does not matter and is almost always set to 0. Sometimes an app doesn’t even allow you to change this. For models like that are based on differential equations, the time step does not affect the internal time steps taken by the numeric solver, only the times for which results are returned. Thus, for some apps, it is also not an input. It is here, so it can run together with a discrete-time model. Only for discrete-time models is this the actual time step that’s being taken.

The tasks below are described in a way that assumes everything is in units of days (rate parameters, therefore, have units of inverse days). If any quantity is not given in those units, you need to convert it first (e.g. if it says a week, you need to convert it to 7 days).

Task 1

• Run the simulation with 1000 susceptible individuals and 1 initially infected host, no recovered. Set simulation duration to 100 days, start time and time step can remain at 0 and 0.1. Set recovery rate g=0.5 per day, and infectiousness b=0.001. You will get an outbreak of some - currently unspecified - infectious disease. If you did it correctly, your outbreak should end with around 203 uninfected individuals still remaining.
• Also take a look at the final number of infected. You’ll see that it is at 3E-9, i.e. much less than 1 but not 0. That’s the issue discussed above in the notes, that for ODE models, numbers can drop below 1, and then can slowly go to 0 but not fully reach 0. Often, this biologically unreasonable feature of these kinds of models can be ignored, but sometimes it matters. You will learn more about this in the apps under the “Stochastic models” section.

Task 2

• Switch the plotting to have x-axis, y-axis or both plotted on a log scale.
• Leave all other settings as before.
• Note that while the look of the plot changes a lot, nothing has changed about the underlying simulation. The results are exactly the same in each case, only plotted differently. This is something to be aware of when you see plots in papers or produce your own. The best plot to use is the one that shows results of interest in the clearest form. Usually, the x-axis is linear and the y-axis is either linear or logarithmic.
• Try again switching between ggplot and plotly as plotting engine.

Task 3

• From the graph, get a (rough) estimate of the day at which the outbreak peaks. That’s easiest done when you make a plotly plot.
• Contemplate the fact that the outbreak ends even though there are still susceptible remaining, i.e. not everyone got infected. Do you find that surprising? How could you maybe explain that? This topic is discussed in more detail in the reproductive number app.
• You have no infected left at the end of the outbreak. Figure out how you can use the information from either the susceptibles or recovered left at the end of the outbreak to figure out how many people in total got infected during the outbreak.
• Rerun the simulation, with the same values you just had. Does anything change? Why (not)?

Task 4

• Set “Models to run” to “both”", which runs and shows both the continuous-time and discrete-time models.
• Start with a discrete-time step of 0.01.
• Leave all other settings as before.

Run the simulation, see what you get. You should see the results from the 2 models essentially on top of each other and barely distinguishable.

Task 5

• Now try different values for dt.
• Leave all other settings as before.

You should notice that as dt gets larger, the differences between discrete-time and continuous-time models increase. At some point when the time-step gets too large, the discrete-time simulation ‘crashes’ and you get an error message. This doesn’t impact the continuous/ODE simulation since it chooses its time-step during the simulation internally and dt only affects the times for which the results are returned.

Task 6

• Go back to running the ODE model only.
• Double the value of the transmission rate, b. Leave everything else the same.
• Compared to what you found in the previous task, how do you expect the min/max and number/fraction of S/I/R at the end of the outbreak to change? What about the time at which the outbreak peaks?
• Run the simulation with the doubled transmission rate, record the same values (final S/I/R and outbreak peak) as above.
• Compare your expectations with the results. How do they agree/disagree? Does it make sense? Anything surprising happening?

Task 7

• Now, double the rate of recovery, g, leave everything as in #2.
• How do you expect the results to change? (Try to make your prediction as precise/quantitative as you can.)
• Run the simulation with these new parameter settings, record the same values as above.
• Compare your expectations with the results. How do they agree/disagree? Does it make sense? Anything surprising happening?

Task 8

• Now, double the number of susceptibles, S₀, leave everything else as you just had.
• How do you expect the results to change?
• Run the simulation with these new parameter settings, record the same values as above.
• Compare your expectations with what results. How do they agree/disagree? Does it make sense? Anything surprising happening?

Task 9

• Keep playing around with changing any of the parameters and starting conditions.
• Every time, think about what you expect to get, then run the simulation, compare your expectations with the results. Then make sense of it.
• What is the minimum and maximum number of outbreaks you can get? Why is that?