Characterizing Infection States - Documentation

Overview

This app allows you to explore a model that has several more compartments than the basic SIR model discussed in the Introduction to ID app. Read about the model in the “Model” tab. Then do the tasks described in the “What to do” tab.

The Model

Model Overview

In the Introduction to ID app, you explored a simple 3-compartment model, the basic SIR model. The model for this app has a few additional compartments, which allows us to include more details/realism into our model. We again focus on tracking individuals with regard to their infection/disease status. For this model, we track the following compartments/stages:

Of course, as with the basic SIR model, we could include further details by extending the number of compartments. In general, for each additional feature you want to track, the existing number of compartments needs to be replicated by the discrete categories you have. For gender, one would need to have 2x the compartments. Similarly if one wanted to stratify according to young/medium/old age, 3x the compartments are required, etc.

In addition to specifying the compartments of a model, we need to specify the dynamics determining the changes for each compartment. In general, more compartments leads to more processes and more parameters governing these processes.

For this model, we include the following processes:

Model Implementation

The flow diagram and the set of ordinary differential equations (ODE) which are used to implement this model are as follows:

Model Diagram

Model Diagram

\[\dot S = -S (b_P P + b_A A + b_I I)\] \[\dot P = S (b_P P + b_A A + b_I I) - g_P P\] \[\dot A = f g_P P - g_A A\] \[\dot I = (1-f) g_P P - g_I I\] \[\dot R = g_A A + (1-d) g_I I\] \[\dot D = d g_I I\]

What to do

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

Start with 1000 susceptibles, 1 initially infected presymptomatic host, P0, simulation duration of 200 days. Assume that only symptomatic individuals transmit, at rate bI = 0.001. Assume that the duration of the presymptomatic, asymptomatic and symptomatic periods are all 5 days long. (Hint: The parameters gP, gA, and gI are the inverses of these periods.) Assume that there are no asymptomatic infections, f=0, and nobody dies due to disease d=0. Make sure the results agree with model settings. For instance you shouldn’t get any deaths, and no asymptomatic infected. If you did it correctly, your outbreak should end with around 7 susceptibles remaining. From the graph, contemplate how you would estimate the day at which the outbreak peaks. What’s the problem? How would you solve it? Rerun the simulation, with the same input settings (i.e. don’t change anything, just hit the Run Simulation button again). Do you see any changes in the results? Why (not)? (This will be different once we move to the stochastic models).

Record

Task 2

Assume now that after the pre-symptomatic stage, half of the infected move on to become symptomatic while the other half is asymptomatic. Set the model parameter controlling the fraction who become asymptomatic accordingly. Don’t change any other assumption. Do you expect to get a change in the outbreak size? Why? Compute the total number of infected as you did for the Basic SIR app. To do so, look at the diagram for the model and make sure you understand how individuals can flow through the system. That information will help you compute the total (cumulative) number infected (both symptomatic and asymptomatic).

Record

Task 3

Now assume that asymptomatic individuals transmit at the same rate as symptomatic individuals. Change bA accordingly. Leave everything else as before. How do you expect the results to change? (Try to make as precise/quantitative a prediction as you can). Compare your results to the first task and make sure you understand why you get the results you do.

Record

Task 4

Next, let’s assume that half the symptomatic infected die. Change the corresponding parameter in the model to reflect this new assumption. Leave everything else as in task 3. How do you expect the results to change?

Record

Task 5

Set all 3 transmission parameters (bP, bA, and bI) to 0.0005, leave everything else as before. Note that this lowers the transmission rate of the asymptomatic and symptomatic groups by a factor of 2, but now pre-symptomatic individuals are also infectious. What do you expect will happen to the peak of P and I as well as the total number of individuals who got infected (i.e., R + D at the end of simulation), compared to the previous task?

Record

Task 6

Further explore how changes in the infectiousness of the different groups (bP, bA, bI) and the average time a person spends in each of those states (gP, gA, gI) affects the infection dynamics. Every time, think about what you expect to get, then run the simulation, compare your expectations with the results. Then make sense of it.

Record

Task 7

Using the settings from task 5, but now assume that 90% of infections are asymptomatic. How do things change?

Record

Task 8

Further explore how changes in the fraction becoming asymptomatic and fraction dying does (or does not) affect the infection dynamics. Every time, think about what you expect to get, then run the simulation, compare your expectations with the results. Then make sense of it.

Record

Further Information

References

Kajita, Emily, Justin T. Okano, Erin N. Bodine, Scott P. Layne, and Sally Blower. 2007. “Modelling an Outbreak of an Emerging Pathogen.” Nature Reviews Microbiology 5 (9): 700–709. https://doi.org/10.1038/nrmicro1660.
Keeling, Matt, and Pejman Rohani. 2007. Modeling Infectious Diseases in Humans and Animals. Princeton University Press.
Milwid, Rachael, Andreea Steriu, Julien Arino, Jane Heffernan, Ayaz Hyder, Dena Schanzer, Emma Gardner, et al. 2016. “Toward Standardizing a Lexicon of Infectious Disease Modeling Terms.” Frontiers in Public Health 4 (September). https://doi.org/10.3389/fpubh.2016.00213.
Vynnycky, Emilia, and Richard White. 2010. An Introduction to Infectious Disease Modelling. Oxford University Press.