Reproductive Number 2 - Practice

Overview

This app covers more aspects of the reproductive number concept. Read about the model in the “Model” tab. Then do the tasks described in the “What to do” tab. If you are not familiar with the reproductive number, you should go through the ‘Reproductive Number 1’ app first.

Learning Objectives

The Model

Model Overview

For this app, we’ll use a basic compartmental SIR model that also includes births, deaths and waning immunity. We allow for 3 different stages/compartments:

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

For our system, we specify the following processes/flows:

Model Implementation

The flow diagram and the set of equations which are used to implement this model are as follows:

Flow diagram for this model.

Flow diagram for this model.

\[\dot S =m - b SI - nS + wR\] \[\dot I = \ S I - g I - nI\] \[\dot R = g I - nR - wR\]

What to do

The tasks below are described in a way that assumes everything is in units of MONTHS (rate parameters, therefore, have units of inverse months). If any quantity is not given in those units, you need to convert it first (e.g. if it says a year, you need to convert it to 12 months).

Task 1

Task 2

Task 3

Task 4

Task 5

Note: The SIR model is not a good model for TB since for TB, the stage where individuals are infected but not yet infectious is long and important. So to really model TB, one would need to include such details. We are only using TB here for illustrative purposes.

Further Information

References

Fine, Paul, Ken Eames, and David L Heymann. 2011. “"Herd Immunity": A Rough Guide.” Clinical Infectious Diseases : An Official Publication of the Infectious Diseases Society of America 52 (7): 911–16. https://doi.org/10.1093/cid/cir007.

Heffernan, J M, R J Smith, and L M Wahl. 2005. “Perspectives on the Basic Reproductive Ratio.” Journal of the Royal Society, Interface 2 (4): 281–93. https://doi.org/10.1098/rsif.2005.0042.

Keeling, Matt J, and Pejman Rohani. 2008. Modeling Infectious Diseases in Humans and Animals. Princeton University Press.

Roberts, M G. 2007. “The Pluses and Minuses of R0.” Journal of the Royal Society, Interface 4 (16): 949–61. https://doi.org/10.1098/rsif.2007.1031.

Vynnycky, Emilia, and Richard White. 2010. An Introduction to Infectious Disease Modelling. Oxford University Press.

Wallinga, J., and M. Lipsitch. 2007. “How generation intervals shape the relationship between growth rates and reproductive numbers.” Proceedings of the Royal Society B 274 (1609): 599–604.