This app explores how one should implement ID control measures for a scenario with multiple consecutive outbreaks. Read about the model in the “Model” tab. Then do the tasks described in the “What to do” tab. Before going through this app, you should go through the ‘Reproductive Number’ apps first. This app closely follows a model and analysis described in (Handel, Longini, and Antia 2007), see the “Further Information” tab for this and other related references.
For this app, we’ll use the basic compartmental SIR model. We include 3 different stages/compartments:
For this app, we specify the following processes/flows:
The flow diagram and the set of equations which are used to implement this model are as follows:
Flow diagram for this model.
\[\dot S = -(1-f) b SI\] \[\dot I = (1-f)b S I\] \[\dot R = g I\]
The parameter f can be non-zero outside of the treatment interval given by tstart and tend. At intervals tnew, the number of infected increases by 1. The simulation for this app also has one more feature. In previous apps, you learned that an unrealistic feature of models implemented by ordinary differential equations (ODE) is that the number of individuals can drop below 1, i.e., there could be a fraction of infected. For this app, the underlying code is such that if the number of infected drops below 1, it is set to 0. This is in some way a bit of a “hack” to deal with this issue. When you work through the stochastic apps, you will learn better ways of handling fractions in models. The advantage of using the “hack” is that we can keep using the ODE model formulation, without stochasticity, which makes things easy.
This app assumes knowledge of the reproductive number concept. If you are not familiar with it, please go through the ‘Reproductive Number’ apps first.
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.
This value for S provides a threshold below which we don’t get an outbreak. This, of course, directly corresponds to the threshold value for R0 which you learned about previously.
The threshold value below which you do not get an outbreak is around 250 for the parameter settings here. However, in the first task, the number of susceptibles left at the end of the outbreak was much less, around 20. The outbreak ‘overshoots’ by depleting more susceptibles than the threshold value. Why is that? Think about what happens at the peak of the outbreak when R0 switches from being larger than 1 to smaller than 1. Does the outbreak stop immediately? Why not?
For a single outbreak, more control is better. We can test that.
For multiple outbreaks, more control is not necessarily better. Let’s explore that.
In the previous task, control ends while there are still a few infected around, which can lead to a second outbreak. An alternative scenario is one where control ends after infected are gone, but then a newly infected person enters the population. We can explore this scenario, as well.
simulate_idcontrolmultioutbreak_ode
. You can call them directly, without going through the shiny app. Use the help()
command for more information on how to use the functions directly. If you go that route, you need to use the results returned from this function and produce useful output (such as a plot) yourself.vignette('DSAIDE')
into the R console.Alexander, Murray E, Christopher S Bowman, Zhilan Feng, Michael Gardam, Seyed M Moghadas, Gergely Röst, Jianhong Wu, and Ping Yan. 2007. “Emergence of drug resistance: implications for antiviral control of pandemic influenza.” Proc Biol Sci 274 (1619): 1675–84. https://doi.org/10.1098/rspb.2007.0422.
Handel, Andreas, Ira M Longini Jr, and Rustom Antia. 2007. “What Is the Best Control Strategy for Multiple Infectious Disease Outbreaks?” Proc Biol Sci 274 (1611): 833–37. https://doi.org/10.1098/rspb.2006.0015.
Handel, Andreas, Ira M Longini, and Rustom Antia. 2009. “Antiviral Resistance and the Control of Pandemic Influenza: The Roles of Stochasticity, Evolution and Model Details.” J Theor Biol 256 (1): 117–25. https://doi.org/10.1016/j.jtbi.2008.09.021.
Lipsitch, Marc, Ted Cohen, Megan Murray, and Bruce R Levin. 2007. “Antiviral Resistance and the Control of Pandemic Influenza.” PLoS Med 4 (1): e15. https://doi.org/10.1371/journal.pmed.0040015.