Evolutionary Dynamics - Practice

Overview

For this module, we will explore a stochastic SIR-type model with 2 different pathogen strains, wild-type and a drug resistant mutant in the presence of drug treatment. Read about the model in the “Model” tab. Then do the tasks described in the “What to do” tab.

The Model

Model Overview

This model tracks susceptibles, wild-type infected untreated, wild-type infected treated, drug-resistant infected and recovered hosts. The following compartments are included:

The included processes/mechanisms are the following:

Model Implementation

The flow diagram for the model implemented in this app is:

Flow diagram for this model.

Flow diagram for this model.

Note that this model is not an ordinary differential equation model. It is instead its stochastic equivalent. We can specify the model by writing down every possible transition/event/reaction that can occur and their propensities (the propensity multiplied with the time step gives the probability that a given event/transition occurs). For our model these are the following:

Event type Transitions Propensity
S turn into Iu S => S-1, Iu => Iu+1 (1-f) * (bu * (1-cu) * Iu + bt * (1-ct) * It) * S
S turn into It S => S-1, It => It+1 f * (bu * (1-cu) * Iu + bt * (1-ct) * It) * S
S turn into Ir S => S-1, Ir => Ir+1 (bu * cu * Iu + bt * ct * It + br * Ir) * S
Recovery of Iu Iu => Iu-1, R => R+1 gu * Iu
Recovery of It It => It-1, R => R+1 gt * It
Recovery of Ir Ir => Ir-1, R => R+1 gr * Ir

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

Set the model parameters such that it corresponds to the following setting:

With parameters set to correspond to the scenario just described, run the simulation. You should see some simulations with outbreaks and some without. For those with outbreaks, you should have around 10-100 susceptible left at the end.

Task 2

Task 3

Task 4

Task 5

Task 6

Keep exploring. For instance, try the following:

Further Information

References

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.