Exploring the impact of parameter changes

Overview

This app allows you to explore the effect of specific model parameters on some outcomes of interest for the basic virus model. Read about the model in the “Model” tab. Then do the tasks described in the “What to do” tab.

The Model

Model Overview

The model used here is the one introduced in the Basic Virus app. If you haven’t done so, check out and explore that app first.

What’s different here is that instead of running the simulation once for a given choice of parameter values and looking at the resulting time-series, this app lets you explore the impact of each model parameter on some outcomes. Specifically, we track both the peak and final numbers for all variables (uninfected cells, infected cells, virus). In general, the number at the end of the simulation is meaningful if it means the model has settled into a steady state. To ensure a steady state, you need to run the simulation for a long enough time. A steady state corresponds biologically to a chronic infection condition.

You can choose one of the model parameters to be varied between some minimum and maximum value. The other parameter values remain fixed. For each parameter value, the model is run and the outcomes of interest computed. The resulting plot is one showing how the outcomes of interest vary with the parameter you investigated.

Model Diagram and equations

As mentioned, this is the same model as used in the Basic Virus app. Details can be found there. For ease of reference, the flow diagram and the equations are shown again here:

Flow diagram for a simple virus infection model.

\[\dot U = n - d_U U - bUV\] \[\dot I = bUV - d_I I\] \[\dot V = pI - d_V V - gb UV\]

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).

Task 1:

Task 2:

Task 3:

The model we have here is so simple that we can in fact figure out how some of the outcomes depend on parameters without having to run simulations but instead by doing some math. Specifically, we can compute the steady states. You saw how that is done in the Basic Virus app. If this is unfamiliar to you or you need a refresher, (re)visit that app and go through the tasks. You found there (if you did the math) that the virus load at steady state is given by \(V_s = (bnp-bd_Ign-d_Id_Ud_V)/bd_Id_V\). This equation suggests that as the rate at which infected cells produce virus, the virus load at steady state goes up, which you found above. It also suggests that the relation between virus load and the rate at which virus infects cell, \(b\), is more complicated, since \(b\) shows up both in the numerator and denominator. If you have practice reading equations, you might be able to visualize how \(V_s\) behaves as \(b\) is varied. You can always explore it numerically. Let’s do that.

We cannot compute similar mathematical expressions for the peak of the variables. In general, as soon as our model reaches a certain level of complexity (maybe around 5 equations and more), getting analytic/mathematical equations for most outcomes of interest is not possible and the numerical approach of running the simulations and looking at the results is the only option we have.

Task 4:

Task 5:

Note that here we focus on a single parameter at a time. With a little bit of coding effort, we could change the underlying simulation to loop over say 2 parameters and produce outcomes for sets of parameter values, e.g. Vpeak as a function of b and p. This can be done by writing 2 loops (or some equivalent way of doing it) to scan over various combinations of parameter values for b and p and run the simulation for each such combination and record the results. These results could be plotted as a 2-dimensional heatmap. While this could be extended to more than 2 parameters, it will become hard to visualize and long to run. If there are many parameters that could change, a different approach is useful, which you’ll learn about in the Uncertainty and Sensitivity App.

Further Information

References