Different Model Formulations

Overview

This app allows exploration of the impact of different model formulations on the results obtained from a simulation. 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 consists of several compartments that capture some of the basic dynamics of virus and immune response during an infection. In this model, we track the following entities, by assigning each to a compartment:

Both the innate and adaptive response are modeled in a rather abstract manner. We could think of them as some kind of cumulative representation of each arm of the immune response, or alternatively a single dominant innate response component, e.g. interferon for the innate and CD8 T-cells for the adaptive response.

In addition to specifying the compartments of a model, we need to specify the dynamics determining the changes for each compartment. Broadly speaking, there are processes that increase the numbers in a given compartment/stage, and processes that lead to a reduction. Those processes are sometimes called in-flows and out-flows.

For the purpose of this app, we specify several alternative processes that allow us to explore different model variants by ‘turning on and off’ specific components of the model.

We specify the following processes/flows:

The idea explored in this app and implemented by this model is that results sometimes, but not always, change depending on different (biologically reasonable) ways the immune response is modeled. We can explore those different models by setting certain parameters describing alternative processes to a non-zero value, and all others to zero. We can then study how different model alternatives affect the outcome.

Obviously, the number of alternative models we could make that are biologically reasonable is virtually endless. The better the underlying biology of a given infection is known, the easier it is to pick one model formulation over another. In the end, for most infections, we still don’t know enough to pick the “right” model. We often have to choose one or a few reasonable model candidates and hope that they approximate the underlying processes reasonably well.

Model Diagram

Flow diagram for the model with different innate and adaptive response variants.

Flow diagram for the model with different innate and adaptive response variants.

Model Equations

\[\dot U =n - d_U U - bVU - k_1FU \] \[\dot I = bVU - d_II - k_2FI - k_4 A I - k_5 \frac{A I}{A+s_A}\] \[\dot V = \frac{pI}{1+k_3 F} - d_VV - bVU - k_6AV\] \[\dot F = p_F - d_F F + f_1 V (F_{max} - F) + f_2 \frac{V}{V+s_V} F + f_3 \frac{VI}{VI+s_V} F\] \[\dot A = a_1 F A + a_2\frac{V}{V+h_V}F + a_3 \frac{F V}{ F V + h_V} A \]

What to do

It is recommended that before you work your way through this app and tasks, you first explore and do the tasks in the “Basic Virus Model” and “Virus and Immune Response Model” apps.

For the tasks below, it is assumed that the model is run in units of days

Task 1:

Task 2:

Task 3:

Task 4:

Task 5:

Task 6:

Further Information

References

Dobrovolny, Hana M, Micaela B Reddy, Mohamed A Kamal, Craig R Rayner, and Catherine AA Beauchemin. 2013. “Assessing Mathematical Models of Influenza Infections Using Features of the Immune Response.” PloS One 8 (2): e57088.
Li, Yan, and Andreas Handel. 2014. “Modeling Inoculum Dose Dependent Patterns of Acute Virus Infections.” Journal of Theoretical Biology 347 (April): 63–73. https://doi.org/10.1016/j.jtbi.2014.01.008.