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:

  • U - uninfected cells
  • I - infected cells
  • V - (free) virus
  • F - innate immune response
  • A - adaptive immune response

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:

  • Uninfected cells are produced at rate n, die naturally at rate dU and become infected at rate b.
  • Infected cells die at rate dI and produce virus at rate p.
  • Free virus is removed at rate dV due to any unmodeled processes, or goes on to infect further uninfected cells at rate b.
  • In the absence of virus, the innate response is produced at a rate pF and removed at a rate dF. In the presence of virus, the innate response additionally grows according to 3 alternative model formulations:
    1. Proportional to virus at rate f1 and saturating at a maximum level Fmax.
    2. Proportional to virus at rate f2, with a growth rate saturating at high levels of virus, determined by the saturation constant sV.
    3. Proportional to both virus and infected cells at rate f3, with a growth rate saturating at high levels of virus and infected cells, determined by the saturation constant sV.
  • The innate response can also act on the system in 3 different ways:
    1. By moving target cells into a “protected” state at rate k1 where those cells can not become infected any longer.
    2. By inducing death of infected cells at rate k2.
    3. By reducing production of virus particles at strength k3.
  • The adaptive response growth is also modeled according to 3 alternative model formulations:
    1. Proportional to the innate response at rate a1.
    2. Proportional to virus at rate a2, with a growth rate saturating at high levels of virus, determined by the saturation constant hV.
    3. Proportional to both virus and innate response at rate a3, with a growth rate saturating at high levels of virus and innate response, determined by the saturation constant h~V~.
  • The adaptive response can act on the system in 3 ways:
    1. By killing infected cells at rate k4.
    2. By killing infected cells at rate k5, with saturation of the maximum killing rate for high adaptive response levels, determined by the saturation constant sA.
    3. By killing virus at rate k6.
  • Adaptive immune response decays at a rate dA.

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:

  • We’ll begin with the basic virus model by turning off any immune response related component.
  • Set all parameter values related to the immune response such that there is no F and A present at any time during the simulation.
  • Set target cell and virus starting conditions to 105 initial uninfected cells, no infected cells, 10 virions.
  • Set infection rate to 10-5, no production or death of uninfected cells, lifespan of infected cells and virus of 1 day and 6 hours respectively. Rate of virus production should be 100 per day.
  • You should get a single acute virus infection with no immune response present and a maximum of 70217 infected cells.

Task 2:

  • Let’s explore different mechanisms for the innate response. Keep initial level of innate response at 0, but set innate production and removal rate to 1. Set all innate growth parameters (the fi) and all innate actions (k1, k2, k3) to 0.
  • Run the simulation, confirm that you get an innate response that settles at a steady state (balance between production and removal), but that there is no further increase and that the rest of the dynamics doesn’t change (i.e. you should get the same maximum number of infected cells as above).

Task 3:

  • Now explore the different types of innate response induction by playing with the fi parameters and the saturation parameter, sV. Relate what you see in the plots to the equations so you get an idea of how different terms in the equations behave. Continue to keep the ki at 0. This means the rest of the variables should not change.

Task 4:

  • Now explore what happens when you have non-zero ki. You’ll find that the action of the innate response impacts the other variables, which in turn can impact further innate activation. Some of the resulting dynamics can get complex.
  • Pay attention to how different processes of innate activation and innate action do or don’t produce different overall dynamics.

Task 5:

  • Finally turn on the adaptive response. Note that in this model, there is no adaptive response without innate response (check it by setting the innate response to 0 while having non-zero adaptive growth rates, ai).
  • Turn the innate response back on and explore how different non-zero adaptive growth rates, ai, affect the adaptive response dynamics. Start by leaving the adaptive action parameters at 0. Then play with those parameters as well and set them to non-zero values. Also explore the impact of the saturation constants.

Task 6:

  • As you’ll notice, some of the specific model choices (i.e. specific parameters/terms in the model being turned on or off) lead to similar results, while other times results are quite different. For a specific system under study, some ways to formulate the immune response and the dynamics one gets might be better than others. Think about a system you are familiar with and consider which (if any) of the possible mechanisms implemented in this model might best describe that system.

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.