Overview
This app allows exploration of an acute virus infection model, with
compartments for uninfected cells, infected cells and (free) virus. The
model also includes abstract versions of the innate and adaptive immune
response. Read about the model in The Model tab. Then, work
through the tasks described in the What To Do tab.
The model used here is a variation of the one introduced in the
Basic Virus Model app. If you haven’t done so,
check out and explore that app first.
Learning Objectives
- Learn about acute virus infection dynamics.
- Understand the impact the immune response can have on outcomes in
this model.
The Model
Model Overview
This model consists of 5 compartments and can capture some of the
basic dynamics of viral infections and the ensuing immune response. 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 (interferon) response
- T - adaptive (T-cell) 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 our system, we specify the following processes/flows:
- Virus infects cells at rate b.
- Infected cells produce new virus at rate p and die at rate
dI.
- Free virus is removed at rate dV or goes on to
infect further uninfected cells at rate gb. To allow conversion
from infectious virus units in the model to some experimental units
(e.g. plaque forming units), an additional conversion factor,
g, is included in the model.
- The innate (interferon) response is induced proportional to the
number of infected cells at rate rF and decays at
rate dF. The innate response lowers the rate of
virus production by infected cells, with strength of reduction
controlled by parameter k.
- The adaptive (CD8 T-cell) response grows exponentially at rate
rT and proportional to the strength of the innate
response, and decays at rate dT. T-cells kill
infected cells at rate kT.
Model Diagram
The diagram illustrating this compartmental model is shown in the
figure.
Model Equations
Implementing this model as a continuous-time,
deterministic model leads to the following set of
ordinary differential equations.
\[\begin{aligned}
\dot U & = - b U V \\
\dot I & = b U V - d_I I - k_T T I \\
\dot V & = \frac{p}{1+k_F F}I - d_V V - gb UV \\
\dot F & = r_F I - d_F F \\
\dot T & = r_T T F - d_T T
\end{aligned}\]
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
Let’s start by considering infection dynamics in the absence of the
immune response. Set the initial conditions to 105 uninfected
cells, no infected cells, and 10 virus particles. Assume also that
infected cells have an average life-span of 1 day, and virus has a
life-span of 6 hours (remember that the inverse of the lifespan is the
rate of death, and make sure you convert to the right units). Set that
the virus production by an infected cell is 100 virions per day and that
the rate at which new cells become infected is 10-6. Assume
there is no need to do any unit adjustment/conversion (i.e. the value of
that parameter is 1). Set the starting values for the immune response
variables F and T to 0, and also set the two
immune response induction rates to 0. With those choices, the values for
the other immune response related parameters do not matter. If you want,
you can also set them all to 0. Run the simulation for 50 days; produce
plots with and without log-scales. You should get a single, acute
infection with virus and infected cells rising at first and then
declining. At the end you should be left with around 11069 uninfected
cells and essentially no infected cells or virus.
Record
- Number of infected cells at the peak of infection
Task 2
In the absence of the immune response, this model is the same as the
one covered in the first part of the Basic Virus app. Convince
yourself that you can get at most one rise, then decline of virus, and
confirm that this acute infection increases as you increase virus
production and infection rates, and that it decreases as you increase
infected and virus death rates. Double the virus production and
infection rate parameters (compared to the previous task), and a half
the lifespans of the infected cells and virus (remember the relation
between durations, such as lifespan, and rates, such as death rates.)
Run the simulation. Compare the number of uninfected cells remaining at
the end with what you found for task 1. If it’s not clear to you why you
get what you find, revisit the Basic Virus app and
especially the task discussing the reproductive number.
Record
- Remaining uninfected cells at the end
Task 3
Now turn on the immune response. Set everything as you did for the
first task. Then change the starting value for T to 1, set
rT = 0.01, dT = 1 and
kT = 0.01. Run the simulation and convince yourself
that in the absence of the innate response, the adaptive response is not
triggered, and the resulting infection thus looks like the one you saw
in the first task. If it is unclear to you why the adaptive response is
not triggered, look at the model. You’ll see that induction/growth of
the adaptive response depends on the innate response.
Record
- Maximum value of infected cells
Task 4
The previous task showed that in the absence of the innate response,
the adaptive response won’t kick in. The reverse is not true. Set
everything back as in task 1. Then set the starting value for F
to 1, rF = 10, dF = 1 and
kF = 0. With these settings, the innate response is
activated, but has no impact yet on the virus kinetics. You should see
the innate response come up and decay, but the same virus and infected
cell trajectories as for task 1. Now, turn on the action of the innate
response by setting kF = 10-5. You should
notice that the innate response reduces the virus load curve and this
leads to fewer total cells that get infected.
Record
- Remaining uninfected cells at the end with innnate action turned
on
Task 5
Finally, we’ll add the adaptive response. Keep settings as in the
previous task. Then set the starting value for T to 1, , set
rT = 2 x 10-5, dT =
0.1 and kT = 0. Run the simulation. You should see
the adaptive response come up, but since we set its killing rate to 0,
it does not yet impact the virus or cell kinetics. Let’s change that by
setting kT = 10-4. You should now see
both the innate and adaptive response being induced, and both act to
reduce the impact of the infection, with a further reduction in virus
load and even fewer infected cells at the end. You will also see that
the adaptive response is lower now, since the faster clearance of virus
and thus reduction of the innate response leads to a reduced adaptive
response. This is an example of the complex interactions one can get
with these kind of models, even fairly simple ones like this.
Record
- Remaining uninfected cells at the end with T-cell killing turned
on
Task 6
Continue to explore how different strengths of innate and adaptive
response kinetics impact the virus and infected cell dynamics. The
kinetics of those immune responses is determined by their induction
rates rF and rT and decay rates
dF and dT. Then explore further
how immune response action, as represented by the parameters
kF and kT, impact the outcomes.
Finally, explore how all of this changes as you change some of the
parameters associated with virus and cell dynamics. You might notice
that the adaptive response is very sensitive to changes in
rT, probably more so than makes biological sense.
This suggests that a different formulation of the model might possibly
be better. Some alternatives are explored and discussed in the
Extended Virus and Immune Response and Model
Variant Exploration apps.
Record
Further Information
This app (and all others) are structured such that the Shiny part
(the graphical interface you see and the server-side function that goes
with it) calls an underlying R script (or several) which runs the
simulation for the model of interest and returns the results.
For this app, the underlying function running the simulation is
called simulate_acutevirusir_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.
You can also download all simulator functions and modify them for
your own purposes. Of course to modify these functions, you’ll need to
do some coding. For examples on using the simulators directly and how to
modify them, read the package vignette by typing
vignette('DSAIRM') into the R console.
This model is based on one of the models discussed in (Li and Handel 2014). See also for instance
(Beauchemin and Handel 2011; Smith and Perelson
2011) and references in the other apps in this section.
References
Beauchemin, Catherine AA, and Andreas Handel. 2011. “A Review of
Mathematical Models of Influenza A Infections Within a Host
or Cell Culture: Lessons Learned and Challenges Ahead.” BMC
Public Health 11 (1): S7.
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.
Smith, Amber M, and Alan S Perelson. 2011.
“Influenza
A Virus Infection Kinetics: Quantitative Data and
Models.” Wiley Interdiscip Rev Syst Biol Med 3 (4):
429–45.
https://doi.org/10.1002/wsbm.129.
