Modeling vaccines and drugs
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Big picture
We generally want to understand pathogen-immune response dynamics with the eventual goal of intervening with drugs, vaccines, etc.
Modeling of drugs is a big area in pharma knows as pharmacometrics (PM/PMX) or pharmacokinetic/pharmacodynamic (PK/PD) modeling. The term Quantitative Systems Pharmacology (QSP) is also used.
General Idea
- Build a model without an intervention that includes the components you are interested in:
- Pathogen
- Immune Response (optional)
- Add drug/treatment into the model
Example 1
- IFN treatment for HCV infection
- Question to answer: By what mechanism does the drug work?
- See “Antiviral Treatment Model” in DSAIRM and also the “Influenza Drug Model”.
Example 1
![]()
\[\begin{align}
\dot U & = n - d_U U - \color{red}{(1-f)}bUV \\
\dot I & = \color{red}{(1-f)}bUV - d_I I \\
\dot V & = \color{blue}{(1-e)}pI - d_V V - \color{red}{(1-f)}gbUV
\end{align}\]
Simplest assumption: Drug effect is constant.
Example 2
![]()
\[\begin{align}
\dot U & = n - d_U U - bUV \\
\dot I & = bUV - d_I I \\
\dot V & = (1-e)pI - d_V V - gb UV \\
\dot C & = - d_C C, \qquad C=C+C_0 \textrm{ at } t = t_{interval} \qquad \textrm{(PK)}\\
e & = E_{max} \frac{C^k}{C^k+C_{50}} \qquad \textrm{(PD)}
\end{align}\]
See the “PK/PD” app in DSAIRM.
Example 3
\[
\begin{aligned}
\dot{B} & = g B(1-\frac{B}{B_{max}}) - d_B B - kBI \color{blue}{- f(C)B}\\
\dot{I} & = r BI - d_I I \\
\color{blue}{\dot{C}} & \color{blue}{= ?}
\end{aligned}
\]
A drug at concentration \(C\) leads to extra killing of bacteria (PD). The drug has some time-course (PK).
Example 3
\[
\begin{aligned}
\dot{B} & = g B(1-\frac{B}{B_{max}}) - d_B B - kBI - eB\\
\dot{I} & = r BI - d_I I \\
\dot C & = - d_C C, \qquad C=C+C_0 \textrm{ at } t = t_{interval} \qquad \textrm{(PK)}\\
e & = E_{max} \frac{C^k}{C^k+C_{50}} \qquad \textrm{(PD)}
\end{aligned}
\]
- The bad news: This model is not part of DSAIRM.
- The good news: We can build it ourselves!
Model Exploration
Get the code/simulation model for the Basic Bacteria model and the PK/PD model, namely simulate_basicbacteria_ode.R and simulate_pkpdmodel_ode.R.
Take a look at both R files. All the stuff at the top (any line that starts with #') is just documentation and you can mostly ignore.
Make a copy of simulate_pkpdmodel_ode.R. You can delete the documentation portion. Modify the code by replacing the \(U/I/V\) part of the model with the \(B/I\) part from the bacteria model.
This might be a bit challenging, but we’ll assist.
\[\begin{align}
\dot U & = n - d_U U - \color{red}{(1-f)}bUV \\
\dot I & = \color{red}{(1-f)}bUV - d_I I \\
\dot V & = \color{blue}{(1-e)}pI - d_V V - \color{red}{(1-f)}gbUV
\end{align}\]
\[\begin{align}
\dot U & = n - d_U U - bUV \\
\dot I & = bUV - d_I I \\
\dot V & = (1-e)pI - d_V V - gb UV \\
\dot C & = - d_C C, \qquad C=C+C_0 \textrm{ at } t = t_{interval} \qquad \textrm{(PK)}\\
e & = E_{max} \frac{C^k}{C^k+C_{50}} \qquad \textrm{(PD)}
\end{align}\]