Modeling vaccines and drugs

Andreas Handel

2025-07-21

Big picture

  • We generally want to understand pathogen-immune response dynamics with the eventual goal of intervening with drugs or vaccines.
  • Modeling of drugs is a big area. In the pharma industry, this is knows as pharmacometrics (PM/PMX) or pharmacokinetic/pharmacodynamic (PK/PD) modeling.
  • For mechanistic models like ours, the term Quantitative Systems Pharmacology (QSP) is often used.
  • PMX/QSP modeling goes beyond infectious diseases. You could build a model where instead of a pathogen, you have cancer cells, or some compartments describing renal function, or some neuro-receptors in the brain…

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
  • Explore the impact of the treatment/intervention on the outcomes of interest

Example 1

  • IFN treatment for HCV infection
  • Question to answer:
    • By what mechanism does the drug work?
    • How effective is the drug?

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 1

\[\begin{align} \dot U & = n - d_U U - \color{red}{(1-f)}bUV \\ \dot I & = \color{red}{(1-f)}bUV - d_I I \\ \dot V & = \color{blue}{(1-e)}pI - d_V V - \color{red}{(1-f)}gbUV \end{align}\]
  • We can explore if a model that acts through either of these two mechanisms (or neither or both) matches the data.
  • Usually, we fit the models to the data and get some statistical answer about which model fits the data better.
  • As a simpler approach, we can run the model and visually compare its predictions to the data.

Example 1 Exercise

  • The “Antiviral Treatment Model” in DSAIRM implements the model and guides you through a a set of tasks where you visually compare the model runs to data.
  • The “Influenza Drug Model” uses the same model, but fits data to the model using a statistical approach (least squares).
  • Explore either or both of these apps. Start by reading the Overview and Model sections. Then go through the tasks in the What to do section.
  • If you are comfortable with R, you can also get and look at the code and interact with the model through writing your own R code. The Further Information section tells you where to find the code.

Example 2

  • Allow the drug concentration to change over time and have an explicit equation for the drug (Pharmacokinetics, PK).
  • Have some more complex mapping from drug concentration to drug impact (Pharmacodynamics, PD).

Example 2

Old:

\[\begin{align} \dot U & = n - d_U U - bUV \\ \dot I & = bUV - d_I I \\ \dot V & = (1-e)pI - d_V V - gb UV \\ \end{align}\]

New

\[\begin{align} e & = E_{max} \frac{C^k}{C^k+C_{50}} \qquad \textrm{(PD)} \\ \dot C & = - d_C C \qquad \textrm{(PK)} \\ C & =C+C_0 \textrm{ at } t = t_{interval} \end{align}\]
  • This is implemented in the Pharmacokinetics and Pharmacodynamics app in DSAIRM. Explore at your own leisure.

Example 3

\[ \begin{aligned} \dot{B} & = g B(1-\frac{B}{B_{max}}) - d_B B - pBI \color{blue}{- f(C)B}\\ \dot{I} & = r BI - d_I I \\ \color{blue}{\dot{C}} & \color{blue}{= ?} \end{aligned} \]

A drug at concentration \(C\) leads to extra killing of bacteria (PD). The drug has some time-course (PK).

Example 3

\[ \begin{aligned} \dot{B} & = g B(1-\frac{B}{B_{max}}) - d_B B - k_IBI - eB\\ \dot{I} & = r BI - d_I I \\ \dot C & = - d_C C, \qquad C=C+C_0 \textrm{ at } t = t_{interval} \qquad \textrm{(PK)}\\ e & = E_{max} \frac{C^n}{C^n+C_{50}} \qquad \textrm{(PD)} \end{aligned} \]

  • The bad news: This model is not part of DSAIRM.
  • The good news: We can build it ourselves! (DSAIRM Level 3.)

Hands-on exercise

  • 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.

  • Definitions/letters for some parameters might have changed between apps. Make sure you call/define them correctly.

  • This might be a bit challenging, but we’ll assist.