Bacterial Infection models

2023-07-07

Basic bacteria infection model

\[ \begin{aligned} \dot{B} & = g B(1-\frac{B}{B_{max}}) - d_B B - kBI\\ \dot{I} & = r BI - d_I I \end{aligned} \]

  • Can also be other extracellular organisms. Probably not ideal for intra-cellular bacteria (e.g., TB).

Extended bacteria infection model

\[\begin{align} \dot B &= g B (1-\frac{B}{B_{max}}) - d_B B - k_I B I - k_A B A \\ \dot I &= r_I B (1 -\frac{I}{I_{max}} ) - d_I I \\ \dot A &= r_A \frac{\log(I)}{h+\log(I)} A - d_A A \end{align}\]

Model uses

Model Exploration

  • Looking at the dynamics (time-series) of a model can be useful.
  • Often, we are not mainly interested in the time series, but instead some more specific quantity, e.g. total number of pathogen/infected cells, steady state values, etc.
  • We usually want to to know how such outcome(s) of interest vary with some parameter(s).
  • What do we need to do to answer that question?

Model Exploration

  1. Choose some parameter values.
  2. Run the simulation model.
  3. Record quantities/outcomes of interest.
  4. Choose another set of parameter values (usually we only vary one at a time).
  5. Repeat steps 2-4 until you got all parameter-outcome pairs of interest.
  6. Report (e.g. plot) your findings.

Model Exploration example 1

  • Question: How dose the antigen dose for a killed (influenza) vaccine affect antibody levels post vaccination?
  • Approach: Build a simple model and explore (after Handel et al 2018 PCB).

\[ \begin{aligned} \dot V &= - d_V V - k_A AV \\ \dot F &= p_F - d_F F + \frac{V}{V+ h_V}g_F(F_{max}-F) \\ \dot B & = \frac{F V}{F V + h_F} g_B B \\ \dot A & = r_A B - d_A A - k_A A V \end{aligned} \] (This is a simpler version of the virus and immune response DSAIRM model.)

Model Exploration example 1

Run model for different antigen doses (\(V_0\)).

Model Exploration example 1

  • Run model for different \(V_0\), record antibodies \(A\) at end of each simulation for each \(V_0\).
  • Use this equation to compute protection as a function of antibody level. \(P= 1 - \frac{1}{e^{k_1(\log(A)-k_2)}}\)

Model Exploration - Example 2

Virus fitness as function of virion binding ( \(k_+\) ) and release ( \(k_-\) ) rates. Handel et al (2014) Proc Royal Soc Interface.

Exploration comments

  • If the system/question is very simple, we might not need a model.
  • Interactions among pathogens and the immune response are often complex. If we know little about our system and its behavior, building and exploring simple models is often a useful first step.

Back to bacteria

  • Assume we think this model is a good approximation for a real system we are interested in.
  • We want to explore/predict the peak burden of bacteria if we were able to increase the induction of the immune response (parameter \(r\)), e.g. by giving a drug.

\[ \begin{aligned} \dot{B} & = g B(1-\frac{B}{B_{max}}) - d_B B - kBI\\ \dot{I} & = r BI - d_I I \end{aligned} \]

Model Exploration

  1. Choose some parameter values.
  2. Run the simulation model.
  3. Record quantities/outcomes of interest. Here: \(B\) at peak.
  4. Choose another set of parameter values (usually we only vary one at a time). Here: \(r\).
  5. Repeat steps 2-4 until you got all parameter-outcome pairs of interest.
  6. Report (e.g. plot) your findings.

Exploration exercise

  • We could do the model exploration by hand through the DSAIRM GUI for the “Basic Bacteria Model” (or any other) app.
  • The Bacteria Model Exploration apps allows you to do some exploration graphically.
  • We could write R code that loops over parameters and repeatedly calls the underlying model (see e.g. ‘Level 2’ in the package tutorial).

Exploration exercise

  • Go to the “Bacteria Model Exploration” app
  • Instead of doing the tasks in the “What to do” section through the GUI, make an R script and write code that runs the basic bacteria model for different parameter values.
  • You can copy/paste/modify the solution from task 6 you looked at as homework.