Bacterial Infection models

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

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 does drug dosing impact the emergence of drug resistance in the presence of different types of immune responses?
• Approach: Build several simple models and explore (after Handel et al 2009 JTB).

\[ \begin{aligned} \dot{B_s} & = (1-\mu)g_s B_s(1-\frac{B_s+B_r}{B_{max}}) -k_I I B_s - \frac{k_s C}{C+C_{50}^s} B_s\\ \dot{B_r} & = (\mu g_s B_s + g_r B_r)(1-\frac{B_s+B_r}{B_{max}}) -k_I I B_r - \frac{k_r C}{C+C_{50}^r} B_r\\ \dot{I} & = r (B_s+B_n)-d_I I \\ \dot C & = - d_C C, \qquad C=C+C_0 \textrm{ at } t = t_{interval} \\ \end{aligned} \]

Model Exploration example 1

Fig 3 of Handel 2009 JTB. IR: dotted lines. Sensitive strain: dashed lines. Resistant strains: solid gray (high fitness) and black (low fitness) lines. 4 different immune response models.

Model Exploration example 1

• Vary drug concentration, explore time for resistance to emerge.

Fig 5 of Handel 2009 JTB

Model Exploration example 2

• 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 2

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

Model Exploration example 1

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

Complete Beginner: Tedious: Run the Basic Bacteria Model app repeatedly for different values of \(r\), record \(B_{max}\) for each run. Then plot the results. Easier: Use the Bacteria Model Exploration app to answer the question.

Advanced Beginner: Start an R script. Write code that implements a loop over a parameter of your choice, for each value calls the simulate_basicbacteria_ode() model function, and computes some outcome of interest. If needed, you can copy/paste/modify the solution from task 6 you looked at as homework.

Intermediate: Use the Extended Bacteria Model app and write some R code to loop over 2 (or more) parameters of your choice and plot the results (whatever outcome you consider interesting to explore).

Advanced: Extend the code of the Extended Bacteria Model (for instance by implementing 2 innate response players, such as NK cells and IFN-gamma). Then explore some outcome of your choice as a function of some of the model parameters.