2020-07-21 14:05:34

## Model Exploration example

• 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

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

## 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.Â total number of infected/pathogens, steady state values, etc.
• We usually want to to know how such outcome(s) of interest vary with some parameter(s).
• For our example, we want to know how antibody levels vary with vaccine dose, and how that affects protection from infection.
• 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.Â plot) your findings.

## Model Exploration

• 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. $$P= 1 - \frac{1}{e^{k_1(\log(A)-k_2)}}$$

## Exploration - summary

• 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.

## Exploration - practice

• We could do the model exploration by hand through the DSAIRM GUI.
• We could automate it by writing R code that loops over parameters and repeatedly calls the underlying model (see e.g.Â ‘Level 2’ in the package tutorial).
• The Model Exploration apps allows you to do such exploration graphically.

## Model Predictions/Virtual Experiments

• We saw how we can use models to explore how outcomes of interest change with parameters.
• Model exploration is often useful to gain general insights into a system early on.
• Once we built up our understanding and have a model that we think approximates reality reasonably well, we can potentially move on to making predictions and explore ‘what-if’ scenarios (virtual experiments).

## Prediction types

• Predictions can be of different types:
• Qualitative: Try to predict shape/direction of an outcome (similar to the ‘exploration’ model use).
• Semi-quantitative: Try to predict the approximate or relative size of an outcome.
• Quantitative: Try to predict (with confidence intervals) the magnitude of an outcome.

## Prediction example

Assume we think this model is a good approximation for a real system we are interested in. We want to predict the peak burden of bacteria if we were able to increase the induction of the immune response (parameter $$r$$), e.g.Â 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}

## Prediction example

Weâ€™ll follow the same approach as for model exploration, the difference is that now we interpret the results as actual predictions instead of suggested findings that need to be further explored.