forcing.qmd

---
title: "Exogenous Forcing"
format: html
---

In SimBA, we have modular software, which opens up the door to take on the question of *forcing* in a very generic way.

# Forcing

What is *exogenous forcing*?

## EIR-PR

A malaria *Pf*PR time series is the outcome of exposure. If we let $E(t)$ denote the daily EIR over time, $\cal X(t)$ the state of the system, and $x(t)$ the *Pf*PR. We have this picture:

$$E(t) \rightarrow {\cal X}(t) \rightarrow x(t)$$ In this equation, the arrow denotes causation: infection is the result of exposure; given a model of malaria infection, the *Pf*PR is an observable state. Given the modular design of SimBA software, we can configure functions describing malaria exposure time series and fit the models to a data set.

## Emergence-PR

In some settings, it is worth thinking about malaria as a system that is *exogenously forced* by adult mosquito emergence. This point of view is particularly important for scenario planning. In these models, we now take $\Lambda(t)$ as our forcing function, we have a model for mosquito ecology (top line); we compute mosquito infection dynamics (middle line); an outcome of the model is the net blood feeding rate of mosquitoes and the EIR; in this model, malaria in humans (bottom) line is part of a model for transmission.

$$ 
\begin{array}{ccccc}
\Lambda(t) \rightarrow & {\cal M}(t) \\
& \downarrow \\
& {\cal Y}(t) &\rightarrow & f q Z(t) & \rightarrow & E(t)   \\
   &  \uparrow &   &&&\downarrow  \\
   &  \kappa(t) &\leftarrow & X(t) & \leftarrow  & {\cal X}(t) & \rightarrow x(t)
\end{array}
$$

## Carrying Capacity - PR

We could go one step back and let the aquatic mosquito population be forced by carrying capacity:\
$$ 
\begin{array}{ccccc}
K(t) \rightarrow &{\cal L}(t) \leftarrow & \Gamma(t) \\
 &\downarrow & \uparrow \\
& \Lambda(t) \rightarrow & {\cal M}
(t) \\
&& \downarrow \\
&& {\cal Y}(t) &\rightarrow & f q Z(t) & \rightarrow & E(t)   \\
&&  \uparrow &   &&&\downarrow  \\
&   &  \kappa(t) &\leftarrow & X(t) & \leftarrow  & {\cal X}(t) & \rightarrow x(t)
\end{array}
$$