Exogenous Forcing

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 PfPR 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 PfPR. 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 PfPR 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:
\[ \begin{array}{ccccc} K(t) \rightarrow &{\cal L}(t) \leftarrow & \Gamma(t) \\ &\downarrow & \uparrow \\ & \Lambda(t) \rightarrow & {\cal M} (t) \\ && \downarrow \\ && {\cal Y}(t) &\rightarrow & f q Z(t) & \rightarrow & E(t) \\ && \uparrow & &&&\downarrow \\ & & \kappa(t) &\leftarrow & X(t) & \leftarrow & {\cal X}(t) & \rightarrow x(t) \end{array} \]