Localizing Transmission

Estimating Local

Suppose that we know PfPR over time, \(x(t)\), and that we have fit a function describing the EIR over time, \(E(t).\) Exposure, in this case, is the weighted sum of local exposure, \(E_\ell(t)\), exposure while traveling, \(E_\delta(t)\). If we let \(\delta\) denote time spent traveling, then: \[E(t) = \mbox{diag} \left(\delta\right) \cdot E_\delta(t) + \mbox{diag}(1-\delta) \cdot E_\ell(t).\]

Given a time at risk matrix, \(\Psi,\) we note that: \[E_\delta(t) = \Psi \cdot E_\ell(t)\]

We substite this and solve to get:

\[E_\ell(t) = \left[ \mbox{diag}\left(\delta\right) \cdot \Psi + \mbox{diag} \left(1-\delta \right) \right]^{-1} \cdot E(t).\]

Since we know \(x(t),\) we can compute net infectiousness (under a model):

\[\kappa = F_\kappa(x).\]

Now, we get a revised formula for estimating the local adjusted reproductive number:

\[R_\ell = b D E_\ell \frac{1 + S \kappa}{\kappa} (1+\alpha)\]

We note that \(R_\ell >1\) is a pseudo-threshold condition for persistence.