Vectorial Capacity in Metapopulation Models
The simple formula for VC describes potential malaria transmission by mosquitoes. With spatial dynamics, the concept of VC can be extended into a form that is useful for understanding the effects of targeting interventions spatially. Here, we define vectorial capacity for autonomouos metapopulation models.
See Related for links to closely related vignettes.
In patch-based spatial models, also known as meta-population models, we define three closely related concepts:
VC or \(V\) — on average, for the full system, how many infectious bites would arise from all the mosquitoes blood feeding on a single fully infectious human on a single day.
Patch VC or \(\vec V\) — how many infectious bites would arise from all the mosquitoes blood feeding in a single patch on a single fully infectious human on a single day.
The VC Matrix or \([V]\) — how many infectious bites would arise in every patch from all the mosquitoes blood feeding in each patch on a single fully infectious human on a single day.
The last formula is the most general. While it is tempting to try and devise similar formulae for the EIR, we have got a problem. To make the math work, the framework treats human mobility and mosquito mobility differently. The patches are defined around mosquito populations. Humans live in one of the patches, but they spend time at risk somewhere else.
The Spatial Model
Here we compute the VC for the spatial analogue of Macdonald’s model: To compute the VC matrix, we need the mosquito demographic matrix \(\Omega\):
\[ \frac{d\vec M}{dt} = \vec \Lambda - \Omega \cdot \vec M \] \[\frac{d \vec Y}{dt} = fq\kappa (M-Y) - \Omega \cdot \vec Y\] \[\frac {d \vec Z}{dt} = e^{-\Omega \tau} fq\kappa (M_\tau -Y_\tau) - \Omega \cdot \vec Z\]
the patch HBR, \(\vec B\) is
\[\vec B = \mbox{diag}\left(\frac{fq\bar M}W\right)\] where \(W\) is the availability of humans.
Note that
\[\bar M = \Omega^{-1} \cdot \vec \Lambda\] We can understand \(\Omega^{-1}\) as a measure of time spent by the mosquitoes emerging from one patch in every other patch:
\[\Omega = \int_0^\infty e^{-\Omega t} dt\] so the number if human blood meals per mosquito, summed over the lifespan of a mosquito, is:
\[[s] = fq \Omega^{-1}\]
mosquito survival and dispersal through the EIP
\[\Upsilon = e^{-\Omega \tau}\]
Now we can write:
\[[V] = fq \Omega^{-1} \cdot e^{-\Omega \tau} \cdot \mbox{diag}\left(\frac{fq\bar M}W \right) = [s] \cdot \Upsilon \cdot \vec B\]