Vectorial Capacity with Forcing
When mosquito bionomic parameters are forced by weather or vector control, vectorial capacity (VC) is different every day, \(V(t)\). Here, we develop theory to compute VC and its temporal dispersion when VC is forced.
See Related for links to closely related vignettes.
Vectorial capacity (VC) is defined as the number of infective bites arising from all the mosquitoes blood feeding on a single human on a single day. If we want to develop theory to discuss VC in forced systems, we must be able to compute VC as a function of time. In general, we define the number of infectious bites arising at time \(\ell\) from all the mosquitoes biting a single fully infectious human at time \(t\)
\[V_\ell(t, \ell) = f(\ell)\]
and
\[V(t) = \int_t^\infty V_\ell(t, \ell) d \ell\]
If we want to compute VC and its temporal dispersion in forced systems, we will need to develop some notation for some of the bionomic parameters.
Bionomics
Let \(f(t)\) be the blood feeding rate at time \(t\)
Let \(q(t)\) be the human fraction at time \(t\)
EIP
Here, we consider the case where the EIP is a fixed lag, and it is a function of temperature. If the EIP varies over time, we need notation for to describe the lag with respect to the moment when a mosquito becomes infected and when it becomes infectious.
Let \(\tau(t)\) denote the lag with respect to the moment a mosquito becomes infected: a mosquito becoming infected at time \(t\) would become infectious at time \(t+\tau(t).\)
let \(\tau'(t)\) denote the lag for a mosquito at the point in time when it becomes infectious: a mosquito becoming infectious at time \(t\) would have become infected at time \(t-\tau'(t).\)
The two are related by the identities \[\tau(t) = \tau'(t+\tau(t))\] and \[\tau'(t) = \tau(t-\tau'(t)).\]
Also, see EIP
Demography
If \(g(t)\) is the mosquito mortality rate, then we let \(G\) represent cumulative mortality: \[\frac{dG}{dt} = g\] Then the probability of surviving any interval \((t_1, t_2)\) is \[G(t_2) - G(t_1)\]
In spatial models, \(\Omega(t)\) is the demographic matrix. Then we let \(U\) denote cumulative survival and dispersal:
\[\frac{dU}{dt} = \Omega\] Survival and dispersal through an interval \((t_1, t_2)\) is \[U(t_2) - U(t_1)\]
VC in Simple Models
We can also describe the temporal dispersion of those bites at a point in time, \(\ell\), after mosquitoes become infected.
\[V_\ell(t, \ell) = \begin{cases} 0 & \text{if } \ell < t + \tau(t) \\ B\left(t\right) \Upsilon\left(t, \tau\left(t\right)\right) f(\ell) q(\ell) G\left(t+\tau\left(t\right), \ell\right) & \text{if } \ell > t+\tau(t) \end{cases}\] So \[[V](t) = B\left(t\right) \Upsilon\left(t, \tau\left(t\right)\right) \int_{t+\tau(t)}^\infty f(\ell) g (\ell) U(t+\tau(t), \ell) d\ell\]