Time Spent at Risk
Notation:
Simple Trip
One way to develop a time spent for mobility in the domestic modality is by modeling a simple trip: hosts travel temporarily to other locations before returning home [1–3]. Here, we define the model for time spent by a single stratum.
Let \(H_j\) denote the density of humans in the \(j^{th}\) patch who reside in the \(i^{th}\) patch, out of a total of \(p\) patches, where they are at risk. We also consider how much time humans spend in places where they are not at risk, so we add two patches: time spent traveling, and time spent in places where there is no risk. Time is spent either at home or away in a sequence of trips: \[\begin{equation} \begin{split} \frac{d H_i}{dt} &= -\sum_{j=1}^{N_p+2} \phi_j H_i + \sum_{j=1}^{N_p+2} \tau_j H_j \\ \frac{d H_j}{dt} &= -\tau_j H_j + \phi_i H_i \\ \end{split} \end{equation}\] The constant \(\phi_j\) represents the rate at which hosts travel to \(j\), the constant \(\tau_j\) is the rate at which hosts visiting \(j\) return home, and we assume \(\phi_i = \tau_i = 0\).
When the movement equations reach a steady state, the population \(H_i\) is distributed across the \(p\) metapopulation sites, travel, and not at risk, as follows: \[\begin{equation*} \begin{split} \theta_{i}^* &= \frac{1}{1 + \sum_{j=1}^{N_p+2} \frac{\phi_{j}}{\tau_{j}} } \\ \theta_{j}^* &= \frac{\phi_{j}}{ \tau_{j}} \frac{1}{1 + \sum_{j = 1}^{N_p+2} \frac{\phi_{j}}{\tau_{j}} } \\ \end{split} \end{equation*}\] These \(\theta\) describe the fraction of time spent in each patch, and \(\sum_{j=1}^{N_p+2}=1\)
Time Spent Matrix
NOTE — To avoid double subscripts in these formulas, we let \(p=N_p\)
Let \(\Theta\) denote the time spent matrix. A time spent matrix is constructed by considering the travel patterns of each population stratum. Each column \((i)\) in \(\Theta\) represents the elements of a vector describing time spent in each patch, \(\theta_{j,i}\) where \(j\in 1,2,\ldots, N_p.\) \[\begin{equation} {\Theta} = \left[ \begin{array}{ccccc} i=1&i=2&i=3&\cdots&i=n \\ \boxed{ \begin{array}{c} {\theta}_{1,1} \\ {\theta}_{2,1} \\ {\theta}_{3,1} \\ \vdots \\ {\theta}_{p,1} \\ \end{array}} & \boxed{ \begin{array}{c} {\theta}_{1,2} \\ {\theta}_{2,2} \\ {\theta}_{3,2} \\ \vdots \\ {\theta}_{p,2} \\ \end{array}} & \boxed{ \begin{array}{c} {\theta}_{1,3} \\ {\theta}_{2,3} \\ {\theta}_{3,3} \\ \vdots \\ {\theta}_{p,3} \\ \end{array}} & \boxed{ \begin{array}{c} \cdots \\ \cdots \\ \cdots \\ \ddots \\ \cdots \end{array}} & \boxed{ \begin{array}{c} {\theta}_{1,n} \\ {\theta}_{2,n} \\ {\theta}_{3,n} \\ \vdots \\ {\theta}_{p,n} \\ \end{array}} \end{array} \right] \end{equation}\] By convention:
a time spent matrix is computed for time spent in and around home, but not traveling, so if the columns sum up to less than one, then some time is spent in places where there is no risk.
The rows should sum up to less than one, if humans travel at all, or if they spend any time in locations where they are not at risk, such as automobiles, or office buildings.
Mosquito Activity
Let \(\xi(d)\) describe relative daily blood feeding activity for a malaria vector species over a day.
In some cases, we can consider how \(\xi\) varies over time, and we want functions such that: \[\begin{equation} \int_0^1 \xi(t) dt \approx 1 \end{equation}\] If we measured mosquito activity patterns, then for some large integer value of \(T \gg 1\), \[\begin{equation} \int_0^T \xi(t) dt \approx T \end{equation}\] and \[\begin{equation} \xi(d) = \frac{1}{T} \sum_T \xi(d+T) \end{equation}\]
Time at Risk
Note that mosquito activity is defined in a similar way to the concept of average time spent over a day, \(\Theta(t)\). We note that \(\xi(t)\) describes blood feeding activity rates and not the distribution of blood meals, which should be thought of as the product of activity and host availability.
A time at risk (TaR) matrix, \(\Psi\), is constructed from a time spent matrix. It is like \(\Theta\) in every way, except that it weights time spent throughout the day by mosquito blood feeding activity rates. Notably, this implies that models with more than one mosquito species would weight time spent differently, so that while there is one time spent matrix, \(\Theta\), there could be as many TaR matrices as vector species, with activity patterns defined by \(\xi_i(d)\). The resulting species-specific TaR matrices would be: \[\begin{equation} \Psi_i = \int_0^1 \mbox{diag} \left(\xi_i(d)\right) \cdot \Theta(d) dt \end{equation}\]