Adult Mosquitoes - BQ

Point Sets

We define this class of micro-simulation models on point sets representing the locations of resources: haunts where mosquitoes rest and where a mosquito might find and feed on a vertebrate host; and aquatic habitats where mosquitoes could find aquatic habitats and lay eggs.

• Let $\left\{ b \right\}$ denote the point set where blood feeding could occur;

• Let $\left\{ q \right\}$ denote the point set where egg laying could occur;

Dispersal

Movement among point sets is modeled using matrices that describe where mosquitoes move each time step. We assume that every surviving mosquito ends up somewhere. Mortality while searching is referenced to the point where the search starts.

The proportion moving from a point in one set to another, from $x\in \left\{ x \right\}$ to $y \in \left\{ y \right\}$, is described by a matrix $\Psi_{y\leftarrow x}$ or equivalently $\Psi_{yx}$. Similarly, the proportion moving from a point in one set to a point in the other is $\Psi_{xx}$. Since there are a finite number of destinations, each column is a probability mass function (PMF).

• Let $\Psi_{b \leftarrow q}$ denote a matrix describing the location where mosquitoes end their flights in $\left\{ b \right\}$ starting from each point in $\left\{ q \right\}$.

• Let $\Psi_{b \leftarrow b}$ denote a matrix describing the location where mosquitoes end their flights in $\left\{ b \right\}$ starting from each point in $\left\{ b \right\}$.

• Let $\Psi_{q \leftarrow b}$ denote a matrix describing the location where mosquitoes end their flights in $\left\{ q \right\}$ starting from each point in $\left\{ b \right\}$.

• Let $\Psi_{q \leftarrow q}$ denote a matrix describing the location where mosquitoes end their flights in $\left\{ q \right\}$ starting from each point in $\left\{ q \right\}$.

Variables & Terms

In the simulation models, the number of adult mosquitoes emerging from each aquatic habitat on each day is a vector denoted $\Lambda_t$ that can be either passed as a parameter or simulated using a population dynamic model for aquatic immature population dynamics. Adult mosquitoes are located at one of these points at each point at each point in time (e.g., one day), represented as a set of vectors: $B_t$, the number seeking blood at haunts; or $Q_t$, the number attempting to lay eggs at habitats.

• Let $B_t$ denote the number of blood feeding mosquitoes at $\left\{ b \right\}$

• Let $Q_t$ denote the number of egg laying mosquitoes at $\left\{ q \right\}$

• Let $\Lambda_t$ denote the number of recently emerged mosquitoes at $\left\{ q \right\}$

Parameters

Mosquito bionomics can depend on both behavioral state and location, including daily survival and daily blood feeding or egg laying success. Note that survival is linked to the point where the search started, but foraging success is linked to the point where the mosquito ends up after moving in a time step.

Survival:

• Let $p_b$ denote the probability of surviving for a mosquito at $\left\{ b \right\}$ at time $t$

• Let $p_q$ denote the probability of surviving for a mosquito at $\left\{ q \right\}$ at time $t$

Foraging Success:

• Let $\psi_b$ denote the probability of surviving for a mosquito at $\left\{ b \right\}$ at time $t$ and let $\hat \psi_b = 1-\psi_b$

• Let $\psi_q$ denote the probability of surviving for a mosquito at $\left\{ q \right\}$ at time $t$ and let $\hat \psi_q = 1-\psi_q$

Dynamics

In the basic feeding cycle model over one time step, mosquitoes either attempt to blood feed or attempt to lay eggs. The result of an attempt is either survival or death, and if the mosquito survives, success or failure. A success moves a mosquito to the other state, and if they fail, they must try again. Either way, the mosquito moves to a point in the set for the resource they seek: the diagonal of $\Psi_{x\leftarrow x}$ is the probability of staying. The dynamics are:

$$\begin{equation} \left[ \begin{array}{cc} B_t \\ Q_t \\ \end{array} \right] = \left[ \begin{array}{c} \Psi_{b q} p_q \Lambda_{t-1} \\ 0 \\ \end{array} \right] + \left[ \begin{array}{ccc} \Psi_{b b} \cdot \mbox{diag}\left(\hat \psi_b\right) & \Psi_{b q} \cdot \mbox{diag}\left(\psi_q\right)\\ \Psi_{q b} \cdot \mbox{diag}\left(\psi_b \right)& \Psi_{q q} \cdot \mbox{diag}\left(\hat \psi_q \right) \\ \end{array} \right] \left[ \begin{array}{cc} p_b B_{t-1} \\ p_q Q_{t-1} \\ \end{array} \right] \label{eq_BQ} \end{equation}$$