---
title: "Simple behavioral state mosquito model"
output: rmarkdown::html_vignette
vignette: >
%\VignetteIndexEntry{Simple behavioral state mosquito model}
%\VignetteEngine{knitr::rmarkdown}
%\VignetteEncoding{UTF-8}
---
```{r, include = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
```
```{r setup}
library(MicroMoB)
library(ggplot2)
library(data.table)
library(parallel)
```
The simple behavioral state mosquito model has two behavioral states which mosquitoes can exist in: blood feeding ($B$) and
oviposition ($Q$). When mosquitoes are in $B$ they will attempt to blood feed until they are successful, at which point they
transition to $Q$ and attempt to oviposit an egg batch. Upon emergence, mosquitoes are primed for blood feeding and are in $B$.
They transition between these states until they die, which occurs according to the state dependent probabilities $p_{B}$ and $p_{Q}$
(these may also vary by location and time). The model does not consider male mosquitoes.
The model also considers infection. Uninfected (susceptible) mosquitoes $M$ may become infected if they are in $B$, successfully
take a blood meal, and are infected (with probability $\kappa$). They then transition to the infected class $Y$, in behavioral state $Q$.
The extrinsic incubation period (EIP) may vary with time, and they advance until they become infectious (if they survive), where they
remain until death. Both dynamics operate simultaneously.
## Deterministic model
The deterministic behavioral state model has the following form:
\begin{equation}
\left[
\begin{array}{cc}
B_{t+1} \\
Q_{t+1} \\
\end{array}
\right]
=
\left[
\begin{array}{ccc}
(1 - \psi_b) \Psi_{b b} & \psi_q \Psi_{q b} \\
\psi_b \Psi_{b q} & (1 - \psi_q) \Psi_{q q} \\
\end{array}
\right]
\left[
\begin{array}{cc}
p_b B_{t} \\
p_q Q_{t} \\
\end{array}
\right]
+
\left[
\begin{array}{c}
\Lambda_{t} \\
0 \\
\end{array}
\right]
\end{equation}
The state is a column vector $\left[\begin{array}{cc} B \\ Q \\ \end{array}\right]$. We assume that there are $p$ locations where mosquitoes
go to seek blood hosts, so that the first $p$ elements correspond to the number of mosquitoes in the $B$ state at those places. There are $l$ locations
where mosquitoes go to oviposit (aquatic habitats), so the last $l$ elements in the vector are mosquitoes in the $Q$ state. There is no requirement
that the set of points where mosquitoes blood feed and oviposit be distinct, although they may be.
The infection states are similar to the Ross-Macdonald model, see `vignette("RM_mosquito")` for more details.
The parameters in the state updating equation are:
* $\psi_b$: probability of successful blood feeding (vector of length $p$); this parameter is computed from $f, q$ (themselves calculated during the bloodmeal algorithm) as $1-e^{-fq}$.
* $\psi_q$: probability of successful oviposition (vector of length $l$).
* $\Psi_{b b}$: transition probability matrix for movement among blood feeding haunts. It has dimension $p\times p$, and has _columns_ that sum to 1 (note state vectors are on the right).
* $\Psi_{q b}$: transition probability matrix for movement from aquatic habitats to blood feeding haunts. It has dimension $p\times l$.
* $\Psi_{b q}$: transition probability matrix for movement from blood feeding haunts to aquatic habitats. It has dimension $l\times p$.
* $\Psi_{q q}$: transition probability matrix for movement among aquatic habitats. It has dimension $l\times l$.
* $p_{B}$: daily survival probability for blood feeding mosquitoes.
* $p_{Q}$: daily survival probability for ovipositing mosquitoes.
## Stochastic model
The stochastic model has similar updating dynamics to the deterministic implementation,
except that all survival and success probabilities are used in binomial draws and
movement is drawn from a multinomial distribution.
## Simulation
We assume that $p = l = 1$ and that the total mosquito density $M = B + Q$ is known, and that we want to solve
for the emergence rate $\Lambda$ such that the system is at equilibrium. Rewriting the
equations when we substitute $Q = M - B$ and $B = M - Q$ we solve the state variables as:
\begin{equation}
Q = \frac{Mp_{B}\Psi_{B}}{p_{B}\Psi_{B} - p_{Q}(1-\Psi_{Q}) + 1} \\
B = \frac{M-Mp_{Q}(1-\Psi_{Q})}{p_{B}\Psi_{B} - p_{Q}(1-\Psi_{Q}) + 1}
\end{equation}
Then the first equation can simply be rearranged to yield:
\begin{equation}
\Lambda = B - p_{B}(1-\Psi_{B})B - p_{Q}\Psi_{Q}Q
\end{equation}
And now the model with 1 point of each type can be set up at equilibrium. We will
use the Beverton-Holt model of aquatic ecology demonstrated in `vignette("BH_aqua")`,
which will be parameterized to provide the correct equilibrium $\Lambda$.
```{R}
p <- l <- 1
tmax <- 1e2
M <- 120
pB <- 0.8
pQ <- 0.95
PsiB <- 0.5
PsiQ <- 0.85
B <- (M - (M*pQ*(1-PsiQ))) / ((pB*PsiB) - (pQ*(1-PsiQ)) + 1)
Q <- (M*pB*PsiB) / ((pB*PsiB) - (pQ*(1-PsiQ)) + 1)
lambda <- B - (pB*(1-PsiB)*B) - (pQ*PsiQ*Q)
nu <- 25
eggs <- nu * PsiQ * Q
# static pars
molt <- 0.1
surv <- 0.9
# solve L
L <- lambda * ((1/molt) - 1) + eggs
K <- - (lambda * L) / (lambda - L*molt*surv)
```
Let's set up the model. We use `make_MicroMoB()` to set up the base model object, and
`setup_aqua_BH()` for the Beverton-Holt aquatic model with our chosen parameters.
`setup_mosquito_BQ()` will set up a behavioral state model of adult mosquito dynamics.
We run a deterministic simulation and store output in a matrix. Note that we calculate
the `f` and `q` parameters to achieve the correct `PsiB` probability; normally these
would be updated dynamically during the bloodmeal but we are running a mosquito-only
simulation so we set these deterministically.
```{R}
# deterministic run
mod <- make_MicroMoB(tmax = tmax, p = p, l = l)
setup_aqua_BH(model = mod, stochastic = FALSE, molt = molt, surv = surv, K = K, L = L)
setup_mosquito_BQ(model = mod, stochastic = FALSE, eip = 5, pB = pB, pQ = pQ, psiQ = PsiQ, Psi_bb = matrix(1), Psi_bq = matrix(1), Psi_qb = matrix(1), Psi_qq = matrix(1), nu = nu, M = c(B, Q), Y = matrix(0, nrow = 2, ncol = 6))
out_det <- data.table::CJ(day = 1:tmax, state = c('L', 'A', 'B', 'Q'), value = NaN)
out_det <- out_det[c('L', 'A', 'B', 'Q'), on="state"]
data.table::setkey(out_det, day)
mod$mosquito$q <- 0.3
mod$mosquito$f <- log(1 - PsiB) / -0.3
while (get_tnow(mod) <= tmax) {
step_aqua(model = mod)
step_mosquitoes(model = mod)
out_det[day == get_tnow(mod) & state == 'L', value := mod$aqua$L]
out_det[day == get_tnow(mod) & state == 'A', value := mod$aqua$A]
out_det[day == get_tnow(mod) & state == 'B', value := mod$mosquito$M[1]]
out_det[day == get_tnow(mod) & state == 'Q', value := mod$mosquito$M[2]]
mod$global$tnow <- mod$global$tnow + 1L
}
```
Now we run the same model, but using the option `stochastic = TRUE` for our dynamics,
and draw 10 trajectories.
```{R}
# stochastic runs
out_sto <- mclapply(X = 1:10, FUN = function(runid) {
mod <- make_MicroMoB(tmax = tmax, p = p, l = l)
setup_aqua_BH(model = mod, stochastic = TRUE, molt = molt, surv = surv, K = K, L = L)
setup_mosquito_BQ(model = mod, stochastic = TRUE, eip = 5, pB = pB, pQ = pQ, psiQ = PsiQ, Psi_bb = matrix(1), Psi_bq = matrix(1), Psi_qb = matrix(1), Psi_qq = matrix(1), nu = nu, M = c(B, Q), Y = matrix(0, nrow = 2, ncol = 6))
out <- data.table::CJ(day = 1:tmax, state = c('L', 'A', 'B', 'Q'), value = NaN)
out <- out[c('L', 'A', 'B', 'Q'), on="state"]
data.table::setkey(out, day)
mod$mosquito$q <- 0.3
mod$mosquito$f <- log(1 - PsiB) / -0.3
while (get_tnow(mod) <= tmax) {
step_aqua(model = mod)
step_mosquitoes(model = mod)
out[day == get_tnow(mod) & state == 'L', value := mod$aqua$L]
out[day == get_tnow(mod) & state == 'A', value := mod$aqua$A]
out[day == get_tnow(mod) & state == 'B', value := mod$mosquito$M[1]]
out[day == get_tnow(mod) & state == 'Q', value := mod$mosquito$M[2]]
mod$global$tnow <- mod$global$tnow + 1L
}
out[, 'run' := as.integer(runid)]
return(out)
})
```
Now we process the output and plot the results. Deterministic solutions are solid lines
and each stochastic trajectory is a faint line.
```{R}
out_sto <- data.table::rbindlist(out_sto)
ggplot(data = out_sto) +
geom_line(aes(x = day, y = value, color = state, group = run), alpha = 0.35) +
geom_line(data = out_det, aes(x = day, y = value, color = state)) +
facet_wrap(. ~ state, scales = "free")
```