--- 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") ```