In cmhoove14/DDNTD: Density Dependence in Neglected Tropical Diseases

knitr::opts_chunk$set(echo = TRUE)

require(tidyverse)
require(viridis)

devtools::load_all()

Inputs

Observational data

• Egg burden data on an individual or aggregated community/school level $\rightarrow$ infection intensity ($W$)
• Prevalence of individuals with sign of infection $\rightarrow$ prevalence of mated (egg producing) female worms ($\Omega$)
• Human population size, $H$
• Infected snail prevalence ($I_P$)
• Snail environmental carrying capacity ($K$)

Known model parameters

• Snail mean lifespan, $\mu_N=1/60$ days
• Infected snail mean lifespan, $\mu_I=1/10$ days
• Patency conversion rate from exposed to infected, $\sigma=1/40$ days

• Maximum reproduction rate, $r=0.1$

• Mean urine produced per day per individual (in units of 10mL), $U=100$

• Egg viability (miracidia hatched from eggs produced), $v=0.08$

Estimates from equilibrium inputs

Snail FOI from infected snail prevalence

r latexImg("\\Lambda=\\frac{\\mu_I(\\mu_N+\\sigma)}{\\frac{\\sigma}{I_P}-\\mu_I-\\sigma}")

snail_dat <- data.frame(I_P = seq(0.01,0.15, length.out = 50),
                        Lambda = sapply(seq(0.01,0.15, length.out = 50), I_get_Lambda,
                                        mu_N = base_pars["mu_N"],
                                        mu_I = base_pars["mu_I"],
                                        sigma = base_pars["sigma"]))

snail_dat %>% 
  ggplot(aes(x = I_P, y = Lambda)) +
    geom_line(size = 1.2) +
    theme_classic() +
    scale_x_continuous(breaks = c(0.01, 0.05, 0.1, 0.15)) +
    labs(x = expression(Infected~Snail~Prevalence~(I[P])),
         y = expression(Snail~FOI~(Lambda)),
         title = expression(Snail~FOI~as~fx~of~equilibirum~infected~snail~prevalence~I[P]))

Equilibrium snail population size from infected snail prevalence (via estimation of $\Lambda$) and snail environmental carrying capacity

r latexImg("N^*(\\Lambda)=K\\Big(1-\\frac{\\mu_N+\\Lambda}{r\\big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}\\big)}\\Big)")

snail_dat2 <- as.data.frame(expand.grid(I_P = seq(0.01,0.15, length.out = 50),
                                        K = c(100, 1000, 10000))) %>% 
  mutate(Lambda = map_dbl(I_P, I_get_Lambda,
                          mu_N = base_pars["mu_N"],
                          mu_I = base_pars["mu_I"],
                          sigma = base_pars["sigma"]),
         N_eq = map2_dbl(Lambda, K, Lambda_get_N_eq,
                         mu_N = base_pars["mu_N"],
                         r = base_pars["r"],
                         sigma = base_pars["sigma"]))

snail_dat2 %>% 
  ggplot(aes(x = I_P, y = N_eq, col = as.factor(K))) +
    geom_line() +
    scale_y_continuous(trans = "log",
                       breaks = c(10,100,1000,10000),
                       limits = c(10, 10000)) +
    theme_classic() +
    labs(x = expression(Infected~Snail~Prevalence~(I[P])),
         y = expression(Equilibrium~Snail~Population~Size~N[eq]),
         col = "K",
         title = expression(Equilibrium~Snail~Population~Size),
         subtitle = expression(as~fx~of~equilibirum~infected~snail~prevalence~(I[P])~and~carrying~capacity~(K)))

Per contact probability of snail infection as function of infected snail prevalence (and resulting $\Lambda$), mean egg output, human population size, and exposure/contamination parameter ($\omega$)

$\Lambda$ and $N^*(\Lambda)$ derived from $I_P$ and $K$, $\mathcal{E}$ and $H$ are observational, $U$ and $v$ are assumed known, and $\omega$ is the main unknown

r latexImg("\\beta=\\frac{\\Lambda N^*(\\Lambda)}{\\mathcal{E}HUv\\omega}=\\Lambda K\\Big(1-\\frac{\\mu_N+\\Lambda}{r\\big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}\\big)}\\Big)")

beta_dat <- as.data.frame(expand.grid(I_P = seq(0.01,0.15, length.out = 15),
                                      omega = seq(0.01,0.15, length.out = 15),
                                      egg_output = c(3,30,300),
                                      H = c(100, 500, 1000))) %>% 
  mutate(Lambda = map_dbl(I_P, I_get_Lambda,
                          mu_N = base_pars["mu_N"],
                          mu_I = base_pars["mu_I"],
                          sigma = base_pars["sigma"]),
         N_eq = map_dbl(Lambda,Lambda_get_N_eq,
                        K = 1000,
                        mu_N = base_pars["mu_N"],
                        r = base_pars["r"],
                        sigma = base_pars["sigma"]))

beta_dat$beta <- apply(beta_dat, 1, function(x){
  beta_from_eggs(egg_output = x["egg_output"],
                 H = x["H"],
                 Lambda = x["Lambda"],
                 N_eq = x["N_eq"],
                 U = base_pars["U"],
                 v = base_pars["v"],
                 omega = x["omega"])
})

beta_dat %>% 
  mutate(beta = if_else(beta>1, NA_real_, beta)) %>% 
  ggplot(aes(x = I_P, y = omega)) +
    geom_tile(aes(fill = beta)) +
    theme_classic() +
    facet_grid(egg_output~H, labeller = label_bquote("E"==.(egg_output), "H"==.(H))) +
    scale_fill_viridis() +
    labs(x = expression(Infected~Snail~Prevalence~(I[P])),
         y = expression(Contamination~fraction~(omega)),
         fill = expression(Snail~infection~probability~(beta)))

Worm acquisition rate or human FOI as function of adult worm mean lifespan and equilibirum infection intensity

r latexImg("\\lambda^*=\\mu_W W^*")

W_dat <- expand.grid(mu_W = c(1/(2*365), 1/(3*365), 1/(4*365), 1/(5*365)),
                     W_star = c(1:100)) %>% 
  mutate(lambda_star = mu_W*W_star)

W_dat %>% 
  ggplot(aes(x = W_star, y = lambda_star, col = as.factor(mu_W))) +
    geom_line(size = 1.2) +
    theme_classic() +
    scale_color_discrete(name = "Mean adult worm\nlifespan",
                         labels = c(2:5)) +
    labs(x = expression(Endemic~Equilibrium~Worm~Burden~(W[eq])),
         y = expression(Equilibirum~Human~FOI~(lambda[eq])))

cmhoove14/DDNTD documentation built on Nov. 23, 2019, 7:04 p.m.