Analysis/Parameter_estimates_from_observed_inputs_assuming_equilibirum_conditions.md

In cmhoove14/DDNTD: Density Dependence in Neglected Tropical Diseases

Main parameters, outcomes, and determinants

Inputs

Observational data

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

Known model parameters

• Snail mean lifespan, μN = 1/60 days
• Infected snail mean lifespan, μI = 1/10 days
• Patency conversion rate from exposed to infected, σ = 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

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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 Λ) and snail environmental carrying capacity

%3DK%5CBig(1-%5Cfrac%7B%5Cmu_N+%5CLambda%7D%7Br%5Cbig(1+%5Cfrac%7B%5CLambda%7D%7B%5Cmu_N+%5Csigma%7D%5Cbig)%7D%5CBig))

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 Λ), mean egg output, human population size, and exposure/contamination parameter (ω)

Λ and N*(Λ) derived from IP and K, ℰ and H are observational, U and v are assumed known, and ω is the main unknown

%7D%7B%5Cmathcal%7BE%7DHUv%5Comega%7D%3D%5CLambda%20K%5CBig(1-%5Cfrac%7B%5Cmu_N+%5CLambda%7D%7Br%5Cbig(1+%5Cfrac%7B%5CLambda%7D%7B%5Cmu_N+%5Csigma%7D%5Cbig)%7D%5CBig))

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

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.