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S3.Rmd
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---
title: "S3"
output:
html_document:
toc: true
toc_float:
smooth_scroll: FALSE
number_sections: true
---
```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, message = FALSE, warning = FALSE)
library(dplyr)
library(gt)
library(Metrics)
library(pomp)
library(readr)
library(readsdr)
library(stringr)
library(tictoc)
library(tidyr)
source("./R_scripts/plots.R")
source("./R_scripts/optim_est.R")
pop_val <- 17475415
beta_vals <- 0.724637681 * c(1, 1.5, 2)
zeta_val <- beta_vals[[2]] * 10 # Contact rate
phi_val <- 0.65 # Vaccine effectiveness
tau_v_val <- 180 # Vaccine availability
iota_v_val <- 0.75 # Willingness to vaccinate
iota_d_val <- 0.51 # Willingness to isolate/quarantine
delta_val <- 0.6 # Willingness to take a test
par_tau_m <- 0 # Day at which mobility restrictions start
bio_df <- read_csv("./Data/bio_params.csv",
show_col_types = FALSE)
bio_params <- as.list(bio_df$Value)
names(bio_params) <- bio_df$Parameter
test_cap_pars <- read_csv("./Data/Testing_capacity_pars.csv",
show_col_types = FALSE) |>
tibble::deframe() |> as.list()
ct_cap_pars <- read_csv("./Data/Tracing_capacity_pars.csv",
show_col_types = FALSE) |>
tibble::deframe() |> as.list()
Qd_0 <- test_cap_pars$Qd
Qk_0 <- ct_cap_pars$Qk
Qm_0 <- 90
```
This electronic supplementary material supports the results presented in the
main text regarding vaccination and the comprehensive intervention.
Specifically, this HTML file is the rendered version of a dynamic document (R
markdown) containing the *R* code that simulates the models and produces the
plots shown in the main text. Additionally, this file includes supplementary
information to complement the discussion in the main text.
# Vaccination sub-model
## Equations
Compared to the SEI3R model (*1A_SEI3R.stmx*), the vaccination sub-model
(*3C_V.stmx*) modify the dynamics of disease transmission in the following way:
\begin{equation} \label{eq:4}
\begin{aligned}
\dot{S_t} &= -\lambda_t S_t - \psi_t S_t\\
\dot{S^h_t} &= -\lambda_t S^h_t \\
\dot{S^v_t} &= (1 - \phi)\psi_t S_t -\lambda_t S^v_t\\
\dot{E_t} &= \lambda_t (S_t + S^v_t) - \sigma E_t\\
\dot{R_t} &= \gamma I_t + \kappa A_t + \phi \psi_t S_t
\end{aligned}
\end{equation}
As with the *TTI* structure, we model vaccination capacity using the logistic
growth structure. Furthermore, considering that vaccines are not available at
the start of the pandemic, we introduce an if-else function to account for
this feature.
\begin{equation} \label{eq:5}
\begin{aligned}
\psi_t &= \frac{\min(S_t, f(Q^v_t, \tau^v))}{S_t} \\
f(Q_t, \tau) &= \begin{cases}
0, & \text{for } t < \tau \\
\frac{Q_t N}{1000}, & \text{for } t \geq \tau
\end{cases}\\
\dot{Q^v_t} &= \begin{cases}
0, & \text{for } t < \tau \\
\rho^v Q^v_t (1 - \frac{Q^v_t} {\alpha^v}), & \text{for } t \geq \tau
\end{cases}\\
\end{aligned}
\end{equation}
## Partial calibration
To configure the logistic structure in the vaccination sub-model, we use data
from the COVID-19 in the Netherlands and model calibration. This data indicates
that vaccination's initial capacity in Feb 2021 was 230.000 per week. From April
onwards, it was between 400.000-500.000 per week, and 6 weeks were needed to
scale up to 1.500.000 per week^[https://www.medicalfacts.nl/wp-content/uploads/2021/02/Presentatie_de_heer_Jaap_van_Delden_programmadirecteur_vaccinatie_COVID-19_bij_het_RIVM1.pdf].
```{r}
data_vacc <- data.frame(time = c(0, 60, 102),
value = 1000 * (c(230000, 400000, 1500000) / 7) / pop_val)
est_obj <- estimate_rho(data_vacc)
par_rho_v <- est_obj$estimate
Qv_0 <- data_vacc$value |> head(1)
par_alpha_v <- data_vacc$value |> tail(1)
```
```{r, fig.cap = "Fig 1. Vaccination capacity"}
plot_fit(est_obj$sim, data_vacc)
```
# Comprehensive intervention (TTI + MR + V)
All interventions are integrated into a single model (3B_TTI_PLUS_MR_PLUS_V.stmx).
## Validation
Before exploring the impact of a comprehensive approach, we verify that the
integrated model and intervention-specific models produce identical output under
equivalent configurations.
```{r}
fp_synthesis <- "./models/3B_TTI_PLUS_MR_PLUS_V.stmx"
mdl_sth <- read_xmile(fp_synthesis,
const_list = c(bio_params,
par_zeta_0 = zeta_val,
par_rho_v = par_rho_v,
par_alpha_v = par_alpha_v,
par_phi = phi_val,
par_iota_v = iota_v_val,
par_iota_d = iota_d_val,
par_delta = delta_val,
par_rho_d = test_cap_pars$rho_d,
par_alpha_d = test_cap_pars$alpha_d,
par_rho_k = ct_cap_pars$rho_k,
par_alpha_k = ct_cap_pars$alpha_k,
par_tau_m = par_tau_m,
par_tau_v = tau_v_val,
par_xi = 0.8),
stock_list = list(Qv = Qv_0,
Qd = test_cap_pars$Qd,
Qk = ct_cap_pars$Qk,
Qm = Qm_0))
```
```{r}
par_tau_v_test <- 0
# Scenarios data frame
# First scenario: No intervention
# Second scenario: Testing and Isolation (TI)
# Third scenario: Contact tracing (CT)
# Fourth scenario: Mobility restrictions (MR)
# Fifth scenario: Vaccination (V)
s_df <- data.frame(Qd = c(0, Qd_0, Qd_0, 0, 0),
Qk = c(0, 0, Qk_0, 0, 0),
Qm = c(0, 0, 0, Qm_0, 0),
Qv = c(0, 0, 0, 0, Qv_0),
iter = 1:5,
itv = c("Base case", "TI", "CT", "MR", "V"),
par_tau_v = par_tau_v_test)
sim_output_syn <- sd_sensitivity_run(mdl_sth$deSolve_components,
stocks_df = s_df |> select(-iter, - itv),
consts_df = s_df |> select(par_tau_v),
start_time = 0,
stop_time = 1000,
timestep = 1 / 32,
reporting_interval = 1/ 8,
multicore = TRUE, n_cores = 5) |>
select(time, iter, C_in, Qv, Qd, Qk, Qm) |>
mutate(model = "Synthesis") |>
left_join(s_df[, c("iter", "itv")])
```
```{r}
mdl_base <- read_xmile("./models/1A_SEI3R.stmx",
const_list = c(bio_params,
par_beta = beta_vals[[2]]))
temp <- sd_simulate(mdl_base$deSolve_components,
start_time = 0,
stop_time = 1000,
timestep = 1 / 32)
sim_output_b <- temp |> select(time, C_in) |>
mutate(itv = "Base case",
model = "Single intervention")
```
```{r}
mdl_test <- read_xmile("./models/2B_TI_Logistic.stmx",
stock_list = list(Qd = Qd_0),
const_list = c(bio_params,
par_beta = beta_vals[[2]],
par_alpha_d = test_cap_pars$alpha_d,
par_rho_d = test_cap_pars$rho_d,
par_iota_d = iota_d_val,
par_delta = delta_val))
sim_output_TI <- sd_simulate(mdl_test$deSolve_components,
start_time = 0,
stop_time = 1000,
timestep = 1 / 32) |>
select(time, C_in) |>
mutate(itv = "TI",
model = "Single intervention")
```
```{r}
fp_ct <- "./models/2C_TTI_Logistic.stmx"
mdl_ct <- read_xmile(fp_ct, stock_list = list(Qd = test_cap_pars$Qd,
Qk = ct_cap_pars$Qk),
const_list = c(bio_params,
par_zeta = beta_vals[[2]] * 10,
par_alpha_d = test_cap_pars$alpha_d,
par_alpha_k = ct_cap_pars$alpha_k,
par_rho_d = test_cap_pars$rho_d,
par_rho_k = ct_cap_pars$rho_k,
par_iota_d = iota_d_val,
par_delta = delta_val))
sim_output_CT <- sd_simulate(mdl_ct$deSolve_components,
start_time = 0,
stop_time = 1000,
timestep = 1 / 32) |>
select(time, C_in) |>
mutate(itv = "CT",
model = "Single intervention")
```
```{r}
fp_mob <- "./models/3C_MR.stmx"
mdl_mob <- read_xmile(fp_mob,
const_list = c(bio_params,
par_zeta_0 = beta_vals[[2]] * 10,
par_tau_m = 0),
stock_list = list(Qm = Qm_0))
sim_output_MR <- sd_simulate(mdl_mob$deSolve_components,
start_time = 0,
stop_time = 1000,
timestep = 1 / 32) |>
select(time, C_in) |>
mutate(itv = "MR",
model = "Single intervention")
```
```{r}
v_fp <- "./models/3A_V.stmx"
mdl_v <- read_xmile(v_fp,
const_list = c(bio_params,
par_beta = beta_vals[[2]],
par_rho_v = par_rho_v,
par_alpha_v = par_alpha_v,
par_phi = phi_val,
par_iota = iota_v_val,
par_tau_v = par_tau_v_test),
stock_list = list(Qv = Qv_0))
sim_output_v <- sd_simulate(mdl_v$deSolve_components,
start_time = 0,
stop_time = 1000,
timestep = 1 / 32) |>
select(time, C_in) |>
mutate(itv = "V",
model = "Single intervention")
```
```{r, warning = FALSE, fig.height = 8, fig.cap = "Fig 2. Validation"}
sgl_int <- bind_rows(sim_output_b, sim_output_TI, sim_output_CT, sim_output_MR,
sim_output_v)
df <- bind_rows(sim_output_syn, sgl_int) |>
mutate(itv = factor(itv, levels = c("Base case", "TI", "CT", "MR", "V"),
ordered = TRUE))
plot_output_comparison(df)
```
# Simulation
This code simulates the comprehensive intervention.
```{r}
sim_sth <- sd_simulate(mdl_sth$deSolve_components, start_time = 0,
stop_time = 365, timestep = 1/64,
integ_method = "euler") |>
filter(time %in% seq(0, 365, 1/8))
```
The code below performs a scenario analysis. In this analysis, we combine two
levels of vaccination willingness and capacity each, resulting in four
scenarios.
```{r}
iota_v_low <- 0.45
# It is assumed that one individual triggers the outbreak
S1 <- (pop_val - 1) * iota_v_val # High willingness
S2 <- (pop_val - 1) * iota_v_low # Low willingness
S_vals <- rep(c(S1, S2), each = 2)
Sh1 <- (pop_val - 1) * (1 - iota_v_val) # High willingness
Sh2 <- (pop_val - 1) * (1 - iota_v_low) # Low willingness
Sh_vals <- rep(c(Sh1, Sh2), each = 2)
# Vaccination scenarios
vacn_scn <- data.frame(Qv = rep(c(Qv_0, 12.26), 2),
S = S_vals,
Sh = Sh_vals,
par_rho_v = rep(c(par_rho_v, 0), 2),
par_alpha_v = rep(c(par_alpha_v, 12.26), 2),
cap_scn = rep(c("C1", "C2"), 2),
wln_scn = c("High", "High", "Low", "Low")) |>
mutate(iter = row_number())
```
```{r}
vacn_scn |> select(-iter) |>
mutate(across(where(is.numeric),~round(.x, 2)),
S = format(S, big.mark = " "),
Sh = format(Sh, big.mark = " ")) |>
gt() |>
cols_label(
par_alpha_v = html("α<sup>v</sup>"),
par_rho_v = html("ρ<sup>v</sup>"),
Qv = html("Q<sub>0</sub><sup>v</sup>"),
S = html("S<sub>0</sub>"),
Sh = html("S<sub>0</sub><sup>h</sup>"),
cap_scn = "Capacity scenario",
wln_scn = "Willingness scenario")
```
```{r}
sim_vacn_scn <- sd_sensitivity_run(mdl_sth$deSolve_components,
consts_df = dplyr::select(vacn_scn,
par_rho_v, par_alpha_v),
stocks_df = dplyr::select(vacn_scn, Qv, S, Sh),
start_time = 0,
stop_time = 365, timestep = 1/64,
integ_method = "euler",
multicore = TRUE, n_cores = 4) |>
filter(time %in% seq(0, 365, 1/8)) |>
left_join(select(vacn_scn, iter, cap_scn, wln_scn))
```
For comparison purposes, we recreate the base-case scenario...
```{r}
base_mdl <- read_xmile("./models/1A_SEI3R.stmx",
const_list = c(bio_params,
par_beta = beta_vals[[2]]))
base_sim <- sd_simulate(base_mdl$deSolve_components,
integ_method = "euler", start_time = 0,
stop_time = 365, timestep = 1 / 64) |>
filter(time %in% seq(0, 365, 1/8))
```
and the TTI + MR scenario.
```{r}
mdl_npi_fp <- "./models/2F_TTI_PLUS_MR.stmx"
mdl_npi <- read_xmile(mdl_npi_fp,
const_list = c(bio_params,
par_zeta_0 = zeta_val,
par_alpha_d = test_cap_pars$alpha_d,
par_alpha_k = ct_cap_pars$alpha_k,
par_rho_d = test_cap_pars$rho_d,
par_rho_k = ct_cap_pars$rho_k,
par_tau_m = par_tau_m,
par_xi = 0.8,
par_iota_d = iota_d_val,
par_delta = delta_val),
stock_list = list(
Qd = test_cap_pars$Qd,
Qk = ct_cap_pars$Qk,
Qm = 90))
sim_npi <- sd_simulate(mdl_npi$deSolve_components, start_time = 0,
stop_time = 365, timestep = 1/64,
integ_method = "euler") |>
filter(time %in% seq(0, 365, 1/8))
```
This code produces Fig 11 in the main text.
```{r, fig.height = 7}
g <- plot_fig_11(sim_sth, base_sim, sim_npi, sim_vacn_scn)
ggsave("./plots/Fig_11_Vaccination_sim.pdf", plot = g,
height = 7, width = 7, device = cairo_pdf)
```