--- title: "SPARSE-MOD COVID-19 Model" author: "JR Mihaljevic" date: "July 2022" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{SPARSE-MOD COVID-19 Model} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r setup, include = FALSE} knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) ``` ```{r, message=FALSE} library(SPARSEMODr) library(future.apply) library(tidyverse) library(viridis) library(lubridate) # To run in parallel, use, e.g., plan("multisession"): future::plan("sequential") ``` ## The COVID-19 Example Model Here we present a walk-through of using the SPARSE-MOD COVID-19 Model, which represents simplified dynamics of transmission of SARS-CoV-2 and COVID-19 progression of patients through the hospital system. The model does not include multiple viral variants, nor does it include vaccination. Therefore, we personally recommend using this R package in an exploratory and educational capacity, not as a mechanism with which to forecast or project disease dynamics. See [our vignette on key features of SPARSEMODr](key-features.html) for more general details of the SPARSEMODr package. And see the end of this vignette for model equations. ### Generating a synthetic meta-population First, we will simulate data that describes the meta-population^[A set of distinct, focal populations that are connected by migration] of interest. ```{r, fig.show='hold'} # Set seed for reproducibility set.seed(5) # Number of focal populations: n_pop = 100 # Population sizes + areas ## Draw from neg binom: census_area = rnbinom(n_pop, mu = 50, size = 3) # Identification variable for later pop_ID = c(1:n_pop) # Assign coordinates, plot for reference lat_temp = runif(n_pop, 32, 37) long_temp = runif(n_pop, -114, -109) # Storage: region = rep(NA, n_pop) pop_N = rep(NA, n_pop) # Assign region ID and population size for(i in 1 : n_pop){ if ((lat_temp[i] >= 34.5) & (long_temp[i] <= -111.5)){ region[i] = "1" pop_N[i] = rnbinom(1, mu = 50000, size = 2) } else if((lat_temp[i] >= 34.5) & (long_temp[i] > -111.5)){ region[i] = "2" pop_N[i] = rnbinom(1, mu = 10000, size = 3) } else if((lat_temp[i] < 34.5) & (long_temp[i] > -111.5)){ region[i] = "4" pop_N[i] = rnbinom(1, mu = 50000, size = 2) } else if((lat_temp[i] < 34.5) & (long_temp[i] <= -111.5)){ region[i] = "3" pop_N[i] = rnbinom(1, mu = 10000, size = 3) } } pop_local_df = data.frame(pop_ID = pop_ID, pop_N = pop_N, census_area, lat = lat_temp, long = long_temp, region = region) # Plot the map: pop_plot = ggplot(pop_local_df) + geom_point(aes(x = long, y = lat, fill = region, size = pop_N), shape = 21) + scale_size(name = "Pop. Size", range = c(1,5), breaks = c(5000, 50000, 150000)) + scale_fill_manual(name = "Region", values = c("#00AFBB", "#D16103", "#E69F00", "#4E84C4")) + geom_hline(yintercept = 34.5, colour = "#999999", linetype = 2) + geom_vline(xintercept = -111.5, colour = "#999999", linetype = 2) + guides(size = guide_legend(order = 2), fill = guide_legend(order = 1, override.aes = list(size = 3))) + # Map coord coord_quickmap() + theme_classic() + theme( axis.line = element_blank(), axis.title = element_blank(), plot.margin = unit(c(0, 0.1, 0, 0), "cm") ) pop_plot # Calculate pairwise dist ## in meters so divide by 1000 for km dist_mat = geosphere::distm(cbind(pop_local_df$long, pop_local_df$lat))/1000 hist(dist_mat, xlab = "Distance (km)", main = "") # We need to determine how many Exposed individuals # are present at the start in each population E_pops = vector("numeric", length = n_pop) # We'll assume a total number of exposed across the # full meta-community, and then randomly distribute these hosts n_initial_E = 20 # (more exposed in larger populations) these_E <- sample.int(n_pop, size = n_initial_E, replace = TRUE, prob = pop_N) for(i in 1:n_initial_E){ E_pops[these_E[i]] <- E_pops[these_E[i]] + 1 } pop_local_df$E_pops = E_pops ``` ### Setting up the time-windows One of the benefits of the SPARSEMODr is that the user can specify how the values of certain parameters of the model change over time (see [our vignette on key features of SPARSEMODr](key-features.html) for more details). We demonstrate this below, where we allow the time-varying transmission rate, $\beta_{t}$, to change in a step-wise fashion due to, for instance, changes in the behavior of the host population. In this case, we assume the transmission rate changes during discrete blocks of time, or time windows. When the parameter values change between two time windows, the model imputes a linear change over the number of days in that window. In other words, the user specifies the value of the parameter achieved on the *last day* of the time window. In other vignettes, we show how the user can instead specify daily values of the time-varying parameters, which allows for more flexibility. Here, we also assume host migration dynamics are invariable across time. We'll use the $\texttt{time_windows()}$ function to generate a pattern of $\beta_{t}$ that looks like the following: ```{r, , fig.width=5, echo=FALSE} # Set up the dates of change. 5 time windows n_windows = 5 # Window intervals start_dates = c(mdy("1-1-20"), mdy("2-1-20"), mdy("2-16-20"), mdy("3-11-20"), mdy("3-22-20")) end_dates = c(mdy("1-31-20"), mdy("2-15-20"), mdy("3-10-20"), mdy("3-21-20"), mdy("5-1-20")) # Date sequence date_seq = seq.Date(start_dates[1], end_dates[n_windows], by = "1 day") # Time-varying beta changing_beta = c(0.3, 0.1, 0.1, 0.15, 0.15) #beta sequence beta_seq = NULL beta_seq[1:(yday(end_dates[1]) - yday(start_dates[1]) + 1)] = changing_beta[1] for(i in 2:n_windows){ beta_temp_seq = NULL beta_temp = NULL if(changing_beta[i] != changing_beta[i-1]){ beta_diff = changing_beta[i-1] - changing_beta[i] n_days = yday(end_dates[i]) - yday(start_dates[i]) + 1 beta_slope = - beta_diff / n_days for(j in 1:n_days){ beta_temp_seq[j] = changing_beta[i-1] + beta_slope*j } }else{ n_days = yday(end_dates[i]) - yday(start_dates[i]) + 1 beta_temp_seq = rep(changing_beta[i], times = n_days) } beta_seq = c(beta_seq, beta_temp_seq) } beta_seq_df = data.frame(beta_seq, date_seq) date_breaks = seq(range(date_seq)[1], range(date_seq)[2], by = "1 month") ggplot(beta_seq_df) + geom_path(aes(x = date_seq, y = beta_seq)) + scale_x_date(breaks = date_breaks, date_labels = "%b") + labs(x="", y=expression("Time-varying "*beta*", ("*beta[t]*")")) + # THEME theme_classic()+ theme( axis.text = element_text(size = 10, color = "black"), axis.title = element_text(size = 12, color = "black"), axis.text.x = element_text(angle = 45, vjust = 0.5) ) ``` Importantly, SPARSEMODr allows the user to assign unique patterns of time-varying $\beta$ *for each population*. Below, we will assume that the pattern of $\beta_{t}$ is unique for each region on the map, above. Correspondingly, each population within the region of interest will have the same pattern of $\beta_{t}$. In this scenario, then, each region has different transmission dynamics, and movement of hosts among regions can influence the local (i.e., within a single population) and regional patterns of disease. ```{r} # Set up the dates of change. 5 time windows n_windows = 5 ## Specify the start and end dates of the time intervals start_dates = c(mdy("1-1-20"), mdy("2-1-20"), mdy("2-16-20"), mdy("3-11-20"), mdy("3-22-20")) end_dates = c(mdy("1-31-20"), mdy("2-15-20"), mdy("3-10-20"), mdy("3-21-20"), mdy("5-1-20")) ### TIME-VARYING PARAMETERS ### # beta pattern per region region_beta = list( "1"=c(0.30, 0.10, 0.10, 0.15, 0.15), "2"=c(0.30, 0.08, 0.08, 0.10, 0.10), "3"=c(0.30, 0.12, 0.12, 0.19, 0.19), "4"=c(0.30, 0.03, 0.03, 0.12, 0.12) ) ## Assign the appropriate, regional pattern of beta ## to each population changing_beta = vector("list", length = n_pop) for (this_pop in 1:n_pop) { this_region <- pop_local_df$region[this_pop] changing_beta[[this_pop]] <- region_beta[[this_region]] } # Migration rate changing_m = rep(1/10.0, times = n_windows) # Migration range changing_dist_phi = rep(150, times = n_windows) # Immigration (none) changing_imm_frac = rep(0, times = n_windows) # Create the time_window() object tw = time_windows( beta = changing_beta, m = changing_m, dist_phi = changing_dist_phi, imm_frac = changing_imm_frac, start_dates = start_dates, end_dates = end_dates ) # Create the covid19_control() object covid19_control <- covid19_control(input_N_pops = pop_N, input_E_pops = E_pops) # Date Sequence for later: date_seq = seq.Date(start_dates[1], end_dates[n_windows], by = "1 day") ``` ### Running the COVID-19 model in parallel Now we have all of the input elements needed to run SPARSEMODr's COVID-19 model. Below we demonstrate a workflow to generate stochastic realizations of the model in parallel. ```{r} # How many realizations of the model? n_realz = 75 # Need to assign a distinct seed for each realization ## Allows for reproducibility input_realz_seeds = c(1:n_realz) # Run the model in parallel model_output = model_parallel( # Necessary inputs input_dist_mat = dist_mat, input_census_area = pop_local_df$census_area, input_tw = tw, input_realz_seeds = input_realz_seeds, control = covid19_control, # OTHER MODEL PARAMS trans_type = 1, # freq-dependent trans stoch_sd = 2.0 # stoch transmission sd ) glimpse(model_output) ``` ### Plotting the output First we need to manipulate and aggregate the output data. Here we show an example just using the 'new events' that occur each day. ```{r} # Grab the new events variables new_events_df = model_output %>% select(pops.seed:pops.time, events.pos:events.n_death) # Simplify/clarify colnames colnames(new_events_df) = c("iter","pop_ID","time", "new_pos", "new_sym", "new_hosp", "new_icu", "new_death") # Join the region region_df = pop_local_df %>% select(pop_ID, region) new_events_df = left_join(new_events_df, region_df, by = "pop_ID") # Join with dates (instead of "time" integer) date_df = data.frame( date = date_seq, time = c(1:length(date_seq)) ) new_events_df = left_join(new_events_df, date_df, by = "time") # Aggregate outcomes by region: ## First, get the sum across regions,dates,iterations new_event_sum_df = new_events_df %>% group_by(region, iter, date) %>% summarize(new_pos = sum(new_pos), new_sym = sum(new_sym), new_hosp = sum(new_hosp), new_icu = sum(new_icu), new_death = sum(new_death)) glimpse(new_event_sum_df) # Now calculate the median model trajectory across the realizations new_event_median_df = new_event_sum_df %>% ungroup() %>% group_by(region, date) %>% summarize(med_new_pos = median(new_pos), med_new_sym = median(new_sym), med_new_hosp = median(new_hosp), med_new_icu = median(new_icu), med_new_death = median(new_death)) glimpse(new_event_median_df) ``` Now we'll start creating a rather complex figure to show the different time intervals. We'll layer on the elements. For this example, we'll just look at the number of new hospitalizations per region. ```{r, fig.height=3.7, fig.width=7, fig.align='center'} # SET UP SOME THEMATIC ELEMENTS: ## Maximum value of the stoch trajectories, for y axis ranges max_hosp = max(new_event_sum_df$new_hosp) ## Breaks for dates: date_breaks = seq(range(date_seq)[1], range(date_seq)[2], by = "1 month") ####################### # PLOT ####################### # First we'll create an element list for plotting: plot_new_hosp_base = list( # Date Range: scale_x_date(limits = range(date_seq), breaks = date_breaks, date_labels = "%b"), # New Hosp Range: scale_y_continuous(limits = c(0, max_hosp*1.05)), # BOXES AND TEXT TO LABEL TIME WINDOWS annotate("rect", xmin = start_dates[1], xmax = end_dates[1], ymin = 0, ymax = max_hosp*1.05, fill = "gray", alpha = 0.2), annotate("rect", xmin = start_dates[3], xmax = end_dates[3], ymin = 0, ymax = max_hosp*1.05, fill = "gray", alpha = 0.2), annotate("rect", xmin = start_dates[5], xmax = end_dates[5], ymin = 0, ymax = max_hosp*1.05, fill = "gray", alpha = 0.2), # THEME ELEMENTS labs(x = "", y = "New Hospitalizations Per Day"), theme_classic(), theme( axis.text = element_text(size = 12, color = "black"), axis.title = element_text(size = 14, color = "black"), axis.text.x = element_text(angle = 45, vjust = 0.5) ) ) ggplot() + plot_new_hosp_base ``` Ok, now we have our plotting base, and we'll layer on the model output. We'll add the stochastic trajectories as well as the median model trajectory. ```{r, fig.height=5, fig.width=7, fig.align='center'} # region labels for facets: region_labs = paste0("Region ", sort(unique(region_df$region))) names(region_labs) = sort(unique(region_df$region)) # Regional beta labels region_beta_df = data.frame( beta_lab = paste0("beta = ",format(unlist(lapply(region_beta, function(x){x[c(1,3,5)]})),nsmall = 1)), region = as.character(rep(c(1:4), each=3)), date = rep(start_dates[c(1,3,5)],times=4), new_hosp = max_hosp*1.05 ) # Create the plot: plot_new_hosp = ggplot() + # Facet by Region facet_wrap(~region, scales = "free", labeller = labeller(region = region_labs)) + # Add our base thematics plot_new_hosp_base + # Add the stoch trajectories: geom_path(data = new_event_sum_df, aes(x = date, y = new_hosp, group = iter, color = region), alpha = 0.05) + # Add the median trajectory: geom_path(data = new_event_median_df, aes(x = date, y = med_new_hosp, color = region), size = 2) + # Add the beta labels: geom_text(data = region_beta_df, aes(x = date, y = new_hosp, label = beta_lab), color = "#39558CFF", hjust = 0, vjust = 1, size = 3.0) + # Colors per region: scale_color_manual(values = c("#00AFBB", "#D16103", "#E69F00", "#4E84C4")) + guides(color="none") plot_new_hosp ``` ## Model equations The version of the model with frequency-dependent transmission that is implemented in each population is below. Note that there is also simulated movement dynamics of the $S$, $I_a$, $I_p$ and $I_s$ classes, moderated by a rate parameter $m$. However, these dynamics are not explicitly represented in these equations. \begin{align} \frac{dS}{dt} &= -\beta_{t} \lambda_{t} S \\ \frac{dE}{dt} &= \beta_{t} \lambda_{t} S - \delta_{1} E \\ \frac{dI_a}{dt} &= \delta_1 \rho_1 E - \gamma_{a} I_a \\ \frac{dI_p}{dt} &= \delta_1 (1 - \rho_1) E - \delta_2 I_p \\ \frac{dI_s}{dt} &= \delta_2 I_p - \delta_3 I_s \\ \frac{dI_b}{dt} &= \delta_3 (1 - \rho_2 - \rho_3) I_s - \gamma_{b} I_b \\ \frac{dI_h}{dt} &= \delta_3 \rho_2 I_s - \delta_4 I_h \\ \frac{dI_{c1}}{dt} &= \delta_3 \rho_3 I_s + \delta_4 \rho_4 I_h - \delta_5 I_{c1} \\ \frac{dI_{c2}}{dt} &= \delta_5 (1 - \rho_5) I_{c1} - \gamma_{c} I_{c2} \\ \frac{dD}{dt} &= \delta_5 \rho_5 I_{c1} \\ \frac{dR}{dt} &= \gamma_{a} I_a + \gamma_{b} I_b + \gamma_{c} I_{c2} + \delta_4 (1 - \rho_4) I_h \end{align} And the time-varying force of infection: $$\lambda_{t} = \frac{\omega_{1} I_a + I_p + I_s + I_b + \omega_2 \left( I_h + I_{c1} + I_{c2} \right)}{N - D}$$ | State Variable | Description | | :------------- | :---------- | | $S$ | Number of susceptible individuals | | $E$ | Number of exposed individuals | | $I_a$ | Number of asymptomatic individuals | | $I_p$ | Number of pre-symptomatic individuals | | $I_s$ | Number of mildly symptomatic individuals | | $I_b$ | Number of mildly symptomatic individuals on bed rest at home | | $I_h$ | Number of hospitalized individuals | | $I_{c1}$ | Number of individuals in the ICU| | $I_{c2}$ | Number of individuals in the recovery (step-down) ICU| | $D$ | Number of deceased individuals | | $R$ | Number of susceptible individuals | | Parameter | Description | Corresponding model input | | :--- | :---------------- | :------- | | $\beta_t$ | Time-varying transmission rate | $\texttt{beta}$ | | $\omega_1$ | Proportion reduction in transmission for asymptomatic folks | $\texttt{frac_beta_asym}$ | | $\omega_2$ | Proportion reduction in transmission for hospitalized folks | $\texttt{frac_beta_hosp}$ | | $N$ | Total number of individuals in population | $\texttt{input_N_pops}$ | | $\delta_1$ | Transition rate: exposed to pre-symptomatic | $\texttt{delta}$ | | $\delta_2$ | Transition rate: pre-symptomatic to symptomatic | $\texttt{recov_p}$ | | $\delta_3$ | Transition rate: symptomatic to home or regular hospital bed or ICU | $\texttt{recov_s}$ | | $\delta_4$ | Transition rate: regular hospital bed to home or ICU | $\texttt{recov_hosp}$ | | $\delta_5$ | Transition rate: ICU to step-down ICU | $\texttt{recov_icu1}$ | | $\gamma_{a}$ | Recovery rate: asymptomatic | $\texttt{recov_a}$ | | $\gamma_{b}$ | Recovery rate: home bed | $\texttt{recov_home}$ | | $\gamma_{c}$ | Recovery rate: step-down ICU | $\texttt{recov_icu2}$ | | $\rho_1$ | Fraction of exposed that transition to asymptomatic | $\texttt{asym_rate}$ | | $\rho_2$ | Fraction of symptomatic that transition to hospital bed | $\texttt{hosp_rate}$ | | $\rho_3$ | Fraction of symptomatic that transition to icu bed | $\texttt{sym_to_icu_rate}$ | | $\rho_4$ | Fraction of hospitalized that transition to ICU | $\texttt{icu_rate}$ | | $\rho_5$ | Fraction of patients in ICU that die of disease | $\texttt{death_rate}$ |