# Extended state-space SIR with a subset of the population showing antibody positivity In this function we allow it to characterize time-varying immunization among a subset of the population that have been tested positive in an antibody assessment. We expanded the SIR model by adding a time-varying antibody-positive proportion $\alpha_t$. ```r eSAIR( Y, R, alpha0 = NULL, change_time = NULL, begin_str = "01/13/2020", T_fin = 200, nchain = 4, nadapt = 10000, M = 500, thn = 10, nburnin = 200, dic = FALSE, death_in_R = 0.02, casename = "eSAIR", beta0 = 0.2586, gamma0 = 0.0821, R0 = beta0/gamma0, gamma0_sd = 0.1, R0_sd = 1, file_add = character(0), add_death = FALSE, save_files = FALSE, save_mcmc = FALSE, save_plot_data = FALSE, eps = 1e-10 ) ``` ## Arguments - `Y`: the time series of daily observed infected compartment proportions. - `R`: the time series of daily observed removed compartment proportions, including death and recovered. - `alpha0`: a vector of values of the dirac delta function $\alpha_t$. Each entry denotes the proportion that will be immunized at each change time point. Note that all the entries lie between 0 and 1, its default is `NULL`. - `change_time`: the change points over time corresponding to `alpha0`, to formulate the dirac delta function $\alpha_t$; its defalt value is `NULL`. - `begin_str`: the character of starting time, the default is "01/13/2020". - `T_fin`: the end of follow-up time after the beginning date `begin_str`, the default is 200. - `nchain`: the number of MCMC chains generated by `rjags`, the default is 4. - `nadapt`: the iteration number of adaptation in the MCMC. We recommend using at least the default value 1e4 to obtained fully adapted chains. - `M`: the number of draws in each chain, with no thinning. The default is M=5e2 but suggest using 5e5. - `thn`: the thinning interval between mixing. The total number of draws thus would become `round(M/thn)*nchain`. The default is 10. - `nburnin`: the burn-in period. The default is 2e2 but suggest 2e5. - `dic`: logical, whether compute the DIC (deviance information criterion) for model selection. - `death_in_R`: the numeric value of average of cumulative deaths in the removed compartments. The default is 0.4 within Hubei and 0.02 outside Hubei. - `casename`: the string of the job's name. The default is "eSAIR". - `beta0`: the hyperparameter of average transmission rate, the default is the one estimated from the SARS first-month outbreak (0.2586). - `gamma0`: the hyperparameter of average removed rate, the default is the one estimated from the SARS first-month outbreak (0.0821). - `R0`: the hyperparameter of the mean reproduction number R0. The default is thus the ratio of `beta0/gamma0`, which can be specified directly. - `gamma0_sd`: the standard deviation for the prior distrbution of the removed rate $\gamma$, the default is 0.1. - `R0_sd`: the standard deviation for the prior disbution of R0, the default is 1. - `file_add`: the string to denote the location of saving output files and tables. - `add_death`: logical, whether add the approximate death curve to the plot, default is false. - `save_files`: logical, whether to save plots to file. - `save_mcmc`: logical, whether save (`TRUE`) all the MCMC outputs or not (`FALSE`).The output file will be an `.RData` file named by the $casename$. We include arrays of prevalence values of the three compartments with their matrices of posterior draws up to the last date of the collected data as `theta_p[,,1]` and afterwards as `theta_pp[,,1]` for $\theta_t^S$, `theta_p[,,2]` and `theta_pp[,,2]` for $\theta_t^I$, and `theta_p[,,3]` and `theta_pp[,,3]` for $\theta_t^R$. The posterior draws of the prevalence process of the antibody-immunized compartment can be obtained via `thetaA_p` and `thetaA_pp`. Moreover, the input and predicted proportions `Y`, `Y_pp`, `R` and `R_pp` can also be retrieved. The prevalence and prediceted proportion matrices have rows for MCMC replicates, and columns for days. The MCMC posterior draws of other parameters including `beta`, `gamma`, `R0`, and variance controllers `k_p`, `lambdaY_p`, `lambdaR_p` are also available. - `save_plot_data`: logical, whether save the plotting data or not. - `eps`: a non-zero controller so that all the input `Y` and `R` values would be bounded above 0 (at least `eps`). Its default value is 1e-10 ## Returns - **casename**: the predefined `casename`. - **incidence_mean**: mean cumulative incidence, the mean prevalence of cumulative confirmed cases at the end of the study. - **incidence_ci**: 2.5%, 50%, and 97.5% quantiles of the incidences. - **out_table**: summary tables including the posterior mean of the prevalance processes of the 3 states compartments ($\theta_t^S,\theta_t^I,\theta_t^R,\theta_t^H$) at last date of data collected (($t^\prime$) decided by the lengths of your input data `Y` and `R`), and their respective credible inctervals (ci); the respective means and ci's of the reporduction number (R0), removed rate ($\gamma$), transmission rate ($\beta$). - **plot_infection**: plot of summarizing and forecasting for the infection compartment, in which the vertial blue line denotes the last date of data collected ($t^\prime$), the vertial darkgray line denotes the deacceleration point (first turning point) that the posterior mean first-derivative of infection prevalence $\dot{\theta}_t^I$ achieves the maximum, the vertical purple line denotes the second turning point that the posterior mean first-derivative infection proportion $\dot{\theta}_t^I$ equals zero, the darkgray line denotes the posterior mean of the infection prevalence $\theta_t^I$ and the red line denotes its posterior median. - **plot_removed**: plot of summarizing and forecasting for the removed compartment with lines similar to those in the `plot_infection`. The vertical lines are identical, but the horizontal mean and median correspond to the posterior mean and median of the removed process $\theta_t^R$. An additional line indicates the estimated death prevalence from the input `death_in_R`. - **spaghetti_plot**: 20 randomly selected MCMC draws of the first-order derivative of the posterior prevalence of infection, namely $\dot{\theta}_t^I$. The black curve is the posterior mean of the derivative, and the vertical lines mark times of turning points corresponding respectively to those shown in `plot_infection` and `plot_removed`. Moreover, the 95% credible intervals of these turning points are also highlighted by semi-transparent rectangles. - **first_tp_mean**: the date t at which $\ddot{\theta}_t^I=0$, calculated as the average of the time points with maximum posterior first-order derivatives $\dot{\theta}_t^I$; this value may be slightly different from the one labeled by the "darkgreen" lines in the two plots `plot_infection` and `plot_removed`, which indicate the stationary point such that the first-order derivative of the averaged posterior of $\theta_t^I$ reaches its maximum. - **first_tp_mean**: the date t at which $\ddot{\theta}_t^I=0$, calculated as the average of the time points with maximum posterior first-order derivatives $\dot{\theta}_t^I$; this value may be slightly different from the one labeled by the "darkgreen" lines in the two plots `plot_infection` and `plot_removed`, which indicate the stationary point such that the first-order derivative of the averaged posterior of $\theta_t^I$ reaches its maximum. - **first_tp_ci**: fwith `first_tp_mean`, it reports the corresponding credible interval and median. - **second_tp_mean**: the date t at which $\theta_t^I=0$, calculated as the average of the stationary points of all of posterior first-order derivatives $\dot{\theta}_t^I$; this value may be slightly different from the one labeled by the "pruple" lines in the plots of `plot_infection` and `plot_removed`. The latter indicate stationary t at which the first-order derivative of the averaged posterior of $\theta_t^I$ equals zero. - **second_tp_ci**: with `second_tp_mean`, it reports the corresponding credible interval and median. - **dic_val**: the output of `dic.samples()` in `dic.samples`, computing deviance information criterion for model comparison. - **gelman_diag_list**: Since version 0.3.3, we incorporated Gelman And Rubin's Convergence Diagnostic using `gelman.diag`. We included both the statistics and their upper C.I. limits. Values substantially above 1 indicate lack of convergence. Error messages would be printed as they are. This would be only valid for multiple chains (e.g. nchain > 1). Note that for time dependent processes, we only compute the convergence of the last observation data (`T_prime`), though it shows to be `T_prime+1`, which is due to the day 0 for initialization. ## Examples ```r NI_complete <- c( 41, 41, 41, 45, 62, 131, 200, 270, 375, 444, 549, 729, 1052, 1423, 2714, 3554, 4903, 5806, 7153, 9074, 11177, 13522, 16678, 19665, 22112, 24953, 27100, 29631, 31728, 33366 ) RI_complete <- c( 1, 1, 7, 10, 14, 20, 25, 31, 34, 45, 55, 71, 94, 121, 152, 213, 252, 345, 417, 561, 650, 811, 1017, 1261, 1485, 1917, 2260, 2725, 3284, 3754 ) N <- 58.5e6 R <- RI_complete / N Y <- NI_complete / N - R # Jan13->Feb 11 change_time <- c("02/08/2020") alpha0 <- c(0.2) # 20% of the susceptible population were found immunized res.antibody <- eSAIR(Y, R, begin_str = "01/13/2020", death_in_R = 0.4, alpha0 = alpha0, change_time = change_time, casename = "Hubei_antibody", save_files = TRUE, save_mcmc = FALSE, M = 5e2, nburnin = 2e2 ) res.antibody$plot_infection change_time <- c("01/16/2020") alpha0 <- c(0.2) NI_complete2 <- c(41, 45) RI_complete2 <- c(1, 1) N2 <- 1E3 res3 <- eSAIR( RI_complete2 / N2, NI_complete2 / N2, begin_str = "01/13/2020", T_fin = 4, alpha0 = alpha0, change_time = change_time, dic = FALSE, casename = "Hubei_q", save_files = FALSE, save_mcmc = FALSE, save_plot_data = FALSE, M = 50, nburnin = 1 ) closeAllConnections() ```

eSAIR function