bayesian-transmission

library(bayestransmission)

Introduction

This package provides a Bayesian framework for transmission modeling on an individual patient level. Modeling is conducted through Markov Chain Monte Carlo (MCMC) methods.This document will explain the basic usage of the package, specification of parameters, and the output of the model.

Data Structure

The algorithms expect a longitudinal data set with the following columns: * facility: The facility where the event occurred. * unit: The unit within the facility where the event occurred. * time: The time at which the event occurred. * patient: The patient involved in the event. * type: The type of event.

The package includes a simulated dataset, simulated.data.

pillar::glimpse(simulated.data)
#> Rows: 8,360
#> Columns: 5
#> $ facility <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
#> $ unit     <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
#> $ time     <dbl> 0.060978, 0.061978, 1.560978, 2.883323, 2.884323, 1.422631, 1…
#> $ patient  <int> 1, 1, 1, 1, 1, 10, 10, 10, 10, 19, 19, 19, 23, 23, 23, 28, 28…
#> $ type     <int> 0, 1, 10, 1, 3, 0, 1, 1, 3, 0, 1, 3, 0, 1, 3, 0, 1, 1, 3, 0, …

There are 12 different types of events that can be specified in the type column. These are, expected numerical codes shown in parentheses:

Admission (0) and Discharge (3)
Surveillance Testing Results
- Negative Test (1)
- Positive Test (2)
Clinical Testing Results
- Negative Test (4)
- Positive Test (5)
Generic Testing
- Negative Test (7)
- Positive Test (8)
Antibiotic Use
- single dose (9)
- Start (10)
- Stop (11)
Isolation Procedures
- Start (6)
- Stop (7)

Not all events need to be used in every data set, but the model selected should reflect the data that is available. Care should be taken to correctly code the data. The EventToCode and CodeToEvent functions can be used to convert between.

table(CodeToEvent(simulated.data$type))
#> 
#>      abxoff       abxon   admission   discharge negsurvtest possurvtest 
#>         297         725        2183        2183        2749         223

Model Specifiction

Model Choice

Since the model is implemented in C++ for speed and efficiency, only the specified models can be used. The currently implemented models are:

"LinearAbxModel", A Linear model with antibiotic use as a covariate.
"MixedModel", and
"LogNormalModel"

Model specification and all parameters are controlled through constructor functions of the same name, or generically through the LogNormalModelParams() function.

For all models there is the choice of either a 2 state (susceptible and colonized) or 3 state (susceptible, colonized, and recovered or latent) model. Number of states is set through the nstates parameter, and the number of states in the model overrides what may be specified in any individual component.

Parameters

The remainder of the parameters are grouped into the following categories:

Abx, Antibiotic use,
AbxRate, Antibiotic rates,
InUnit, In unit infection rates,
OutOfUnitInfection, Out of unit infection rates,
Insitu, In situ parameters,
Testing:
- SurveilenceTest, Surveillance testing,
- ClinicalTest, Clinical testing.

Unless otherwise specified the parameters are all distributed gamma with specified shape and rate parameters. Each parameter can also be left as fixed or be sampled at each iteration of the MCMC.

Specifying parameters

Parameters for the model may be specified by the Param() function. This function takes up to four arguments:

1. `init`, is the initial value of the parameter.
2. `weight`, is the weight of the prior distribution in updates.
3. `update`, a flag of if the parameter should be sampled in the MCMC algorithm. 
    `FALSE` indicates that the parameter should be fixed, and is by default `TRUE` when `weight` is greater than zero.
4. `prior`, the mean of the prior distribution.  Taken with the weight will fully parameterize the distribution.

# Fully specified parameter.
Param(init = 0, weight = 1, update = TRUE, prior = 0.5)
# Fixed parameter
# Weight = 0 implies update=FALSE and prior is ignored.
Param(0, 0)
# Update parameter that starts at zero.
Param(0, weight =1, update=TRUE)
# Parameters specified at zero implies fixed.
Param(0)

`Abx` Antibiotic use

Antibiotic use is specified by the Abx parameter. This parameter is a list constructed with the AbxParams() function with the following components:

onoff, If antibiotics are being used or not. The two following parameters are only used if onoff is TRUE.
delay, the delay for the antibiotic to take effect.
life, the duration where the antibiotic to be effective.

abx <- AbxParams(onoff = TRUE, delay = 0.3, life = 1)

Currently, all antibiotics are assumed to be equally effective and have the same duration of effectiveness.

`AbxRate` Antibiotic rates

The AbxRate parameter control the antibiotic administration rates.

abxrate <- AbxRateParams(
  # Uncolonized patients do use antibiotics but at a low rate.
  uncolonized = 0.05,  
  # Colonized patients use antibiotics at a high rate.
  colonized = 0.7      
)

Here since both parameters are non-zero both will be updated. A rate of zero for either would indicate that group would never be on antibiotics.

`InUnit` In unit infection rate

Transmission within unit is the main defining characteristic that differentiates models. For example the linear antibiotic model, LinearAbxModel(), is differentiated from the log normal model, LogNormalModelParams() by the use of a ABXInUnitParams() for the InUnit argument rather than the LogNormalInUnitAcquisition() which does not take into account antibiotic use. All in unit transmission is defined in terms of acquisition, progression, and clearance.

Aqcuisition Model

In the base log normal antibiotic model, LogNormalABXInUnitParameters() log acquisition probability is a linear function.

$\log(P(\mathrm{Acq(t)})) = \beta_0 + \beta_t(t-t_0) + \beta_c N_og(\mathrm{Colonized(t)})a_{ca}N_{ca}(t) + \beta_A A_i(t) + \beta_E E_i(t)$ Where $\beta_\star$ represents the coefficient corresponding to the amounts, $N_c(t)$ represent the total number of colonized patients at time $t$ , $N_{ca}(t)$ the number of colonized on antibiotics, and $A_i(t)$ and $E_i(t)$ represents if patient $i$ is currently or ever on antibiotics.

The linear antibiotic (LinearAbxAcquisitionParams) takes a more complicated form for the acquisition model. $$ P(\mathrm{Acq(t)}) = \left[e^{\beta_\mathrm{time}(t-t_0)}\right]\\ \left\{e^{\beta_0} \left[ \left(\frac{\beta_\mathrm{freq}}{P(t)}+(1 - e^{\beta_\mathrm{freq}})\right) e^{\beta_\mathrm{mass}}\left( (N_c(t) - N_{ca}(t)) + e^{\beta_\mathrm{col\_abx}}N_{ca}(t) \right) + 1 - e^{\beta_\mathrm{mass}} \right] \right\}\\ \left[ N_S(t) - N_E(t) + e^{\beta_\mathrm{suss\_ever}}\left(\left(E_i(t)-A_i(t)\right) +A_i(t)e^{\beta_\mathrm{suss\_abx}}\right) \right] $$

acquisition <- LinearAbxAcquisitionParams(
    base = Param(0.01),     #< Base acquisition rate (Updated)
    time = Param(1, 0),     #< Time effect (Fixed)
    mass = Param(1),        #< Mass Mixing (Updated)
    freq = Param(1),        #< Frequency/Density effect (Updated)
    col_abx = Param(1, 0),  #< Colonized on antibiotics (Fixed)
    suss_abx = Param(1, 0), #< Susceptible on antibiotics (Fixed)
    suss_ever = Param(1, 0) #< Ever on antibiotics (Fixed)  
)

Progression Model

In the 3 state model there is a latent state and the progression model controls how patient transition out of the latent state. The base rate can be affected by currently being on antiboitics or ever being on antbiotcs.

$\log(P(\mathrm{progression})) = \delta_0+\delta_AA_i(t) + \delta_EE_i(t)$ the linear antibiotic model is:

$P(\mathrm{progression}) = e^{\delta_0}\left[1-E_i(t)+e^{\delta_2}\left(E_i(t)-A_i(t)+e^{\delta_1}A_i(t)\right)\right]$

Where here we use $\delta$ for the coefficients, but the notation is the same.

progression <- ProgressionParams(
    rate = Param(0.01),     #< Base progression rate (Updated)
    abx  = Param(1, 0),     #< Currently on antibiotics (Fixed)
    ever = Param(1, 0)      #< Ever on antibiotics (Fixed)
)

Clearance Model

The clearance model is the same as the progression model in both the log normal and the linear cases, the coefficients however are independent.

clearance <- ClearanceParams(
    rate = Param(0.01),     #< Base clearance rate (Updated)
    abx  = Param(1, 0),     #< Currently on antibiotics (Fixed)
    ever = Param(1, 0)      #< Ever on antibiotics (Fixed)
)

inunit <- ABXInUnitParams(
  acquisition = acquisition,
  progression = progression,
  clearance   = clearance
)

Out of Unit Importation

The out of unit parameters control the rate at which admissions come in, and which state they enter in.

$\log(P(\mathrm{state}_i \rightarrow \mathrm{state}_j)|t) = P_j - Q_{i,j} e^{-t \sum_i r_i}$

outcol <- OutOfUnitInfectionParams(
  acquisition = 0.1,
  clearance = 0.5
)

In Situ

I’m not sure what these parameters do. It’s a set of gamma distributed parameters one for each state. The updates and probabilities are not time dependent.

When updating the rates for each state are sampled from a $gamma(N_i, 1)$ distribution. Then all three are normalized to sum to 1.

insitu <- InsituParams(
  # Starting 90/10 split uncolonized to colonized  
  probs = c(uncolonized = 0.90, #< names are unnecessary.
              colonized = 0.10)
)

Testing

There are two types of testing, surveillance, which is conducted routinely at regular intervals such as on admission then every 3 days after, and clinical, where the testing is precipitated by staff, and thus the timing is informative.

Surveillance Testing

The timing of surveillance testing is assumed to not be informative. Therefore, surveillance testing is only parameterized in terms of probability of a positive test given the underlying status. Surveillance test parameters are updated with a sample from a $Beta(N_{s,1}, N_{s,0})$ distribution where $N_{s,1}$ and $N_{s,0}$ are the number of positive and negative tests respectively for state $s$ .

surv <- SurveillanceTestParams(
    # Probability of a positive test when uncolonized
    uncolonized = Param(0),
    # Probability of a positive test when colonized
    colonized = Param(0.9, 1, TRUE)
)

Clinical Testing

Since clinical testing time is informative, clinical testing is assumed to be at random within infection stage. The rate of testing within each stage is sampled from a gamma distribution. Sensitivity/Specificity are handled the same as surveillance testing and the likelihood is multiplicative between rate and effectiveness.

clin <- ClinicalTestParams(
    # Rate of testing when uncolonized
    uncolonized = ParamWRate(Param(0, 0), rate = Param(1)),
    # Rate of testing when colonized
    colonized = ParamWRate(Param(0, 0), rate = Param(1))
)

All Together

params <- LinearAbxModel(
  nstates = 2,
  Insitu = insitu,
  SurveillanceTest = surv,
  ClinicalTest = clin,
  OutOfUnitInfection = outcol,
  InUnit = inunit,
  Abx = abx,
  AbxRate = abxrate
)

Running the Model

The model is run through the runMCMC() function. This function takes the following arguments:

results <- runMCMC(
  data = simulated.data,
  MCMCParameters = list(
    nburn = 100,
    nsims = 1000,
    outputparam = TRUE,
    outputfinal = TRUE
  ),
  modelParameters = params,
  verbose = TRUE
)
#> Initializing Variables
#> Creating RNG...Done
#> Setting up System...Done
#> Creating model...
#> (In LinearAbxModel specialization)
#>   * Setting up Abx...Done
#>   * Setting up Insitu...Done
#>   * Setting up Surveillance Test...Done
#>   * Setting up Clinical Test...Done
#>   * Setting up Out of Unit...Done
#>   * Setting up In Unit...Done
#>   * Setting up Abx Rates...Done
#> Done
#> Set time origin
#> Building history structure...Done
#> Finding tests for WAIC.
#> Building sampler.
#> 
#> === INITIAL PARAMETERS ===
#> 0.9000000000 0.0000000000    0.0000000000    0.8000000000    0.0000000000    0.8000000000    0.0000000000    0.0000000000    1.0000000000    0.0000000000    0.1000000000    0.5000000000    0.0100000000    1.0000000000    0.9999999999    0.9999999999    1.0000000000    1.0000000000    1.0000000000    0.0100000000    1.0000000000    1.0000000000    0.0500000000    0.7000000000            LogLike=-inf
#> === END INITIAL PARAMETERS ===
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#> MCMC done.
#> WAIC 1 2 =   -14.1094    14.1111
#> Writing complete form of final state.

Analyzing MCMC Results

Converting Parameters to Data Frame

The results$Parameters object contains the MCMC chain of all model parameters. To create trace plots and posterior distributions, we need to convert this nested list structure into a tidy data frame format.

# Function to safely extract parameter by name
get_param <- function(vec, name, default = NA_real_) {
  if (name %in% names(vec)) {
    return(as.numeric(vec[name]))
  }
  return(default)
}

# Convert parameters to long-format data frame
param_df <- data.frame()

for (i in seq_along(results$Parameters)) {
  iter_params <- results$Parameters[[i]]
  
  # Extract key parameters based on actual structure
  row <- data.frame(
    iteration = i,
    
    # In-situ probabilities
    insitu_uncolonized = get_param(iter_params$Insitu, "Insit.P(unc)"),
    insitu_colonized = get_param(iter_params$Insitu, "Insit.P(col)"),
    
    # Surveillance test parameters
    surv_test_uncol_neg = get_param(iter_params$SurveillanceTest, "ATest.P(+|unc-)"),
    surv_test_col_neg = get_param(iter_params$SurveillanceTest, "ATest.P(+|col-)"),
    surv_test_uncol_pos = get_param(iter_params$SurveillanceTest, "ATest.P(+|unc+)"),
    surv_test_col_pos = get_param(iter_params$SurveillanceTest, "ATest.P(+|col+)"),
    
    # Clinical test parameters
    clin_test_uncol = get_param(iter_params$ClinicalTest, "RTest.P(+|unc)"),
    clin_test_col = get_param(iter_params$ClinicalTest, "RTest.P(+|col)"),
    clin_rate_uncol = get_param(iter_params$ClinicalTest, "RTest.rateUnc"),
    clin_rate_col = get_param(iter_params$ClinicalTest, "RTest.rateCol"),
    
    # Out of unit parameters
    outunit_acquisition = get_param(iter_params$OutCol, "Out.acq"),
    outunit_clearance = get_param(iter_params$OutCol, "Out.clr"),
    
    # In unit parameters (first few unnamed parameters)
    inunit_param1 = if(length(iter_params$InCol) >= 1) iter_params$InCol[1] else NA_real_,
    inunit_param2 = if(length(iter_params$InCol) >= 2) iter_params$InCol[2] else NA_real_,
    inunit_param3 = if(length(iter_params$InCol) >= 3) iter_params$InCol[3] else NA_real_,
    inunit_param4 = if(length(iter_params$InCol) >= 4) iter_params$InCol[4] else NA_real_,
    
    # Antibiotic rate parameters
    abxrate_uncolonized = get_param(iter_params$Abx, "Abx.rateUnc"),
    abxrate_colonized = get_param(iter_params$Abx, "Abx.rateCol"),
    
    # Log likelihood
    loglikelihood = results$LogLikelihood[i]
  )
  
  param_df <- rbind(param_df, row)
}

# Display first few rows
head(param_df)
#>                                  iteration insitu_uncolonized insitu_colonized
#> PARAMETER NAMES NOT IMPLEMENTED          1          0.9302875                0
#> PARAMETER NAMES NOT IMPLEMENTED1         2          0.9998525                0
#> PARAMETER NAMES NOT IMPLEMENTED2         3          0.9908906                0
#> PARAMETER NAMES NOT IMPLEMENTED3         4          0.9556866                0
#> PARAMETER NAMES NOT IMPLEMENTED4         5          0.8666393                0
#> PARAMETER NAMES NOT IMPLEMENTED5         6          1.0000000                0
#>                                  surv_test_uncol_neg surv_test_col_neg
#> PARAMETER NAMES NOT IMPLEMENTED                    1                 0
#> PARAMETER NAMES NOT IMPLEMENTED1                   1                 0
#> PARAMETER NAMES NOT IMPLEMENTED2                   1                 0
#> PARAMETER NAMES NOT IMPLEMENTED3                   1                 0
#> PARAMETER NAMES NOT IMPLEMENTED4                   1                 0
#> PARAMETER NAMES NOT IMPLEMENTED5                   1                 0
#>                                  surv_test_uncol_pos surv_test_col_pos
#> PARAMETER NAMES NOT IMPLEMENTED                    1                 0
#> PARAMETER NAMES NOT IMPLEMENTED1                   1                 0
#> PARAMETER NAMES NOT IMPLEMENTED2                   1                 0
#> PARAMETER NAMES NOT IMPLEMENTED3                   1                 0
#> PARAMETER NAMES NOT IMPLEMENTED4                   1                 0
#> PARAMETER NAMES NOT IMPLEMENTED5                   1                 0
#>                                  clin_test_uncol clin_test_col clin_rate_uncol
#> PARAMETER NAMES NOT IMPLEMENTED                0             0        1391.482
#> PARAMETER NAMES NOT IMPLEMENTED1               0             0        5333.154
#> PARAMETER NAMES NOT IMPLEMENTED2               0             0        7545.635
#> PARAMETER NAMES NOT IMPLEMENTED3               0             0        5170.308
#> PARAMETER NAMES NOT IMPLEMENTED4               0             0        4230.554
#> PARAMETER NAMES NOT IMPLEMENTED5               0             0       14083.426
#>                                  clin_rate_col outunit_acquisition
#> PARAMETER NAMES NOT IMPLEMENTED              0         0.002885853
#> PARAMETER NAMES NOT IMPLEMENTED1             0         0.002885853
#> PARAMETER NAMES NOT IMPLEMENTED2             0         0.001218417
#> PARAMETER NAMES NOT IMPLEMENTED3             0         0.001218417
#> PARAMETER NAMES NOT IMPLEMENTED4             0         0.001218417
#> PARAMETER NAMES NOT IMPLEMENTED5             0         0.001218417
#>                                  outunit_clearance inunit_param1 inunit_param2
#> PARAMETER NAMES NOT IMPLEMENTED       0.0012281783   0.003159614             1
#> PARAMETER NAMES NOT IMPLEMENTED1      0.0012281783   0.003116127             1
#> PARAMETER NAMES NOT IMPLEMENTED2      0.0006971535   0.003844998             1
#> PARAMETER NAMES NOT IMPLEMENTED3      0.0006971535   0.004461223             1
#> PARAMETER NAMES NOT IMPLEMENTED4      0.0006971535   0.003907606             1
#> PARAMETER NAMES NOT IMPLEMENTED5      0.0006971535   0.004158357             1
#>                                  inunit_param3 inunit_param4
#> PARAMETER NAMES NOT IMPLEMENTED              1             1
#> PARAMETER NAMES NOT IMPLEMENTED1             1             1
#> PARAMETER NAMES NOT IMPLEMENTED2             1             1
#> PARAMETER NAMES NOT IMPLEMENTED3             1             1
#> PARAMETER NAMES NOT IMPLEMENTED4             1             1
#> PARAMETER NAMES NOT IMPLEMENTED5             1             1
#>                                  abxrate_uncolonized abxrate_colonized
#> PARAMETER NAMES NOT IMPLEMENTED              1226980          636348.4
#> PARAMETER NAMES NOT IMPLEMENTED1             1165656          629777.4
#> PARAMETER NAMES NOT IMPLEMENTED2             1271520          680347.0
#> PARAMETER NAMES NOT IMPLEMENTED3             1150118          666265.7
#> PARAMETER NAMES NOT IMPLEMENTED4             1214577          672626.5
#> PARAMETER NAMES NOT IMPLEMENTED5             1226288          734988.7
#>                                  loglikelihood
#> PARAMETER NAMES NOT IMPLEMENTED           -Inf
#> PARAMETER NAMES NOT IMPLEMENTED1          -Inf
#> PARAMETER NAMES NOT IMPLEMENTED2          -Inf
#> PARAMETER NAMES NOT IMPLEMENTED3          -Inf
#> PARAMETER NAMES NOT IMPLEMENTED4          -Inf
#> PARAMETER NAMES NOT IMPLEMENTED5          -Inf

Trace Plots

Trace plots show the evolution of parameters across MCMC iterations, helping to assess convergence.

library(ggplot2)
library(tidyr)

# Select key parameters for trace plots
trace_params <- param_df[, c("iteration", "insitu_colonized", "surv_test_col_pos", 
                              "outunit_acquisition", "inunit_param1", 
                              "abxrate_colonized", "loglikelihood")]

# Convert to long format
trace_long <- pivot_longer(trace_params, 
                           cols = -iteration,
                           names_to = "parameter",
                           values_to = "value")

# Create trace plots
ggplot(trace_long, aes(x = iteration, y = value)) +
  geom_line() +
  facet_wrap(~parameter, scales = "free_y", ncol = 2) +
  theme_minimal() +
  labs(title = "MCMC Trace Plots",
       x = "Iteration",
       y = "Parameter Value")

Posterior Distributions

Posterior distributions show the estimated distribution of each parameter after the MCMC sampling.

# Remove burn-in if needed (here we already set nburn in the MCMC call)
# For demonstration, let's use all samples since nburn=0 was specified

# Create density plots for posterior distributions
ggplot(trace_long, aes(x = value)) +
  geom_density(fill = "steelblue", alpha = 0.5) +
  geom_vline(aes(xintercept = mean(value, na.rm = TRUE)), 
             color = "red", linetype = "dashed") +
  facet_wrap(~parameter, scales = "free", ncol = 2) +
  theme_minimal() +
  labs(title = "Posterior Distributions",
       subtitle = "Red dashed line shows posterior mean",
       x = "Parameter Value",
       y = "Density")
#> Warning: Removed 1000 rows containing non-finite outside the scale range
#> (`stat_density()`).

Summary Statistics

# Calculate summary statistics for each parameter
library(dplyr)
#> 
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#> 
#>     filter, lag
#> The following objects are masked from 'package:base':
#> 
#>     intersect, setdiff, setequal, union

summary_stats <- trace_long %>%
  group_by(parameter) %>%
  summarise(
    mean = mean(value, na.rm = TRUE),
    median = median(value, na.rm = TRUE),
    sd = sd(value, na.rm = TRUE),
    q025 = quantile(value, 0.025, na.rm = TRUE),
    q975 = quantile(value, 0.975, na.rm = TRUE),
    .groups = "drop"
  )

print(summary_stats)
#> # A tibble: 6 × 6
#>   parameter                    mean       median        sd       q025       q975
#>   <chr>                       <dbl>        <dbl>     <dbl>      <dbl>      <dbl>
#> 1 abxrate_colonized   656449.            6.56e+5   3.39e+4    5.91e+5    7.24e+5
#> 2 insitu_colonized         0             0         0          0          0      
#> 3 inunit_param1            0.000533      1.48e-4   9.40e-4    3.85e-5    4.17e-3
#> 4 loglikelihood         -Inf          -Inf       NaN       -Inf       -Inf      
#> 5 outunit_acquisition      0.000437      3.19e-4   4.88e-4    3.03e-5    1.55e-3
#> 6 surv_test_col_pos        0             0         0          0          0