# Cost-Effectiveness Applications

## Introduction

This vignette outlines how to evaluate a cost-effectiveness model in a manner that allows for dynamic pricing and dynamic uptake. Let us consider in turn:

- the decision problem we are modeling through a cost-effectiveness analysis
- dynamic patient uptake
- dynamic pricing

## Methods and Assumptions

We wish to evaluate the cost-effectiveness, measured as incremental cost per QALY, of a new intervention compared to the standard of care (SoC). The model is a partitioned survival analysis typical in oncology with three health states: progression-free (PF), progressive disease (PD) and death, with additional assumptions as follows.

### General assumptions

- The time horizon is 20 years, a timestep is 1 week, discounting is at 3% per year, with an effective date of calculation is 2025-06-01.
- For SoC, progression-free survival (PFS) is modeled as an exponential distribution with mean of 50 weeks; overall survival (OS) in weeks is modelled according to a lognormal distribution with mean and standard deviation on the log scale as 4 and 1 respectively.
- The effect of the new intervention relative to the SoC is a hazard ratio of 0.5 on PFS and 0.6 on OS.
- Health utility is 0.8 in the PF state and 0.6 in the PD state.
- Drug acquisition costs are $400 and $1500 per week for SoC and the new intervention respectively, incurred throughout a patient’s time in the PF state (treat to progression).
- Drug administration costs are $50 and $75 per week on treatment for SoC and the new intervention respectively.
- Adverse events occur in 0.08% and 0.10% of SoC and new intervention patients per week respectively, while on treatment, at an average cost of $2000 per event.
- Disease management costs are $80 per week in PF and $20 per week in PD.
- Patients receive the treatment of interest (new or SoC) until progression. Subsequent treatment costs are $1200 and $300 per week per patient while in the PD state for SoC and new intervention patients respectively.

### Dynamic pricing

Let us suppose the following assumptions concerning pricing:

- Costs are assumed to increases in line with general inflation (2.5% per year), except for effects on drug acquisition costs due to LoEs.
- The date of calculation is 2025-09-01.
- The LoE for the SoC is assumed to occur first, at 2028-01-01, after which there is anticipated to be a 70% reduction in prices over one year.
- The new intervention has an LoE occuring three years later, at 2031-01-01, after which there would be a 50% reduction in prices over one year.

### Dynamic uptake

Let us suppose the following assumptions concerning patient uptake. The aim here is to estimate the incidence of patients for whom the decision problem applies, i.e. the patients who would receive the new intervention, were it made available. This will be a function of, but not the same as, disease incidence or prevalence. We assume:

- Only newly incident patients with the cancer being modeled would be eligible for the new treatment. Existing/prevalent patients with the condition would not be eligible.
- The disease incidence is 1 patient per week.
- Among these patients, were the new intervention to be made available, uptake of the new intervention would be expected to rise linearly from 0% to 100% after 2 years.

In this way, uptake would be gradually increasing with time, accounting for disease epidemiology and the share of patients who receive the new intervention.

## Implementation

### Set-up

First we load the packages necessary for this vignette.

    library(dplyr)
    library(ggplot2)
    library(lubridate)
    library(flexsurv)
    library(heemod)
    library(tidyr)
    library(dynamicpv)

We code the time constants, time horizon, discount rates and inflation rates first.

    # Time constants
    days_in_year <- 365.25
    days_in_week <- 7
    cycle_years <- days_in_week / days_in_year # Duration of a one week cycle in years

    # Time horizon (years) and number of cycles
    thoz <- 20
    Ncycles <- ceiling(thoz/cycle_years)

    # Real discounting
    disc_year <- 0.03 # Per year
    disc_cycle <- (1+disc_year)^(cycle_years) - 1 # Per cycle

    # Price inflation
    infl_year <- 0.025 # Per year
    infl_cycle <- (1+infl_year)^(cycle_years) - 1 # Per cycle

    # Nominal discounting
    nomdisc_year <- (1+disc_year)*(1+infl_year) - 1
    nomdisc_cycle <- (1+nomdisc_year)^(cycle_years) - 1 # Per cycle

### Cost-effectiveness model

The cost-effectiveness model may then be coded in heemod as follows.

    # State names
    state_names = c(
      progression_free = "PF",
      progression = "PD",
      death = "Death"
      )

    # PFS distribution for SoC with Exp() distribution and mean of 50 weeks
    surv_pfs_soc <- heemod::define_surv_dist(
      distribution = "exp",
      rate = 1/50
    )

    # OS distribution for SoC with Lognorm() distribution, meanlog = 4.5, sdlog = 1
    # This implies a mean of exp(4 + 0.5 * 1^2) = exp(4.5) = 90 weeks
    surv_os_soc <- heemod::define_surv_dist(
      distribution = "lnorm",
      meanlog = 4,
      sdlog = 1
    )

    # PFS and OS distributions for new
    surv_pfs_new <- heemod::apply_hr(surv_pfs_soc, hr=0.5)
    surv_os_new <- heemod::apply_hr(surv_os_soc, hr=0.6)

    # Define partitioned survival model, soc
    psm_soc <- heemod::define_part_surv(
      pfs = surv_pfs_soc,
      os = surv_os_soc,
      terminal_state = FALSE,
      state_names = state_names
      )

    # Define partitioned survival model, soc
    psm_new <- heemod::define_part_surv(
      pfs = surv_pfs_new,
      os = surv_os_new,
      terminal_state = FALSE,
      state_names = state_names
      )

    # Parameters
    params <- heemod::define_parameters(
      # Discount rate
      disc = disc_cycle,
      # Disease management costs
      cman_pf = 80,
      cman_pd = 20,
      # Drug acquisition costs - the SoC regime only uses SoC drug, the New regime only uses New drug
      cdaq_soc = dispatch_strategy(
        soc = 400,
        new = 0
      ),
      cdaq_new = dispatch_strategy(
        soc = 0,
        new = 1500
      ),
      # Drug administration costs
      cadmin = dispatch_strategy(
        soc = 50,
        new = 75
      ),
      # Adverse event risks
      risk_ae = dispatch_strategy(
        soc = 0.08,
        new = 0.1
      ),
      # Adverse event average costs
      uc_ae = 2000,
      # Subsequent treatments
      csubs = dispatch_strategy(
        soc = 1200,
        new = 300
      ),
      # Health state utilities
      u_pf = 0.8,
      u_pd = 0.6,
    )

    # Define PF states
    state_PF <- heemod::define_state(
      # Costs for the state
      cost_daq_soc = discount(cdaq_soc, disc_cycle),
      cost_daq_new = discount(cdaq_new, disc_cycle),
      cost_dadmin = discount(cadmin, disc_cycle),
      cost_dman = discount(cman_pf, disc_cycle),
      cost_ae = risk_ae * uc_ae,
      cost_subs = 0,
      cost_total = cost_daq_soc + cost_daq_new + cost_dadmin + cost_dman + cost_ae + cost_subs,
      # Health utility, QALYs and life years
      pf_year = discount(cycle_years, disc_cycle),
      life_year = discount(cycle_years, disc_cycle),
      qaly = discount(cycle_years * u_pf, disc_cycle)
      )

    # Define PD states
    state_PD <- heemod::define_state(
      # Costs for the state
      cost_daq_soc = 0,
      cost_daq_new = 0,
      cost_dadmin = 0,
      cost_dman = discount(cman_pd, disc_cycle),
      cost_ae = 0,
      cost_subs = discount(csubs, disc_cycle),
      cost_total = cost_daq_soc + cost_daq_new + cost_dadmin + cost_dman + cost_ae + cost_subs,
      # Health utility, QALYs and life years
      pf_year = 0,
      life_year = heemod::discount(cycle_years, disc_cycle),
      qaly = heemod::discount(cycle_years * u_pd, disc_cycle)
      )

    # Define Death state
    state_Death <- heemod::define_state(
      # Costs are zero
      cost_daq_soc = 0,
      cost_daq_new = 0,
      cost_dadmin = 0,
      cost_dman = 0,
      cost_ae = 0,
      cost_subs = 0,
      cost_total = cost_daq_soc + cost_daq_new + cost_dadmin + cost_dman + cost_ae + cost_subs,
      # Health outcomes are zero
      pf_year = 0,
      life_year = 0,
      qaly = 0,
    )

    # Define strategy for SoC
    strat_soc <- heemod::define_strategy(
        transition = psm_soc,
        "PF" = state_PF,
        "PD" = state_PD,
        "Death" = state_Death
      )

    # Define strategy for new
    strat_new <- heemod::define_strategy(
      transition = psm_new,
      "PF" = state_PF,
      "PD" = state_PD,
      "Death" = state_Death
    )

    # Create heemod model
    heemodel <- heemod::run_model(
      soc = strat_soc,
      new = strat_new,
      parameters = params,
      cycles = Ncycles,
      cost = cost_total,
      effect = qaly,
      init = c(1, 0, 0),
      method = "life-table"
    )

### Dynamic pricing

The dynamic pricing assumptions can be codified as follows.

    # Dates
    # Date of calculation = 1 September 2025
    doc <- lubridate::ymd("20250901")
    # Date of LOE for SoC = 1 January 2028
    loe_soc_start <- lubridate::ymd("20280101")
    # Maturation of SoC prices by LOE + 1 year, i.e. = 1 January 2029
    loe_soc_end <- lubridate::ymd("20290101")
    # Date of LOE for new treatment = 1 January 2031
    loe_new_start <- lubridate::ymd("20310101")
    # Maturation of new treatment prices by LOE + 1 year, i.e. = 1 January 2032
    loe_new_end <- lubridate::ymd("20320101")

    # Effect of LoEs on prices once mature
    loe_effect_soc <- 0.7
    loe_effect_new <- 0.5

    # Calculation of weeks since DoC for LoEs and price maturities
    wk_start_soc <- floor((loe_soc_start-doc) / lubridate::dweeks(1))
    wk_end_soc <- floor((loe_soc_end-doc) / lubridate::dweeks(1))
    wk_start_new <- floor((loe_new_start-doc) / lubridate::dweeks(1))
    wk_end_new <- floor((loe_new_end-doc) / lubridate::dweeks(1))

    # Price maturity times
    wk_mature_soc <- wk_end_soc - wk_start_soc
    wk_mature_new <- wk_end_new - wk_start_new

    # Create a tibble of price indices of length 2T, then pull out columns as needed
    # We only need of length T for now, but need of length 2T for future calculations later
    pricetib <- tibble(
      model_time = 1:(2*Ncycles),
      model_year = model_time * cycle_years,
      static = 1,
      geninfl = (1 + infl_cycle)^(model_time - 1),
      loef_soc = pmin(pmax(model_time - wk_start_soc, 0), wk_mature_soc) / wk_mature_soc,
      loef_new = pmin(pmax(model_time - wk_start_new, 0), wk_mature_new) / wk_mature_new,
      dyn_soc = geninfl * (1 - loe_effect_soc * loef_soc),
      dyn_new = geninfl * (1 - loe_effect_new * loef_new)
    )

    # Price indices required for calculations
    prices_oth <- pricetib$geninfl
    prices_static <- pricetib$static
    prices_dyn_soc <- pricetib$dyn_soc
    prices_dyn_new <- pricetib$dyn_new

### Dynamic uptake

The dynamic uptake assumptions can be codified as follows.

    # Time for uptake to occur
    uptake_years <- 2

    # Uptake vector for non-dynamic uptake
    uptake_single <- c(1, rep(0, Ncycles-1))

    # Uptake vector for dynamic uptake
    uptake_weeks <- round(uptake_years / cycle_years)
    share_multi <- c((1:uptake_weeks)/uptake_weeks, rep(1, Ncycles-uptake_weeks))
    uptake_multi <- rep(1, Ncycles) * share_multi

## Results

### Without dynamic pricing or dynamic uptake

The conventional cost-effectiveness model is static.

    heemodel
    #> 2 strategies run for 1044 cycles.
    #>
    #> Initial state counts:
    #>
    #> PF = 1
    #> PD = 0
    #> Death = 0
    #>
    #> Counting method: 'life-table'.
    #>
    #> Values:
    #>
    #>     cost_daq_soc cost_daq_new cost_dadmin cost_dman   cost_ae cost_subs
    #> soc     19455.25          0.0    2431.906  4601.624  8000.267  42634.43
    #> new         0.00     141997.2    7099.860  8666.132 19999.582  16394.22
    #>     cost_total   pf_year life_year     qaly
    #> soc   77123.47 0.9321475  1.613053 1.154261
    #> new  194156.99 1.8142467  2.861562 2.079786
    #>
    #> Efficiency frontier:
    #>
    #> soc -> new
    #>
    #> Differences:
    #>
    #>     Cost Diff. Effect Diff.   ICER Ref.
    #> new   117033.5    0.9255249 126451  soc

Let’s examine each payoff more closely. The next steps extract a payoff vector from the model object. The model/object contains several payoffs accumulated in each timestep, calculated as at time zero:

- drug acquisition cost of the standard of care (cost_daq_soc),
- drug acquisition cost of the new intervention (cost_daq_new),
- cost of drug administration (cost_dadmin),
- cost of disease management (cost_dman),
- cost of treating adverse events (cost_ae),
- cost of subsequent lines of treatment (cost_subs),
- total cost (cost_total),
- progression-free life years (pf_year),
- life years (life_year), and
- QALYs (qaly).

The `dynamicpv::get_dynfields()` function extracts these parameters from the heemod model object, and calculates ‘rolled-up’ values as at the start of each timestep rather than discounted to time zero. The rolled-up values are what `dynamicpv::dynpv()` requires.

    # Pull out the payoffs of interest from oncpsm
    payoffs <- get_dynfields(
        heemodel = heemodel,
        payoffs = c("cost_daq_new", "cost_daq_soc", "cost_total", "qaly", "life_year"),
        discount = "disc"
        ) |>
        mutate(
          model_years = model_time * cycle_years,
          # Derive costs other than drug acquisition, as at time zero
          cost_nondaq = cost_total - cost_daq_new - cost_daq_soc,
          # ... and at the start of each timestep
          cost_nondaq_rup = cost_total_rup - cost_daq_new_rup - cost_daq_soc_rup
        )

    # Create and view dataset for SoC
    hemout_soc <- payoffs |> filter(int=="soc")
    head(hemout_soc)
    #> # A tibble: 6 × 16
    #>   model_time cost_daq_new cost_daq_soc cost_total   qaly life_year int      vt
    #>        <int>        <dbl>        <dbl>      <dbl>  <dbl>     <dbl> <chr> <dbl>
    #> 1          1            0         396.       695. 0.0153    0.0192 soc   1
    #> 2          2            0         388.       705. 0.0152    0.0191 soc   0.999
    #> 3          3            0         380.       714. 0.0151    0.0191 soc   0.999
    #> 4          4            0         372.       721. 0.0150    0.0191 soc   0.998
    #> 5          5            0         365.       726. 0.0149    0.0190 soc   0.998
    #> 6          6            0         357.       730. 0.0148    0.0189 soc   0.997
    #> # ℹ 8 more variables: cost_daq_new_rup <dbl>, cost_daq_soc_rup <dbl>,
    #> #   cost_total_rup <dbl>, qaly_rup <dbl>, life_year_rup <dbl>,
    #> #   model_years <dbl>, cost_nondaq <dbl>, cost_nondaq_rup <dbl>

    # Create and view dataset for new intervention
    hemout_new <- payoffs |> filter(int=="new")
    head(hemout_new)
    #> # A tibble: 6 × 16
    #>   model_time cost_daq_new cost_daq_soc cost_total   qaly life_year int      vt
    #>        <int>        <dbl>        <dbl>      <dbl>  <dbl>     <dbl> <chr> <dbl>
    #> 1          1        1493.            0      1847. 0.0153    0.0192 new   1
    #> 2          2        1477.            0      1831. 0.0153    0.0192 new   0.999
    #> 3          3        1461.            0      1815. 0.0152    0.0191 new   0.999
    #> 4          4        1446.            0      1799. 0.0152    0.0191 new   0.998
    #> 5          5        1431.            0      1783. 0.0151    0.0190 new   0.998
    #> 6          6        1416.            0      1766. 0.0150    0.0190 new   0.997
    #> # ℹ 8 more variables: cost_daq_new_rup <dbl>, cost_daq_soc_rup <dbl>,
    #> #   cost_total_rup <dbl>, qaly_rup <dbl>, life_year_rup <dbl>,
    #> #   model_years <dbl>, cost_nondaq <dbl>, cost_nondaq_rup <dbl>

### Scenario 1: No dynamic uptake or pricing

With non-dynamic uptake, we use `uptakes`=`uptake_single`=1. Drug acquisition costs are constant in real terms (`prices=prices_static`) and are discounted at the risk-free real rate (`discrate=disc_cycle`). Other costs rise in line with general price inflation (`prices=prices_oth`) and are discounted at nominal discount rates (`discrate=nomdisc_cycle`). QALYs are not affected by price inflation (`prices=prices_static`) and are discounted at the risk-free real rate (`discrate=disc_cycle`).

    # SOC, costs other than drug acquisition
    s1_soc_othcost <- dynamicpv::dynpv(
        uptakes = uptake_single,
        payoffs = hemout_soc$cost_nondaq_rup,
        prices = prices_oth,
        discrate = nomdisc_cycle
        )

    # SOC, drug acquisition costs
    s1_soc_daqcost <- dynamicpv::dynpv(
        uptakes = uptake_single,
        payoffs = hemout_soc$cost_daq_soc_rup,
        prices = prices_static,
        discrate = disc_cycle
        )

    # SOC, total costs
    s1_soc_cost <- s1_soc_daqcost + s1_soc_othcost

    # SOC, QALYs
    s1_soc_qaly <- dynamicpv::dynpv(
        uptakes = uptake_single,
        payoffs = hemout_soc$qaly_rup,
        prices = prices_static,
        discrate = disc_cycle
        )

    # New intervention, costs other than drug acquisition
    s1_new_othcost <- dynamicpv::dynpv(
        uptakes = uptake_single,
        payoffs = hemout_new$cost_nondaq_rup,
        prices = prices_oth,
        discrate = nomdisc_cycle
        )

    # New intervention, drug acquisition costs
    s1_new_daqcost <- dynamicpv::dynpv(
        uptakes = uptake_single,
        payoffs = hemout_new$cost_daq_new_rup,
        prices = prices_static,
        discrate = disc_cycle
        )

    # New intervention, total costs
    s1_new_cost <- s1_new_daqcost + s1_new_othcost

    # New intervention, QALYs
    s1_new_qaly <- dynamicpv::dynpv(
        uptakes = uptake_single,
        payoffs = hemout_new$qaly_rup,
        prices = prices_static,
        discrate = disc_cycle
        )

    # Incremental cost
    s1_icost <- s1_new_cost - s1_soc_cost

    summary(s1_icost)
    #> Summary of Dynamic Pricing and Uptake
    #>      Number of cohorts:             1044
    #>      Number of times:               1
    #>      Total uptake:                  0
    #>      Total present value:           117033.5
    #>      Mean present value:            Inf

    # Incremental QALY
    s1_iqaly <- s1_new_qaly - s1_soc_qaly

    summary(s1_iqaly)
    #> Summary of Dynamic Pricing and Uptake
    #>      Number of cohorts:             1044
    #>      Number of times:               1
    #>      Total uptake:                  0
    #>      Total present value:           0.9255249
    #>      Mean present value:            Inf

    # ICER
    s1_icer <- total(s1_icost) / total(s1_iqaly)
    s1_icer
    #> [1] 126451

These results show that the new intervention is associated with $117,034 incremental costs relative to the standard of care (including $141,997 of drug acquisition costs for the new intervention) and 0.926 incremental QALYs. The cumulative ICER (incremental cost per QALY) is $126,451 per QALY at the 20 year time horizon.

### Scenario 2: Dynamic pricing, no dynamic uptake

The costs of drug acquisition in each arm differ in Scenario 2 through applying the relevant dynamic price index (`prices_dyn_soc` and `prices_dyn_new`), with discounting at nominal rates (`discrate = nomdisc_cycle`). Otherwise costs and QALYs are unchanged from Scenario 1.

    # SOC, costs other than drug acquisition are unchanged
    s2_soc_othcost <- s1_soc_othcost

    # SoC, drug acquisition costs
    s2_soc_daqcost <- dynamicpv::dynpv(
        uptakes = uptake_single,
        payoffs = hemout_soc$cost_daq_soc_rup,
        prices = prices_dyn_soc,
        discrate = nomdisc_cycle
        )

    # SoC, total costs
    s2_soc_cost <- s2_soc_daqcost + s2_soc_othcost

    # SoC, QALYs are unchanged
    s2_soc_qaly <- s1_soc_qaly

    # New intervention, costs other than drug acquisition are unchanged
    s2_new_othcost <- s1_new_othcost

    # New intervention, drug acquisition costs
    s2_new_daqcost <- dynamicpv::dynpv(
        uptakes = uptake_single,
        payoffs = hemout_new$cost_daq_new_rup,
        prices = prices_dyn_new,
        discrate = nomdisc_cycle
        )

    # New intervention, total costs
    s2_new_cost <- s2_new_daqcost + s2_new_othcost

    # New intervention, QALYs are unchanged
    s2_new_qaly <- s1_new_qaly

    # Incremental cost
    s2_icost <- s2_new_cost - s2_soc_cost

    summary(s2_icost)
    #> Summary of Dynamic Pricing and Uptake
    #>      Number of cohorts:             1044
    #>      Number of times:               1
    #>      Total uptake:                  0
    #>      Total present value:           114820
    #>      Mean present value:            Inf

    # Incremental QALY
    s2_iqaly <- s2_new_qaly - s2_soc_qaly

    summary(s2_iqaly)
    #> Summary of Dynamic Pricing and Uptake
    #>      Number of cohorts:             1044
    #>      Number of times:               1
    #>      Total uptake:                  0
    #>      Total present value:           0.9255249
    #>      Mean present value:            Inf

    # ICER
    s2_icer <- total(s2_icost) / total(s2_iqaly)
    s2_icer
    #> [1] 124059.3

Under scenario 2, the new intervention has an incremental cost-effectiveness of $124,059 per QALY (incremental costs of $114,820, incremental QALYs of 0.926).

### Scenario 3: Dynamic uptake, not dynamic pricing

The calculation for Scenario 3 is the same as for Scenario 1 except for dynamic uptake, which is handled by setting `uptakes = uptake_multi`.

    # SOC, costs other than drug acquisition
    s3_soc_othcost <- dynamicpv::dynpv(
        uptakes = uptake_multi,
        payoffs = hemout_soc$cost_nondaq_rup,
        prices = prices_oth,
        discrate = nomdisc_cycle
        )

    # SoC, drug acquisition costs
    s3_soc_daqcost <- dynamicpv::dynpv(
        uptakes = uptake_multi,
        payoffs = hemout_soc$cost_daq_soc_rup,
        prices = prices_static,
        discrate = disc_cycle
        )

    # SoC, total costs
    s3_soc_cost <- s3_soc_daqcost + s3_soc_othcost

    # SoC, QALYs
    s3_soc_qaly <- dynamicpv::dynpv(
        uptakes = uptake_multi,
        payoffs = hemout_soc$qaly_rup,
        prices = prices_static,
        discrate = disc_cycle
        )

    # New intervention, costs other than drug acquisition
    s3_new_othcost <- dynamicpv::dynpv(
        uptakes = uptake_multi,
        payoffs = hemout_new$cost_nondaq_rup,
        prices = prices_oth,
        discrate = nomdisc_cycle
        )

    # New intervention, drug acquisition costs
    s3_new_daqcost <- dynamicpv::dynpv(
        uptakes = uptake_multi,
        payoffs = hemout_new$cost_daq_new_rup,
        prices = prices_static,
        discrate = disc_cycle
        )

    # New intervention, total costs
    s3_new_cost <- s3_new_daqcost + s3_new_othcost

    # New intervention, QALYs
    s3_new_qaly <- dynamicpv::dynpv(
        uptakes = uptake_multi,
        payoffs = hemout_new$qaly_rup,
        prices = prices_static,
        discrate = disc_cycle
        )

    # Incremental costs
    s3_icost <- s3_new_cost - s3_soc_cost

    summary(s3_icost)
    #> Summary of Dynamic Pricing and Uptake
    #>      Number of cohorts:             1044
    #>      Number of times:               1
    #>      Total uptake:                  0
    #>      Total present value:           78010854
    #>      Mean present value:            Inf

    # Incremental QALYs
    s3_iqaly <- s3_new_qaly - s3_soc_qaly

    summary(s3_iqaly)
    #> Summary of Dynamic Pricing and Uptake
    #>      Number of cohorts:             1044
    #>      Number of times:               1
    #>      Total uptake:                  0
    #>      Total present value:           520.8296
    #>      Mean present value:            Inf

    # ICER
    s3_icer <- total(s3_icost) / total(s3_iqaly)
    s3_icer
    #> [1] 149781.9

Under scenario 3, the new intervention has an incremental cost-effectiveness of $149,782 per QALY (incremental costs of $78,010,854, incremental QALYs of 521 from a cohort comprising 992 patients).

### Scenario 4: Dynamic pricing and uptake

The costs of drug acquisition in each arm differ in Scenario 4 from Scenario 3 through applying the relevant dynamic price index (`prices_dyn_soc` and `prices_dyn_new`). Otherwise costs and QALYs are unchanged from Scenario 3.

    # SOC, costs other than drug acquisition are unchanged
    s4_soc_othcost <- s3_soc_othcost

    # SoC, drug acquisition costs
    s4_soc_daqcost <- dynamicpv::dynpv(
        uptakes = uptake_multi,
        payoffs = hemout_soc$cost_daq_soc_rup,
        prices = prices_dyn_soc,
        discrate = nomdisc_cycle
        )

    # SoC, total costs
    s4_soc_cost <- s4_soc_daqcost + s4_soc_othcost

    # SoC, QALYs are unchanged
    s4_soc_qaly <- s3_soc_qaly

    # New intervention, costs other than drug acquisition are unchanged
    s4_new_othcost <- s3_new_othcost

    # New intervention, drug acquisition costs
    s4_new_daqcost <- dynamicpv::dynpv(
        uptakes = uptake_multi,
        payoffs = hemout_new$cost_daq_new_rup,
        prices = prices_dyn_new,
        discrate = nomdisc_cycle
        )

    # New intervention, total costs
    s4_new_cost <- s4_new_daqcost + s4_new_othcost

    # New intervention, QALYs
    s4_new_qaly <- s3_new_qaly

    # Incremental costs
    s4_icost <- s4_new_cost - s4_soc_cost

    summary(s4_icost)
    #> Summary of Dynamic Pricing and Uptake
    #>      Number of cohorts:             1044
    #>      Number of times:               1
    #>      Total uptake:                  0
    #>      Total present value:           49255066
    #>      Mean present value:            Inf

    # Incremental QALYs
    s4_iqaly <- s4_new_qaly - s4_soc_qaly

    summary(s4_iqaly)
    #> Summary of Dynamic Pricing and Uptake
    #>      Number of cohorts:             1044
    #>      Number of times:               1
    #>      Total uptake:                  0
    #>      Total present value:           520.8296
    #>      Mean present value:            Inf

    # ICER
    s4_icer <- total(s4_icost) / total(s4_iqaly)
    s4_icer
    #> [1] 94570.4

Under scenario 4, the new intervention has an incremental cost-effectiveness of $94,570 per QALY (incremental costs of $49,255,066, incremental QALYs of 521 from a cohort comprising 992 patients).

### Summary of results

Total costs and QALYs for each scenario are summarized in the table below. The results above are skewed by the fact that some scenarios represent more than one patient cohort. Presenting results per patient, allows easier comparison between the scenarios.

Cost-effectiveness model results by scenario
|  | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4
--- | --- | --- | --- | --- | ---
Dynamic pricing? |  | No | Yes | No | Yes
Dynamic uptake? |  | No | No | Yes | Yes
Cohort size |  | 1 | 1 | 992 | 992
Total costs (cohort) |  |  |  |  |
| New intervention | 194,157 | 191,248 | 129,401,743 | 91,698,157
| Standard of care | 77,123 | 76,428 | 51,390,888 | 42,443,090
| Incremental | 117,034 | 114,820 | 78,010,854 | 49,255,066
Total costs (per patient) |  |  |  |  |
| New intervention | 194,157 | 191,248 | 130,380 | 92,391
| Standard of care | 77,123 | 76,428 | 51,779 | 42,764
| Incremental | 117,034 | 114,820 | 78,600 | 49,627
Total QALYs (cohort) |  |  |  |  |
| New intervention | 2.08 | 2.08 | 1,309 | 1,309
| Standard of care | 1.154 | 1.154 | 788 | 788
| Incremental | 0.926 | 0.926 | 521 | 521
Total QALYs (per patient) |  |  |  |  |
| New intervention | 2.08 | 2.08 | 1.319 | 1.319
| Standard of care | 1.154 | 1.154 | 0.794 | 0.794
| Incremental | 0.926 | 0.926 | 0.525 | 0.525
ICER |  | 126,451 | 124,059 | 149,782 | 94,570

## Future single cohort ICER

The table above presents the cost-effectiveness results as of the date of calculation, 2025-09-01. However, it is interesting to explore how the ICER will change over time, given the expected evolution of prices. We use `dynamicpv::futurepv()` to calculate present values at future times. This function is a wrapper for `dynamicpv::dynpv()`. This is the single cohort ICER (no dynamic uptake) but with dynamic pricing, so corresponds with Scenario 2 from earlier.

First, we calculate the costs at each time of interest. Then we calculate the ICER, given the incremental QALYs we have already observed - and which are immune from pricing effects.

    # Times at which to plot ICER
    gtimes <- round((0:(2*thoz))/cycle_years/2)
    # SOC drug acquisition costs
    gc_soc_daq <- gtimes |>
        purrr::map_vec(\(l) mean(futurepv(
            tzero = l,
            payoffs = hemout_soc$cost_daq_soc_rup,
            prices = prices_dyn_soc,
            discrate = nomdisc_cycle
        ))
    )
    # SOC other costs
    gc_soc_oth <- gtimes |>
        purrr::map_vec(\(l) mean(futurepv(
            tzero = l,
            payoffs = hemout_soc$cost_nondaq_rup,
            prices = prices_oth,
            discrate = nomdisc_cycle
        ))
    )
    # New drug acquisition costs
    gc_new_daq <- gtimes |>
        purrr::map_vec(\(l) mean(futurepv(
            tzero = l,
            payoffs = hemout_new$cost_daq_new_rup,
            prices = prices_dyn_new,
            discrate = nomdisc_cycle
        ))
    )
    # New other costs
    gc_new_oth <- gtimes |>
        purrr::map_vec(\(l) mean(futurepv(
            tzero = l,
            payoffs = hemout_new$cost_nondaq_rup,
            prices = prices_oth,
            discrate = nomdisc_cycle
        ))
    )
    # Combine in a tibble
    ds <- tibble(
        # Time in weeks and years
        time_weeks = gtimes,
        time_years = time_weeks * cycle_years,
        # Evaluation date
        evaldate = doc + time_weeks * 7,
        # Price/inflation index
        pinfl = prices_oth[gtimes + 1],
        # Total costs for each intervention
        totcost_new = gc_new_daq + gc_new_oth,
        totcost_soc = gc_soc_daq + gc_soc_oth,
        # Scenario 1/2 QALYs
        qaly_soc = total(s1_soc_qaly),
        qaly_new = total(s1_new_qaly)
      ) |>
      mutate(
        # Incremental cost and QALYs
        icost = totcost_new-totcost_soc,
        iqaly = qaly_new-qaly_soc,
        # Nominal ICER, and real (inflation adjusted) ICER
        Nominal = icost/iqaly,
        Real = Nominal / pinfl
      ) |>
      # Pivot to long so can be used in a graphic
      pivot_longer(
        cols = c("Nominal", "Real"),
        names_to = "Type",
        values_to = "ICER"
      )

We should check at this point that the ICER we are starting at ($124,059) matches the scenario 2 ICER from earlier ($124,059).

The following plot shows the nominal ICER calculated at different times, given the pricing, LoE and other assumptions. The horizontal dotted line confirms that the initial ICER matches the value from Scenario 2. The vertical dashed lines mark the timings of the LoEs of first the standard of care, and then the new treatment.

    # Plot real and nominal present value over time
    ggplot(ds,
      aes(x = evaldate, y = ICER, color=Type)) +
      geom_line() +
      labs(x = "Evaluation date") +
      geom_hline(yintercept = ds$ICER[1], linetype='dotted') +
      geom_vline(xintercept = loe_new_start, linetype='dashed') +
      geom_vline(xintercept = loe_soc_start, linetype='dashed') +
      scale_y_continuous(
        labels = scales::comma,
        limits=c(0, 150000)
        )

## Discussion

The findings for this example model are as follows:

- The effect of assuming dynamic pricing rather than flat pricing over time is shown in the difference between results of scenarios 1 and 2. Scenario 1 assumed no increases in drug acquisition costs, but inflationary increases in other costs. Scenario 2 additionally assumed drug prices of the new intervention and SoC comparator would be eroded due to losses of exclusivity. This has a modest impact on costs per patient of the new intervention (reducing from $194,157 to $191,248). Accordingly the ICER reduces from $126,451 to $124,059 per QALY.

- The effect of modeling uptake dynamically rather than just the current cohort of patients is shown in the difference between results of scenarios 1 and 3. Dynamic modeling of uptake has the effect of weighting the costs and QALYs per patient in scenario 3 towards a later cohort of patients, where the accumulation of costs and QALYs over the time remaining in the payer’s 20 year time horizon will be less. The incremental costs per patient reduced from $194,157 in scenario 1 to $130,380 in scenario 3, whereas the incremental QALYs reduced from 2.08 to 1.319 per patient. Overall, the ICER increased from $126,451 to $149,782 per QALY.

- The effects of dynamic pricing and dynamic uptake are extremely synergistic. Scenario 4 illustrates the effects of both dynamic pricing and cohort modeling. The results per patient are identical to scenario 3, except that the cost per patient of the new intervention reduces from $130,380 to $92,391, due to the impact of dynamic pricing. The ICER in this case is reduced considerably to $94,570 per QALY.

- As shown in the Figure, the ICER increases a little at first, as we approach the LoE of the standard of care comparator. After all, a cheaper comparator will increase the incremental cost of the new treatment. However, this is soon dwarfed by the effect of the LoE of the new treatment, which occurs later. This reduces the ICER significantly as the LoE approaches. Thereafter, the ICER is constant in real terms (but increasing with price inflation in nominal terms).