# Introduction

This vignette is an example of an elementary semi-Markov model using the rdecision package. It is based on the example given by Briggs et al1 (Exercise 2.5) which itself is based on a model described by Chancellor et al2. The model compares a combination therapy of Lamivudine/Zidovudine versus Zidovudine monotherapy in people with HIV infection.

# Creating the model

## Model variables

The variables used in the model are all numerical constants, and are defined as follows.

# transition counts
nAA <- 1251
nAB <- 350
nAC <- 116
nBB <- 731
nBC <- 512
nBD <- 15
nCC <- 1312
nCD <- 437
# Healthcare system costs
dmca <- 1701 # direct medical costs associated with state A
dmcb <- 1774 # direct medical costs associated with state B
dmcc <- 6948 # direct medical costs associated with state C
ccca <- 1055 # Community care costs associated with state A
cccb <- 1278 # Community care costs associated with state B
cccc <- 2059 # Community care costs associated with state C
# Drug costs
cAZT <- 2278 # zidovudine drug cost
cLam <- 2087 # lamivudine drug cost
# Treatment effect
RR <- 0.509
# Discount rates
cDR <- 6 # annual discount rate, costs (%)
oDR <- 0 # annual discount rate, benefits (%)

## Model structure

The model is constructed by forming a graph, with each state as a node and each transition as an edge. Nodes (of class MarkovState) and edges (class Transition) may have various properties whose values reflect the variables of the model (costs, rates etc.). Because the model is intended to evaluate survival, the utility of states A, B and C are set to 1 (by default) and state D to zero. Thus the incremental quality adjusted life years gained per cycle is equivalent to the survival function.

# create Markov states for monotherapy (zidovudine only)
sAm <- MarkovState$new("A", cost=dmca+ccca+cAZT) sBm <- MarkovState$new("B", cost=dmcb+cccb+cAZT)
sCm <- MarkovState$new("C", cost=dmcc+cccc+cAZT) sDm <- MarkovState$new("D", cost=0, utility=0)
# create transitions
tAAm <- Transition$new(sAm, sAm) tABm <- Transition$new(sAm, sBm)
tACm <- Transition$new(sAm, sCm) tADm <- Transition$new(sAm, sDm)
tBBm <- Transition$new(sBm, sBm) tBCm <- Transition$new(sBm, sCm)
tBDm <- Transition$new(sBm, sDm) tCCm <- Transition$new(sCm, sCm)
tCDm <- Transition$new(sCm, sDm) tDDm <- Transition$new(sDm, sDm)
# construct the model
m.mono <- SemiMarkovModel$new( V = list(sAm, sBm, sCm, sDm), E = list(tAAm, tABm, tACm, tADm, tBBm, tBCm, tBDm, tCCm, tCDm, tDDm), discount.cost = cDR/100, discount.utility = oDR/100 ) ## Transition rates and probabilities Briggs et al1 interpreted the observed transition counts in 1 year as transition probabilities by dividing counts by the total transitions observed from each state. With this assumption, the annual (per-cycle) transition probabilities are calculated as follows and applied to the model via the set_probabilities function. nA <- nAA + nAB + nAC + nAD nB <- nBB + nBC + nBD nC <- nCC + nCD Pt <- matrix( c(nAA/nA, nAB/nA, nAC/nA, nAD/nA, 0, nBB/nB, nBC/nB, nBD/nB, 0, 0, nCC/nC, nCD/nC, 0, 0, 0, 1), nrow=4, byrow=TRUE, dimnames=list(source=c("A","B","C","D"), target=c("A","B","C","D")) ) m.mono$set_probabilities(Pt)

More usually, fully observed transition counts are converted into transition rates (rather than probabilities), as described by Welton and Ades3. This is because counting events and measuring total time at risk includes individuals who make more than one transition during the observation time, and can lead to rates with values which exceed 1. In contrast, the difference between a census of the number of individuals in each state at the start of the interval and another at the end is directly related to the per-cycle probability. As Miller and Homan4, Welton and Ades3, Jones et al5 and others note, conversion between rates and probabilities for multi-state Markov models is non-trivial5 and care is needed when modellers calculate probabilities from published rates for use in SemiMarkoModels.

# Checking the model

## Diagram

A representation of the model in DOT format (Graphviz) can be created using the as_DOT function of SemiMarkovModel. The function returns a character vector which can be saved in a file (.gv extension) for visualization with the dot tool of Graphviz, or plotted directly in R via the DiagrammeR package. The Markov model for monotherapy is shown in Figure 1.

## Model states

The states in the model can be tabulated with the function tabulate_states. For the monotherapy model, the states are tabulated below. The cost of each state includes the annual cost of AZT (Zidovudine).

Name Cost
A 5034
B 5330
C 11285
D 0

## Per-cycle transition probabilities

The per-cycle transition probabilities, which are the cells of the Markov transition matrix, can be extracted from the model via the function transition_probabilities. For the monotherapy model, the transition matrix is shown below. This is consistent with the Table 1 of Chancellor et al2.

A B C D
A 0.7215 0.2018 0.0669 0.009804
B 0 0.5811 0.407 0.01192
C 0 0 0.7501 0.2499
D 0 0 0 1

# Running the model

Model function cycle applies one cycle of a Markov model to a defined starting population in each state. It returns a table with one row per state, and each row containing several columns, including the population at the end of the state and the cost of occupancy of states, normalized by the number of patients in the cohort, with discounting applied.

Multiple cycles are run by feeding the state populations at the end of one cycle into the next. Function cycles does this and returns a data frame with one row per cycle, and each row containing the state populations and the aggregated cost of occupancy for all states, with discounting applied. This is done below for the first 20 cycles of the model for monotherapy, without half cycle correction, with discount. In addition, the proportion of patients alive at each cycle (the Markov trace) is added to the table. The populations and discounted costs are consistent with Briggs et al, Table 2.31, and the QALY column is consistent with Table 2.4 (without half cycle correction). No discount was applied to the utilities.

# create starting populations
N <- 1000
populations <- c(A = N, B = 0, C = 0, D = 0)
m.mono$reset(populations) # run 20 cycles MT.mono <- m.mono$cycles(ncycles=20, hcc.pop=FALSE, hcc.cost=FALSE)
Years A B C D Cost QALY
0 1000 0 0 0 0 0
1 721 202 67 10 5153 0.99
2 520 263 181 36 5393 0.964
3 376 258 277 89 5368 0.911
4 271 226 338 165 5055 0.835
5 195 186 364 255 4541 0.745
6 141 147 361 350 3929 0.65
7 102 114 341 444 3301 0.556
8 73 87 309 531 2708 0.469
9 53 65 272 610 2179 0.39
10 38 49 234 679 1727 0.321
11 28 36 198 739 1350 0.261
12 20 26 165 789 1045 0.211
13 14 19 136 830 801 0.17
14 10 14 111 865 609 0.135
15 7 10 90 893 460 0.107
16 5 8 72 915 346 0.085
17 4 5 57 933 258 0.067
18 3 4 45 948 192 0.052
19 2 3 36 959 142 0.041
20 1 2 28 968 105 0.032

# Model results

## Monotherapy

The estimated life years is approximated by summing the proportions of patients left alive at each cycle (Briggs et al1, Exercise 2.5). This is an approximation because it ignores the population who remain alive after 21 years, and assumes all deaths occurred at the start of each cycle. For monotherapy the expected life gained is 7.991 years at a cost of 44663 GBP.

## Combination therapy

For combination therapy, a similar model was created, with adjusted costs and transition rates. Following Briggs et al1 the treatment effect was applied to the probabilities, and these were used as inputs to the model. More usually, treatment effects are applied to rates, rather than probabilities.

# annual probabilities modified by treatment effect
pAB <- RR*nAB/nA
pAC <- RR*nAC/nC
pBC <- RR*nBC/nB
pBD <- RR*nBD/nB
pCD <- RR*nCD/nC
# annual transition probability matrix
Ptc <- matrix(
0, (1-pBC-pBD),     pBC, pBD,
0,           0, (1-pCD), pCD,
0,           0,       0,   1),
nrow=4, byrow=TRUE,
dimnames=list(source=c("A","B","C","D"), target=c("A","B","C","D"))
)
# create Markov states for combination therapy
sAc <- MarkovState$new("A", cost=dmca+ccca+cAZT+cLam) sBc <- MarkovState$new("B", cost=dmcb+cccb+cAZT+cLam)
sCc <- MarkovState$new("C", cost=dmcc+cccc+cAZT+cLam) sDc <- MarkovState$new("D", cost=0, utility=0)
# create transitions
tAAc <- Transition$new(sAc, sAc) tABc <- Transition$new(sAc, sBc)
tACc <- Transition$new(sAc, sCc) tADc <- Transition$new(sAc, sDc)
tBBc <- Transition$new(sBc, sBc) tBCc <- Transition$new(sBc, sCc)
tBDc <- Transition$new(sBc, sDc) tCCc <- Transition$new(sCc, sCc)
tCDc <- Transition$new(sCc, sDc) tDDc <- Transition$new(sDc, sDc)
# construct the model
m.comb <- SemiMarkovModel$new( V = list(sAc, sBc, sCc, sDc), E = list(tAAc, tABc, tACc, tADc, tBBc, tBCc, tBDc, tCCc, tCDc, tDDc), discount.cost = cDR/100, discount.utility = oDR/100 ) # set the probabilities m.comb$set_probabilities(Ptc)

The per-cycle transition matrix for the combination therapy is as follows:

A B C D
A 0.8585 0.1027 0.03376 0.00499
B 0 0.7868 0.2072 0.006069
C 0 0 0.8728 0.1272
D 0 0 0 1

In this model, lamivudine is given for the first 2 years, with the treatment effect assumed to persist for the same period. The state populations and cycle numbers are retained by the model between calls to cycle or cycles and can be retrieved by calling get_populations. In this example, the combination therapy model is run for 2 cycles, then the population is used to continue with the monotherapy model for the remaining 8 years. The reset function is used to set the cycle number and elapsed time of the new run of the mono model.

# run combination therapy model for 2 years
populations <- c('A'=N, 'B'=0, 'C'=0, 'D'=0)
m.comb$reset(populations) # run 2 cycles MT.comb <- m.comb$cycles(2, hcc.pop=FALSE, hcc.cost=FALSE)
# feed populations into mono model & reset cycle counter and time
populations <- m.comb$get_populations() m.mono$reset(
populations,
icycle=as.integer(2),
elapsed=as.difftime(365.25*2, units="days")
)
# and run model for next 18 years
MT.comb <- rbind(
MT.comb, m.mono\$cycles(ncycles=18, hcc.pop=FALSE, hcc.cost=FALSE)
)

The Markov trace for combination therapy is as follows:

Years A B C D Cost QALY
0 1000 0 0 0 0 0
1 859 103 34 5 6912 0.995
2 737 169 80 14 6736 0.986
3 532 247 178 43 5039 0.957
4 384 251 270 96 4998 0.904
5 277 223 330 170 4713 0.83
6 200 186 357 258 4245 0.742
7 144 148 357 351 3684 0.649
8 104 115 337 443 3102 0.557
9 75 88 307 530 2551 0.47
10 54 66 271 609 2057 0.391
11 39 49 234 678 1633 0.322
12 28 37 198 737 1279 0.263
13 20 27 165 787 990 0.213
14 15 20 136 829 760 0.171
15 11 14 111 864 579 0.136
16 8 11 90 892 437 0.108
17 6 8 72 914 329 0.086
18 4 6 58 933 246 0.067
19 3 4 46 947 183 0.053
20 2 3 36 959 136 0.041

## Comparison of treatments

The ICER is calculated by running both models and calculating the incremental cost per life year gained. Over the 20 year time horizon, the expected life years gained for monotherapy was 7.991 years at a total cost per patient of 44,663 GBP. The expected life years gained with combination therapy for two years was 8.94 at a total cost per patient of 50,607 GBP. The incremental change in life years was 0.949 years at an incremental cost of 5,944 GBP, giving an ICER of 6264 GBP/QALY. This is consistent with the result obtained by Briggs et al1 (6276 GBP/QALY), within rounding error.

# References

1.
Briggs, A., Claxton, K. & Sculpher, M. Decision modelling for health economic evaluation. (Oxford University Press, 2006).
2.
Chancellor, J. V., Hill, A. M., Sabin, C. A., Simpson, K. N. & Youle, M. Modelling the cost effectiveness of Lamivudine/Zidovudine combination therapy in HIV infection. Pharmacoeconomics 12, 54–66 (1997).
3.
4.
Miller, D. K. & Homan, S. M. Determining Transition Probabilities: Confusion and Suggestions. Med Decis Making 14, 52–58 (1994).
5.
Jones, E., Epstein, D. & García-Mochón, L. A Procedure for Deriving Formulas to Convert Transition Rates to Probabilities for Multistate Markov Models. Med Decis Making 37, 779–789 (2017).