The most simple Markov models in health economic evaluation are models were transition probabilities between states do not change with time. Those are called homogenous or time-homogenous Markov models.

If you are not familiar with heemod, first consult the introduction vignettevignette("introduction", package = "heemod").

Model description

In this example we will model the cost effectiveness of lamivudine/zidovudine combination therapy in HIV infection (Chancellor, 1997) further described in Decision Modelling for Health Economic Evaluation, page 32.

This model aims to compare costs and utilities of two treatment strategies, monotherapy and combined therapy.

Four states are described, from best to worst healtwise:

Transition probabilities

Transition probabilities for the monotherapy study group are rather simple to implement:

mat_mono <-
  define_matrix(
    .721, .202, .067, .010,
    .000, .581, .407, .012,
    .000, .000, .750, .250,
    .000, .000, .000, 1.00
  )
mat_mono
## An unevaluated matrix, 4 states.
## 
##   A     B     C     D    
## A 0.721 0.202 0.067 0.01 
## B 0     0.581 0.407 0.012
## C 0     0     0.75  0.25 
## D 0     0     0     1

The combined therapy group has its transition probabilities multiplied by rr, the relative risk of event for the population treated by combined therapy. Since \(rr < 1\), the combined therapy group has less chance to transition to worst health states.

The probabilities to stay in the same state are equal to \(1 - \sum p_{trans}\) where \(p_{trans}\) are the probabilities to change to another state (because all transition probabilities from a given state must sum to 1).

rr <- .509

mat_comb <-
  define_matrix(
    1-(.202*rr+.067*rr+.010*rr), .202*rr,   .067*rr, .010*rr,
    .000, 1-(.407*rr+.012*rr),   .407*rr,   .012*rr,
    .000, .000,                  1-.250*rr, .250*rr,
    .000, .000,                  .000,      1.00
  )
mat_comb
## An unevaluated matrix, 4 states.
## 
##   A                                         B                            
## A 1 - (0.202 * rr + 0.067 * rr + 0.01 * rr) 0.202 * rr                   
## B 0                                         1 - (0.407 * rr + 0.012 * rr)
## C 0                                         0                            
## D 0                                         0                            
##   C             D         
## A 0.067 * rr    0.01 * rr 
## B 0.407 * rr    0.012 * rr
## C 1 - 0.25 * rr 0.25 * rr 
## D 0             1

State values

The costs of lamivudine and zidovudine are defined:

cost_zido <- 2278
cost_lami <- 2086

In addition to drugs costs (called cost_drugs in the model), each state is associated to healthcare costs (called cost_health). Cost are discounted at a 6% rate with the discount function.

Efficacy in this study is measured in terms of life expectancy (called life_year in the model). Each state thus has a value of 1 life year per year, except death who has a value of 0. Life-years are not discounted in this example.

For example state A can be defined with define_state:

A_mono <-
  define_state(
    cost_health = 2756,
    cost_drugs = cost_zido,
    cost_total = discount(cost_health + cost_drugs, .06),
    life_year = 1
  )
A_mono
## An unevaluated state with 4 values.
## 
## cost_health = 2756
## cost_drugs = cost_zido
## cost_total = discount(cost_health + cost_drugs, 0.06)
## life_year = 1

The other states for the monotherapy treatment group can be specified in the same way:

B_mono <-
  define_state(
    cost_health = 3052,
    cost_drugs = cost_zido,
    cost_total = discount(cost_health + cost_drugs, .06),
    life_year = 1
  )
C_mono <-
  define_state(
    cost_health = 9007,
    cost_drugs = cost_zido,
    cost_total = discount(cost_health + cost_drugs, .06),
    life_year = 1
  )
D_mono <-
  define_state(
    cost_health = 0,
    cost_drugs = 0,
    cost_total = discount(cost_health + cost_drugs, .06),
    life_year = 0
  )

Similarly, for the the combined therapy treatment group, only cost_drug differs from the monotherapy treatment group:

A_comb <-
  define_state(
    cost_health = 3052,
    cost_drugs = cost_zido + cost_lami,
    cost_total = discount(cost_health + cost_drugs, .06),
    life_year = 1
  )
B_comb <-
  define_state(
    cost_health = 3052 + cost_lami,
    cost_drugs = cost_zido,
    cost_total = discount(cost_health + cost_drugs, .06),
    life_year = 1
  )
C_comb <-
  define_state(
    cost_health = 9007 + cost_lami,
    cost_drugs = cost_zido,
    cost_total = discount(cost_health + cost_drugs, .06),
    life_year = 1
  )
D_comb <-
  define_state(
    cost_health = 0,
    cost_drugs = 0,
    cost_total = discount(cost_health + cost_drugs, .06),
    life_year = 0
  )

State lists

All states from a treatment group must be combined in a state list with define_state_list:

states_mono <-
  define_state_list(
    A_mono,
    B_mono,
    C_mono,
    D_mono
  )
## No named state -> generating names.
states_mono
## A list of 4 unevaluated states with 4 values each.
## 
## State names:
## 
## A
## B
## C
## D
## 
## State values:
## 
## cost_health
## cost_drugs
## cost_total
## life_year

Similarly for combined therapy:

states_comb <-
  define_state_list(
    A_comb,
    B_comb,
    C_comb,
    D_comb
  )
## No named state -> generating names.

Model definition

Models can now be defined by combining a transition matrix and a state list with define_model:

mod_mono <- define_model(
  transition_matrix = mat_mono,
  states = states_mono
)
mod_mono
## An unevaluated Markov model:
## 
##     0 parameter,
##     4 states,
##     4 state values.

For the combined therapy model:

mod_comb <- define_model(
  transition_matrix = mat_comb,
  states = states_comb
)

Running models

Both models can then be run for 20 years with run_model. Models are given simple names (mono and comb) in order to facilitate result interpretation:

res_mod <- run_models(
  mono = mod_mono,
  comb = mod_comb,
  cycles = 20
)

By default models are run for one person starting in the first state (here state A).

Model values can then be compared with summary:

summary(res_mod)
## 2 Markov models run for 20 cycles.
## 
## Initial states:
## 
##   N
## A 1
## B 0
## C 0
## D 0
##      cost_health cost_drugs cost_total life_year
## mono    45479.45   18176.56   44613.85  7.979173
## comb    89433.47   43596.75   81026.56 13.864239

The incremental cost-effectiveness ratio of the combiend therapy strategy is thus:

\[ \frac{81026.56 - 44613.85}{13.864239 - 7.979173} = 6187.307 \]

6187GBP per life-year gained.