EQUILIBRIUM POPULATION METHODS FOR MARKOV MODELS OF HEALTH INTERVENTIONS
Gordon Hazen, Ph.D., Min Huang. IEMS Department, Northwestern University
Purpose and methods. In Markov models of healthcare
interventions, cohort methods are often used to calculate
population QALYs and costs. However, analysts may also
wish to determine equilibrium population levels before and
after an intervention. We review methods from the
stochastic processes literature that can be used for this
purpose.
Results. We consider several real and hypothetical
examples. One is a simple model of individuals at risk for a
hypothetical disease with annual incidence 12 per thousand,
disease mortality 10 times the healthy annual mortality of 4
per thousand, and quality of life 0.20 in the disease state.
Consider a seemingly beneficial intervention that halves
disease mortality. Cohort methods yield an increase of
3.75 QALYs on average for an initially healthy individual.
Equilibrium methods yield an increase in disease
prevalence from 23.1% to 37.5%, a decrease of 0.115 yr in
population average QALYs accrued per year, and a
decrease of 4.6 yr in population average lifetime QALYs.
Intervention that reduces disease mortality
rate
Results (cont). The problem with these equilibrium
measures is that they do not account for population size
increase due to better survival. The corresponding total
population measures behave reasonably: Population total
QALYs accrued per year increases by 5.7% and population
total lifetime QALYs increases by 12.2%. In a second
example, we show, however, that there are interventions
that increase initially-healthy individual QALYs and
nevertheless decrease population total lifetime QALYs. It is
only population total QALYs accrued per year that behaves
consistently with initially-healthy individual QALYs. We
also apply these equilibrium methods to a published
analysis of tamoxifen use for breast cancer prevention (Col
et al 2002), where we find no comparable anomalies.
Conclusions. For Markov models of healthcare
interventions, equilibrium methods from the stochastic
processes literature may be a useful adjunct to cohortbased QALY calculations. However, caution is advised in
choosing measures of effectiveness, as seemingly intuitive
equilibrium measures of population health may conflict with
each other and with individual-level measures.
Discounted Total QALYs (DTQ): Mean total discounted QALYs for
this and all subsequent generations of the population.
(1) Intervention to decrease disease mortality.
What measures of quality are possible at the
population level?
406.5 yr x 1.00 =
406.5041
406.5
176.1518
176.2
yr x 0.75 =
132.1
538.6 person-QALYs
Average Lifetime QALY (ALQ): Mean QALY of a randomly
selected individual from the equilibrium population
Conventional QALY calculations for an individual
Example anomalies.
406.5
yr x 1.00 =
406.5041
406.5
400.3
yr x 0.75 =
400.3449
300.3
25.41
x
=
10.4
41%
706.8 person-QALYs
12.20 yr
x1=
Discount rate = 3%
12.2
59%
5.28 yr
x 0.75 =
20.83
x
3.96
=
12.3
22.7 QALY
16.16 QALY
12.20 yr
17.62 yr
x1=
12.2
x 0.75 = 13.21
Relationships
Total Lifetime QALY (TLQ): Mean total QALYs of all individuals in
the equilibrium population.
Relationships between average and total measures
25.41 QALY
25.41
x
=
310
12.2
17.6
Augment model by allowing “births”
20.83
x
The result: Population no longer dies out – instead, it
reaches a new equilibrium after the intervention
= 367.1
677 person-QALYs
 Equilibrium 


TLQ   population   ALQ


size


 Equilibrium 


TQ / yr   population   AQ / yr


size


(2) Intervention to decrease healthy mortality also
increases disease mortality.
Implication: An intervention that has a beneficial effect on average
measures may have a detrimental effect on total measures (or vice
versa) if the intervention changes equilibrium population size.
Relationships between individual and population measures
TQ / yr 
Average QALYs per Year (AQ/yr): One-year QALY of a randomly
selected individual from the equilibrium population
41% x 1.00 =
0.41
59% x 0.75 =
0.44
0.85 QALY/yr
Total QALYs per Year (TQ/yr): One-year QALY of all individuals in
the equilibrium population
12.2
x 1.00 =
12.2
17.62 x 0.75 =
13.22
25.42 person-QALY/yr
Society for Medical Decision Making Annual Meeting, San Francisco, October 2005
 "Birth" rate   Mean individual QALYs 



into s   beginning in state s 
All states s 

Implication: If “births” enter the model only in the initial “Well” state,
then an intervention that does not change “birth” rates will have the
same relative effect at the population level as measured by TQ/yr as
it does at the individual level as measured by QALYs.
 "Birth" rate   Mean individual QALYs 



into s   beginning in state s 
All states s 
AQ / yr 
 "Birth" rate   Mean individual lifetime 




into s   beginning in state s 
All states s 

Implication: Even if “births” enter the model only in the initial “Well”
state and the intervention does not change “birth” rates, the
intervention may have the opposite effect at the population level as
measured by AQ/yr than at the individual level as measured by
QALYs.
Relationships between discounted total QALYs and total
QALYs per year: If the population is initially in equilibrium, and DTQ
is calculated at interest rate r, then
1
DTQ   TQ / yr
r
Implication: Interventions that do not change population equilibrium
will have the same relative effect on DTQ as on TQ/yr.
(3) Col N.F., Orr R.K., Fortin J.M. Survival impact of tamoxifen use
for breast cancer risk reduction: projections from a patient-specific
Markov model, Med Decis Making 2002; 22: 386-393.