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5.4.1 Podgor and Leske (1986)’s framework for a single-stage chronic disease

As mentioned in section 5.3.1, in a population in steady state, there is an in- tuitive relationship between incidence and prevalence. Additionally, assuming the disease is stable over time, one can treat the age-dimension of prevalence as a time dimension, whereby static age-specific prevalence represents the in- dividual’s probability of having the disease once reaching a certain age.

Relying on the two assumptions of: (1) stable population, i.e. popula- tion with stable age composition and (2) stable epidemic, Podgor & Leske (1986) outlined the relationship between incidence, prevalence and mortal- ity for an irreversible one-stage chronic disease characterised by differential mortality. This relationship was modelled via a 3-state closed unidirectional system, whereby individuals of a population of size N were either healthy (H),

infected by the irreversible disease (I) or dead (D).

The authors defined the number of “infected” people of age a + t in the population as the sum of: (i) the number of individuals who were healthy at age a times the joint-probability PHI that they became infected during the t-

year period and survived and (ii) the number of individuals who were already infected at age a times the probability PII that they survived during the t-

year period, given that they were infected. Similarly, the number of “healthy” people of age a + t in the population was defined as the number of people who were healthy at age a times the joint probability PHH that they did not

become infected nor died during the t-year period. These two relationships are expressed by equations 5.1 and 5.2:

Na+tΠa+t = Na(1 − Πa)PHI + NaΠaPII (5.1)

Na+t(1 − Πa+t) = Na(1 − Πa)PHH (5.2)

Where Na+t and Na represent the total number of alive individuals of age

a + t and a at any point in time in the system and Πa+t and Πa represent the

static prevalence of the disease among population individuals of age a+t and a. Combining equations 5.1 and 5.2 eliminates the population parameters and provides a single equation with one unknown PHI (5.3).

(1 − Πa)Πa+t

PHH

(1 − Πa+t)

= ΠaPII+ (1 − Πa)PHI (5.3)

PHI represents the joint probability of developing the disease and surviving

during the t-year period given that one is infected, whereas the parameter of interest is the probability of becoming infected among the total at-risk population. To derive this parameter, Podgor & Leske (1986) expressed the joint-probabilities PHH, PHI and the conditional probability PII as a function

of the rates of mortality among respectively those infected and those healthy and of the rate of disease incidence, assuming that rates followed indepen- dent exponential distributions. Solving for the rate of disease incidence how- ever, required a rather sophisticated and computationally-intensive method (Newton-Raphson method).

5.4.2 Extension of Podgor and Leske (1986)’s framework to a k- stage chronic disease

For COPD, the modelling of the relationship between prevalence, incidence and mortality also need to account for disease progression and the impact of disease severity on survival. Podgor & Leske (1986)’s model was therefore extended to a k-stage chronic disease.

To reduce the computational burden associated with probabilities estima- tion, the extension of Podgor & Leske (1986)’s model was carried out in a discrete time setting. Importantly the extension of the model is also under- pinned by the two assumptions of stable population and stable epidemic.

The modelling of the relationship between incidence, prevalence, survival and disease progression was based on the structural assumptions described in section 5.3.2. In particular, whilst individuals were allowed to enter the disease pathway in any state to reflect late diagnosis, once diagnosed with the disease it was assumed that, during each t-year period they could either: (i) stay in the disease state they were in or (ii) progress to the next severe state or (iii) die. The t-year probabilities to transit between severity states i are therefore of the form Pi,i+1.

To extend the model in a discrete-time setting, PHI, PHH and PII were

expressed in terms of t-year transition probabilities, where PX,Y denotes the

probability of transiting to state Y during a t-year period conditional on being in state X at age a. It follows that for i = 1, ..., k representing disease stages:

PHH = (1 − PH,i)(1 − PH,D)

PHI = PH,i(1 − Pi,D)

PII = (1 − Pi,D)

In a stable population with stable disease, the total number of people aged a + t who are in stage i of the disease at any point in time is the sum of: (i) individuals who were healthy at age a and got diagnosed with COPD in stage i during the t-year period and survived, (ii) individuals who were in stage i at age a and did not move to the next severe stage nor died during the t-year

period and (iii) individuals who were in stage i − 1 at age a and moved to stage i and survived during the t-year period. This relationship is expressed in equation 5.4, using notations for population size and disease prevalence introduced in section 5.4.1.

Na+tΠi,a+t = Na(1 − k

X

i=1

Πi,a)PH,i(1 − Pi,D)

+ NaΠi,a(1 − Pi,i+1)(1 − Pi,D)

+ NaΠi−1,aPi−1,i(1 − Pi,D)

(5.4)

For i=1, equation 5.4 simplifies to: Na+tΠi,a+t = Na(1 −

k

X

i=1

Πi,a)PH,i(1 − Pi,D) + NaΠi,a(1 − Pi,i+1)(1 − Pi,D)

For i=k, equation 5.4 equals to: Na+tΠi,a+t = Na(1−

k

X

i=1

Πi,a)PH,i(1−Pi,D)+NaΠi,a(1−Pi,D)+NaΠi−1,aPi−1,i(1−Pi,D)

Similarly, the number of people aged a + t who are healthy at any point in time is simply the sum of individuals who were healthy at age a and did not develop the disease nor died during the t-year period.

Na+t(1 − Πi,a+t) = Na(1 − k X i=1 Πi,a)(1 − k X i=1 PH,i)(1 − PH,D) (5.5)

Combining equations 5.4 and 5.5 to eliminate the population size param- eters Na+t and Na would however, generate an equation with two unknowns:

PH,i and

Pk

i=1PH,i. To address this issue, the total size of the population of

age a + t was expressed as a function of: (i) the total size of the population of age a and (ii) the probabilities of survival for individuals of age a conditional

on their health status : Na+t = Na(1 − k X i=1 Πi,a)(1 − PH,D) + Na k X i=1 Πi,a(1 − Pi,D) (5.6)

Substituting the expression for Na+t into equation 5.5 enabled to express

the t-year probabilities for individuals of age a to transit within the closed- system from the state “healthy” to the each i disease state (i.e. PH,i,i=1,...,k)

as a function of:

(i) t-year death probabilities conditional on health status: Pi,D,i=1,...,k, PH,D

(ii) t-year probabilities of transiting to the next more severe state: Pi,i+1,i=1,...,k−1

(iii) disease prevalence by severity stage for t-year age-groups: Πi,a, Πi,a+t

For i=2,...,k-1: PH,i= Πi,a+t[(1 − Pk i=1Πi,a)(1 − PH,D) + Pk

i=1Πi,a(1 − Pi,D)] − Πi,a(1 − Pi,D)(1 − Pi,i+1)

(1 −Pk

i=1Πi,a)(1 − Pi,D)]

− Πi−1,a(1 − Pi,D)(1 − Pi−1,i) (1 −Pk

i=1Πi,a)(1 − Pi,D)]

(5.7) For i=1: PH,i= Πi,a+t[(1 − Pk i=1Πi,a)(1 − PH,D) + Pk

i=1Πi,a(1 − Pi,D)] − Πi,a(1 − Pi,D)(1 − Pi,i+1)

(1 −Pk

i=1Πi,a)(1 − Pi,D)]

(5.8) For i=k:

PH,i=

Πi,a+t[(1 −Pk_i=1Πi,a)(1 − PH,D) +Pk_i=1Πi,a(1 − Pi,D)] − Πi,a(1 − Pi,D)

(1 −Pk

i=1Πi,a)(1 − Pi,D)]

− Πi−1,a(1 − Pi,D)(1 − Pi−1,i) (1 −Pk

i=1Πi,a)(1 − Pi,D)]

5.5

Estimating COPD diagnosis probabilities by sever-