### Three-state model

The US adult population is modeled at 1-year intervals starting at year *t* = 28 (2007) and ending at *t* = 71 (2050). Specifically, define the following numbers of people in various states, rates, and flows. All rates are annual and flows occur during year *t*, i.e., in the interval (*t*-1,*t*].

*X*(*t*) = the number of adults without diabetes at time = *t*.

*Z*(*t*) = the number of adults with undiagnosed diabetes at time = *t*.

*Y*(*t*) = the number of adults with diagnosed diabetes at time = *t*.

*b*(*t*) = Census projection of the number of adults turning 18 (births) during year *t*.

*m*(*t*) = Census projection of the number of adults migrating into the United States during year *t*.

*d*(*t*) = Census projected death rate for the US resident population aged 18-79 years.

*γ*(*t*) = nondiabetes death rate among *X*(*t*-1).

*r*
_{
1
}
*γ*(*t*) = diabetes death rate among *Z*(*t*-1); *r*
_{
2
}
*γ*(*t*) = diabetes death rate among *Y*(*t*-1); *r*
_{
1
}and *r*
_{
2
}are relative risks.

*i*(*t*) = incidence rate of diagnosed diabetes among *X*(*t*-1) or *Z*(*t-* 1), denoted earlier by .

*I*(*t*)= incidence rate of all diabetes (undiagnosed or diagnosed) among *X*(*t*-1).

*f*
_{
x
}(*t*) = proportion of *b*(*t*) without diabetes.

*f*
_{
z
}(*t*) = proportion of *b*(*t*) with undiagnosed diabetes.

*f*
_{
y
}(*t*) = proportion of *b*(*t*) with diagnosed diabetes.

*g*
_{
x
}(*t*) = proportion of *m*(*t*) without diabetes.

*g*
_{
z
}(*t*) = proportion of *m*(*t*) with undiagnosed diabetes.

*g*
_{
y
}(*t*) = proportion of *m*(*t*) with diagnosed diabetes.

The relations *f*
_{
x
}(*t*)+ *f*
_{
z
}(*t*)+ *f*
_{
y
}(*t*) = 1 and *g*
_{
x
}(*t*)+ *g*
_{
z
}(*t*)+ *g*
_{
y
}(*t*) = 1 ensure consistency with the census projections. Define , the prevalence of diabetes at time *t*; , the prevalence of undiagnosed diabetes at time *t*; , the prevalence of diagnosed diabetes at time *t*. Clearly θ(*t*) = θ_{1}(*t*) + θ_{2}(*t*).

Consider the following transition matrix:

.

Note that this matrix displays the distribution of the beginning year stocks (rows) to the ending year stocks (the columns), and thus, the transition rates in each row must be nonnegative and add to unity for each year *t*. Some assumptions about transition rates are apparent. First, people cannot move from diabetes to nondiabetes; this assumption is reasonable because remission is extremely rare. Second, the relative risks of death for the two diabetes states are constant over time. No data were found to support time varying relative risks, but published results [17] lead to estimates *r*
_{1} = 1.77 and *r*
_{2} = 2.11. We also set *r*
_{1} = 1.00 and *r*
_{2} = 4.08 in a sensitivity analysis, consistent with projections from Narayan et al [6] aggregated to the US adult population aged 18-79 years. Third, the transition rates to diagnosed or undiagnosed diabetes for nondiabetics are constant multiples of the transition rate to diagnosed diabetes for undiagnosed diabetics. General time varying rates for these transitions were not available, but, as detailed below, estimates of ξ_{1} and ξ_{2} could be obtained. This assumption implies the proportion of diagnosed diabetics among all new diabetics in any given year *t* is constant over time, i.e., or . To calculate η, assume that 95% of the time a person spends less than seven years in the undiagnosed diabetes state [22] and that the hazard rate for moving from undiagnosed to diagnosed diabetes is constant over time. This equates to a 0.19 probability of moving to the diabetes state within six months. The transition matrix leads to the system of first order difference equations

with initial conditions *X*(28), *Y*(28), *Z*(28).

Consistency with census projections of the number of US adults *N*(*t*) requires *X*(*t*) *+ Z*(*t*) *+ Y*(*t*) *= N*(*t*) where

This is guaranteed if the following two equations are satisfied:

But the second equation is equivalent to

(with a little algebra). The first is a consequence of

where the values .398 and .129 come from [23]. These equations imply

with *N*(28) = 215,750,418, the census estimate of the 2007 US adult population.

Now, given a projection *i*(*t*), then

and *β*(*t*) is

Noting that, in general,

then

where the value .0106 is derived in Appendix 2. Solving this equation for x_{1} yields

.

Thus, *ξ*
_{1}, *ξ*
_{2} are determined.

Finally, the distributions of births and net migration across the three subpopulations for each year are determined by *f*
_{
x
}
*(t)* = 1, *f*
_{
z
}
*(t)* = 0, *f*
_{
y
}
*(t)* = 0and

The first set of equations reflects our baseline assumption that all incoming births are nondiabetic. The second set of equations simply distributes the net migration for year *t* according to the proportions in each state at the beginning of the year. All parameters in the model are thus determined and the system of equations described earlier can be used to calculate the model projections.

### Four-state model

The four-state model expands the three-state model by splitting *X*(*t*) = *HX*(*t*) + *LX*(*t*), where *HX(t) = p(t)X(t)* is the number of adults in a high-risk group (e.g., those with IFG) and *LX(t) =* [1-*p(t)*] *X(t)* is the number in a low-risk group (e.g., those without IFG). This must be done so that *X*(*t*), *Z*(*t*), and *Y*(*t*) are as in the three-state model.

To this end, consider the transition matrix

Note that the death rates are equal to those in the three-state model when

Also, transition rates to diabetes are the same as in the three-state model if the following relation holds:

One additional assumption is made to determine *h*(*t*) and *l*(*t*), that the ratio of the incidence rates to any diabetes of the high-risk group to the low-risk group is constant over time (constant relative incidence). Specifically,

The two equations above yield the expressions

The constant *c* can be calculated as follows. Let λ equal the 2008 incidence rate from the high-risk group to any diabetes and *p*(28) the proportion of the nondiabetic population at high risk in 2007. For example, for IFG λ = .0287 and *p*(28) = .257 from Appendix 2. Then, (ξ_{1} + ξ_{2})*h*(29)*p*(28)*X*(28) = λ*p*(28)*X*(28) implies . But and .

For the risk strata IFG, this gives *c* = 6.6. All of this ensures that the relevant transition rates from the nondiabetic population to the diabetic population are the same as in the three-state model. Since death rates for these populations are as in the three-state model, the four-state model properly expands the three-state model when

and

To complete the model, the α's must be chosen. The constraints are

Let

Then there exists *q(t)* in the interval (0,1) and *s(t)* in (0,1) such that

For the high-risk population with IFG, we set *q*(*t*) = *q* = .93 to get α_{1}(29) = .89 from [23]. Finally, assuming the proportion *p*(*t*) = .257 for all *t* ≥ 28, it is easy to derive (details omitted)

The actual computation of the four-state model can be implemented through the system of difference equations

with initial conditions *HX*(28) *= p*(*28*) *X*(28), *LX*(28)=[1-*p*(28)] *X*(28), *Y*(28), *Z*(28).

### Five state model (preventive intervention)

Given the outputs from the two previous baseline models, consider the following transition matrix reflecting intervention on the high-risk population. Note that *IX*(*t*) now denotes the number of people in the high-risk or intervention group. Also, a new state has been added with *LXI*(*t*) equals the number of people in the intervention group who have regressed to low risk. The transition matrix is

where rows are labeled as columns with times *t*-1, and

State variables are relabeled because we expect the intervention model to deviate from the previous models. The initial conditions are the same as in the four-state model, with *LXI*(28) = 0. The associated system of difference equations is omitted and can be derived in the same way as in the previous two models.