A Population Dynamic Model to Assess the Diabetes Screening and Reporting Programs and Project the Burden of Undiagnosed Diabetes in Thailand

Diabetes mellitus (DM) is rising worldwide, exacerbated by aging populations. We estimated and predicted the diabetes burden and mortality due to undiagnosed diabetes together with screening program efficacy and reporting completeness in Thailand, in the context of demographic changes. An age and sex structured dynamic model including demographic and diagnostic processes was constructed. The model was validated using a Bayesian Markov Chain Monte Carlo (MCMC) approach. The prevalence of DM was predicted to increase from 6.5% (95% credible interval: 6.3–6.7%) in 2015 to 10.69% (10.4–11.0%) in 2035, with the largest increase (72%) among 60 years or older. Out of the total DM cases in 2015, the percentage of undiagnosed DM cases was 18.2% (17.4–18.9%), with males higher than females (p-value < 0.01). The highest group with undiagnosed DM was those aged less than 39 years old, 74.2% (73.7–74.7%). The mortality of undiagnosed DM was ten-fold greater than the mortality of those with diagnosed DM. The estimated coverage of diabetes positive screening programs was ten-fold greater for elderly compared to young. The positive screening rate among females was estimated to be significantly higher than those in males. Of the diagnoses, 87.4% (87.0–87.8%) were reported. Targeting screening programs and good reporting systems will be essential to reduce the burden of disease.

Information S1. Solved a large set of Ordinary Differential Equations (ODE) of Demographic submodel We solved a large set of Ordinary Differential Equations (ODE) of demographic deterministic sub-model. Let ( ) be the number of people of at age, a, at time, t and be the fertility rate in female aged a years old [2] The number of newborn babies at any time t is shown as follows: Death [3] among male and female population were calculated from the age-specific mortality rate dra: Net migration [4] among male and female population were calculated from the migration rate mra: Aging is a rate at which individuals move to the next age group were also represented as at rate ( . ) per year where age.diff represented the difference between two age classes. In this model the age.diff is always equal to 1 year. We generated the matrix equation for individual dynamics as follow: All the parameters included in the model was shown in table 1; Since the case fatality data were not stratified by gender, it was assumed that the rates were the same in both genders.
Case fatality of diabetic ( ℎ ) [5] among diagnosed individuals in each age group were a sum of the deaths from natural causes ( ) and the deaths occurred from DM itself with the case fatality rates : The diabetes incidence of each gender and age group ( ) was taken to be a function of nondiabetic ( ) with corresponding diabetes incidence rate which represents the rates of total diabetes both diagnosed and undiagnosed of each age group: The positive diabetes screening of each age group ( ) was taken to be a function of undiagnosed diabetes ( _ ) with corresponding diabetes screening rate of each gender and age group: The positive diagnosed diabetes of each age group ( ) was taken to be a function of undiagnosed diabetes ( _ ) with corresponding diagnosed diabetes rate of gender and each age group: Rates of change in , _ and within the gender g and age group a were represented by ordinary differential equations. For example, the rate of change of the nondiabetic females aged 1 year old was represented by the following equation which describes the balance between birth inflows, diabetes incidence, aging, and death as follows: Similarly, rate of change in the undiagnosed diabetic compartment was calculated as a balance between screening, diagnosis, diabetes incidence and death outflows as follows: Rate of change in the diabetic compartment was calculated as a balance between screening rate, diagnosis, aging, and death outflows as follow: , _ and were determined by numerical integration of the corresponding differential equations. Diabetes prevalence in any gender g and age group a ( ) was finally calculated as following: Cumulative incidence was analyzed by numerical integration of the corresponding healthy ( ) and diabetes incidence of each age group ( ) calculated as following: Equation 14) and similarly, for other reported measures. Note that the report parameters were used to calculate reported incidence and prevalence diabetes by multiplication of diagnosed incidence and prevalence diabetes and reporting proportion.
Information S3. The Bayesian framework.
Bayesian inference of diabetes dynamic model provides a framework for estimating parametric uncertainty in terms of probabilistic distributions, and allowing a direct quantification of parameter uncertainty.
Bayes theorem states that the best estimate (posterior uncertainty ( | )) for a parameter vector given data y is given by: Here, ( ) is the prior information and, is the likelihood ratio. Markov Chain Monte Carlo (MCMC) algorithms were applied to approximate these distributions which used a sampling scheme to estimate the posterior distribution [8,9].

Prior distribution
Uniform distribution was chosen to be the prior distribution for all parameter values given little information about these parameters was measured and reported. The minimum and maximum values were initiated and narrowed down from the iterative model fitting procedure.

Likelihood function
The likelihood of parameters given the data is equal to probability of the data given the parameters including incidence rates, screening and reporting. We defined the likelihood as the product of likelihood terms for each data point. The data arise from the diabetes annual epidemiological surveillance report between 2005 and 2015, and are linked to the summation of expected age and gender rates via a Poisson distribution. The log-likelihood (used as the target in the MCMC algorithm) is: Where θ is the annual diabetes data at each age class a and time t and DM is the expected incidences from the model at each age class a and time t.

Posterior estimation
We used Differential-Evolution MCMCzs (DE-MCzs) to estimate the posterior distributions.
We consider Markov chain methods of sampling that are proposed by Ter Braak and Vrugt et al, 2008 [10], which has been used for numerical problems, implemented in the Bayesian Tools R package.