Introduction

Accurate estimates of the number of people with hearing loss and dementia are essential for planning services that meet their needs. However, data sources often diverge in their age groups, making it difficult to develop precise estimates.

Global research has established that dementia is one of the most prevalent conditions in older people (Adeloye et al. 2019; GBD 2019 Dementia Forecasting Collaborators 2022; Prince et al. 2013, 2016; Radford et al. 2019), posing many challenges to individuals and their families (Leicht et al. 2013). A combination of population ageing and population growth is predicted to result in a significant increase in the number of people living with dementia (GBD 2019 Dementia Forecasting Collaborators 2022). Similarly, hearing loss is linked to the ageing process and is a leading cause of global years lived with disability for individuals older than 70 years (Cruickshanks et al. 1998; GBD 2019 Hearing Loss Collaborators 2021; Mick et al. 2014). There is a known association between hearing loss and incident dementia (Lin et al. 2011, 2013; Thomson et al. 2017), with untreated hearing loss being a risk factor for cognitive decline (Dawes 2019; Dawes et al. 2015; Fritze et al. 2016; Orgeta et al. 2019; Shen et al. 2018). In 2017, the Lancet Commission suggested that mid-life hearing loss should be considered a potentially modifiable risk factor for dementia (Livingston et al. 2017). In July 2020, the Lancet Commission reconvened and again, mid-life hearing loss was deemed to be a potentially modifiable risk factor. They estimated that 8% of dementia cases globally can be attributed to this factor (Livingston et al. 2020). There are several potential explanations for the correlation between hearing loss and the onset of dementia. These include shared underlying pathology, potentially related to issues with blood flow; reduced sensory input having a negative impact on the structure and function of the brain; cognitive resources being overtaxed by efforts to hear clearly, leaving fewer available for more complex cognitive processes; and a possible interaction between hearing function and the onset of dementia (Griffiths et al. 2020). It is important to note that these possible explanations are not necessarily independent of one another. The treatment of hearing loss with hearing aids is encouraged and appears to reduce the risk of developing dementia (Jiang et al. 2023; Livingston et al. 2020).

The International Standards Organization standard 7029 (ISO Central Secretary 2017), classifies the median hearing for ontologically normal people, except for the effect of a specific agent like noise, in decibel per age for the frequencies 125 Hz to 8000 Hz. The formula provided is \(\Delta {H}_{md,Y}={\alpha}_md{\left(Y-18\right)}^{\beta_{md}}\) where αmd and βmd are dimensional quantities that are given in the ISO norm and Y is the age in years. This multidimensional mathematical model may be too precise over frequencies and too imprecise in age ranges that are relevant for dementia and therefore not particularly helpful for service delivery planners (see Fig. 1).

Fig. 1
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ISO 7029 ΔHmd, Y @ 770Hz with confidence interval (md = median, Y = Age in years)

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Those responsible for service delivery need information regarding disabling levels of hearing loss (World Health Organization 2021) and more detailed information for the age groups most affected by hearing loss and dementia. Studies to determine regional prevalence of these co-morbidities are becoming available. Patient data was gathered from the AOK, Germany’s largest public health insurer. Analysis of this data made it possible to detail regional prevalence rates for both deafness and dementia in Germany, from the individual to the regional level (Teipel et al. 2015). Recent research sought to provide regional prevalence figures for hearing loss and dementia in Scotland (McMenemy and Johnson 2020; McMenemy et al. 2020). Hearing loss prevalence data is not collected at GP level in Scotland; therefore, the prevalence rate of hearing loss cannot be calculated from primary data. This also affects the establishment of prevalence for the co-morbidities of hearing loss and dementia from primary data (McMenemy et al. 2020). To estimate the prevalence of hearing loss and dementia in Scotland, it has been necessary to compensate for this lack of data by using existing models to estimate figures. In the United Kingdom the model used for estimating prevalence of hearing loss was developed by Davis (Davis 1989, 1995) and dementia estimates, as used by Alzheimer Scotland are derived from the Eurocode model (Alzheimer Europe 2019). Both models are grounded in population statistics. Although these models were utilized in recent reports on hearing loss and deafness/dementia prevalence in Scotland (McMenemy and Johnson 2020; McMenemy et al. 2020), it was a time-consuming process due to the lack of flexibility and data. Additionally, combining the two models proved to be challenging as they do not use the same age groupings. By employing the Stannard Growth Model, this study will showcase the resolution of these issues, ultimately increasing efficiency and producing nuanced and reliable prevalence estimates. It has enabled the creation of a table, specific to age, that details the prevalence of both hearing loss and dementia. It is envisaged that this could be used as an efficient estimation tool for those involved in the provision of health and welfare services.

Materials and Methods

The exact numbers of deaf people in Scotland varies depending upon which source is consulted. This lack of clarity is problematic, and there is a need to establish better reporting procedures to capture relevant statistics. Although published in 1995, Davis’ comprehensive study on the prevalence of deafness in the UK population is still the benchmark for statistics in this field (Davis 1995). This book was made available online as a fully searchable PDF (Akeroyd et al. 2019). The criteria Davis set out in his study was synthesized by Akeroyd et al. (2014) as a loss of hearing of 35 dB minimum in the better ear at 0.5, 1, 2 and 4 kHz of respondents. Using this data and population level estimates from the Scottish National Records office, it was possible to gain general estimates of the number of people with hearing loss based on the age groups presented.

It was found that data on dementia (and other morbidities) exists in Scotland at GP level (excluding practices with fewer than 5 cases due to General Data Protection Regulations) and can be obtained from the Scottish Primary Care Information Resource (SPIRE), but this data is not structured by age. However, structured age group models based on data collection do exist via the European Collaboration on Dementia (EuroCoDe). The age group model was acquired from EuroCoDe (Table 1) and the Davis Model (Table 2) (Akeroyd et al. 2019) based on Davis chapter 3 (Davis 1995).

Table 1 Prevalence numbers for age related dementia in percent and mean age points of the EuroCoDe “model”
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Table 2 Prevalence numbers for age related hearing loss in percent and mean age points of the Davis “model”
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Although both relate age and gender with prevalence, the age groups in both models do not correspond. Age groups in statistical sources can vary dramatically. Take, for example, the Scottish Census Table KS102SC – Age Structure (see Table 3). This table uses age classifications as noted below. There is considerable variation in the age groupings, with age 15 being in a separate group. Whether this is by accident or design, this table structure is demonstrative of the wide range of variation seen in data presentation. The group of 90+ years combines all remaining ages into one category. Presently, the oldest person who was born and died in Scotland was 111 (Hall 2009). While this is an outlying age, if we consider that as the uppermost point in the longevity scale, it means the 90+ age group can potentially span 21 years. Likewise, the Davis Model groups those age 81+ into one category, giving a potential age span of 31 years and the 100+ age grouping of the Eurocode data could potentially span 12 years. As such, age groups do not fit well and make it difficult to analyze population data in Scotland (statistics.gov.scot or the National Records office).

Table 3 Age groups in Scottish Census Table KS102SC – Age Structure
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Using Growth Models

To develop a robust framework for estimating the correlation between the prevalence of deafness and dementia with age, and to estimate these rates for different age groups based on population numbers, we utilized a well-established growth model that was parameterized for this purpose. Mid-life hearing loss and dementia are ageing processes that can be mathematically modelled just as with other ageing processes (Novoseltsev and Mikhalskii 2011). Mathematical Modelling has a class of models that describe growth processes (Fekedulegn et al. 1999; Kaps et al. 2000; Khamiz et al. 2005; Panik 2014; Tsoularis and Wallace 2002). Models of this class are often logistic based, meaning they start off slowly, go through rapid expansion and then level off at an asymptote. The aim was to determine a formula that could be applied to all possible age groups, expressed in years at the hearing loss level ≥ 35 dB and dementia. We decided to investigate formulas that could describe the advancement of prevalence of hearing loss and dementia with age.

To get a good fit to the data in Tables 1 and 2, we used parameter optimization in R 4.0.2 on the following growth models: Brody (Kaps et al. 2000), Generalized Richards (Day and Tinney 1969; Richards 1959), Generalized Logistic (Panik 2014), Logistic, Gompertz, Richard, Chapman–Richards, von Bertalanffy, Weibull, negative Exponential, Mitcherlich , Monomolecular (Fekedulegn et al. 1999), Blumberg (Tsoularis and Wallace 2002), Loglogistic, Schnute, Morgan–Mercer–Flodin and Stannard growth models (Khamiz et al. 2005).

The Davis Model – Hearing Loss

For the Davis Model we had data for men, women and both genders pooled with 95% confidence intervals. To generate a point at which a prevalence rate was fixed we took the midpoint between the lower and the upper age group boundary. For example (18+30)/2. For the 81+ we treated this age group as per the preceding age groups and set 90 as the upper age group boundary.

EuroCoDe – Dementia

The EuroCoDe dementia data supplied from Alzheimer Scotland only included prevalence rates for men and women, with no confidence intervals or pooled data. We took the mean of male and female rates to generate a parameterization for the overall gender model in dementia. To generate a point at which a prevalence rate was fixed, we took the midpoint between the lower and the upper age group boundary. For people above 100 years of age, we set the upper age group boundary to the maximum completed full year (111 years old).

Combining Models – Hearing Loss and Dementia

We then combined the two parameterizations in one formula. This combined formula can be applied to obtain estimates of prevalence rates for hearing loss and dementia for men and women.

Results

While some of the models had an adequate fit, the best fit proved to be the Stannard Model [Khamiz et al. 2005; Kyurkchiev and Iliev 2016), see Eq. 1.

The Stannard model is defined as

$${y}_t=\frac{M}{{\left(1-{e}^{\frac{-\left(\beta + kt\right)}{m}}\right)}^m}$$
(1)

where M = upper asymptote, β = growth displacement, k = growth rate, m = slope of growth and t = time.

The curves for the parameterization for hearing loss can be found in Fig. 2.

Fig. 2
figure 2

Results of parameter optimization for a Stannard growth model for (A) women, (B) men, (C) both genders. t = time in years for age related hearing loss

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The curves for the parameterization for dementia can be found in Fig. 3.

Fig. 3
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Results of parameter optimization for a Stannard growth model for (A) women, (B) men, (C) both genders. t = time in years for age related dementia

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The individual results can be seen in Table 4.

Table 4 Results of parameterization for the Stannard model equation for dementia (Dem) and hearing loss (HL) at the ≥ 35 dB level for 0.5, 1, 2, 4 kHz in the better ear
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The people with hearing loss of ≥ 35 dB at 0.5, 1, 2, 4 kHz in the better ear in each year, times the number of people with dementia, can be calculated where year on year data in population estimates are available as follows.

$${y}_t=\frac{M_{HL}}{100\times {\left(1-{e}^{\frac{-\left({\beta}_{HL}+{k}_{HL}\times t\right)}{m_{HL}}}\right)}^{m_{HL}}}\times \frac{M_D}{100\times {\left(1-{e}^{\frac{-\left({\beta}_D+{k}_D\times t\right)}{m_D}}\right)}^{m_D}}\times {n}_t$$
(2)

where MHL = upper asymptote, βML = growth displacement, kHL = growth rate, mML = slope of growth, MD= upper asymptote, βD = growth displacement, kD = growth rate, mD = slope of growth, t = time and nt = the number of people at age t. HL = Hearing Loss and D = Dementia.

To show how this can be applied for cases where age group population data is available, we run through the following example. We could sum all values in Table 5 for an age group and divide it through the number of ages in that age group. As an example, if we have the age group 16–18 for women, we add (0.00000222901311, 0.0000000281633776, 0.000000355841711) the prevalence values as proportions and divide it through 3 which we would multiply by the number of people in that age group we got from a population estimate of an area (like a federal state or a local authority or a city) for that age group. This would give one the prevalence of people having dementia and hearing loss in that area at the same time.

Table 5 Lookup table in percent for age classification of hearing loss at ≥ 35 dB in the better ear at 0.5, 1, 2, 4 kHz together with dementia in the UK
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This would be expressed as

$${y}_t^{area}=\frac{\sum_{t=a}^b{x}_t}{m}\times {n}^{area}$$
(3)

where a = 16 and b = 18, m = 3 the number of ages in an age group, xt are the values at each step (0.0000000222901311, 0.0000000281633776, 0.000000355841711), narea the population estimate in that area for that age group and \({y}_t^{area}\) the prevalence of people having dementia and hearing loss in that area.

Quick Reference Table

A reference tool has been provided in Table 5.

The prevalence rate for hearing loss is similar for men and women, while women have a higher dementia prevalence rate than men. This result is then reflected in the co-morbidity of hearing loss and dementia, where women are more affected than men (see Fig. 4)

Fig. 4
figure 4

Comparing the prevalence as proportion of men and women for hearing loss ≥ 35 dB at 0.5, 1, 2 and 4 kHz in the better ear, dementia and the combined effect and the linear model results indicating the strength of the greater prevalence of men compared to women. All figures were created with R (Statistical software package) https://www.R-project.org/

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Discussion

The degree of fidelity of the models to biological veracity is evidenced by their ability to depict nuanced aspects of the phenomenon under study. For instance, our findings reveal a higher prevalence of dementia among women as compared to men. This is consistent with earlier research that reports higher rates of dementia in women, with a female-to-male ratio of 1.69 (GBD 2019 Dementia Forecasting Collaborators 2022). Applying our model to the EuroCoDe Data, we find that dementia is 1.48 times more prevalent in women than men, this figure assumes men and women would have the same life expectancy (see Fig. 4). While men and women have about the same prevalence in hearing loss (see Fig. 4), the combined prevalence does not alter the relationship between men and women as compared to dementia. Approximately 1.51 times more women than men have hearing loss and dementia, if men would live as long as women. The sex ratio in humans varies (Jacobsen et al. 1999) and women tend to live longer than men (Austad 2006). This longevity is not because women age more slowly, but because they are more robust at every age. Paradoxically, although women have lower mortality rates, they have higher overall rates of physical illness than men (Austad 2006).

The use of mathematical modelling techniques in biology, ecology and epidemiology is well established (Bies et al. 2008; Grima 2008; Koesters 2005; Wolkenhauer et al. 2009). As biological decline is the opposite of growth, logistic type models are often used in modelling (Ali et al. 2013; Ovaskainen et al. 2010). The Stannard growth model is a logistic, sigmoidal type model and has been used recently in Alzheimer’s disease progression research (Ghazi et al. 2021). Examples of the application of the model can be found in diverse areas of research, from its use in bacterial growth (Zwietering et al. 1990), agricultural research (Korkmaz and Üçkardeş 2013), oil palm growth yield (Khamiz et al. 2005) and bioethanol production from carob extract (Germec et al. 2021). The Gompertz curve (a logistic growth model like the Stannard Growth Model) was derived by Benjamin Gompertz (1825) to estimate human mortality. As such, it is not surprising that two human ageing processes, hearing loss and dementia and their combination, fit a sigmoidal curve.

The parameterized models developed here can be used to both model prevalence of these human ageing processes and the table developed (Table 5) can be applied in a practical way in any Excel sheet to estimate the prevalence of dementia, hearing loss and the combination of both in any age category. Previous models are improved upon by adding age-group independence and highlighting the nature of decay of this biological process as being sigmoidal. Additionally, it provides a way to combine data that has been collected with different age categories, that are often out of sync, leading to imprecise estimates. The present formula does not account for projecting future dementia incidences attributed to varying degrees of hearing impairment, which is acknowledged as a modifiable dementia risk factor. Nonetheless, it facilitates an estimation of the concurrent prevalence of hearing loss and dementia within a given population at a particular point in time (such as during a census).

Although the Stannard growth model for hearing loss was specifically devised for Scotland, the Davis model utilized UK data and the EuroCoDe data drew upon European data. Consequently, the combination of these models is suitable for estimating the prevalence of hearing loss and dementia in the UK. It would be of interest to investigate whether the age-related relationship between hearing loss and dementia extends beyond the UK or Europe, encompassing smaller geographic regions or specific dementia types such as Alzheimer’s, or even different frequency spectra. This would permit a more comprehensive and precise assessment of local and global prevalence rates for hearing loss and/or dementia.

In conclusion, our study highlights the importance of using an epidemiological model to estimate the number of people with hearing loss and dementia in Scotland given diverse age-group data. By using our model, public health planners can develop appropriate services that meet the needs of people with these conditions.