# Leveraging knowledge graphs to update scientific word embeddings using
latent semantic imputation

Jason Hoelscher-Obermaier, Edward Stevinson11footnotemark: 1
Valentin Stauber, Ivaylo Zhelev
Victor Botev, Ronin Wu22footnotemark: 2, Jeremy Minton22footnotemark: 2
Iris AI, Bekkestua, Norway
<EMAIL_ADDRESS>Co-first authors Co-PIs

###### Abstract

The most interesting words in scientific texts will often be novel or rare.
This presents a challenge for scientific word embedding models to determine
quality embedding vectors for useful terms that are infrequent or newly
emerging. We demonstrate how latent semantic imputation (LSI) can address this
problem by imputing embeddings for domain-specific words from up-to-date
knowledge graphs while otherwise preserving the original word embedding model.
We use the Medical Subject Headings (MeSH) knowledge graph to impute embedding
vectors for biomedical terminology without retraining and evaluate the
resulting embedding model on a domain-specific word-pair similarity task. We
show that LSI can produce reliable embedding vectors for rare and out of
vocabulary (OOV) terms in the biomedical domain.

##  1 Introduction

Word embeddings are powerful representations of the semantic and syntactic
properties of words that facilitate high performance in natural language
processing (NLP) tasks. Because these models completely rely on a training
corpus, they can struggle to reliably represent words which are infrequent, or
missing entirely, in that corpus. The latter will happen for any new
terminology emerging after training is complete.

Rapid emergence of new terminology and a long tail of highly significant but
rare words are characteristic of technical domains, but these terms are often
of particular importance to NLP tasks within these domains. This drives a need
for methods to generate reliable embeddings of rare and novel words. At the
same time, there are efforts in many scientific fields to construct large,
highly specific and continuously updated knowledge graphs that capture
information about these exact terms. Can we leverage these knowledge graphs to
mitigate the short-comings of word embeddings on rare, novel and domain-
specific words?

We investigate one method for achieving this information transfer, latent
semantic imputation (LSI) (Yao et al., 2019). In LSI the embedding vector for
a given word, $w$, is imputed as a weighted average of existing embedding
vectors, where the weights are inferred from the local neighborhood structure
of a corresponding embedding vector, $\mathbf{w}_{d}$, in a domain-specific
embedding space. We study how to apply LSI in the context of the biomedical
domain using the Medical Subject Headings (MeSH) knowledge graph (Lipscomb,
2000), but expect the methodology to be applicable to other scientific
domains.

##  2 Related work

Embeddings for rare/out of vocabulary (OOV) words. Early methods for embedding
rare words relied on explicitly provided morphological information
(Alexandrescu and Kirchhoff, 2006; Sak et al., 2010; Lazaridou et al., 2013;
Botha and Blunsom, 2014; Luong and Manning, 2016; Qiu et al., 2014). More
recent approaches avoid dependence on explicit morphological information by
learning representations for fixed-length character n-grams that do not have a
direct linguistic interpretation (Bojanowski et al., 2017; Zhao et al., 2018).
Alternatively, the subword structure used for generalization beyond a fixed
vocabulary can be learnt from data using techniques such as byte-pair encoding
(Sennrich et al., 2016; Gage, 1994) or the WordPiece algorithm (Schuster and
Nakajima, 2012). Embeddings for arbitrary strings can also be generated using
character-level recurrent networks (Ling et al., 2015; Xie et al., 2016;
Pinter et al., 2017). These approaches, as well as transformer-based methods
mentioned below, provide some OOV generalization capability but are unlikely
to be a general solution since they will struggle with novel terms whose
meaning is not implicit in the subword structure such as, e.g., eponyms. Note
that we experimented with fastText and it performed worse than our approach.

Word embeddings for the biomedical domain. Much research has focused on how to
best generate biomedical-specific embeddings and provide models to improve
performance on downstream NLP tasks (Major et al., 2018; Pyysalo et al., 2013;
Chiu et al., 2016; Zhang et al., 2019). Work in the biomedical domain has
investigated optimal hyperparameters for embedding training Chiu et al.
(2016), the influence of the training corpus (Pakhomov et al., 2016; Wang et
al., 2018; Lai et al., 2016), and the advantage of subword-based embeddings
(Zhang et al., 2019). Word embeddings for clinical applications have been
proposed (Ghosh et al., 2016; Fan et al., 2019) and an overview was provided
in Kalyan and Sangeetha (2020). More recently, transformer models have been
successfully adapted to the biomedical domain yielding contextual, domain-
specific embedding models (Peng et al., 2019; Lee et al., 2019; Beltagy et
al., 2019; Phan et al., 2021). Whilst these works highlight the benefits of
domain-specific training corpora this class of approaches requires retraining
to address the OOV problem.

Improving word embeddings using domain information. Our problem task requires
improving a provided embedding model for a given domain, without detrimental
effects on other domains.

Zhang et al. (2019) use random walks over the MeSH headings knowledge graph to
generate additional training text to be used during the word embedding
training. Similar ideas have led to using regularization terms that leverage
an existing embedding during training of a new embedding to preserve
information from an original embedding during training on a new corpus (Yang
et al., 2017). Of course, these methods require the complete training of one
or more embedding models.

Faruqui et al. (2014) achieve a similar result more efficiently by defining a
convex objective function that balances preserving an existing embedding with
decreasing the distance between related vectors, based on external data
sources such as a lexicon. This technique has been applied in the biomedical
domain (Yu et al., 2016, 2017), but has limited ability to infer new
vocabulary because without the contribution from the original embedding this
reduces to an average of related vectors.

Another approach is to extend the embedding dimension to create space for
encoding new information. This can be as simple as vector concatenation from
another embedding (Yang et al., 2017), possibly followed by dimensionality
reduction (Shalaby et al., 2018). Alternatively, new dimensions can be derived
from existing vectors based on external information like synonym pairs (Jo and
Choi, 2018). Again, this has limited ability to infer new vocabulary.

All of these methods change the original embedding, which limits applicability
in use-cases where the original embedding quality must be retained or where
incremental updates from many domains are required. The optimal alignment of
two partially overlapping word embedding spaces has been studied in the
literature on multilingual word embeddings (Nakashole and Flauger, 2017;
Jawanpuria et al., 2019; Alaux et al., 2019) and provides a mechanism to patch
an existing embedding with information from a domain-specific embedding.
Unfortunately, it assumes the embedding spaces have the same structure,
meaning it is not suitable when the two embeddings encode different types of
information, such as semantic information from text and relational information
from a knowledge base.

##  3 Latent Semantic Imputation

LSI, the approach pursued in this paper, represents embedding vectors for new
words as weighted averages over existing word embedding vectors with the
weights derived from a domain-specific feature matrix (Yao et al., 2019). This
process draws insights from Locally Linear Embedding (Roweis and Saul, 2000).
Specifically, a local neighborhood in a high-dimensional word embedding space
$E_{s}$ ($s$ for semantic) can be approximated by a lower-dimensional manifold
embedded in that space. Hence, an embedding vector $\mathbf{w}_{s}$ for a word
$w$ in that local neighborhood can be approximated as a weighted average over
a small number of neighboring vectors.

This would be useful to construct a vector of a new word $w$ if we could
determine the weights for the average over neighboring terms. But since, by
assumption, we do not know $w$’s word embedding vector $\mathbf{w}_{s}$, we
also do not know its neighborhood in $E_{s}$. The main insight of LSI is that
we can use the local neighborhood of $w$’s embedding $\mathbf{w}_{d}$ in a
domain-specific space, $E_{d}$, as a proxy for that neighborhood in the
semantic space of our word-embedding model, $E_{s}$. The weights used for
constructing an embedding for $w$ in $E_{s}$ are calculated from the domain
space as shown in Fig. 1: a k-nearest-neighbors minimum-spanning-tree (kNN-
MST) is built from the domain space features. Then the L2-distance between
$\mathbf{w}_{d}$ and a weighted average over its neighbors in the kNN-MST is
minimized using non-negative least squares. The resulting weights are used to
impute the missing embedding vectors in $E_{s}$ using the power iteration
method. This procedure crucially relies on the existence of words with good
representations in both $E_{s}$ and $E_{d}$, referred to as anchor terms,
which serve as data from which the positions of the derived embedding vectors
are constructed.

Figure 1: Latent Semantic Imputation. $\mathbb{R}^{d}$ is the domain space and
$\mathbb{R}^{s}$ is the semantic space.

##  4 Methodology

We extend the original LSI procedure described above in a few key ways.
Instead of using a numeric data matrix as the domain data source of LSI, we
use a node embedding model trained on a domain-specific knowledge graph to
obtain $E_{d}$. As knowledge graphs are used as a source of structured
information in many fields, we expect our method to be applicable to many
scientific domains. Knowledge graphs are prevalent in scientific fields as
they serve as a means to organise and store scientific data, as well as to aid
downstream tasks such as reasoning and exploration. Their structure and
ability to represent different relationship types makes it relatively easy to
integrate new data, meaning they can evolve to reflect changes in a field and
as new data becomes available.

We use the 2021 RDF dump of the MeSH knowledge graph (available at
https://id.nlm.nih.gov/mesh/). The complete graph consists of 2,327,188 nodes
and 4,272,681 edges, which we reduce into a simpler, smaller, and undirected
graph to be fed into a node embedding algorithm. We extract a subgraph
consisting of solely the nodes of type "ns0__TopicalDescriptor" and the nodes
of type "ns0__Concept" that are directly connected to the topical descriptors
via any relationship type. The relationship types and directionality were
removed. This results in 58,695 nodes and 113,094 edges.

We use the node2vec graph embedding algorithm (Grover and Leskovec, 2016) on
this subgraph to produce an embedding matrix of 58,695 vectors with dimension
200 (orange squares in Fig. 2). The hyperparameters are given in Appendix 8.1.
These node embeddings form the domain-specific space, $E_{d}$, as described in
the previous section. We note that in preliminary experiments, the adjacency
matrix of the knowledge graph was used directly as $E_{d}$ but this yielded
imputed embeddings that performed poorly.

To provide the mapping between the MeSH nodes and the word embedding
vocabulary we normalize the human-readable "rdfs__label" node property by
replacing spaces with hyphens and lower-casing. The anchor terms are then
identified as the normalized words that match between the graph labels and the
vocabulary of the word-embedding model; resulting in 12,676 anchor terms. As
an example, "alpha-2-hs-glycoprotein" appears as both a node in the reduced
graph and in the word-embedding model, along with its neighbors in the kNN-
MST, which include "neoglycoproteins" and "alpha-2-antiplasmin". These serve
to stabilise the positions of unknown word embedding vectors for domain space
nodes which did not have corresponding representations in the semantic space
during the LSI procedure.

LSI has one key hyper-parameter: the minimal degree of the kNN-MST graph, $k$.
The stopping criterion of the power iteration method is controlled by another
parameter, $\eta$, but any sufficiently small value should allow adequate
convergence and have minimal impact on the resulting vectors. Following Yao et
al. (2019) we set $\eta=10^{-4}$ but we use a larger $k=50$ since initial
experiments showed a better performance for larger values of $k$.

Figure 2: Extended latent semantic imputation pipeline. A knowledge graph is
simplified to a smaller, undirected graph. This is used to derive the node
embedding model used in LSI (see Fig. 1) to impute missing terms in the
semantic space.

##  5 Experiments

We aim to answer two questions to evaluate our imputation approach: Do the
imputed embeddings encode semantic similarity and relatedness information as
judged by domain experts? And, can the imputed embeddings be reliably used
alongside the original, non-imputed word embeddings?

We use the UMNSRS dataset to answer these questions (Pakhomov et al., 2010).
It is a collection of medical word-pairs annotated with a relatedness and
similarity score by healthcare professionals, such as medical coders and
clinicians; some examples are shown in Table 1. For each word-pair we
calculate the cosine similarity between the corresponding word embedding
vectors and report the Pearson correlation between these cosine similarities
and the human scores.

Term1 | Term2 | Similarity | Relatedness
---|---|---|---
Acetylcysteine | Adenosine | 256.25 | 586.50
Anemia | Coumadin | 623.75 | 926.50
Rales | Lasix | 742.00 | 1379.50
Tuberculosis | Hemoptysis | 789.50 | 1338.50
Table 1: Examples of UMNSRS word pairs. Scores range from 0 to 1600 (larger =
more similar/related).

To obtain additional insight into the performance of the imputation procedure
we split the words in the UMNSRS dataset into two groups of roughly the same
size: one group of words (trained) which we train directly as part of the word
embedding training and another group of words (imputed) which we obtain via
imputation. This split results in three word-pair subsets that contain
imputed/imputed word pairs, trained/trained word pairs, and imputed/trained
word pairs. Note that due to an incomplete overlap of the UMNSRS test
vocabulary with both the MeSH node labels and our word embedding vocabulary we
cannot evaluate on every word pair in UMNSRS (see Table 4 for more details).
Applying the UMNSRS evaluation to these three groups of word pairs we aim to
measure the extent to which the imputation procedure encodes domain-specific
semantic information.

For word embedding training we prepare a corpus of 74.4M sentences from open
access publications on PubMed (from
https://ftp.ncbi.nlm.nih.gov/pub/pmc/oa_bulk/; accessed on 2021-08-30). To
simulate the problem of missing words as realistically as possible we then
prepare a filtered version of this corpus by removing any sentence containing
one of the imputed terms (in either singular or plural form). This filtering
removes 2.36M of the 74.4M sentences (3.2%).

We then train 200-dimensional skip-gram word embedding models on both the full
and the filtered version of the training corpus. In addition, we also train
fastText embeddings (Bojanowski et al., 2017) on both the full and the
filtered corpus. For details on the hyper-parameters see Appendix 8.2. Since
fastText, which represents words as n-grams of their constituent characters,
has been shown to give reasonable embedding vectors for words which are rare
or missing in the training corpus it represents a suitable baseline to which
we can compare our imputation procedure.

We check that the embedding models (both skip-gram and fastText) trained on
the filtered corpus perform roughly on par with those trained on the full
corpus when evaluated using the trained/trained subset of the UMNSRS test
data. We also check that the skip-gram model trained on the full corpus
performs comparable to the BioWordVec model (Zhang et al., 2019) across all
subsets of UMNSRS. See Appendix 8.3 for details.

LSI is a means of leveraging the domain space to create OOV embedding vectors.
As a simple alternative baseline, we directly use the domain space embeddings
for the OOV words. We need to align the domain space onto the semantic space,
which we do with a rotation matrix derived from the anchor term embeddings in
the two spaces via singular value decomposition.

###  5.1 Results

The main results are displayed in Fig. 3 which shows the Pearson correlation
between cosine similarities and human annotator scores for UMNSRS similarity
and relatedness. The error bars are standard deviations across 1,000 bootstrap
resamples of the test dataset. From left to right we show results for the
trained/trained, imputed/trained, and imputed/imputed subsets.

We compare two models trained on the filtered corpus (which does not contain
any mentions of the imputed words): a skip-gram model extended by LSI and a
fastText model. For reference we also show the correlation strengths obtained
when directly using the MeSH node embeddings which form the basis of the
imputation. Note that for this last model, the test cases we evaluate are
different, since the MeSH model cannot represent all word pairs in UMNSRS (see
appendix 8.3 for details). Uncertainties on the MeSH model are high for the
trained/trained subset due to the limited overlap of the MeSH model with the
words in the trained subset (see Table 4).

In Fig. 3 the imputed/trained group also includes the performance of the
simple baseline, Skip-gram (filtered) + MeSH, formed of a mixture of aligned
embeddings. We do not show the performance of this baseline on the other two
groups since, by construction, it is identical to that of Skip-gram (filtered)
+ LSI for trained/trained and that of MeSH node2vec for imputed/imputed.

(a) UMNSRS similarity.

(b) UMNSRS relatedness.

Figure 3: Correlation with UMNSRS scores.

Three things stand out:

  1. 1.

The LSI-based model is competitive on novel vocabulary: it performs
significantly better than the fastText model on word pairs containing only
imputed terms (imputed/imputed) and modestly better on mixed word pairs
(imputed/trained). It also outperforms the simple but surprisingly strong
baseline, Skip-gram (filtered) + MeSH.

  2. 2.

There is a significant difference in Pearson correlation between the different
word pair categories. Note that the same trend in correlation across word pair
categories can be seen in the word embedding model trained on the full corpus
without imputation (see Fig. 6).

  3. 3.

The LSI-based model obtains better scores than the underlying MeSH node
embeddings across most categories. This proves that the similarity and
relatedness information directly encoded in the domain embedding does not
limit the similarity and relatedness information encoded in the resulting
imputed model.

###  5.2 Discussion

In this paper we use a significantly larger subset of the MeSH graph compared
to related work on MeSH-based embeddings (Guo et al., 2021; Zhang et al.,
2019) by including more than just the topical descriptor nodes. Using a larger
graph for the imputation allows us to impute a wider variety of words and
evaluate the imputation procedure on a larger subset of UMNSRS. The graph we
use for imputation is also much larger than the domain data used in previous
work on LSI (Yao et al., 2019). This shows that LSI can apply to knowledge
graphs and scale to larger domain spaces which is crucial for real-world
applications.

We observe that the UMNSRS similarity and relatedness correlations of the MeSH
node embedding models do not constitute an upper bound on the correlations
obtained for the imputed word embeddings. This is intuitively plausible since
LSI combines the global structure of the trained word embedding vectors with
the local structure of the domain embeddings. This is in contrast to the
original LSI paper in which the domain data alone was sufficient to obtain
near perfect scores on the evaluation task and, as such, could have been used
directly which obviates the need for LSI. This observation reduces the
pressure for an optimal knowledge graph and associated embedding, although a
systematic search for better subgraphs to use is likely to yield improved
imputation results.

It is also of note that most of the trends displayed by the LSI model hold for
both the similarity and relatedness scores, despite these being distinctly
separate concepts. Relatedness is a more general measure of association
between two terms whilst similarity is a narrower concept tied to their
likeness. This might not be the case if the graph construction had been
limited to particular relationship types or if direction of the relations had
been retained.

There are noteworthy differences between our experiment and the use cases we
envisage for LSI. The words we impute in our experiment are taken from the
constituent words of the UMNSRS word pairs rather than being solely defined by
training corpus statistics. This is a necessary limitation of our evaluation
methodology. It remains a question for further research to establish ways of
evaluating embedding quality on a larger variety of OOV words and use this for
a broader analysis of the performance of LSI.

##  6 Strengths and weaknesses of LSI

Our experiments highlight several beneficial features of LSI. It is largely
independent of the nature of the domain data as long as embeddings for the
domain entities can be inferred. It does not rely on retraining the word
embedding and is therefore applicable to cases where retraining is not an
option due to limitations in compute or because of lack of access to the
training corpus. It allows word embeddings to be improved on demand for
specific OOV terms, thus affording a high level of control. In particular, it
allows controlled updates of word embeddings in light of new emerging
research.

The current challenges we see for LSI are driven by limited research in the
constituent steps of the imputation pipeline. Specifically, there is not yet a
principled answer for the optimal selection of a subgraph from the full
knowledge graph or the optimal choice of node embedding architecture. The
answer to these may depend on the domain knowledge graph. Also, there are not
yet generic solutions for quality control of LSI. This problem is likely
intrinsically hard since the words which are most interesting for imputation
are novel or rare and thus exactly the words for which little data is
available.

##  7 Conclusion

In this paper, we show how LSI can be used to improve word embedding models
for the biomedical domain using domain-specific knowledge graphs. We use an
intrinsic evaluation task to demonstrate that LSI can yield good embeddings
for domain-specific out of vocabulary words.

We significantly extend the work of Yao et al. (2019) by showing that LSI is
applicable to scientific text where problems with rare and novel words are
particularly acute. Yao et al. (2019) assumed a small number of domain
entities and a numeric domain data feature matrix. This immediately yields the
metric structure required to determine the nearest neighbors and minimum
spanning tree graph used in LSI. We extend this to a much larger number of
domain entities and to domain data which does not have an a priori metric
structure but is instead given by a graph structure. We demonstrate that LSI
can also work with relational domain data thus opening up a broader range of
data sources. The metric structure induced by node embeddings trained on a
domain knowledge graph provides an equally good starting point for LSI.

This shows that LSI is a suitable methodology for controlled updates and
improvements of scientific word embedding models based on domain-specific
knowledge graphs.

##  8 Future work

We see several fruitful directions for further research on LSI and would like
to see LSI applied to other scientific domains, thereby testing the
generalizability of our methodology. This would also provide more insight on
how the domain knowledge graph as well as the node embedding architecture
impact the imputation results.

The use of automatic methods for creating medical term similarity datasets
(Schulz and Juric, 2020) would facilitate the creation of large-scale test
sets. The UMNSRS dataset, along with the other human-annotated, biomedical
word pair similarity test sets used in the literature, all consist of fewer
than one thousand word pairs (Pakhomov et al., 2016, 2010; Chiu et al., 2018).
The use of larger test sets would remove the aforementioned evaluation
limitations.

Further research could elucidate how to best utilize the full information of
the domain knowledge graph in LSI. This includes information about node and
edge types, as well as literal information such as human-readable node labels
and numeric node properties (such as measurement values). It also remains to
be studied how to optimally choose the anchor terms (to be used in the
imputation step) to maximize LSI performance. Our methodology could also be
generalized from latent semantic imputation to what might be called latent
semantic information fusion where domain information is used for incremental
updates instead of outright replacement of word embedding vectors.

Finally, LSI could also be extended to provide alignment between knowledge
graphs and written text by using the spatial distance between imputed vectors
of knowledge graph nodes and trained word embedding vectors as an alignment
criterion.

## Acknowledgements

This paper was supported by the AI Chemist funding (Project ID: 309594) from
the Research Council of Norway (RCN). We thank Shibo Yao for helpful input and
for sharing raw data used in (Yao et al., 2019) and Dr. Zhiyong Lu and Dr.
Yijia Zhang of the National Institute of Health for sharing their word
embedding models. We thank the three anonymous reviewers for their careful
reading and helpful comments.

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## Appendix

###  8.1 Hyper-parameters for MeSH node2vec

We train node2vec (https://github.com/thibaudmartinez/node2vec) embeddings
with the hyperparameters shown in Table 2 from a subgraph of MeSH containing
58,695 nodes and 113,094 edges.

Hyperparameter | Variable name | Value
---|---|---
Training epochs | epochs | 50
No. of random walks | n_walks | 10
Return parameter | p | 0.5
Inout parameter | q | 0.5
Context window | context_size | 15
Dimension | dimension | 200
Table 2: Hyperparameters for MeSH node2vec training

###  8.2 Hyper-parameters for word embeddings

We use gensim (https://radimrehurek.com/gensim; version 4.1.2.) for training
skipgram and fastText word embedding models with the hyperparameters provided
in Table 3. All other hyperparameters are set to the default values of the
gensim implementation. For the skipgram model we use the hyperparameters from
Chiu et al. (2016), which are reported to be optimal for the biomedical
domain. For fastText we are not aware of literature on optimal hyperparameters
for the biomedical domain so we use the default values except for the
embedding dimension which we set to 200 to ease comparison with the skipgram
model. We trained the fastText models for 10 epochs but found that the
performance of the fastText model on UMNSRS saturates after epoch 1. We use
the fastText model after the first epoch for the remainder of our experiments
and analysis.

Variable name |  fastText |  skipgram
---|---|---
epochs | 1 | 10
negative | 5 | 10
vector_size | 200 | 200
alpha | 0.025 | 0.05
sample | 1E-03 | 1E-04
window | 20 | 30
Table 3: Hyperparameters for skipgram and fastText training. See the gensim
documentation for the definition of the hyperparameters.

(a) UMNSRS similarity.

(b) UMNSRS relatedness.

Figure 4: UMNSRS correlations for skipgram models.

(c) UMNSRS similarity.

(d) UMNSRS relatedness.

Figure 5: UMNSRS correlations for fastText models.

(a) UMNSRS similarity.

(b) UMNSRS relatedness.

Figure 6: UMNSRS correlations for BioWordVec. | UMNSRS relatedness | UMNSRS similarity
---|---|---
Model |  | trained/
---
trained
| imputed/
---
trained
| imputed/
---
imputed
| trained/
---
trained
| imputed/
---
trained
| imputed/
---
imputed
MeSH node2vec | 28 | 70 | 133 | 30 | 72 | 135
all other models | 83 | 99 | 124 | 84 | 101 | 126
Table 4: Number of test cases per model and test set split for UMNSRS
evaluation.

###  8.3 Details on the UMNSRS evaluation

Table 4 shows the number of test cases per model and UMNSRS test data split.
All models have been evaluated on the same subsets of UMNSRS except for the
MeSH node embeddings model where limited overlap with the UMNSRS test
vocabulary prevents us from evaluating on exactly the same subsets.

The embedding models (both skip-gram and fastText) trained on the filtered
corpus perform roughly on par with those trained on the full corpus when
evaluated using the trained/trained subset of the UMNSRS test data (see Fig. 6
and 6). When comparing the performance of the filtered skipgram model + LSI to
the full skipgram model on the subset of test data involving imputed words
(imputed/trained and imputed/imputed) the full model outperforms LSI (see Fig.
6). This suggests that, if training text for the OOV words were available, we
should make use of it. Similarly, and as expected, when comparing the
performance of the filtered and full fastText models on the subset of test
data involving imputed words (imputed/trained and imputed/imputed) the full
model again outperforms the filtered model (see Fig. 6).

As a sanity check, we also compare the skip-gram model trained on the full
corpus to BioWordVec, a recent state-of-the-art word embedding model for the
biomedical domain (Zhang et al., 2019) and find similar performance across all
subsets of UMNSRS (see Fig. 6).


# Modelling Social Care Provision in
An Agent-Based Framework with Kinship Networks

Umberto Gostoli
MRC/CSO Social and Public Health Sciences Unit
University of Glasgow, Glasgow, UK G2 3AX
<EMAIL_ADDRESS>
Eric Silverman
MRC/CSO Social and Public Health Sciences Unit
University of Glasgow, Glasgow, UK G2 3AX
<EMAIL_ADDRESS>

###### Abstract

Current demographic trends in the UK include a fast-growing elderly population
and dropping birth rates, and demand for social care amongst the aged is
rising. The UK depends on informal social care – family members or friends
providing care – for some 50% of care provision. However, lower birth rates
and a graying population mean that care availability is becoming a significant
problem, causing concern amongst policy-makers that substantial public
investment in formal care will be required in decades to come. In this paper
we present an agent-based simulation of care provision in the UK, in which
individual agents can decide to provide informal care, or pay for private
care, for their loved ones. Agents base these decisions on factors including
their own health, employment status, financial resources, relationship to the
individual in need, and geographical location. Results demonstrate that the
model can produce similar patterns of care need and availability as is
observed in the real world, despite the model containing minimal empirical
data. We propose that our model better captures the complexities of social
care provision than other methods, due to the socioeconomic details present
and the use of kinship networks to distribute care amongst family members.

_K_ eywords Social Care, Kinship Networks, Agent-Based Modelling

##  1 Introduction

As human lifespans continue to lengthen and birth-rates drop throughout much
of the developed world, many nations are experiencing an increase in demand
for social care – the provision of personal and medical care for people in
need of assistance due to age, disability or other factors. In the UK, the
elderly consume the largest share of social care, and lower birth-rates mean
that the supply of available carers is decreasing over time even as demand is
growing rapidly [1]. As a result, social care is a frequent topic in UK policy
debates, with widespread concern that the country will be unable to afford the
significant public investment needed to fulfill this growing care need.

Age UK notes that nearly 50% of those aged 75 and over are living with a long-
term limiting health condition – and the fastest-growing age group in the UK
is the over-85s, so the problem will only get worse in the coming decades [2].
As a result, social care need is rising at a pace that outstrips the growth of
public and private care supply. Unmet social care need is therefore an
increasingly important and widespread social problem in the UK. The 2017 Ipsos
MORI report _Unmet Need for Care_ showed that “more than half of those with
care needs had unmet need for at least some of their needs” [3]. This means
that less than half of those elderly individuals in need of assistance with
activities of daily living (ADLs) were able to receive sufficient care. Large
numbers of UK citizens are thus living without sufficient care; an estimated
1.2 million people did not receive sufficient care for their needs in 2017, an
increase of 48% since 2010 [2].

The increased pressure on the social care system also has an impact on the
health care system, with delayed discharges from hospital being a particularly
expensive consequence of unmet care need. Patients in need of social care
frequently stay in hospital longer than necessary due to a lack of care
availability, leading to a shortage of beds for other patients and increased
costs to hospitals. According to Age UK, between 2010 and 2016 the number of
additional bed-days attributable a lack of available home-care packages
increased by 181.7%, while wait times for residential care placements
increased by 40% [2]. In 2016 the National Audit Office estimated the total
delays attributable to care shortages amounted to 2.7 million bed-days per
year, resulting in an annual cost to the National Health Service (NHS) of
approximately £820 million [4].

The UK is largely dependent on informal social care, or care provided free-of-
charge by family members and loved ones, in order to meet the needs of the
population. Informal care is enormously widespread in the UK and is much
larger than the formal care infrastructure. The Family Resources Survey
2013/14 showed that there were 5.3 million informal carers in the UK [5],
while projections indicate that the number of people receiving informal care
will increase by 60% in the period 2015-2035 [6]. The increase in demand is
such that Carers UK proposes that carer numbers would need to increase 40%
over the next two decades to meet demand [7]. At the same time, the number of
people using privately-funded social care is expected to rise by almost 50% by
2035, and private expenditure for social care is projected to increase from
£6.8 billion to almost £20 billion in the same period, almost a three-fold
increase [6].

Understanding how the need for social care evolves and the process through
which informal and formal care is provided is a vital component in developing
sustainable social care policies. In this regard, the importance of support
and care-giving networks has long been recognized [8]. In particular, research
has shown that informal care is provided mostly through care networks with an
average of three to five members [9]. These networks are predominantly
composed of an individual’s close relatives. Aldridge and Huges [5] report
that 72% of carers provide care to a member of their immediate family, whether
a parent (40%), partner (18%), son or daughter (14%), while Petrie and Kirkup
[10] show that 51% of carers provide care to someone in their household.
Wettstein and Zulkarnain[11] show that, in 2011, 31% of informal care in the
US was provided by spouses, 47% by children and 18% by other relatives (sons-
in-law, daughters-in-law and grandchildren), with only 4% being provided by
non-relatives.

Moreover, empirical research has shown that the kind of social care provided
is affected by socioeconomic status. Petrie and Kirkup [10] report that people
working in routine occupations and those with lower qualifications are more
likely to provide informal care. Moreover, Laing and Buisson [12] find that,
while across the UK an estimated 24% of care home residents are funded through
private ‘top-ups’, in North-East England only 18% are funded privately, as
opposed to 54% in the wealthier South-East. Overall, these findings suggest
that informal care becomes less common in higher socioeconomic status groups
than in lower groups, while formal care becomes more common.

Finally, the social care literature also reveals a significant gender gap in
social care provision. Wettstein et al. [11] reported that in the US daughters
are almost twice as likely (31%) to provide care as compared to sons (16%).
These findings are confirmed by Petrie and Kirkup [10], who find that 59% of
family carers in the UK are women. At the population level, 16% of women
provide informal care as compared to 12% of men.

In this paper, we propose an agent-based model of the UK informal and
privately-funded formal social care system. This model reflects the complexity
of a system where demographic, social and economic processes interact to
determine the dynamics of social care demand and supply. Our aim is to provide
a theoretical framework that allows us to to improve our understanding of the
mechanisms driving unmet social care need. Moreover, using ABMs enables us to
model scenarios of economic and social policy change in virtual populations,
providing a means to test complex policies targeted at multiple levels of
society. Such models can allow policy-makers to experiment with potential
policy interventions and reveal any possible unintended side-effects of those
policies prior to implementing them in the real world.

Previous work has attempted to address the social care problem using agent-
based simulation approaches [13] [14]. Here we present a simulation that
significantly extends these previous efforts. The model provides a more
comprehensive simulation of social care provision behaviour, via the inclusion
of a detailed socioeconomic model and the representation of care provision as
not just a simple one-to-one exchange of resources, but a complex negotiation
taking place between members of the care receivers’ kinship networks.

##  2 The Model

###  2.1 Motivations

Here we present an agent-based model as a formal representation of the complex
demographic and social processes affecting informal social care demand and
supply, and the dynamics of the social care outcomes resulting from their
interaction over time. Our primary concern at this stage, was not the precise
replication of empirical, real-world data but, rather, the development of a
theoretical framework potentially capable of representing the full complexity
of the social care system. In our results we aim for _qualitative similarity_
to real-world social care trends, rather than precise numerical replication,
in order to determine if our modelling of the underlying processes is
producing appropriate outcomes.

With these motivations in mind, at this initial stage the behavioural
assumptions made in this paper should not be considered necessarily accurate,
but rather should be seen as a first approximation, i.e. a necessary point of
departure allowing us to provide proof-of-concept results and demonstrate the
model’s potential as a policy-making tool. Further work and more specialized
expertise will be needed to refine and revise the model’s behavioural
assumptions in order for the model to be used in this way.

Ultimately, our aim is for this simulation to facilitate the development and
evaluation of alternative social care policies, including complex
interventions aimed at behavioural change. Our addition of health care costs
and a detailed economic model to the simulation can also clarify the impact of
interventions on other key areas of policy, enable more robust policy
evaluation, and reduce the incidence of unintended consequences derived from
otherwise well-meaning policy prescriptions. Future versions of this framework
can also be applied to nations other than the UK, simply by altering the
simulated geography and population data.

###  2.2 Basics of the model

The model itself is complex and contains many detailed economic and social
processes and sub-processes; this section provides a high-level overview of
the model’s functionality. For details of the previous models which formed the
initial basis for this simulation, please refer to Noble et al. [13] and
Silverman et al. [14]. 111For those who wish to examine the model more
closely, or run it for yourself, you can find the annotated Python code
available at https://github.com/UmbertoGostoli/ABM-for-social-
care/releases/tag/v.09

The simulated agents occupy a space roughly based on UK geography. Agents live
in houses which form towns containing clusters of up to 1225 houses. These
clusters vary in size in rough proportion to real-world population density,
which varies across the 8$\times$12 grid composing the model’s geography. The
agent population is scaled down from real UK levels at a factor of roughly
1:10,000.

The model updates in one-year time steps. Initial populations are generated
and randomly distributed in the year 1860, and the model then runs until 2040
when final social care costs are calculated and recorded. We start the model
in 1860 to ensure the population dynamics have time to settle before empirical
population data (UK Census data) is integrated into the model in 1951.

###  2.3 Agent Life-Course

Newborn agents are classed as dependent children until age 16, at which point
they reach adulthood (the minimum working age in the UK). The agents then
decide whether to continue to study or to start looking for a job. This choice
is repeated every two years until the age of 24. This is a probabilistic
choice that depends on the household’s income level and on the parent’s
education level. When the agent decides to look for a job (or when it reaches
age 24 at the latest), the agent then enters the workforce. During their
working life, agents can be hired, fired and can change jobs. When an agent is
unemployed, they find a job with a certain probability which depends on the
unemployment rate (which is an input of the model), then start earning an
income and paying tax. At a certain age, agents retire from the workforce (at
age 65 when default parameter settings are used), at which point they stop
paying tax and begin receiving a pension.

This version of the simulation uses a Gompertz-Makeham mortality model to
approximate mortality rates until 1951, as in Noble et al. [13]. From 1951 the
simulation uses mortality rates from the Human Mortality Database [15]. After
2009 a Lee-Carter model is used to generate future mortality rates, as in
Silverman et al. [14].

####  2.3.1 Partnership Formation

Upon reaching adulthood, agents can form partnerships with other agents
222Hereafter we use ‘partnership’ as shorthand for relationships capable of
producing children.. Employed male agents and adult female agents are randomly
paired with probabilities based on the agents’ socioeconomic difference, age
difference and geographical distance (with the relative weight of these three
factors depending on the model’s parameters). Age-specific annual divorce
probabilities determine whether a couple dissolves their partnership.
Fertility rates follow the procedures outlined in Silverman et al. [14], in
which data from [16] and the Office for National Statistics [17] are used from
1950–2009, at which point Lee-Carter projections are used.

####  2.3.2 Migration

As in the real world, agents can migrate for a variety of reasons. An agent
becomes independent when they leave the parental home. This can happen when
they form a partnership, when they find a job in a different town from the
family home, or with a certain probability when they find a job in the same
town of their parental home. When a partnership dissolves, the male agent will
move elsewhere on the map, while any dependent children resulting from that
partnership will stay with the mother. A family will relocate to a new house
if one of the two parents finds a job in another town or when the family needs
a larger house due to the increased size of the family. Retired agents with
social care needs may elect to move into the household of one of their
children, with a probability directly proportional to the amount of care
provided by that household. Rarely, dependent children will be orphaned before
reaching adulthood; in that case the agent will be adopted by a household of
their kinship network or by a randomly-selected couple if no such household is
available.

###  2.4 Health status and care need

Agents begin in a healthy state and have no need of additional care. They may
enter different categories of care need depending on age- and gender-specific
probabilities. Table 1 shows the five possible categories of care need and the
amount of care required per week at each level of need. Agents who enter a
state of care need do not recover and return to normal health, but instead
continue to progress to higher levels of need over time. The probability of
progressing to higher care need levels depends positively on the agent’s age
and on the discounted sum of the agent’s unmet care need in past periods
333The assumption is that a prolonged period of unmet care will increase the
agents’ frailty and, therefore, the probability that their health will further
deteriorate..

Table 1: The different care need categories, with the number of hours of care required per week Care need category | Weekly hours of care required
---|---
None | 0
Low | 8
Moderate | 16
Substantial | 32
Critical | 80

Social care provision is linked to informal care availability in an agent’s
kinship network. An agent’s kinship network’s nodes consist of the households
of agents with a familial relationship with the agent; the degree of kinship
is defined as the network distance between the household and the agent. If
they have time or available income, agents will provide informal or formal
care to anyone in their kinship network with care need. The amount of care
agents are available to provide depends on their status (through their
income), their kinship relationship with the receiver and their geographical
distance from the receiver (see the Kinship Network sub-section below for
details).

###  2.5 Model Enhancements

This updated version of the Linked Lives model presented in Silverman et al.
[14] has been rewritten from the ground up and substantially extended for
greater detail and realism. The following features have been introduced:

  * •

The population is composed of five socioeconomic status groups.

  * •

The care supply is provided by the agent’s kinship network, (a network of
households which have a kinship relationship to the agent).

  * •

Households allocate part of their income to care provision, which can be in
the form of both informal and formal care.

  * •

A salary function implying an inverse relationship between time taken off work
to provide informal care and the agent’s hourly wage.

  * •

Unmet social care needs affect the agents’ hospitalization probability (and
the associated health care costs).

####  2.5.1 Socioeconomic Status Groups

The population is composed of 5 distinct socioeconomic status (SES) groups.
These categories follow the Approximated Social Grade, a socioeconomic
classification produced by the Office for National Statistics, which is
composed of 6 categories (A, B, C1, C2, D and E). For convenience, we
redistributed category E (state pensioners, casual and lowest grade workers,
unemployed with state benefits only) into the categories D (semi-skilled and
unskilled manual workers) and C2 (skilled manual workers), to maintain a
unimodal distribution. We initialised the groups’ distribution to roughly
reflect the 2016 UK distribution. The SES groups are characterized by
different education levels, different career paths (represented by income
growth curves) and different unemployment rates.

The introduction of SES groups has a number of effects on the various stages
of agent life-courses. A higher socioeconomic position is associated with
lower mortality and fertility rates, and to a lower probability of developing
care need. In the marriage market, the probability that two opposite-sex
individuals will form a couple depends on their SES distance (which determine
the marriage probability together with the partners’ geographical distance and
age difference)444The relationship between SES distance and probability of
forming a couple is asymmetric: the probability of getting married decreases
less rapidly with the SES distance if the higher-status individual is male
rather than female.. In the job market, a higher SES is associated with higher
starting and maximum salaries (but a lower salary growth rate); a higher
probability to find a job and a lower probability to be fired (probabilities
which are reflected in a lower unemployment rate); and a higher probability to
change jobs if the job offer comes from a different town from their current
hometown.

Given that in this model care supply and relocation decisions depend on income
level, the socioeconomic position of an agent affects its behaviour (and that
of the household it belongs to) through the agent’s income, as a higher
socioeconomic position is associated with a higher income level. For example,
the share of income the household allocates to care supply depends positively
on the household’s per capita income (see the Formal Care section below).

We included an inter-generational mobility process which allows agents to move
to a different SES group from their parents’. Each SES group is associated
with an education level. From the age of 16, an agent can decide whether to
continue its studies (a choice that will allow the agent to reach a higher
education level and therefore a higher SES group) or start searching for a job
(in which case the agent is assigned the SES group associated with the
education level reached). This choice is made by the agents every two years,
until the age of 24 (i.e. at ages 16, 18, 20 and 22) 555We assume each
education step lasts 2 years. The educational stages correspond to A-level,
Higher National Diploma, Degree and Higher Degree.. The probability that an
agent keeps studying depends on three factors: the per capita available income
(i.e, net of social care costs) of the agent’s household; the difference
between the maximum education level reached by the agent’s parents and the
agent’s current education level; and the amount of time the agent allocates to
informal social care supply. In this highly stylized inter-generational
mobility process, an agent’s SES group is determined by the education level
the agent reaches666Meaning the SES group of agents ending their studies when
they are 16, 18, 20, 22 and 24 year-old, will be, respectively, D, C2, C1, B
and A..

####  2.5.2 Kinship Networks

Each agent is associated with a kinship network – a network of households
containing at least one agent with a kinship relation to them. This network
includes:

  * •

the agent’s household (distance 0);

  * •

the households of the agents’ parents and of children who are not part of the
agent’s household (distance I);

  * •

the households of the agents’ grandparents, grandchildren, brothers and
sisters who are not part of the agent’s household nor of the households at
distance I (distance II);

  * •

the households of the agents’ uncles, aunts, nephews and nieces who are not
part of the agent’s household nor of the households at distance I and II
(distance III).

The kinship network has two functions. First, the network defines the total
care supply available for a particular agent in need. The care receiver’s
total care supply is the sum of the available care supply of all the members
of all the households that are part of the kinship network which met certain
conditions. These conditions represent kinship- and space-specific limitations
in the supply of care. Kinship relationship and distance determine the
quantity of informal care that each member of the receiver’s kinship network
is available to supply, as shown in Table 2.5.2:

[htb] Amount of care agents can provide depending on their status and distance
from the receiver Agent status Household (D-0) D-I D-II D-III Teenager 16 0 0
0 Student 24 16 8 4 Employed 28* 20* 12* 8* Unemployed 32 24 16 8 Retired 48
36 20 10

  * •

*These are the minimum amounts the employed can offer, representing the informal care provided outside of the working hours. If needed, additional informal care can be supplied by these agents taking time off their working hours. Moreover they can use their income to pay for formal care. See the Formal Care section for details.

In particular, we assume first that the amount of care which an agent is
available to provide to another agent depends positively on their kinship’s
closeness. Agents living with the care receiver are an exception, as these
agents’ available care supply is assumed to be equal to that of the receiver’s
next of kin (i.e. spouse, children and parents), independently from the
kinship degree (although most of the time the members of the receiver’s
household are his/her next of kin). The second factor affecting the informal
care supply availability is the physical distance from the receiver’s
household. With regards to the physical distance we can distinguish three
classes of care suppliers:

  * •

agents living in the receiver’s household;

  * •

relatives living in another household in the same town as the care receiver;

  * •

relatives living in a different town.

We assume that only households living in the same town of the care receiver
can provide informal care. Moreover, in the case of formal care we assume that
only the care receiver’s household and the households at distance I (i.e., the
care receiver’s parents or children) are available to provide formal care.

The care allocation process proceeds in a series of steps in which a 4-hour
‘quantum’ of care supply is transferred to an agent needing care from one of
the households with available supply in its kinship network. The care
allocation function first samples a care receiver from the pool of people with
unmet social care need who have a kinship network with available care supply.
The probability of each care receiver being sampled is directly proportional
to the care receivers’ quantity of unmet care need. Then the allocation
function samples a household from the selected care receiver’s kinship network
with a probability that is directly proportional to a distance-weighted
measure of the household’s available supply 777For an household at kinship
network’s distance x from the care receiver, the weight is the reciprocal of
the exponential function of the product of x and a parameter.. Once the
supplying household has been selected, a ’quantum’ of care is transferred from
one of the household’s members with available supply to the care receiver.

The member who is to provide care within the selected household is determined
in two steps. First, one of the household’s six possible care sources is
selected with a probability proportional to the residual available care of
each source. The six care sources consist of the five groups that can provide
informal care as shown in Table 2: teenager, student, retired, unemployed and
employed; plus a sixth source which represents the amount of care which the
household is available to supply by allocating part of the household income (a
category we call out-of-income care). The household’s out-of-income care
supply is the share of income that the household has available to allocate to
care, either directly in the form of formal care, or indirectly in the form of
informal care provided by employed household members (who provide care by
taking unpaid time off work). If one of the first five sources is selected,
the household member with the greater residual available care within that
category will provide care. If the out-of-income category is selected, the
household will provide formal care if the lowest hourly wage among the hourly
wages of the employed household members is higher than the hourly cost of
social care. Otherwise, the employed household member with the lowest hourly
wage will provide the ‘quantum’ of care888Given that the quantity of out-of-
income care supply depends on the household’s per-capita income, this care
allocation mechanism implies that the probability that members of the
households will spend time providing informal care is inversely related to the
household’s per-capita income. In other words, the wealthier the supplying
household, the more likely that it will provide formal care.

At the end of this step, both the care receiver’s care need and the supplying
household’s availability are reduced by the amount of care transferred (i.e.
the 4-hour ‘quantum’ of care). The allocation process is repeated until the
set of care receivers with unmet care need and available care supply is empty
(i.e. there are no more care receivers with outstanding care need and
available care supply). The ultimate result of this care allocation function
is that units of social care are distributed from potential providers to
receivers across the receiving agents’ kinship network, and decisions about
who provides care and choosing informal or formal care provision are made
according to the composition, socioeconomic position and employment status of
potential care providers. We suggest this detailed decision-making process
better represents the complexities of care decisions that families need to
navigate. The specifics of this complex process can be adjusted further by
incorporating insights received from qualitative and quantitative data on
informal care-givers and their decision-making.

The kinship network also allows households to compute the informal care
attraction associated with each town, which represents a rough measure of the
informal care that a household expects to get or supply in a given town. The
households use this town-specific attraction to make relocation decisions
999In this model, agents face the choice of relocating to other towns either
if they receive a job offer or partner with someone from another town.. More
precisely, for a particular household $H_{i}$, the informal care attraction of
a town $T$, is a function of the sum of the members of the households with a
kin relationship with H, living in the town $T$ 101010The household’s kinship
network is obtained by joining the kinship networks of the household’s
members.. The contribution of a household $H_{j}$ to the social care
attraction is weighted according to the degree of kinship between $H_{i}$ and
$H_{j}$. This weighted sum is then multiplied by the complement to one of the
share of care provided by the government. Two assumptions characterise the
towns’ informal care attraction: first, the larger the local kinship network
in a town (in terms of number of people who are part of it), the higher the
amount of social care the agent can expect to receive (or to supply) and the
higher the social care ‘value’ associated with a town; second, the higher the
share of care supplied by the government, the lower the importance of the
potential informal care in the relocation decision.

Apart from this ‘network size’ factor, a household’s probability of relocating
depends on the relocation cost, which increases with the household’s size and
the number of years the household’s members lived in the current town
111111The relocation cost R can be thought of as a measure of the social
capital developed by the household in their current town. This social capital
would be lost by relocating to another town, so it acts as a barrier to
relocation. Formally, relocation cost is computed as:
$R=K\sum\limits_{i=1}^{n}y_{i}^{p}$ where n is the set of the household’s
members, yi is the number of years the household member i spent in the current
town and p is a parameter with value smaller than 1 (as additional years will
increase the member’s social capital by increasingly smaller amounts). K is a
scaling parameter., and the town’s homophily attraction, which depends on the
town’s share of people belonging to other SES groups. Towns with a larger
share of unoccupied houses are more likely to be chosen for relocation.

####  2.5.3 Formal Care

Both informal and formal care are allocated through the care recipient’s
kinship network. Each household allocates a share of its income to care, which
increases with the household’s per capita income 121212More precisely, with x
being the household’s per-capita income, the share allocated to care supply is
the complement to one of the reciprocal of the exponential function of the
product of x and a parameter.. We assume that formal care can be provided only
by the care receiver’s own household or from the households in the care
receiver’s kinship network with distance equal to 1131313In other words, only
the care receiver’s parents and children are available to pay for formal care
if they cannot provide informal care.. As explained above, the choice between
the informal and formal care is stochastic, with probabilities equal to the
relative availability of the two kinds of care. In order for the household to
supply formal care, the hourly wage of the working member with the lowest wage
(and available time left) must be higher than the price of formal social care.
If the hourly wage of this member is lower than the price of formal social
care, they will take time off work to provide informal care (as in this case
it is cheaper for the household to give up the agent’s salary than pay for
formal care). However, the supplying household can provide informal care only
if it is in the same town of the care receiver, otherwise it will provide only
formal care.

####  2.5.4 Salary function.

We model the hourly salary that an agent receives (on average) when it finds a
job using the Gompertz function. This function is a double exponential that
takes the following three SES-specific arguments:

  * •

the initial salary level;

  * •

the final salary level;

  * •

the salary growth rate.

The salary growth rate is multiplied by the agent’s cumulative work
experience, which is the discounted sum of all the fractions of the working
week allocated to work (if an agent works full time, this fraction is equal to
1) 141414Formally, the salary function is:  $w=Fe^{ce^{-rt}}$ where:
$c=ln{\frac{I}{F}}$ and F is the maximum hourly wage, I is the initial hourly
wage, r is the wage growth rate and t is the discounted cumulative work
experience..

This equation implies that if an agent takes time off work to provide informal
care, this will result in less work experience and, therefore a lower hourly
salary. Given the properties of the care allocation mechanism, the agents
employed with the lowest hourly salary will be more likely to provide informal
care in the future (if there are not retired/students/unemployed agents or if
the household’s retired/students/unemployed agents do not have available
informal care supply). In this model new mothers devote their time to caring
for their newborn, meaning that we see a gender pay gap emerge due to the
interaction between:

  * •

the allocation of part of the household’s income to care supply;

  * •

the choice of the care giver within the supplying household;

  * •

the salary function.

Retired agents receive a pension which is proportional to their final income
level. We assume that when an agent needs care they retire due to sickness and
thereafter receive a pension. If their care need is low or moderate, their
ill-health pension is their normal pension reduced in accordance with the
ratio between lost working years and the maximum number of working years (in
other words, the working years of a person in good health). If the agent’s
care need is substantial or critical, the ratio is computed by reducing the
number of lost working years by 50%.

####  2.5.5 Hospitalisation

We assume that agents with care need will spend additional time in the
hospital, time whose duration is a function of the agents’ care need level and
the average discounted share of unmet care need. The higher the agent’s care
need level and the higher her average share of unmet care need, the greater
the number of days the agent is expected to spend in the hospital each year.
By multiplying the sum of the hospitalisation’s duration by the
hospitalisation cost per day, we can determine the cost of unmet care need for
the public health service.

####  2.5.6 Simulation steps

More specifically, the simulation unfolds through the following eleven steps:

  1. 1.

deaths: with a given probability which depends on age, SES and care need
level, some agents are removed from the population;

  2. 2.

adoptions: children without parents are adopted;

  3. 3.

births: with a probability which depends on age and SES, married woman give
birth to new agents;

  4. 4.

divorces: some couples dissolve (and the male relocates);

  5. 5.

marriages: with a certain probability which depends on age, SES and
geographical location, some singles get married and go to live together.

  6. 6.

social care allocation: social care is transferred from agents with available
hours (or income) to agents with social care needs;

  7. 7.

age transitions: the age of the agents is incremented, and their age-related
status is updated.

  8. 8.

social transitions: students decide whether to start working or keep studying,
depending on their family income per capita, their parents’ SES and their care
responsibilities. In case they start looking for a job, they are assiged the
SES associated to the education level they have reached.

  9. 9.

job market: employed and unemployed agents receive new job offers (and
eventually accept them) and employed agents are fired, with probabilities
which depend on the age and SES-specific unemployment rate.

  10. 10.

relocations: agents who have accepted job offers from towns different from
their current town, relocate.

  11. 11.

care transitions: with a probability which depends on age, SES, current care
need level and unmet care need, agents pass to higher care need levels or are
hospitalised.

##  3 Social Policy Experiments

ABM allows us to investigate the effects of virtual social or economic
policies. This model is characterised by various policy-related parameters,
the value of which can be directly or indirectly related to specific measures
of social care policy. Therefore, by varying these parameters, we can examine
‘what if’ scenarios and simulate social care outcomes and costs under
alternative social care policies.

For illustrative purposes, we investigated the effect on social care of two
policies: the introduction of tax-deductible social care expenses, and direct
governmental funding of care for people above a certain level of care need.
For the first policy, we assume that the carers are allowed to deduct 100% of
social care expenses from their tax. In the second policy experiment, we
assume that the government directly pays for the social care expenses of those
agents with the highest social care need (i.e., care need level at ‘critical’,
see Table 1).

We assume that the two policies are implemented from simulation year 2020 and
compare the outputs of these two policy scenarios with the benchmark no-policy
scenario in the period 2020-2040. We present two results: the hours of unmet
care need in each scenarios and the incremental cost-effectiveness ratio
(ICER) of the two policies considered 151515In this paper we formally define
the ICER as:  $e=\frac{C_{p}-C_{b}}{U_{b}-U_{p}}$ where C is the total
additional cost of policy implementation, U is the total unmet care need and
the subscripts p and b indicate two scenarios (i.e. the policy scenario and
the benchmark scenario, respectively). In this case, being b the no-policy
scenario, Cb is equal to $0$. .

Figure 1: Hours of informal and formal care received and of unmet care need in
the simulated population.

Figure 2: Hours of informal and formal care received and unmet care need per
recipient.

##  4 Results

As described above, this simulation is highly complex with numerous processes
at play, and a number of parameters governing system behaviour. For these
early-stage results, we present figures from a representative single run at
default parameter values plus a comparison of this run, taken as the
benchmark, with two simulations where we introduce two alternative policy
interventions in the year 2020. The single-run charts displayed here were
chosen to highlight the key features of this new simulation framework, namely
the modelling of formal care, additional economic and labour market details
such as socioeconomic status groups, interaction between social care need and
health care demand, and agent kinship networks.

Figure 2 shows the evolution of hours of informal and formal care delivered
and of unmet social care need per week for the whole population over the
period 1990-2040. In our simulations, the total population reaches a peak of
around 10,000 agents in the period 2025, and then decreases slowly to around
8000 by 2040. As in Bijak et al. [18], the simulation’s population projections
roughly match those of the Office for National Statistics in the UK if we
assume no international migration.

Figure 3: Hours of informal and formal care received and unmet care need per
recipient by care need level.

Figure 4: Hours of informal and formal care received and unmet care need per
recipient by SES group (with I being the poorest SES group).

Figure 5: Informal care supplied by women as share of total informal care
supplied, for the population.

Figure 6: Ratio of women’s to men’s income, for the population as a whole.

In Figure 2 we can see that from 2010 informal care cannot keep pace with the
growth in care need, with a consequent rapid increase of unmet care need,
which reaches a peak in 2025. In Figure 2 we show the weekly hours of informal
and formal care delivered and the unmet social care need per recipient. Here
we can see that the negative trend of the unmet care need after 2025 in Figure
2 is entirely due to the decrease of population: in fact, the unmet care need
per recipient keeps growing quite steadily up to 2033.

Figure 4 shows the mean informal and formal care received and the unmet care
need by care need level over the period 2020-2040 (note that the heights of
the bars correspond to the weekly hours of care required as shown in Table 1).
We can see that most of the unmet care need is due to the agents with the
highest care need. Figure 4 shows the mean informal and formal care received
and the unmet care need by SES group (with I being the poorest SES group). We
can see that the relative weight of informal care with respect to formal care
decreases from the poorest to the wealthiest SES group, a result which
reflects empirical findings. As expected, by comparing the green bars in
Figure 4, we can see that the poorest SES group has a somewhat higher level of
unmet care need compared to the wealthiest group but also that unmet care need
is a significant share of total care need across SESs.

Figure 6 shows the share of total informal care provided by women. At the
population level this figure starts from just below 75% in 1990, then
decreases steadily until 2020, when it starts fluctuating just above 60%, a
value which is in line with empirical findings. Moreover, there is a marked
difference between SES groups, with the poorest SES group having the highest
(and growing) gender inequality. As shown in Figure 6, this inequality in care
provision affects income inequality between genders. At the population level,
female agents’ income is about 5% lower than the male’s income. Therefore our
model can explain at least part of the gender pay gap.

Figure 8 shows the total informal and formal care provided in the period
2020-2040 according to the kinship distance between caregiver and receiver:
household (distance 0); parents and children (distance 1); grandparents,
grandchildren and brothers (distance 2); aunts, uncles and nephews (distance
3). We can see that most social care is supplied within the household, however
a significant part comes from outside the household, especially formal care.
Figure 8 shows annual per capita health care cost due to the hospitalization
of people with care needs. We can see that although the total unmet care need
levels off after 2020 (as shown in Figure 2), the per capita health care cost
keeps growing until the late 2030s. This is due to the decrease in population
after 2015, resulting in the per capita health care cost reaching its peak 15
years after the unmet social care need.

Figure 7: Total informal and formal care supplied by different groups of
receivers’ suppliers: household, parents and children (I), grandparents,
grandchildren and brothers (II) and aunts, uncles and nephews (III).

Figure 8: Per capita health care cost due to the hospitalisation of people
with social care needs (per year).

In the last two charts we investigate and compare the efficacy of two policy
interventions. These two policy experiments are illustrative examples that
show how this model may be used as a tool for the design and evaluation of
social care policies. In these examples we consider a tax deduction policy, in
which the households are allowed to deduct all the social care expenses from
their tax base, and a direct funding policy, in which the social care needs of
people with the highest need level (the ‘critical’ level in Table 1) are taken
care of directly by the state. The policies are implemented starting in
simulation year 2020, and the two scenarios are compared to the benchmark no-
policy scenario.

In Figure 10 we can see that the two policies have a positive impact on total
unmet social care need, as expected. However, while the tax deduction policy
has a marginal effect, decreasing the total unmet care need by about 11%, the
direct funding policy has a drastic effect, reducing the total unmet care to
23% of the original level. Although the tax deduction policy has a smaller
effect on the unmet social care need, Figure 10 shows that the tax deduction
policy is more cost-effective than the direct funding policy.

Figure 9: Unmet care need per recipient in the no-policy scenario and with the
two alternative social policies.

Figure 10: Cost-effectiveness of ‘tax deduction’ and ‘direct funding’
policies.

This is because, while the effect of direct funding on total unmet social care
is seven times the effect of the tax deduction, the cost of the former policy
compared to the latter is much higher, as shown in Figure 12. We note however
that notwithstanding its cost-effectiveness, the effect of the tax deduction
policy on unmet care demand is limited and thus it is not a solution to the
growing social care demand on its own. Finally, Figure 12 shows the discounted
value of the hospitalization costs in the period 2020-2040 in the three policy
scenarios. We can see that, apart from the direct costs associated with each
policy, our model allows us to estimate the policies’ spillover effects (in
this case, the effect on hospitalization cost), effects which we need to take
into account to assess the relative advantages and disadvantages of
alternative policies.

Figure 11: Unmet care need per recipient in the no-policy scenario and the two
alternative social policy scenarios.

Figure 12: Cost-effectiveness of ‘tax deduction’ and ‘direct funding’
policies.

##  5 Discussion

While the results presented here are still early, the outcomes of these
simulation runs suggest that this model can produce broadly realistic
portraits of the coming trends in UK social care. The overall population
dynamics largely mirror those in evidence in the real world, with the notable
exception of international migration which is not modelled here. The
simulation also produces inequalities in care provision by both SES and
gender; while the data presently available is not sufficient to determine if
the socioeconomic inequalities are accurate, the gender pay gap is reflective
of current UK norms.

The simulation results on unmet care need lend credence to the worries
expressed by UK policy-makers. The simulation shows that unmet care need will
continue to grow over time, and given that 1.2 million older people in England
alone are not receiving sufficient care [2], further growth in these figures
could lead to severe consequences for public health. The results also show
marked inequalities in care provision, with the wealthy capable of supporting
their aged relatives in need through privately-funded care while still staying
in work and receiving high wages. Among the lower-income groups women are
providing an overwhelmingly larger share of informal care as compared to men,
meaning that women at the lower end of the socioeconomic scale are more likely
to be pushed out of work and toward unpaid care provision.

Our use of kinship networks as the mechanism for distributing care illustrates
that kinship distance impacts care provision behaviour, and thus is an
important aspect of care decision-making that should be taken into account in
future research. We observed a significant difference in caring behaviours
between within-household and kinship distance I agents, with the latter
providing much more formal care than informal. This suggests that future
models in this area should consider kinship networks and their impact on
caring behaviours. Understanding the negotiation process within families
regarding care provision will be an important aspect for policy-makers to
examine as well, as policies put into place to support and encourage informal
care may need to take account of these complex social aspects of care.

Finally, the speculative policy scenarios we provide here demonstrate the
efficacy of this platform as an aid to policy-makers who wish to examine the
impact and possible unintended side-effects (spillover effects) of their
planned policy interventions. The scenarios showed that while some policies
such as tax deductions may seem like an easy ‘win’ for the policy-maker, the
actual impact on social care need is minimal. For significant reductions in
care need and related health care costs, more expansive – and expensive –
interventions are needed. The direct funding scenario here is a relatively
simplistic example used to demonstrate this point; the model is capable of
simulating the results of much more nuanced policy interventions, due to the
detailed modelling of key social care mechanisms and related economic and
social processes.

##  6 Future Work

While the current simulation does broadly reflect the expected trends in
social care provision, validating the results is difficult. Ongoing projects
in Scotland are linking administrative data sources to develop a clearer
picture of social care provision and receipt across the country. In future
revisions of the model we intend to make use of these data by focusing the
simulation on Scotland rather than the whole UK. As the framework matures and
incorporates more real-world data, we will use Gaussian process emulators to
perform detailed sensitivity analyses, following the example of Silverman et
al. [14].

Apart from the validation process, this model can be expanded in various
directions. First, according to the Office for National Statistics the
proportion of informal childcare to GDP increased from 13.8% to 17.6% in the
period 2005-2014 [19]. Child care represents a significant part of the total
household’s informal care need and, therefore, it affects the care supply
which can be allocated to social care needs. Our next planned update to the
model will include a child care mechanism which will allow us to model the
interaction with social care supply.

Second, according to the Office for National Statistics, in 2016, 28.2% of
births in England and Wales were to women who were not born in the UK [20].
Moreover, projections show that post-2016 immigration accounts for 77% of
total population growth until 2041 [21]. The addition of international
migration to the model will allow us to understand UK demographic dynamics in
future decades.

Following on from the simplistic policy comparison presented here, in future
work we will consult with social care policy-makers in Scotland to model
proposed social care policies in more detail. Social care policy is very
complex, with the programmes available varying not only by region (England,
Wales, Scotland and Northern Ireland), but by individual local councils. The
model is capable of representing these details by varying relevant policy
parameters across the simulated UK geography. Once we are able to replicate
the current state of social care policy in the UK, we will then be able to
develop detailed evaluations of the possible impacts of new policy
interventions. Our model’s ability to capture detailed individual-level care
decision-making will enable policy-makers to examine policies aimed at
behavioural change as well as broader economic and social policies.

Finally, while the current simulation is very UK-centric, the core simulation
engine can be altered easily to examine the situation in other countries.
Ultimately we hope to produce a simulation framework that is capable of
modelling informal and formal social care across a variety of cultural and
economic contexts.

##  7 Data Availability

The code is available on GitHub:

https://github.com/UmbertoGostoli/ABM-for-social-care/tree/v.09

The output data used for the figures is available in this Dryad repository:

https://datadryad.org/review?doi=doi:10.5061/dryad.6tm6183

##  8 Authors’ Contributions

Umberto Gostoli developed the model (based on a previous version, developed by
Eric Silverman), run the simulations, gathered the data and produced the
charts. Eric Silverman conceived the research and helped develop the model.
Both authors contributed to draft the paper. All authors gave final approval
for publication.

##  9 Funding

The authors are part of the Complexity in Health Improvement Programme
supported by the Medical Research Council (MC_UU_12017/14) and the Chief
Scientist Office (SPHSU14).

##  10 Acknowledgements

We would like to thank Chris Patterson and Lauren White from the MRC/CSO
Social and Public Health Sciences Unit for their helpful comments.

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