1.1 Key concepts and results in mathematical epidemiology

This section provides a very brief overview of the formal derivation of the basic reproductive ratio in mathematical epidemiology. For an excellent and certainly more exhaustive review see Heffernan et al. [23] (2005) (as well as [35] for criticism). Epidemiologists are interested in the growth of the number of infected individuals within a population. Let us assume that we can divide the whole population into several pairwise disjoint subpopulations, or compartments, some of which consist of infected individuals. These compartments can reflect, e.g., social structure, age structure, or stages of a disease. Assuming that there are n compartments, let x≥0 be a n-dimensional vector that lists the number of individuals in each of the compartments. Let I and S denote the set of infected and not yet infected (susceptible) compartments, respectively.

In what follows, we describe a way of deriving the basic reproductive ratio referred to as the ‘next generation method’ [36]. Let F_i(x) denote the rate of new infections that enter an infected compartment i∈I and let T_i(x) measure the rate of net-transitions that go out of compartment i. The rate of change of x_i then is given by .

Given all rates F_i(x) and T_i(x), we can compute the ‘next generation matrix’ FT⁻¹, where and . Here, x₀ is the population dynamic equilibrium if there is no disease. The basic reproductive ratio then is given by the dominant eigenvalue ρ of the next generation matrix, i.e., R₀ = ρ(FT⁻¹). The intuition behind this definition is that reproduction depends (i) on the number of newly infected individuals and (ii) on how long individuals are infectious. The former is captured by F. The latter is captured by T, which collects all net-transition rates. Then, T⁻¹ can be interpreted as measuring all waiting times before an individual exits all infected compartments.

The derivation becomes easy and intuitively clear if there is only a single infected compartment. In the ‘susceptible-infectious-susceptible’ (short: SIS) model of infectious diseases the population is split into two compartments, susceptible and infected individuals [32]; for a review of the basic reproductive ratio in other model setups see van den Driessche (2017). In that case, S and I consist of one compartment each. Individuals can become infected or recover, in which case they switch back to the susceptible compartment. With a slight abuse of notation, let S and I denote the respective fractions of individuals. We assume that both subpopulations are homogeneously mixed. Then, SIS-dynamics can be formulated by means of a pair of ordinary differential equations (1) where we assume I+S = 1, i.e., overall population size is normalized and kept constant. Here, β is the (per capita) rate of disease transmission and γ is the rate at which infected individuals recover. The demographic parameter μ denotes mortality rate (i.e., the duration 1/μ corresponds to one generation). Note that the term +μ in the first equation ensures that population size is constant. The disease-free equilibrium is (S,I) = (1,0), so that . The transition rate is T = γ+μ and the inverse T⁻¹ = 1(γ+μ) is the expected waiting time until recovery takes place. Hence, R₀ = β/(γ+μ).

There are several ways of estimating the basic reproductive ratio empirically [23, 37, 38]. To begin with, it could be determined by estimating the model parameters it is defined by. Clearly, in the SIS model, estimates of β,γ, and μ suffice to determine R₀. In practical terms, however, it is not always easy to obtain reliable estimates for β and γ, and this more generally holds true for more complex models [23]. There are three other interesting ways for obtaining R₀ estimates.

First, R₀ can be estimated based on the prevalence of the disease at its population-dynamic equilibrium [32]. The argument goes like this: Let p_I denote prevalence, i.e., the fraction of all infected individuals. Then for an infected individual, 1−p_I is the probability of interacting with a susceptible individual, so that the number of infection events induced by that individual is R₀(1−p_I). At population-dynamic equilibrium when there is no change in the number of infected individuals, however, this number is exactly one and hence R₀ = 1/(1−p_I).

We can retrace this derivation in the SIS-model by computing prevalence p_I = I* (the star denoting a population-dynamic equilibrium). Thus, setting we find that , which implies the above formula.

Second, we can use age of first infection AoI to estimate R₀ [22, 38]. Assume that in the population, all ages are homogeneously mixed and equally frequent. That is, the population follows a rectangular age distribution. Let LE denote expected lifetime (life expectancy). If we assume that, once infected, individuals cannot become susceptible again, then at population-dynamics equilibrium, AoI/LE individuals are not yet infected, and p_I = 1−AoI/LE are already infected. The above considerations about equilibrium prevalence entail that R₀ = LE/AoI.

Third, the basic reproductive ratio can be estimated based on the intrinsic (exponential) growth rate r of a pathogen [23, 39]. In the early phase of an outbreak, intrinsic growth rate is implicitly given by . That is, given a trajectory of prevalence, one can estimate r. The basic reproductive ratio then is R₀ = 1+rL, where L is the duration of the infected period.

Irrespective of how the basic reproductive ratio is estimated, the herd immunity threshold can be computed as which turns out to equal equilibrium prevalence [29–32]. If the fraction of individuals in the population that cannot transmit the disease exceeds this threshold, the disease cannot spread and the fraction of infected individuals will approach zero.

1.2 Transferring concepts and results to linguistics

The epidemiological concepts and results outlined above can be transferred to the linguistic domain. Here, in place of pathogens, there are linguistic constituents like words, phonemes, or constructions that are passed from one individual to another [4, 6, 8, 20]. Crucially, this is done through multiple interactions among individuals. Instead of infected and susceptible individuals, let us divide the speaker population into heterogeneously mixed users and non-users of a linguistic constituent [25, 28, 40]. Users already know and use the constituent of interest while non-users still have to acquire it. Let us define the respective fractions as U (for user) and N (for non-user). We can then define the system (2)

Here, β defines the per capita rate at which the constituent is transmitted from a user to a non-user. Let us refer to such events as instances of propagation. Upon successful propagation, the non-user switches to the user class. Note that β measures propagation rate under the condition that users and non-users interact, and not that a user utters the constituent. In fact, Nowak et al. (2000) [26] define this rate as β = bqϕ, where b is the number of contact events, i.e., linguistic interactions, in which learning can take place, q is the probability to adopt a constituent if encountered once, and ϕ is relative utterance frequency of the constituent. Hence, β can expected to be a function of utterance frequency, learnability, and the number of contact events. Note that the number of contact events can be much higher in linguistic propagation than in the transmission of pathogens since linguistic communication is restricted to physical contacts to a lesser extent. Due to the assumption of heterogenous mixing, the rate at which non-users switch to the user class is the product of propagation rate, the fraction of users, and the fraction of non-users (mass-action transmission; [23, 35]). The parameter γ defines the rate at which users stop using the constituent (for instance, because they forget it). It does not depend on contacts among individuals. As above, μ defines per capita mortality rate. Note that we assume all users subject to death to be added to the non-user class. This represents that new-born individuals are not yet equipped with the linguistic constituent. Apart from μ, age structure is not reflected in the model. In particular, heterogeneous mixing implies that interaction events are not conditioned by age. Propagation can occur in interactions with young individuals (reflecting the scenario of L1 acquisition in children that interact with their caretakers) as well as in interactions among old individuals (reflecting the adoption of words in everyday speech). All model parameters are summarized in Table 1. The model is schematically represented in Fig 1 (left).

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Fig 1. Compartment model.

Left: schematic representation of the compartment model. Right: trajectories of logistic growth (green) and decline (magenta). Parameters: U(0) = 0.25 and γ+μ = 1 in both cases; β = 4 and β = 0.1 for the growing (green) and declining (magenta) case, respectively.

https://doi.org/10.1371/journal.pone.0312336.g001

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Table 1. Epidemiological and linguistic model parameters.

https://doi.org/10.1371/journal.pone.0312336.t001

As above, the basic reproductive ratio of this system is R₀ = β/(γ+μ), and if R₀>1 then it asymptotically approaches a stable population-dynamic equilibrium with . That is, the basic reproductive ratio and the number of users at equilibrium increase as propagation rate increases, and as the rate of abandoning a constituent and mortality rate decrease.

The ODE above models logistic, i.e., sigmoid, growth. Since N+U = 1, the system can be reformulated as (3)

It can be solved analytically to yield (4) where U₀ = U(t)>0 is the initial fraction of users [40]. Here, the intrinsic growth rate is given by r: = β−γ−μ, and defines the carrying capacity of the system. The logistic equation was used abundantly to model sigmoid (S-shaped) growth in linguistics on various levels of linguistic analysis [2, 10, 13]. Clearly, if U₀>U* and r>0 then the solution approaches the equilibrium from above, i.e., the curve is falling, and if U₀<U* and r>0 then the curve is growing. If r<0 then the solution approaches U = 0 (Fig 1, right).

The considerations about transferring epidemiological concepts above suggest that standard results from mathematical epidemiology can be transferred to the linguistic domain as well [41]. More specifically, this means the following: First, the basic reproductive ratio of linguistic constituents can be estimated given estimates of β,γ and μ, or intrinsic growth rate r and any two of the three model parameters. Mortality rate μ can be estimated from demographic data (as μ = 1/LE). Adoption rate β and abandoning rate γ, however, are much harder to determine.

Second, for linguistic constituents at their population dynamic equilibrium, the basic reproductive ratio can be estimated by means of prevalence, i.e., the fraction of users. That is, R₀ = 1/(1−p_U), where p_U = U* is prevalence at population dynamic equilibrium. Third, for linguistic constituents at their population dynamic equilibrium, the basic reproductive ratio can be estimated by means of age of acquisition, i.e., the age at which the constituent was learned, so that R₀ = LE/AoA, where AoA is the age of acquisition of the constituent.

Fourth, we can use intrinsic growth rate to estimate the basic reproductive ratio as R₀ = 1+r∙L where L = LE−AoA approximates the period during which an individual uses a constituent. This approximation only holds well if the constituent repertoire of individuals does not shrink substantially as they age.

Note that, theoretically, all ways of estimating the basic reproductive ratio should yield identical estimates of R₀. In particular, this should hold true for estimates based on age of acquisition and prevalence, since both share the same assumption, namely that the population dynamics rest at their equilibrium. Table 2 juxtaposes epidemiological and linguistic variables relevant for estimating the basic reproductive ratio.

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Table 2. Epidemiological and linguistic variables relevant to the estimation of R₀.

https://doi.org/10.1371/journal.pone.0312336.t002

Like the herd immunity threshold, we can derive, once R₀ is estimated, the stability threshold H = 1−1/R₀. In linguistic terms it denotes the minimum fraction of individuals which could potentially acquire a linguistic constituent that need to be excluded from adoption for that constituent to vanish in the long run. If there were, for instance, hypothetical political measures for banning a certain word, one would need to ensure that at least H individuals are not able to adopt that word. Conversely, if, in this hypothetical scenario, 1−H individuals would refuse to adhere to these measures, the spread of the word could not be prevented. Note that this does not consider social structure in any way. We will come back to this point in the discussion section.

2.1 Data

We test the predictions outlined above on the lexical level, i.e., by inspecting dynamics of words. We restrict our diachronic analysis to words that have increased in frequency during the past two centuries, and we will refer to such words as lexical innovations. We do so irrespective of whether these words have already been present initially at low frequency, whether they are associated with new concepts or are involved in onomasiological competition, whether they are new word formations or have been adopted from a different language. Moreover, we only consider periphery words, which we define, somewhat simplistically, as words that are not used by everyone, based on prevalence estimates.

For obtaining estimates of the basic reproductive ratio for these words, we combine different data sets in different languages: English (eng), Spanish (spa), German (ger), and Italian (ita). We use AoA estimates for a large set of English from Kuperman et al. [42]. AoA norms for about 4600 Spanish verbs were taken from Alonso et al. [43]. For German, we used data from Birchenough et al. [44]; 3,200 words), and for Italian, data from Montefinese et al. [45] was employed. All of these estimates refer to word forms (there are sometimes separate AoA norms for inflectional and/or derivational forms of a base form such as accelerate, accelerated, acceletation, etc. in [42]) and they were obtained by averaging subjective age-of-acquisition ratings collected via (crowdsourcing) surveys. Although ratings are subjective, estimates obtained by means of this methodology were shown to correlate with age-of-acquisition estimates obtained under controlled conditions [42, 46].

Lexical prevalence was proposed as a relevant covariate for psycholinguistic research [47], so that prevalence estimates are already available for a set of languages. Prevalence estimates for English word forms were taken from Brysbaert et al. (2019). These estimates were aggregated via crowdsourcing as well. Note that prevalence is not probit transformed as in Brysbaert et al. [48]. Rather, for a word, the prevalence estimate measures the fraction of individuals participating in the survey knowing the word. Baumann and Sekanina (2022) show that there is a slight difference between the fraction of individuals knowing a word (only passive knowledge) and the fraction of individuals using a word (active knowledge). The latter operationalization, as a matter of fact, fits better to prevalence as introduced in the previous section as the fraction of users. Overall, however, both estimates correlate strongly so that the estimates from Brysbaert et al. (2019) will serve as good enough proxies for the present study. For the other three languages (ger, spa, ita), we approximated prevalence by computing, for each word, the fraction of participants reporting a subjective AoA rating [43–45]. If no rating was given or if participants reported that the word was unknown this was interpreted as neither knowing nor using the word. Note that the accuracy of the prevalence estimates substantially depends on sample size, i.e., on the number of participants they are based on. The respective sample sizes are about 300 for English, about 23 for German, about 50 for Spanish, and 25 for Italian. Hence, maximal margins of error (for p = 0.5 at a 95% confidence level) range from about ±0.03 (for English) to slightly over ±0.1 (for German).

For measuring growth, we use word frequencies derived from diachronic corpus data. For English, we derive frequency trajectories from the Corpus of Historical American English (COHA; Davies 2010) for all words in the intersection of Kuperman et al. [42] and Brysbaert et al. (2019). This corpus spans the period from 1820 to 2000. Trajectories have a resolution of one frequency estimate per decade. Frequencies are normalized with respect to the decade-wise sub-corpus size. The main advantage of COHA clearly is that it is balanced with respect to genre. For German, Spanish, and Italian, we needed to resort to normalized frequency trajectories extracted from Google books (ngrams; https://books.google.com/ngrams/). For the sake of consistency, exactly those word forms represented in the respective AoA lists were chosen in all queries for both COHA and Google books. This is, because the AoA list in [42] features estimates for separate inflectional/derivational word forms and does not differentiate between word classes. Thus, there are, e.g., separate AoA estimates for abbreviate and abbreviated, but it is not clear whether the latter corresponds to the past-tense form of the verb or to the adjectival participle. Hence, we did not differentiate between word classes in the corpus data either and aggregated frequencies of all word forms matching the entries in the AoA list per decade in COHA. In Google books, frequency trajectories have a resolution of one estimate per year. Frequencies were aggregated per decade, however, so that the resolution matches that of COHA. Table 3 shows a summary of all data sources used in this study. Fig 2 displays prevalence estimates, age-of-acquisition estimates, and historical frequency trajectories of four lexical examples.

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Fig 2. Lexical innovations.

Examples for all considered languages: blasé (eng), europeizar (spa, ‘europeanize’), Charta (ger, ‘charter’), and cracker (ita, ‘cracker’): trajectories of normalized frequency (cf. 2.2.1) and pie charts illustrating age of acquisition relative to life expectation (top) and prevalence (bottom).

https://doi.org/10.1371/journal.pone.0312336.g002

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Table 3. Data summary for all languages (eng, spa, ger, ita).

https://doi.org/10.1371/journal.pone.0312336.t003

2.2 Methods

2.2.1 Operationalization.

The formulae in Section 1.2 are used to estimate the basic reproductive ratio based on (i) age of acquisition, (ii) prevalence, and (iii) diachronic trajectories. We refer to (i-iii) as estimation methods. For (i-iii) we restrict ourselves to words with a prevalence smaller than 0.95, i.e., we exclude the core lexicon. In addition, for (iii), we focus on lexical innovations, which we conceptualize as words that have been increasing significantly in frequency throughout the observation period (see below for more details). We use the collected age-of-acquisition estimates, an estimate of life expectancy based on demographic data (https://data.worldbank.org/), and estimate the basic reproductive ratio as (i) . Note that life expectancy has increased through the past two centuries and that women have, on average, a higher life expectation than men, so that the assumption of a single point estimate is clearly simplistic. In the UK and the US, for instance, life expectancy has increased from about 71 in the early 1960s to about 80 in the 2000s (Fig 3, left). Notably, UK/US displays the smallest increase in life expectancy. For all populations, mean LE is roughly 75 with an average margin of error of 1 year (95% confidence interval) in this time period. This margin of error will be considered in the error analysis in 2.2.2. Furthermore, the estimation of the basic reproductive ratio depends on the age distribution being roughly rectangular (i.e., relatively evenly distributed; see above). Such a distribution is typically adopted in high-income countries (Fig 3, right) [31, 32]. In particular, this assumption seems to be met for the English-speaking population. More straight forwardly, we can use prevalence estimates p_U to derive (ii) .

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Fig 3. Demographic information for the relevant speaker populations.

Left: life expectancy (LE) depending on time with mean LE (from left to right: eng, spa, ger, ita). Data taken from https://data.worldbank.org/. Right: age distributions for UK and US combined (eng), Spain (spa), Germany (ger), and Italy (ita) as representatives of the respective speaker populations. Data taken from https://data.un.org/.

https://doi.org/10.1371/journal.pone.0312336.g003

Both of these estimates rest on the assumption that the population dynamics are at their equilibrium, i.e., that the fraction of users neither grows nor falls substantially. For that reason, we compute for each word the absolute value of the Spearman correlation coefficient of time and frequency, ρ≥0 from 1950 to 2000. We use S = 1−ρ as a measure of contemporary stability. If S it is close to 1 for a word this indicates that the dynamics of that word are relatively stable and rest at their dynamic equilibrium. We will use this to examine the impact of S on the quality of the estimates of the basic reproductive ratio.

Using frequency data to estimate the basic reproductive ratio requires a bit more work. The goal is to use the approximation R₀≌1+r∙L where r and L denote intrinsic growth rate and the duration of the prolific period, respectively. As a matter of fact, to get r we need a prevalence trajectory, i.e., a trajectory of the number of users U(t) over time , as opposed to a frequency trajectory. Here, is the observation period divided into 18 decades (from 1820 to 2000). It is not at all easy to obtain word prevalence based on corpus data. This is because corpora often feature only a small number of authors and because the amount of text per author is likely not representative of their linguistic knowledge (let alone the fact that the author population in linguistic corpora is not necessarily representative of the whole population, in particular, as far as the social dimension is concerned). See Johns et al. [51] for an exploration of ways to estimate prevalence based on corpus data.

In the present study, however, we adopt a different approach to reconstruct prevalence trajectories for a subset of words. For each word, we take its frequency trajectory f(t) and we restrict our set of words to those that have a present-day prevalence p_U<0.95. For each word we check if it has increased significantly based on a Spearman’s correlation test (correlating U(t) and t at α = 0.05). Furthermore, we derive the stability coefficient S as above. Subsequently, we normalize each frequency trajectory with respect to its maximum, i.e., , and we define a word’s prevalence trajectory as . That is, we do as if the frequency trajectory would mirror the trajectory of prevalence. We only considered words with U(t)≤1 for all t and significantly increasing prevalence trajectories (i.e., words that we consider lexical innovations in this study).

In a final step, we fit a logistic model , a,b>0, as defined in Section 1.2 to the trajectory U(t) in order to estimate intrinsic growth rate r and define (iii) , where we take L = LE−AoA as the prolific period of the word. This estimate was not computed for words with 1+r∙L<0, since the basic reproductive ratio is conceptually non-negative. For each model, we compute (5) to assess goodness of fit, where denotes mean prevalence across all . Finally, the inflection point t_I is extracted from each model. This point divides the trajectory into an initial (accelerating) and final (decelerating) period and helps to identify when the spreading phenomenon took place, at least approximately.

Of course, this approximation is not unproblematic. First, it is likely that the prevalence trajectory of a word in fact precedes that of frequency because a word can also spread, language internally, through various possible linguistic contexts. That is, r and hence is potentially underestimated (because r should be greater than frequency developments suggest). On another account, life expectancy has changed considerably over the past two centuries so that LE and hence is in fact lower than assumed. Perhaps these biases balance each other, but this still needs to be tested. Finally, age of acquisition is likely not constant over time. See Section 4.1 for discussion. Table 4 lists the sample sizes used for all ways of measuring the basic reproductive ratio in the four languages, considering all restrictions mentioned above (periphery words; lexical innovations).

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Table 4. Sample sizes for all languages (eng, spa, ger, ita) and all estimation methods (i-iii).

https://doi.org/10.1371/journal.pone.0312336.t004

4.1 Generalizability and limitations

The observation that R₀ estimates derived through AoA and prevalence correlate for all languages (except Italian) is reassuring but not particularly surprising, for that matter. We will discuss the theoretical implications of this relationship below. What is surprising, however, is that the positive relationship between both estimates is not substantially modulated by the stability of the recent development. This is interesting because contemporarily unstable, i.e., growing or declining words, are not expected to yield reliable and estimates. Contemporarily unstable words do not fulfil a crucial assumption for deriving these measures (cf. 1.2 and 2.2.1). However, at least for English and German the interaction with stability points into the theoretically expected direction (albeit at a low effect size). The general lack of a positive correlation seems surprising given that was also defined by means of prevalence (because trajectories were normalized to fit present-day prevalence) and age of acquisition (because the prolific period is constrained by age of acquisition) in the first place.

The observation that most average R₀ estimates fall into a similar range, across languages as well as across estimation methods, is reassuring as well and potentially more revealing. While it can be expected that, across languages, scores derived with the same method fall into similar ranges, it is potentially more surprising that empirically estimated ranges indeed roughly coincide when making comparisons across methods. It can be seen that different estimation methods produce more similar results for English, followed by Spanish. They show certain discrepancies for German and Italian where prevalence-based estimates are higher than the other two estimates (large effect sizes in all cases for German and Italian). It is plausible that this can be attributed to the size and quality of the German and Italian data. Both datasets are relatively small, perhaps biased towards the core lexicon, and prevalence estimates were derived only indirectly (by considering missing AoA scores) and based on a smaller number of participants per word (about 25). Indeed, prevalence-based estimates for German and Italian are almost rendered useless by their, on average, large margins of error. We learn from this that using prevalence for the estimation of the basic reproductive ratio seems to only make sense if prevalence estimates are based a sufficiently large number of participants (about at least 50 as in the case of Spanish).

The approach of estimating employed here suffers from a multitude of problems. First, it is very likely that age of acquisition, just like life expectancy, was not constant through the past two centuries. This is so, because age of acquisition is a function of frequency even if one controls for prevalence [48]. It could be that individuals living, say, one century ago were exposed much more rarely to words that are nowadays used frequently; not because these words were employed by fewer individuals but because they displayed low utterance frequency per se (for instance because there was a relatively more frequent competing synonym available). By consequence, historical age of acquisition would be higher, hence lowering for low-frequency words (because the prolific period is shortened in this way). That is, our approach potentially overestimates for low-frequency words.

The only way of mitigating this issue is to obtain reliable historical age-of-acquisition estimates. Evidently, subjective ratings are not available for early periods so that one needs to rely on reconstructed estimates, ideally working with evidence independent from frequency. An approach that might prove useful in this regard is that of estimating age of acquisition through simulating semantic learning. Botarleanu et al. [56] have exploited word embeddings trained on incrementally increasing corpora to approximate age of acquisition through semantic trajectories. Since this approach could be applied to historical corpora as well it could be used to approximate historical age of acquisition norms. Note, though, that word-embedding similarities were shown to depend on frequency as well [57] so that this factor needs to be considered carefully.

It is work pointing out, however, that judging from the range of inflection points of the estimated diachronic trajectories (Fig 5, left), the majority of all innovations covered in this study seem to represent relatively recent phenomena, mostly confined to the 20^th century. The interquartile range of inflection points largely overlaps with the range of birth dates of the participants in the three age-of-acquisition surveys. In the case of English and German, participants seem to have witnessed the emergence of the examined innovations (in the case of Spanish, inflection points largely precede participant birth dates, though). Note that the actual inflection points could be overestimated due to the time lag between writing and publishing texts [58]. This affects Google Books (spa, ger) and, to a lesser extent, COHA (eng).

Second, and related to that, we used normalized frequency as a proxy of prevalence for reconstructing prevalence trajectories. While it was shown that lexical prevalence and frequency are correlated significantly [47, 59], the two measures do not match (for instance, computing Pearson’s r for frequency and prevalence for all words in Brysbaert et al. [48] known by less than 95% of all individuals yields a correlation of R = 0.28. For prevalence thresholds of 80% and 50%, one yields R = 0.16 and Rr = 0.12, respectively). Notably, this relationship becomes weaker for lower prevalence values so that using frequency as a proxy does not seem to be a very reliable strategy when examining dynamics of lexical innovations that are initially rare by definition. More reliable historical prevalence estimates could be obtained by inspecting the fraction of authors in large-scale corpora that employ a particular word, an approach already pursued by Johns et al. [51] and Feltgen [60]. Although promising, this approach evidently suffers from data scarcity in earlier periods: for estimating prevalence in this way, one ideally needs both a large number of authors and a large and representative selection of texts per author, and prevalence will be necessarily underestimated [60].There is an interesting mismatch between the lacking correlation between and on the lexical level, and that on the phonological level that we have demonstrated to hold for the dynamics for English word-final consonant clusters [28]. The obvious advantage of phonological diachronic analyses is that prevalence estimates are more accurate: before the onset of certain sound changes in the history of English, prevalence of (most) English word-final consonant clusters was exactly zero, and upon completion prevalence can be safely assumed to be close to one for the majority of such sound sequences. Estimating intrinsic growth then amounts to inspecting the steepness of the growth curve; a task involving fewer degrees of freedom.

Third, the nature of the employed historical corpus data could have affected the estimates. In contrast to COHA, which consists of a genre-stratified sample of texts throughout (most of) the observation period, the Google Books corpus was shown to be increasingly biased towards scientific texts [58]. As a result, growth rates r and hence could be overestimated for scientific words and, consequently, underestimated for non-scientific words for Spanish and German (Italian was not included in the diachronic analysis, anyway). This also entails, that some non-scientific words were missed out in the present analysis due to not displaying significantly increasing trajectories. On another account, this bias might also explain why is significantly lower than and in the case of Spanish and German. A post-hoc comparison of English frequency trajectories in COHA and Google Books (see attached code for details) suggests that the great majority of trajectory pairs (78.8%) are relatively similar (strong and significant correlation between the pair of trajectories) while only a few cases (1.6%) yield opposing trends (significant negative correlation between the pair of trajectories). However, biases towards scientific vocabulary could still affect the estimates. With that in mind, conclusions drawn from English corpus data (i.e., COHA) seem to be more robust.

Fourth, the approximation of the prolific duration as L = LE−AoA rests on the assumption that individuals do not forget words. Estimates of vocabulary size depending on age could make the approximation more robust (and, at the same time, inform us about the rate γ in the model given in Eq (2)). Extant research on the relationship between age and vocabulary size, however, suggests that the number of known words increases at least up to the age of 70 [61, 62] so that the assumption made, although certainly simplifying, does have some empirical support.

The mathematical model is evidently simplistic. First, it assumes constant homogeneous mixing, and hence the three ways of estimating the basic reproductive ratio rest on this assumption as well. Problems that come along with this assumption are reviewed in [35]. It was shown that the invasion boundary in models that assume a more realistic population structure (scale-free networks) is lowered, so that R₀ estimates could be effectively underestimated [7, 63]. This is because in real-world social networks there are individuals that have a disproportionally large number of contacts [64, 65], an observation that was suggested to be responsible for why cultural evolution progresses faster than biological evolution [66]. Mean-field models provide a good way of integrating a more realistic network structure [7, 63]. Meyers [67] shows that in an epidemiological contact network, (9) where T is transmissibility (i.e., mean probability of transmission given a contact event), and 〈k〉 and 〈k²〉 are mean degree and mean squared degree of the contact network. Since degree variance is given by 〈k²〉−〈k〉², more heterogeneous contact networks are expected to display larger R₀ values. Network models could be informed by social-media data [68] or by extrapolating from contact surveys conducted for epidemiological purposes [69]. It would be interesting to see if a more realistic population structure would also lead to stronger correlations among the three R₀ estimates.

The model does not feature age structure. That is, adoption is assumed to be independent from age, which does not reflect how cognitive-semantic plasticity changes over time [70, 71]; clearly, the lexicon grows as age increases which in turn affects learning of new words. Likewise lexical processing and retrieval changes with age [72, 73]. Implementing several age-classes with differential adoption, abandoning, as well as mortality rates could mitigate this issue. Related to that, the procedure of estimating the basic reproductive ratio from age of acquisition assumes a rectangular age distribution. While this approximately applies to the age distributions of the present speaker populations considered in this study, this might not be the case for other speaker populations. On that note, the fact that English data mostly yield expected outcomes in this study (in contrast to the other three languages) might not necessarily be a reflex of larger sample size. It could also result from the fact that the age-distribution of the English-speaking population comes closer to a rectangular (i.e., equal) distribution than that of the other three languages (Fig 3). Likewise, life expectancy for the English-speaking population shows the smallest increase of all languages covered in this study. Both is in favor of the model assumptions. Hence, it could be that the higher quality of R₀ estimates (as is evident from the smaller differences between estimation methods and higher correlations among the respective estimated values) is grounded in demographic properties of the speaker population as well.

The model does not feature social structure, nor does it reflect geo-political patterns or other population demographics. Evidently, this can be problematic given that the investigated languages are spoken in several countries. However, compartment models are flexible enough to integrate information like this if contact rates among social and geo-political compartments are known. Likewise, the simplistic assumption of constant population size can be relaxed (note that work by May and Anderson [32] implies that similar approximations of the basic reproductive ratio based on age of acquisition can be used for increasing populations). Finally, the model assumes constant demographic and transmission rates. In particular, it excludes any type of stochasticity [16, 74]. Based on analytical work on stochastic SIS-models [75], we have shown earlier [76] that stochasticity in transmission decreases the basic reproductive ratio. This could be examined empirically based on diachronic corpus data.

Finally, when it comes to language-internal dynamics, the model is simplistic in that it assumes the transmission of one word to be independent from the rest of the lexicon. That is, it ignores effects of the semantic neighborhood [77] and semantic competition [78]. Such interactions would require a version of the model in which multiple words can explicitly interact with each other.

All the limitations discussed in this section point at avenues for improvement of the model and empirical analysis. Note, though, that similarly abstract models and concepts (in particular, the basic reproductive ratio) have been employed fruitfully in the field of epidemiology. This is by no means to be interpreted as an excuse for further improvement, but it tells us that we can learn something about contact-driven dynamics even from abstract models.

4.2 Implications

Given the limitations discussed in the previous section, does the approach adopted here represent an accurate model of lexical spread? Certainly not–as it depends on simplistic assumptions. If the model assumptions are met, however, our analysis suggests that periphery words, on average, show a basic reproductive ratio of about R₀≅6, with inter-quartile ranges going from about 2 to 17, depending on the language and estimation method. This puts spreading periphery words–in principle–roughly on par with infectious diseases like Rubella (‘three day measles’, [79]) Poliomyelitis [24, 32], or the COVID-19 Delta variant [80, 81] (‘in principle’ because the basic reproductive ratio is dimensionless; see discussion below). This figure defines a threshold that an average lexical innovation needs to pass to spread successfully [40]. That not every linguistic variant will spread is plausible, and the issue that new variants require a certain frequency of users in order to be of communicative value (which is rarely the case for new variants to begin with) was dubbed the “threshold problem” by Nettle [82]. Importantly, though, the epidemiological model adopted here does not feature any mechanism of positive frequency dependence modeling increased communicative value for higher user frequency. Rather, the model entails that a threshold for successful spread exists even if such frequency-dependent pressures are absent. Note that while the three estimation methods are expected to yield similar estimates, differences between languages might in fact reflect differential social, geographic, and demographic structures of the respective speaker populations. A larger set of languages is required, though, to investigate such effects.

The estimate also implies that the herd-immunity threshold of an average lexical innovation equals about 83% (; minimal and maximal bounds of observed inter-quartile ranges yield estimates for this threshold that go from 50% to 94%). That is, 83% of all individuals would need to be prevented from adopting that word for it to vanish in the long run. Whichever hypothetical measures of language policy a ruling body would implement, they would need to be fierce to show an effect. Likewise, only a fraction of 17% resisting said measures would suffice to maintain that word, even in the lack of social or geographic structure that would enforce stable usage in a smaller subpopulation. The observation has interesting consequences for the study of language policy and censorship ([83]; see also [84] for a recent example). What the model entails is that, from this admittedly simplistic epidemiological point of view, top-down prohibitive measures of language policy are likely not to have good chances of success (and note that in this argument we do not even consider the preserving effect of written language). Integrating social structure into the model (and potentially differential language policies) would be crucial for a detailed analysis of the matter.

Finally, the estimate R₀≅6 also tells us about the role that individuals play in the propagation of new words through contact networks. Mean personal network size was shown to equal about 600 [85]. This reflects all individuals that one gets to know during one’s life time. This means that a single individual that picks up a lexical innovation passes it on to only about 1% of their personal contacts–notably, throughout the whole period during which that individual knows and uses that word. If one considers an average offspring count of about 1 (per individual), the impact that a single individual has on the spread of words decreases even further: literally only a handful of other people will on average pick up the lexical innovation due to interactions with that individual. That is, the basic reproductive ratio of words can shed light on how information is transmitted through speaker networks [86, 87].

A notable difference between linguistics and epidemiology in this context is that the prolific, i.e., infectious, period of words L is much longer than that of the above-mentioned infectious diseases (once acquired, words are kept for a much longer time span than, say, COVID-19 pathogens that get dealt with by the immune system over the course of a few weeks, on average). It follows from the approximation R₀≅1+r∙L that words (with lower r but higher L) can plausibly display similar R₀ values as the mentioned diseases (with higher r but lower L). Low growth rates do not entail that lexical change per se is a rare phenomenon (we look at each individual word separately, and there can by multiple lexical innovations that spread in parallel), nor that lexical change is inert (the basic reproductive ratio, as a dimensionless quantity, does not measure rate of growth). However, the finding informs us about the relative impact that each individual has on such change phenomena. Importantly, R₀ is defined in such a way that it only considers propagation events in which one individual already knows and uses a certain constituent while another individual does not do so. It does not capture events in which both individuals already know a word and where the usage of the word is promoted in one individual as a reaction to the other one using it, say, through psychological priming effects or socio-pragmatic factors like conformism and prestige [82, 88, 89]. See [90] for a socio-pragmatically more elaborated version of the model including conformity and non-conformity biases.

On a more general level, it is evident that the model, if taken for granted, together with the three ways for deriving the basic reproductive ratio provides links between three different domains: (i) lexical acquisition, (ii) synchronic lexical distribution, and (iii) lexical change. In particular, it provides a mechanistic link relevant to three lines of research. First, the link between language acquisition and change was subject to multiple studies in the field of evolutionary linguistics [91]. Generally speaking, words that are learned early are diachronically more stable in that they are less likely to change their form [28, 71, 92] or meaning [93]. One mechanism that was suggested to explain this link is that of cognitive entrenchment [94–96]. Words that are learned early display a higher degree of routinization (e.g., in their production, retrieval, or perception), remain stable in linguistic usage, and are therefore less likely to get ousted. In fact, the epidemiological model discussed in this contribution is agnostic with respect to such mechanisms: it predicts that early acquired words are diachronically more stable simply because early acquisition increases the length of the prolific (i.e., infectious) period in individuals, thus promoting successful propagation. In that sense, the model can also function as a–let us call it: population dynamic–base line against which hypotheses about the effects of cognitive entrenchment need to be tested.

Second, and connected to this, the basic reproductive ratio links synchronic lexical distribution (prevalence) and diachronic lexical change (growth). In short, the prediction is that what is contemporarily prevalent has been successfully growing in the past. While this might sound trivial at first sight, the mathematical intricacies involved in that link provide more information. The idea is that a word’s prevalence at its population dynamic equilibrium (i.e., neither when it is growing nor declining) entails how fast it has been growing in the past, and vice versa. Quantitative research on diachronic lexical change largely draws on investigating frequency trajectories, or (trajectories of) dispersion measures based on token frequency and document frequency [97–99]. As discussed before, it is not trivial to derive lexical prevalence from corpus data [51], so that using a combination of frequency trajectories and contemporary prevalence estimates (as in this study) or conducting apparent-time diachronic analyses [100, 101] (Boberg 2004; Bailey 2004) remain the only feasible options. Both approaches come with their own problems. Most notably, they can only be reasonably applied to recent phenomena (words that have been prevalent a century ago but got ousted afterwards cannot be captured by contemporary surveys). Likewise, both approaches are challenged by speaker internal changes [102]. We conclude that more research is needed to reliably test the link between equilibrium prevalence and growth suggested by the model.

Third, and finally, the basic reproductive ratio links acquisition with synchronic lexical distributions: words that are acquired early are used by a large share of the population [48], a link that was subject to research on language learning. It was shown that learning from multiple informants via distributed exposure promotes acquisition, both in infants [103, 104] and adults [105]. Again, the epidemiological model does not necessarily need to rely on mechanisms like this. Even if all individuals learn words at a constant rate, independent of their age, it trivially predicts that early acquired words are more widely spread and vice versa because (a) early acquisition increases the prolific period and hence the number of users of a word and (b) because words are more quickly adopted by a non-users if there are many users around. What is potentially more interesting than the correlation between ease of acquisition and prevalence is how exactly they are linked to each other through the basic reproductive ratio (more specifically: 1/(1−p_U) = R₀ = LE/AoA and hence p_U = 1−AoA/LE): we have seen that straightforward transformations of age of acquisition and prevalence lead to estimates of the basic reproductive ratio that fall into similar ranges. Crucially, these transformations are theoretically motivated and follow from relatively basic epidemiological considerations (as opposed to assumptions about how age, learning, and cognition hang together). That is, the basic reproductive ratio functions as a theoretically grounded way of normalizing data from diverse domains to map them onto the same scale.