A power law (PL) describes a non-linear functional relationship between two variables—one varies as a power of another (e.g., f(x) = ax^b) and has certain properties, such as scale invariance, lack of well-defined average value, and universality [1–4]. The scale invariance is exhibited by a simple log-transformation of PL into a straight-line (linear) on log-log scale {e.g., ln [f(x)] = ln (a) + bln (x)}, and it also specifies that all PLs with a particular scaling exponent are equivalent up to constant factors, e.g., f(cx) = a(cx)^b = c^bf(x)∝f(x). The lack of well-defined average value refers to a reality that arithmetic mean or average is a poor indicator for the majority of the power-law variables (e.g., the average income of a population that includes a billionaire). A PL usually has a well-defined mean only for a certain range of its scaling exponents, and the variance of PL seems disproportionally large and is frequently not well defined, which explains the association between PL phenomena and black swan behavior. This also makes many classic statistical methods that are based on the normal distribution and/or on the homogeneity of variance inapplicable to data of PL phenomena. The third property of PL is the universality that is to do with the scale invariance or the equivalence of PLs with a particular scaling exponent. In dynamic systems, diverse systems with the same power-law scaling exponents (also known as critical exponents) can exhibit identical scaling behavior and share the same fundamental dynamics as they approach criticality, such as phase transitions. Systems with the same critical exponents are classified as belonging to the same universality class [1–6].

Taylor's power law (TPL), first discovered by entomologist and ecologist L. R. Taylor [7] in investigating the spatial distribution of insect populations more than a half-century ago [5, 8–12], has been expanded far beyond its original domains of agricultural entomology and population ecology [1, 2, 5, 6, 13–19]. The TPL is one form of PLs that describe the distributions of a wide variety of natural and man-made phenomena over a wide range of scales [20–22]. PL patterns have been discovered/rediscovered in astronomy, biology and ecology, computer science, criminology, economics, finance, geology, mathematics, meteorology, physics, statistics, and especially in inter-disciplinary fields [3, 4].

Taylor's power law, as one of the most well-known PLs in ecology and biology, shares the three general properties of PLs mentioned above. It differs from other PLs in choosing its two variables: the mean (M) and variance (V) of population abundances (counts) [5, 7, 11], i.e.,V = aM^b. It has also been rediscovered in many other fields beyond its original domain of population ecologies, such as epidemiology, genomics and metagenomics, and computer science [5, 6, 14, 15, 20–23]. It was extended to community ecology, especially the community and landscape ecology of human microbiomes [6, 23, 24]. In the present study, we take the advantage of TPL in modeling the relationship between mean and variance for quantifying the uncertainty of natural phenomenon. This should be feasible because variance is arguably the most commonly used statistic moment for characterizing the uncertainty (variation) of random variables. The approach is particularly advantageous if the distribution of random variable follows PL distribution, but it should still be applicable otherwise since TPL holds across a wide range of mean-variance relationships as signaled by a wide range of its scaling parameter (b).

Species-area relationship (SAR) is another classic PL in ecology, which relates the number of species (species richness: S) and the area (A) of species habitat, in the form of S = cA^z. Ma [25, 26] further extended the SAR to a general diversity-area relationship (DAR) by replacing species number (richness) with the general diversity measured in Hill numbers. Ma [25–27] further introduced PL with exponential cutoff (PLEC) model to describe DAR and proposed the concept of maximal accrual diversity (MAD). Based on the PLEC model for DAR, Ma derived the estimation of MAD. MAD can be considered as a proxy of potential (dark) diversity, which includes both local diversity and the portion of diversity that are absent locally but present regionally (or in regional species pools). In other words, potential diversity measures both visible and invisible (dark) diversities and is of obvious significance for biodiversity conservation. Similar to SAR/DAR, there is so-called species-time relationship (STR) or diversity-time relationship (DTR) [26]. The PLEC version of DTR was successfully applied to predict the inflection points (tipping points) of COVID-19 infections [28].

Power law with exponential cutoff, as a variant of PL, has more general applications beyond the abovementioned SAR/DAR/STR/DTR/COVID-19 predictions [25–28]. PL behaves (grows or declines) exponentially, especially at late stages, and the PLEC possesses an exponential cutoff parameter that ultimately tapers off the unlimited growth or decline ultimately. Therefore, the PLEC model is of important practical significance when prediction or estimation is needed. However, existing PLEC modeling can only provide point estimation and not the interval of the estimation (i.e., uncertainty quantification of the estimation).

The present article is aimed to integrate the TPL with the PLEC model with the objective to improve the predictive power of the PLEC model by quantifying the uncertainty of estimation (prediction) with TPL. Specifically, by harnessing the capacity of TPL in estimating the variance (SD), we develop an approach to offering CIs for the estimation of PLEC quantities (see Figure 1). We demonstrate our method with the estimations of potential American gut microbiome (AGM) diversity and COVID-19 fatalities. The demonstrated approach can be potentially suitable for a predictive mathematical model as long as the variance and mean of its dependent variable can be quantified with the TPL model.

The following findings can be summarized from previous sections:

In perspective, we expect that the power-law coupling approach possesses great promises for a wide range of important problems whenever both TPL and PLEC models can be successfully applied. The precondition that both PL models must be reliably built also reminds us that the approach cannot be a silver-bullet solution. For example, in the case of PLEC-DAR modeling for the gut microbiome diversity, we only presented the results for species richness (i.e., the Hill numbers when diversity order q = 0). The reason was that TPL was failed to fit the mean and variance of the Hill numbers at other diversity orders. This made it infeasible to estimate the CIs for other diversity orders. TPL has been found applicable in many natural and man-made systems; however, there are situations where it may fail. Taylor [5] conjectured that TPL might work poorly for ratios and very poorly for bounded ratios. The Hill number at diversity q = 0 (i.e., species richness) is an integer, but at other diversity orders, such as q = 1, 2, or 3, the Hill numbers are indeed bounded ratios. Taylor's [5] conjecture may explain the limitation of TPL in fitting the mean-variance relationship in measuring biodiversity.

Furthermore, the universality property of PLs hints great promises for our coupling approach, although there have been occasional debates on proving universally in practical data fitted to PLs {e.g., [4]}. The universality refers to the equivalence of PLs with a particular scaling parameter (exponent), such as b in TPL, z in SAR (DAR), or w in STR (DTR), which are termed critical exponents. Critical exponents are termed so because the PL distributions of certain quantities are associated with phase transitions in dynamic systems as they approach criticality. The hallmark of universality is therefore the sharing of dynamics, and the systems with precisely the same critical exponents are said to belong to the same universality class. In the field of TPL, the transitions between aggregated (heterogeneous), random (Poisson), and uniform distribution of biological population or species abundance distribution can be characterized by the population aggregation critical density (PACD) [13] or community heterogeneity critical diversity (CHCD) [23], which could be generated by self-organizations in the ecosystems (e.g., population or community). Different from physics, the processes, such as self-organization in biology and ecology, are difficult to prove rigorously. Nevertheless, there are indeed observations of the equivalence of TPL scaling exponents, such as the apparent invariance (constancy) of TPL scaling parameter (b) of global hot spring microbiomes across wide ranges of pH values and temperatures [40]. If these observations are found general in ecosystems, then the predictions based on our coupling approach of PLs can be not only feasible but also be reliable. Unlike the events that are governed by the normal (Gaussian) distribution, the events governed by the highly skewed PL distribution are particularly challenging to predict. In particular, some PL-governed events often lack of well-defined average values, but with potentially unbounded variance, tend to be black-swan and/or catastrophic. This also makes our proposed coupling method particularly valuable potentially.

Finally, we would like to present a very brief discussion on the general modeling strategy that is related to the two demonstrative case studies for illustrating the applications of the proposed coupling PLs. Since modeling strategy may be influenced by domain-specific knowledge, the discussion below is conducted in the context of ecological modeling {e.g., [41]} and COVID-19 prediction {e.g., [28, 37]}, to keep relevant to the two demonstrative examples of this article. According to Levins [42], it is ideal to operate with manageable models that maximize generality, realism, and precision toward the overlapping, but not identical goals of understanding, predicting, and modifying nature. Levins [42] distinguished three alternative strategies, namely, [1] sacrifice generality to realism and precision (which is the approach of most simulation models); [2] sacrifice realism to generality and precision (most physicists who work in population ecology follow this tradition; the Lotka-Volterra model is an example); and [3] sacrifice precision to realism and generality, an example of this strategy is the theory of island biogeography by MacArthur and Wilson [43], which we have briefly discussed in the final paragraphs of this article. It is noted that the term “precision” here, more precisely, refers to more specific or detailed factors (information) used in modeling works.

Although Darwin's evolutionary theory answered the question of where and how biological species are originated and evolved on the earth's planet, the evolutionary theory did not explain how and why species co-exist and form diverse communities of species. Indeed, the competition or struggle for living, one essential aspect of evolutionary theory, would predict that the earth could be dominated by a handful of ultimate winners from competitions, which is obviously not consistent with the reality that the earth is cohabited by diverse species that usually coexist. In fact, biodiversity has been studied and paid attention by both scientists and the general public extensively in modern societies [44]. The study of biodiversity distribution, known as biogeography, was stuck in a “natural history phase” until the 1960s, due to the dominance by the collection of data and description of species, which were necessary but not sufficient. MacArthur and Wilson [43] demonstrated in their landmark monograph “The Theory of Island Biogeography” that the first principles of population ecology and genetics can be applied to explain how distance and area combine to regulate the balance between immigration and extinction in island populations. They were motivated to stimulate new forms of theoretical and empirical studies, rather than synthesizing and unifying existing theories or establishing a general new theory. Somewhat contrary to their unassuming start, their work does lead to a stronger theory of biodiversity. Today, even a half of a century has passed, the monograph continues to be inspiring and remains at the center of discussions about the geographic distributions of species in biodiversity research. Here are mentioned the above historical episodes for two reasons. First, MacArthur and Wilson's [43] island biogeography theory is well recognized as a landmark breakthrough in biogeography and community ecology. It can be considered as an extremely successful example of the modeling strategy of sacrificing precision (details) to realism and generality. Second, one of the key elements of their theory is the SAR PL, which is one of the PLs coupled in this study, i.e., the DAR extended by Ma [25–27]. Both factors should have contributed to the success of the biodiversity and COVID-19 predictions demonstrated in this study.

Besides the frequent infeasibility in simultaneously maximizing generality, realism, and precision of mathematical models, another commonly encountered dilemma for modelers is the complex vs. simple models. According to Jewell et al. [37], intuitively, simpler models may offer less valid predictions due to their limited capacity in capturing complex and unobserved human mixing patterns and other time-varying properties of infectious disease spread. However, complex models can be no more reliable than simple ones if they fail to capture key aspects of the problem. In addition, complex models may produce the illusion of realism and make it prone to omit crucial points. Furthermore, outputs of complex models are usually more sensitive to changes in parametric assumptions and/or the estimations of external disease or environmental factors, such as the lengths of latent/infectious periods due to mutation of a pathogen [37]. Of course, the disadvantages of complex models are not necessarily the advantages of simpler models. On the other hand, simpler models are usually inexpensive to construct and manage, and they may provide adequate solutions under certain circumstances. We hope that this work proposes and demonstrates a simple modeling approach for certain problems where PLs are applicable.

Finally, one may wonder how accurate the prediction of our coupling power laws is in forecasting the worldwide COVID-19 fatality. Compared with the worldwide COVID-19 fatality number on January 24th, 2022 (when this paper is formally accepted and online), the error rate of the prediction with our coupled power laws, made in the May 2021 (based on the fatality data then alone) is approximately 7% only (i.e., the precision level is 93%).

Specifically, we computed the worldwide fatality on Jan 24, 2022 with the following parameters and formula: F = CT^w exp(dT) + F₀, where C = 4957.140, w = 1.248, T = 308, d = −0.003, F₀ = 2716229 (the fatality number at the starting date of the model-building, i.e., March 21st, 2021 in the case of the world model). We obtain F = 5226117, i.e., the predicted fatality number on January 24, 2022, and the prediction is based on the power law model established with the worldwide fatality numbers before May 21st, 2021 (Table 1, Figure 3). According to the publicly released COVID-19 fatality (https://github.com/CSSEGISandData/COVID-19), the actual worldwide fatality number is 5610729 on Janurary 24th, 2022. The precision of point estimation is then 92.6 or 93% approximately. Furthermore, the 95% confidence interval of the estimation can be computed with Eqns. (12, 13) and is [4713112, 5739122].Therefore, the point estimation of the worldwide COVID-19 fatality number on January 24th 2022 does fall within the confidence interval with a precision level of 92.6%. In fact, these results (including Table 1 and Figure 3) had already been released on May 23rd 2021 in the preprint of this article Ma [45].

Our model (Table 1, Figure 3) also predicted that the turning point (inflection point) of worldwide COVID-19 fatality would not occur until the July of 2022, which contrasts with the recent prediction made by Murray [46] who suggested that the “end of the pandemic is near” by March 2022.

Publicly available datasets were analyzed in this study. Available publicly: https://github.com/CSSEGISandData/COVID-19.

The author confirms being the sole contributor of this work and has approved it for publication.

This study received funding from the following sources: A National Natural Science Foundation (NSFC) grant (no. 31970116), Cloud-Ridge Industry Technology Leader Grant.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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