key: cord-0493740-rzurz5bj
authors: Ma, Zhanshan
title: Spatiotemporal fluctuation scaling law and metapopulation modeling of the novel coronavirus (COVID-19) and SARS outbreaks
date: 2020-03-08
journal: nan
DOI: nan
sha: 0cda048d50c1e093a9c5c6f126836a7e12a14eb7
doc_id: 493740
cord_uid: rzurz5bj

We comparatively analyzed the spatiotemporal fluctuations of the 2019-novel coronavirus (COVID-19) and SARS outbreaks to understand their epidemiological characteristics. Methodologically, we introduced TPL (Taylor power law) to characterize their spatiotemporal heterogeneity/stability and Hubbell (2001) unified neutral theory of biodiversity (UNTB) [specifically Harris et al. (2015) HDP-MSN model (hierarchical Dirichlet process multi-site neutral model)] to approximate the metapopulation of coronavirus infections. First, TPL analysis suggested that the coronaviruses appear to have a specific heterogeneity/stability scaling parameter (TPL-b) slightly exceeding 2 for cumulative infections or exceeding 1 for daily incremental infections, suggesting their potentially chaotic, unstable outbreaks. Another TPL parameter (M0) (i.e., infection critical threshold) depends on virus kinds (COVID-19/SARS), time (disease-stages), space (regions) and public-health interventions (e.g., quarantines and mobility control). M0 measures the infection level, at which infections are random (Poisson distribution) and below which infections follow uniform distribution and may die off if M0 coincides or below the level of Allee effects. It was found that COVID-19 outbreak seems nearly twice more risky than SARS, and the lower infection threshold may be due to its lower lethality than SARS since lower fatality rates can facilitate the survival and spread of pathogen. Second, metacommunity UNTB neutrality testing seems appropriate for approximating metapopulation of coronavirus infections. Specifically, two parameters {theta} and M, borrowed from neutral theory, may be used to assess the relative significance of infection through local contagion vs. infection through migration, both of which may depend on time, space, virus kinds, and particularly public-health interventions.

The spatial and/or temporal distributions of many biological populations including microbes and humans follows Taylor's power law (TPL) (Taylor 1961 , Taylor et al. 1977 , 1983 , 1988 , and recent studies have also revealed its applicability at the community scale (Ma 2012a , Li & Ma 2019 , Taylor 2019 , Ma & Taylor 2020 . TPL has been verified by hundreds if not thousands of field observations in nature (Taylor 2019), and it has also found cross-disciplinary applications beyond its original domains of biology and ecology in disciplines such as computer science, natural disaster modeling, and experimental physics (Eisler et al. 2008 , Ma 2012b , Tippett & Cohen 2016 , Helmrich et al. 2020 . Despite that TPL was proposed more than a half century ago (Taylor 1961) and there is not yet a consensus on the underlying mechanisms generating TPL (Eisler 2008 , Stumpf & Porter 2012 , there seems to be a recent resurgence of interests in this near universal model that captures the relationship between the population mean (m) and variance (V) in the form of a simple power function (i.e., V=am b ) (e.g., Cohen & Schuster 2012 , Cohen & Xu 2015 , Giometto et al. 2015 , Kalinin et al. 2018 , Oh et al. 2016 , Reuman et al. 2017 ). Among the numerous existing studies on TPL, there have already been its applications to the analyses of spatial variation of human population (Cohen et al. 2013) , human mortality (Bohk et al. 2015) , and epidemiology (Rhodes & Anderson 1996) . Given these previous applications to human demography and epidemiology, we postulate that TPL should also be applicable to the outbreak analyses of coronavirus infection diseases such as the still ongoing 2019 novel coronavirus (2019-nCoV) or COVID-19 (coronavirus-infected Pneumonia disease) (https://www.who.int/emergencies/diseases/novel-coronavirus-2019) (Li et al. 2020 , Thompson et al. 2020 , Kucharski et al. 2020 , Zhang et al. 2020 and 2003 SARS (Severe Acute Respiratory Syndrome) (https://www.who.int/csr/sars). In the present report, we test this hypothesis and further explore possible epidemiological processes (mechanisms) underlying the outbreak of COVID-19 infections.

While TPL can be harnessed to investigate the spatiotemporal fluctuations of coronaviruses, specifically, the scaling (changes) law of coronaviruses infections over space and time, we also aim to understand the spread of the virus infections from both local contagion (endemic) and external migration (epidemic and pandemic) perspectives. Nevertheless, this can be rather challenging given the lack of controlled experimental data, which is ethically infeasible to collect obviously. In principle, all of the infections existing globally constitute a metapopulation of people infected by the coronavirus, but constructing standard epidemiological models (e.g., Wang et al. 2018 , Rivers et al. 2019 with existing data is rather difficult. We realized that Hubbell's (2001) neutral theory of biodiversity, which is one of the four major metacommunity models (the other three include species sorting, mass effect, and patch dynamics) (Rosindell et al. 2011 (Rosindell et al. , 2012 Vellend 2010 Vellend , 2016 , might be adapted to approximate the meta-population dynamics. This approximation allows us to obtain, to the minimum, an educated guess for the local contagion spread and global dispersal (migration) parameters of the coronavirus infections.

Overall, this study sets two primary objectives: (i) to investigate the spatiotemporal fluctuation scaling law and (ii) to obtain an educated guess for the local contagion spread and global migration parameters of the COVID-19 infections. In addition, we also perform comparative analyses with the SARS to get more general insights on the epidemiology of coronavirus infections. To the best of our knowledge, this should be the first systematic application of TPL and UNTB in epidemiology, and obtained scaling/contagion/migration parameters should also be of significant biomedical importance.

We collected the worldwide, daily incremental and cumulative infections of 2019 novel coronavirus and SARS, respectively. For the datasets collected in China, the unit of data collections was set to Chinese provinces. In addition, for the COVID-19 infections, we also collected the datasets of 17 cities of Hubei province of China. For the worldwide COVID-19 infections, the unit of data collections was set to country or region recognized by the WHO (world health organization). The date range for collecting the SARS data was between March 17 and August 7 of 2003 (136 days), and that for COVID-19 was between January 19 and Feb 29, 2020 (40 Days). Since the COVID-19 infections are still continuing, the analyses conducted in this report may be updated periodically. Taylor (1961) discovered that the relationship between mean abundance (m) and corresponding variance (V) of biological populations follows the following power function,

where b is termed population aggregation parameter and is thought to be species-specific, and a is initially thought to be related to sampling schemes used to obtain the data. The relationship is known as Taylor 

TPL was initially discovered in fitting the spatial or cross-sectional sampling data (Taylor 1961) and later found that it is equally applicable to temporal or time-series sampling data (Taylor 2019). In the context of time-series modeling, b measures the population stability (variability).

More recently, it was found that TPL can be extended to community level from its original population level (Ma 2015) . At the community level, the four Taylor's power law extensions (TPLE), can be used to measure the community spatial (temporal) heterogeneity ( on both the general principle of TPL (explained above) and system-or data-specific information (such as the biology of COVID or SARS).

Specifically, regarding the first aspect or the fitting of TPL, we adopt two fitting approaches: one is the simple linear regression via log-transformation [Eqn.

(2)] and another is the geometric mean regression (GMR) (Clark and Perry 1995, Warton et al 2007) . The advantage of the first approach is its computational simplicity and the advantage of the second or GMR is that it is more robust for small sample size (N<15 according to Clark and Perry 1995) . Both approaches preserve the scale invariance of power law. Regarding the second aspect or the interpretation of the TPL, there is controversy on the claim that TPL parameter (b) is species specific, in particular when there are changes in sampling method, life stage, environment or spatial scale (Taylor et al. 1988, Clark and Perry 1995) . Our opinion is that, unlike parameter a, parameter b is primarily shaped by evolutionary forces and less influenced by ecological or environmental factors. However, we do not take the "invariance" or "constancy" at ecological time scale or with environmental factors as granted. Instead, we draw conclusions based on rigorous statistical tests of the differences in parameter b among treatments. In the case of this study, we perform the permutation (randomization) test to judge whether or not the TPL parameters are invariant. For further information on the randomization test, readers are referred to Collingridge (2013) .

To further harness the TPL parameters, Ma (1991 Ma ( , 2012a Ma ( , 2015 derived a third parameter (M 0 ) for TPL or its extensions at the community scale, population aggregation critical density at the population scale or community critical heterogeneity at the community scale, which is in the form of:

where a & b are TPL parameter. M 0 is the level of mean population abundance, the COVID-19

or SARS infection level in the case of this study, at which the fluctuation (dynamics) of virus infection is random (following Poisson statistical distribution generally). When m>M 0 , the population (infection) fluctuation (dynamics) is more irregular than random (often following highly skewed distributions such as the negative binomial distribution or power-law statistical distribution). In this case, population is highly unstable and the infection may expand continuously. When m<M 0 , the infection fluctuation is regular and may follow the uniform statistical distribution. In this case, the inflection level should be stabilized or might even die off.

When m=M 0 , the infection is random and should follow Poisson distribution statistically. In the context of this study, we term PACD or M 0 as mean infection critical threshold or mean infection threshold, which is similar to classic Allee effects (Allee 1927 

where € β = (β 1 ,...,β S ) is the relative frequency of each species in the metacommunity.

At the metacommunity level, a Dirichlet process is also applicable and the metacommunity distribution can be modeled with a stick breaking process, i.e., € β~Stick(θ )

Given that both local community and metacommunity follow Dirichlet processes, the problem can be formulated as a hierarchical Dirichlet process (HDP) in the domain of machine learning (Teh et al. 2006 , Harris et al. 2015 .

Furthermore, Dirichlet process (DP) can be formulated as the so-called Chinese restaurant process, from which Antoniak equation (Antoniak 1974 ) can be derived. The Antoniak equation

represents for the number of types (or species) (S) observed following N draws from a Dirichlet process with concentration parameter θ, and is with the following form:

where s(N, S) is the unsigned Stirling number of the first kind and Γ(.) is the gamma function. By treating the coronavirus infections at different sites (e.g., provinces of China, or different countries/regions of the world, in this study) as a "metacommunity" consisting of N local communities (e.g., each local community corresponding to a province), the above-described metacommunity model can be built with the dataset of daily incremental infections, nationally or internationally. Different from traditional metacommunity concept, here the "metacommunity"

is actually a metapopulation consisting of N local populations. However, if we treat the local infections at a particular time point (e.g., day) as a local sub-population, then the virus subpopulations at different time points can be considered as a total population (or "species" in the terminology of community ecology). With this conceptual transformation, the concept and models for metacommunity and Hubbell's UNTB can be readily applied to the metapopulation of coronavirus infections without a need to revise the models. With this adaptive scheme, the fundamental biodiversity number (speciation rate: θ) from the previously introduced HDP-MSN model can be used to approximate the average local (contagion) infection rate. Similarly, the fundamental dispersal number (M) can be used to approximate the average infection rate through migration. The migration probability (m), which is a function of M, has a similar interpretation as M, but simply in the form of probability.

As a side note, we expect that the coronavirus infections should follow Hubbell's (2001) UNTB theory, which is not surprising to us, due to our treatment of metapopulation as metacommunity.

This is because the "species" in our "metacommunity" are, in fact, populations of a single virus species, and they should be "equivalent" in terms of the neutral theory. Therefore, whether or not 

From Tables 1-2, which were summarized from Tables S1-S8 in the OSI (online supplementary information), we obtain the following findings, which are also illustrated with Figs 1-7.

(i) The spatiotemporal fluctuation scaling (changes) of COVID-19 infections follow TPL (Taylor's power law) at all scales (schemes) tested, including world-wide, country-wide, cumulative and daily incremental infections, as well as gradually shrunken partial datasets to test TPL robustness, as evidenced by the p-value<0.001 from the TPL fittings. The brief results were summarized in Table 1 and the detailed results were displayed in Tables S1-S2 and S4-S7. It was discovered that the infections of COVID-19, like populations of other organisms, follow seeming universal power law. This implies that the infection of the novel coronavirus is highly contagious and its outbreak (spread) is chaotically unstable in general, as indicated by the b-values exceeding 1 (for daily incremental infection) or exceeding 2 (for cumulative infections).

(ii) The TPL aggregation (stability) parameter (b) for COVID-19 seems rather stable or even invariant, as evidenced by the randomization tests (Tables S3 & S8 ). The randomization tests were performed by statistically comparing the model parameters fitted with the whole datasets and those fitted with the "partial datasets." The schemes of partial datasets were devised by removing in China is about twice higher than that of the worldwide, but about ½ lower than that of worldwide SARS infections. In other words, the COVID-19 infections seem more "dangerous" than SARS from a public health perspective.

Note that M 0 is the mean infection level; therefore, it can be more meaningful to convert it into absolute value from a biomedical perspective. For example, M 0 =5.758 for COVID-19 in China, when converted to total national infection level, would be 5.758x34 (provinces)=196. This threshold number may suggest that when the number of total infections nationally is at this level, the infections are random. When infections exceed this threshold level, the infections can be nonrandom and highly unstable. When infections are below this threshold level, the infection should be rather stable (following a Uniform distribution statistically) or may even die off if the threshold level coincides with the level of Allee effects. Since the level of Allee effects is still unknown, whether or not the infections under M 0 will die off is still an open question.

(iv) As expected, all datasets passed the neutrality tests of HDP-MSN model as indicated by the p-value>0.05 (Table 2) . We use the ratio of Q=M/θ as a measure of the relative importance of "infection spread via migration" vs. "infections spread via local contagion" in spreading the infections, with larger Q indicating higher migration role and smaller Q indicating higher local contagion. The ratio of Q=133.581/6.325≈21 (Table 2) indicating that spread via migration is approximately 21 times more significant than spread via local contagion on average nationally in China. However, the worldwide Q ratio is approximately Q=1 (8.037/10.801). Therefore, the ratio Q is dependent on time (disease stages), space (regions), and disease-kinds (COVID-19 or SARS), and perhaps most importantly, public-health interventions such as quarantines and/or mobility control.

Compared with COVID-19, SARS appeared to exhibit a different pattern of the relative importance of migration vs. local contagion in spreading the infections. We postulate that this difference might signal the higher risk of pandemics of COVID-19 compared with SARS.

Nevertheless, we cannot exclude the possibility that the range of SARS datasets were complete, while COVID-19 infections are not over yet.

The third parameter (m) or immigration probability suggests the level of infection via migration.

The m=0.052 in China vs. m=0.003 worldwide for COVID-19 suggested that the risk of infection via migration within China is approximately 10 times higher than that of worldwide migration.

This is most likely due to the disruption of international travels. Comparing the m for COVID-19 and SARS for worldwide data (0.003 for COVID-19 vs. 0.040 for SARS) may simply be due to stronger travel restrictions imposed for controlling COVID-19 outbreaks. Similar to the above comparison, this difference may be due to the difference in the data range.

(v) Comparing the PACD (M 0 ) ( Table 1 ) and θ in (Table 2) (Table 1) and θ=6.325 of COVID-19 in China (Table 2) is only approximately 0.6. We postulate that, when the "local speciation (contagion)" (θ) approximates the population aggregation critical density or infection critical threshold (M 0 ), the infections could become random (suggesting a potentially stabilized infection level). That is, without inputs from external migration, local infections via local contagion (measured by θ) could become random (as indicated by M 0 ). If this postulation is true, then it may suggest that the mobility control (such as travel restrictions or quarantines) can be critically effective in stabilizing outbreaks. This finding indicates the complementary nature of the two approaches we adopted in this study. Nevertheless, it is important to reiterate that this closeness between M 0 and θ (M 0 ≈θ) is likely to be an exception, rather than the "norm", for the reasons explained below.

As to the lack of closeness between M 0 and θ of COVID-19 at worldwide scale (M 0 =2.701 vs.

θ=10.801), this may indicate that control mobility is not sufficient to stabilize infections world widely anymore at the current stage. In the case of SARS, the worldwide M 0 =9.475 vs.

θ=27.169, indicated that local contagion (θ) alone already exceed the infection critical threshold (M 0 ). Therefore, θ >> M 0 (PACD) may be the "norm."

We aimed to discover critical insights on the endemic/epidemic/pandemic characteristics of the While TPL-b appears rather stable or even invariant with time and/or space, the population aggregation critical density (PACD) (i.e., M 0 or infection critical threshold), which is the level at which infections are random and below which infections may be stabilized, can depend on disease kinds (COVID-19 or SARS), time (disease or outbreak stages), and space (regions). We postulate that M 0 should also be influenced by public-health interventions such as quarantines and travel restrictions. Therefore, M 0 can be an important epidemiological parameter for evaluating the characteristics of disease outbreaks. For example, COVID-19 exhibited significantly lower threshold (M 0 ) than SARS, suggesting a potentially lower infection "tolerance" threshold of COVID-19. Nevertheless, it should be cautioned that the "tolerance" threshold only means the level of random infections, which may signal the level of stabilized infections. However, whether or not the "tolerance" threshold is biometrically tolerable or safe depends on other biomedical characteristics, among which Allee effects can be a critical factor to determine whether or not the infections will die off or persist. We postulate that if M 0 coincide or is below the level Allee effects in action, the infections may die off.

Second, all datasets we tested easily passed the neutrality test with HDP-MSN and indicated that The ratio of Q=M/θ may be used as a measure of the relative importance of "infection spread via migration" vs. "infections spread via local contagion" in spreading the infections, with larger Q indicating higher migration role and smaller Q indicating higher local contagion.

Third, both the TPL scaling law and metapopulation modeling may complement each other. The difference (closeness) between M 0 (the infection critical threshold from TPL) and θ (local contagion or "speciation" rate from metapopulation modeling) may signal the effectiveness of completely blocking the migration (dispersal) in spreading infections. For example, in the case of COVID-19 infections in China, both parameters (M 0 and θ) are rather close to each other and their difference is only approximately 0.6, suggesting that, without external inputs, the infections from local contagion is only approximately 0.6 higher than the infection critical threshold (M 0 ).

This makes the mission of controlling local contagion for stabilizing infections much less challenging than the mission when θ >> M 0.

Finally, we suggest that the approaches demonstrated previously should be of general applicability for epidemiological research. In particular, we consider TPL-b can be a pathogen specific parameter, primarily shaped by evolutionary forces. Another TPL parameter, Ma (1991 Ma ( , 2015 

The COVID-19 (confirmed) infection datasets during January 19 and February 29 were collected from (https://news.qq.com/zt2020/page/feiyan.htm#/) and (https://news.ifeng.com/c/special/7tPlDSzDgVk). The worldwide SARS infections were from the WHO (https://www.who.int/csr/sars/country/en/). Tables 1-8   Table S1A . Fitting the TPL to the daily cumulative COVID-19 infections in the whole China with two schemes: fitting with the full datasets or fitting with the gradually shrinking datasets 

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