# Epidemiological Model With Anomalous Kinetics: Early Stages of the COVID-19 Pandemic

^{1}Department of Physics, Faculty of Science, Ege University, Izmir, Turkey^{2}Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems, Rio de Janeiro, Brazil^{3}Santa Fe Institute, Santa Fe, NM, United States^{4}Complexity Science Hub Vienna, Vienna, Austria

We generalize the phenomenological, law of mass action-like, SIR and SEIR epidemiological models to situations with anomalous kinetics. Specifically, the contagion and removal terms, normally linear in the fraction *I* of infected people, are taken to depend on *N* for each country.

## 1 Introduction

The classic and still widely used SIR and SEIR epidemiological models [1] represent contagion and removal in analogy with the law of mass action in chemistry, corresponding to a mean-field approach based on the assumption of homogeneous mixing. The latter hypothesis constitutes an oversimplification, particularly for the COVID-19 pandemic, due to strong government intervention (social distancing; lockdown) and underreporting as the number of cases grows beyond testing capacity. Diverse aspects are discussed, assuming homogeneous or nonhomogeneous mixing, in epidemiological models in general [2–4], as well as in the current pandemic [5–15].

Epidemic models can be formulated on varying levels of detail, from individual agents in geographically realistic settings to models of large populations without spatial structure. Each level has its own benefits and costs; the study of an ensemble of models is expected to yield a more reliable description than any single approach in isolation. In chemical kinetics of processes involving anomalous diffusion and/or complex conformational pathways, effective descriptions typically employ noninteger power-law terms where the mean-field or mass-action analysis involves integer powers of concentrations, as in the analysis of reassociation of folded proteins [16, 17]. With this motivation, we consider SIR- and SEIR-like models in which the contagion and removal terms depend on *I*, as they do in mean-field/homogeneous descriptions. Such generalization is consistent with anomalous human mobility and spatial disease dynamics [18, 19] and emerges naturally within statistical mechanics based on nonadditive entropies [20] as we show in what comes next.

Let us now follow along lines close to [16], which provided a satisfactory description of reassociation in folded proteins [17]. Consider the equation

Its solution is given by

with *only then* the coefficient *a* (which characterizes the scale of the evolution of *not* renormalized by the initial condition *a*; the difference can be very important depending on the values of *q* and

## 2 Generalized Models

### 2.1 *q*-SIR Model

The SIR set of equations is (see [1] for instance) as follows:

with *N* is the total population, *removed* means either recovered or dead). Now let us *q*-generalize this model as follows:

with *effective* population *q*-SIR models given by Eqs 3 and 4, respectively, by focusing on the *β* term; i.e., let us compare *increases* with time, a fixed value for *decreases* with time. These tendencies are similarly realistic since they both reflect, each in its own manner, the generic action of pandemic authorities to isolate people in order to decrease the contagion represented by the *β* term in both models.

The particular limit

and hence,

where

Before the peak,

**FIGURE 1**. Time evolution of **(A)** Fixed *β* and various values of *γ*. For **(B)** Fixed *γ* and various values of *β*. For *q*_{down}-exponential function is precisely recovered. **(C)** Time dependence of the slope of **(D)** The maximal slope of

We have checked that the *q*-SIR model provides functions

### 2.2 *q*-SEIR Model

The *q*-SIR model is not capable (for any choice of its parameters) of correctly fitting the epidemiologically crucial function *q*-generalize the SEIR model with no vital dynamics (no births; no deaths), which is given by

with *E* stands for *exposed*. We can generalize it as follows:

where once again we have generalized the bilinear couplings between subpopulations into nonbilinear ones and the linear *q*-SIR model, as defined here above. Notice that the *cumulative function**q*-SEIR set of four equations by allowing in the right hand *S* and *E*, but no need has emerged to increase the number of free parameters of the model, since the allowance for *convex* function of *concave* function of *γ* and *N* of the particular region under focus. For other mathematical aspects of nonlinear models such as the present one, the reader may refer to [26].

**FIGURE 2**. Time evolution of *q*-SEIR equations with *t* to run virtually from zero to infinity (not necessarily during only the typical range of real epidemics, say 1,000 days). **(A)** Fixed *σ*. **(B)** Fixed *β*. **(C)** Time dependence of the slope of *q*-SIR equations, is *not* bounded between **(D)** The maximal slope of *q*-SEIR model recovers the standard SEIR model, which increases exponentially (and not as a power-law) toward the corresponding peak. Notice also that this slope decreases when

## 3 Application of q-SEIR Model to COVID-19 Pandemic

In Figures 3 and 4, we have illustrations of this model for realistic COVID-19 cases. We identify the present variable *I* with the number of *active cases*,^{1} as regularly updated online [27]. We verify that the description provided by the *q*-SEIR model for nonhomogeneous epidemiological mixing is indeed quite satisfactory for the early stages of the pandemic (before an unpredictable but possible second wave). Let us also mention that we have not followed here a road looking for the minimal number of free parameters, but rather a road where various realistic elements are taken into account, even if at the fitting-parameter level some of them might be redundant. Any further model yielding a deeper, or even first-principle, expression of exponents such as *Porous Medium Equation* [28] to Plastino and Plastino nonlinear Fokker-Planck equation [29], which in turn implied the scaling law *q* being the index value of the *q*-Gaussian solution for the nonlinear Fokker-Planck equation). This scaling law recovers, for *D*-dimensional *H*-theorem, and Carnot’s cycle efficiency, with microscopically established analytical equations of state [34–37]. An attempt to follow along similar lines for the present *q*-SEIR model would surely be a very interesting challenge. In summary, we have *q*-generalized, through Eq. (8), the SEIR epidemiological model. By solving this set of deterministic equations given the initial conditions and its parameters, we obtain

**FIGURE 3**. Time evolution of available data for COVID-19 numbers of active cases (probably under-reported in most cases) and deaths per day [27] and their (linear scale) Least Squares Method fittings with *q*-SEIR model; *N* is the population of the country. Notice that, (i) by convention, *reproduction number* or *growth rate*), *exposition time*), *recovering time*), and *incubation time*) are consistent with those currently indicated in the literature [21–25]; (vi) we considered all the data reported until June 13, excluding some very initial transients or sudden anomalous discrepancies (e.g., in China, Turkey, and Brazil).

**FIGURE 4**. Continuation of Figure 3.

From a general perspective, let us stress that the law of mass action, the Arrhenius relaxation law, and the Kramers mechanism [38] of escape over a barrier through normal diffusion constitute pillars of contemporary chemistry. They are consistent with Boltzmann-Gibbs (BG) statistical mechanics and constitute some of its important successes. However, they need to be modified when the system exhibits complexity due to hierarchical space and/or time structures. It is along this line that a generalization has been proposed based on nonadditive entropies [20], characterized by the index *q* (*q*-generalization of the standard SEIR model.

At the level of the numerical performance of the present *q*-SEIR model for the COVID-19 pandemic, it advantageously compares with models including time-dependent coefficients [44–47]. For instance, the SEIQRDP model [44, 45, 47] includes seven equations with several coefficients, two of them phenomenologically being time-dependent. It does fit rather well the COVID-19 reported data until a given date. However, the *q*-SEIR, which includes four (instead of seven) equations with several coefficients, all of them being fixed in time, fits definitively better the same data for all the countries that we have checked: see illustrations in Figure 5.

**FIGURE 5**. Comparison, using precisely the same reported data (green dots), of the SEIQRDP model (left plots) and the *q*-SEIR model (right plots) for the time series of Germany (from February 26^{th} to June 13^{th} 2020) and of Italy (from February 21^{st} to June 13^{th} 2020). To obtain the SEIQRDP fittings (seven linear/bilinear equations satisfying *q*-SEIR fittings (four not necessarily linear/bilinear equations satisfying

At this stage, let us emphasize a rather interesting fact. Neither the SIR nor the SEIR models distinguish the *dead* from the *recovered*, within the *removed* (R) subpopulation. However, the same values for *both* the numbers of active cases and of deaths per day for a given country, as shown in Figures 3 and 4. At this point, it would be worth noting that although the present model, unlike, for example, the SEIRD model [48, 49], does not distinguish deaths from healings, the deceased cases will still be roughly proportional to infected people. It is this proportionality, we believe, which makes us obtain reasonable fits using the variable *I* of the model, again incorporating the proportionality into

## 4 Conclusion

To conclude, let us remind that the *q*-SEIR model recovers, as particular instances, the *q*-SIR model introduced here, as well as the traditional SEIR and SIR ones. It has, however, an important mathematical difference with the usual epidemiological models. Virtually all these models (SIR, SEIR, SAIR, SEAIR, SIRASD, SEAUCR, and SEIQRDP) are defined through equations that are multilinear in their variables; i.e., that are linear in each one of its variables. This multilinearity disappears in models such as the present *q*-SIR and *q*-SEIR ones if either *q*-generalized models; its precise interpretation remains to be elucidated, perhaps in terms of the sociogeographical circumstances of that particular region. Last but not least, let us stress that the aim of the present *q*-SEIR model is to *mesoscopically describe a single epidemiological peak*, including its realistic *power-law* growth and relaxation in the time evolution of the number of active cases, and by no means to qualitatively address possibilities such as the emergence of two or more peaks, a task which is (sort of naturally, but possibly less justified on fundamental grounds) attainable within approaches using traditional (multilinear) models where one or more coefficients are allowed to phenomenologically depend on time by realistically adjusting their evolution along the actual epidemics. Alternatively, it is always possible to approach the two-peak case by proposing a linear combination of two *q*-SEIR curves starting each of them at two different values of the departing time (

## Data Availability Statement

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found below: https://data.humdata.org/dataset/novel-coronavirus-2019-ncov-cases

## Author Contributions

All authors contributed equally to the article.

## Conflict of Interest

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

## Acknowledgments

We have greatly benefitted from very fruitful discussions with R. Dickman, T. Pereira and D. Eroglu, as well as from partial financial support by CNPq and Faperj (Brazilian agencies).

## Footnotes

^{1}At this point, it should be noted that the variable *I* in the model represents both detected plus undetected active cases although, of course, the database includes only detected ones as active cases. However, since these cases are roughly proportional to each other (with a proportionality coefficient which might change from country to country), we use the model variable *I* to fit the active cases reflecting the incorporation of this proportionality into *ρ*.

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Keywords: COVID-19, pandemics, complex systems, nonextensive statistical mechanics, epidemiological models

Citation: Tirnakli U and Tsallis C (2020) Epidemiological Model With Anomalous Kinetics: Early Stages of the COVID-19 Pandemic. *Front. Phys.* 8:613168. doi: 10.3389/fphy.2020.613168

Received: 01 October 2020; Accepted: 02 November 2020;

Published: 05 December 2020.

Edited by:

Matjaž Perc, University of Maribor, SloveniaReviewed by:

Tassos Bountis, University of Patras, GreeceOscar Sotolongo, University of Havana, Cuba

Airton Deppman, University of São Paulo, Brazil

Copyright © 2020 Tirnakli and Tsallis. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Ugur Tirnakli, ugur.tirnakli@ege.edu.tr