Interpreting airborne pandemics spreading using fractal kinetics’ principles

Fits to COVID-19 data

The best fits of Eqs. 4 and 6 to the data³⁰ were obtained by maximizing the R² of the two adjacent periods. By anchoring the date of each country’s lockdown decision (or any similar form of draconian measure) and moving forward in time, the Levenberg–Marquardt algorithm of least squares was implemented. The lockdown dates are close or very close to the transition from herd kinetics to fractal kinetics and vice versa. A minimum value of R² = 0.985 was set as a criterion of goodness of fit and every value higher than that was accepted. The turning time data point at which the best R² value began to diminish was rejected and its prior time data point was accepted. From that time segment and further on the consequent kinetic profile was fitted to the data points until the plateau of quasi-steady state was reached. The fitting discontinuities observed in the kinetics between the distinct periods (e.g., from second to third period for France) are associated with the fact that I(t) values at the boundary of the two periods were not equalized in our fitting methodology. Between the quasi-steady state and the beginning of the second herd period a 10% change of the number of cumulative infected cases at one week interval was sought in order to establish the commencement of a second viral wave and the reproduction of the according fitting procedure. Data acquisition, modelling and simulations were programmatically implemented with Python language³¹ and its respective libraries.

The “Herd-Fuzzy-Fractal-Herd-Fuzzy-Fractal” (HFF)² kinetic motif

Initially, virus transmission takes place under “herd kinetics’” conditions (Eq. 6, Figure 3A). This prevails until the first preventive measures are imposed; these can be followed by a lockdown decision. The preventive measures and the lockdown status induce a gradual reduction in the rate of the disease spread, i.e., “fractal kinetics” starts operating (Eq. 4, h>1, Figure 3B). The transition from herd kinetics (Eq. 6) to fractal kinetics (Eq. 4, h>1) can be gradual during this fuzzy period, with both equations operating concurrently. The prevalence of fractal kinetics during the lockdown period results in an asymptotic approach of I(t) to the steady state, i.e., I(t) = (1+c)^-1 (see Eq. 5, Figure 3B); according to Eq. 4 the higher the value of the fractal exponent of time h, the more rapid is the approach of I(t) to the steady state. This pattern we call “Herd-Fuzzy-Fractal” (HFF) kinetic motif. When the confirmed new cases reach a steady state, governments relax lockdown rules. In theory, when such a decision is taken, the termination of the first wave of the pandemic has been accomplished. However, the relaxation of lockdown measures in conjunction with the large number of infected individuals at steady state can, after a while, initiate a second wave of the pandemic leading to the application of new preventive measures and new lockdown rules. Consequently, a second wave of the disease emerges (Figures 3C and 3D); hence, the (HFF)² kinetic motif.

Figure 3. A schematic of the “Herd-Fuzzy-Fractal-Herd-Fuzzy-Fractal” (HFF)² kinetic motif of COVID-19 pandemic.

The gray line segments indicate fuzzy periods. A–D. Subplots correspond to the four distinct periods of the kinetic motif. Equations and parameter values used: A: Eq. 6, β = 8, c = 3 × 10¹⁶; B: Eq. 4, h = 4.5, α = 0.012 (time)⁻¹, c = 240; C: Eq. 6, β = 1.18, c = 119350; D: Eq. 4, h = 8, α = 3.35 × 10⁻³(time)⁻¹, c = 20.

Analysis of COVID-19 data

We focused on the data³⁰ of four model countries, namely, France, Greece, Italy, and Spain. Figure 4 shows for each one of the four countries, the fittings of Eqs. 6 and 4 to herd- and fractal-kinetics’ periods’ data, respectively. Parameter estimates derived are listed in Table 1. High R² values listed in this Table indicate that the model of Eqs. 4 and 6, for all four countries, is in excellent agreement with the disease data except Italy’s fourth fractal kinetics’ period data.

Figure 4. Best fits (solid lines) of Eqs. 6 and 4 for the herd kinetics periods and fractal kinetics periods, respectively, to data³⁰ (points) for France, Greece, Italy and Spain.

The data correspond from time zero up to 10 December 2020. The gray line segments indicate fuzzy periods.

For the first herd kinetics’ period, the estimate for β_{herd 1} in Greece was found to be 2.38 ± 0.06, which is much smaller than for the other three countries. This is in agreement with the remarkably lower initial I(t) profile of Greece in Figure 4. We should emphasize the valid estimation of the parameter β_herd1 for all countries studied. This is clear proof that the initial phase follows a power of time function (Eq. 6) which is contrary to the general belief that the initial phase increases exponentially. This subexponential increase has been observed in the early phase of COVID-19 spreading in different parts of China.⁶ During the first fractal kinetics’ period, the estimate for α_{fractal 2} in Greece was also higher, 0.02 ± 2 × 10⁻⁴ (days)⁻¹ compared with 0.010–0.012 (days)^-1 found for the other three countries. This leads to a shorter half-life of 42 days for Greece compared with an average of 63 days for the three other countries; this, coupled with the earlier lockdown rules imposed in Greece, explains the more rapid approach to the steady state (Figure 4). The fractal exponent h_{fractal 2} was smaller in Greece, 2.93 ± 0.06, while for France, Italy and Spain it was 4.71 ± 0.09, 4.39 ± 0.02, 5.14 ± 0.06, respectively (Table 1). On the contrary, the estimate for c_{fractal 2} in Greece 3227 ± 30.19 was roughly ten-times higher than in the other three countries, resulting in much lower I(t) steady-state value.

All countries remained in a slightly moving upwards quasi-steady state for 2–3 months (Figure 4). This period was followed by a gradually increasing phase in the number of confirmed infected cases. Relaxed rules led to higher population mobility. All countries re-entered a herd kinetics’ period (blue concaving upwards segment in Figure 4). The estimates for β_{herd 3} were found 1.17 ± 0.02, 3.56 ± 0.04, 3.80 ± 0.04, 5.08 ± 0.08 for Italy, Spain, Greece and France, respectively, in full agreement with the visually increasing “curvature” of the blue concaving upwards segment of the four countries. All countries imposed preventive measures and lockdown rules several times (Figure 4). For France, Greece and Spain a remarkably similar reliable estimate for α_{fractal 4}, 0.003 (days)⁻¹ was found; this is indicative of a slow process with a half-life of 231 days. However, different h_{fractal 4} estimates, 16.11 ± 0.54, 9.01 ± 0.42 and 6.28 ± 0.43 were found for France, Greece, and Spain, respectively. Assuming that the conditions will not change in the next time period, predictions, based on the parameter estimates of the fourth fractal kinetics period for the steady-state value and t_90%, can be made for the three countries (Table 1). On the contrary, the fitting of Eq. 4 to Italy’s fourth fractal kinetics’ period data was not equally successful and reliable parameters estimates for h_{fractal 4}, α_{fractal 4} and c_{fractal 4} were not derived (Table 1). This is due to the fact that the point of inflection has not been reached yet and therefore the fitting algorithm cannot converge to reliable parameters estimates.

The estimates for t_ip reported in Table 1 for France, Greece and Spain correspond to time (days) from the commencement of the fourth fractal kinetics’ period. These estimates were found to be in agreement with the observed values, which is an additional piece of evidence for the validity of the fractal model. An estimate for Italy’s t_ip was not obtained for reasons mentioned above. Besides, the fourth fractal kinetics period data were used to predict the t_90% (expressed in days from the commencement of this period) and the final steady-state (1/(1+c)) for France, Greece and Spain (Table 1).

Analysis of COVID-19 data for countries deviating from the (HFF)² kinetic motif

A large number of countries, besides the four analyzed, followed the (HFF)² kinetic motif shown in Figure 4, e.g., Australia, China, Germany, Austria, United Kingdom.³⁰ Yet, several countries did not exhibit the (HFF)² motif, lacking a second wave and followed a “herd-fuzzy-fractal” (HFF) kinetic motif. Argentina and Brazil are examples of countries where strict/mild preventive measures were either not applied or did not work effectively. Both countries exhibit an (HFF) kinetic motif, Figure 5. Parameter values determined: Argentina, 1^st stage: (h = 1), β = 1.929 ± 0.028; 2^nd stage: h = 2.189 ± 0.043, α = (1.754 ± 0.067) × 10⁻³ (days)⁻¹, c = 3.61 ± 0.38; Brazil, 1^st stage: (h = 1), β = 4.633 ± 0.017; 2^nd stage: h = 2.892 ± 0.023, α = (4.044 ± 0.017) × 10⁻³ (days)⁻¹, c = 21.86 ± 0.24.

Figure 5. I(t) versus time plots for Argentina and Brazil.³⁰

Xs mark the implementation of mild preventive measures. The gray lines indicate the fuzzy period after the X point.

On the other hand, some countries exhibited a more complex pattern, which deviates from the (HFF)² and (HFF) motifs. Infection data for USA did not follow either the (HFF)² or the (HFF) kinetic motif. The I(t) time profile never reached a steady state and the shape of the curve indicates a deformed three-wave like kinetic profile (Figure 6). Probably both types of kinetics (herd and fractal) run concurrently for most of the time throughout the course of the pandemic, with the contribution of each varying with time. This is most likely due to different COVID-19 policy containment measures followed in different states around the country. Sweden intentionally applied the herd immunity strategy³⁰ during the COVID-19 pandemic. An initial herd-kinetics type continuous increase in the number of total infected cases reached a point of inflection around 20 July 2020, followed by a slower rate of increase of infected cases, (Figure 6). Since neither strict measures nor lockdown rules were applied at that time, the shape of the curve should be attributed to a fractal kinetics-like self-organization of the society. A rather sharp increase after 10 September 2020 can be attributed to the increased mobility of the individuals since no relaxation measures were taken close to this date.

Figure 6.

Confirmed infected cases for USA and Sweden.³⁰

The reproductive number

The reproductive number, R₀ is not needed for the initial growth of the disease⁴ being incompatible with the not well-mixed hypothesis, Figure 1. Limitations associated with the estimation of R₀, can be found in numerous publications. Our results show that the time exponent β of Eq. 6 controls the time evolution of the disease throughout the initial herd kinetics’ period. In other words, β drives the initial phase of the disease spreading being the slope of Eq. 7, i.e. a linearized form of Eq. 6. The predominant role of β during the herd kinetics’ period can be also concluded from Eq. 12, which explicitly shows that the infected population fraction at the inflection point, I(t)_ip is solely dependent on β. Although R₀ and β are different, however, they can be used complementary to each other during the initial stages of the pandemics. Estimates for β derived from the analysis of herd kinetics’ period data at two time points from 100 countries are shown in Figure 7 and Table 2. The degree of uncertainty (standard deviation) for the estimates was found in most cases small; this was accompanied with high correlation coefficients (not shown). Overall, the estimates derived from the longer period of 35 days seem to be either similar or higher or significantly higher than these derived from the analysis of the shorter period (10 days) data. For some countries, the small number of confirmed infected cases in the first 10 days did not allow the estimation of β. In view of the diversity and variability of data presented in Figure 7, we quote the median values derived from the analysis of 100 countries, 2.44 (0.25–12.24) and 1.34 (0.20–6.13) for the β estimates corresponding to 35 and 10 days, respectively.

Figure 7. A bar plot of 100 countries based on the estimates with standard deviations for β, derived from the nonlinear regression analysis of data³⁰ using Eq. 6.

Data of 10 and 35 days, after the first reported case, were analyzed. See also Table 2.

Exponential versus power growth

The classical phraseology “the exponential growth of the disease” used by medical doctors, scientists and laymen is questionable. This phrase is related to the approximate solution of the SIR model, which is an exponential function, when the parameter of the recovery rate is equal to zero.⁵ Based on our theoretical results and the good fittings of Eq. 6 to data of herd kinetics’ period (Figure 4), “the herd kinetics’ period seems to obey a power of time function”. According to Eq. 2, β drives the disease spreading when h = 1 and the rate of infection is inversely proportional to time. This is in agreement with the real-life conditions because of the continuous reduction of the probability of infection as a function of time (β/t). However, the resemblance of the I(t) profiles of the classical, h = 0 and the special case h = 1 in Figure 2 makes the discernment of the kinetics of the initial phase difficult.

Herd immunity

Herd immunity⁵ calculations rely on an estimate for R₀ and syllogisms based on the relative magnitude, λ = R(t)/R₀, which is the proportion of the population that is susceptible to catching the disease. If preventive measures are not applied, an estimate for the time needed to reach a certain level of the infected population fraction, e.g., I(t) = 0.6 ensuring herd immunity can be obtained from Eq. 7. Assuming an infected individual at time t=1, i.e., $\frac{1}{I\left(t=1\right)}=N$, where N is the population of the country, then, from Eq. 7, we get c = N − 1 ≈ N. Hence, the time t_hi needed to reach a certain level of herd immunity I (t_hi) under non preventive measures is

Figure 8 shows t_hi as a function of β and N assigning I (t_hi) = 0.6. It can be seen that population size has a mild effect, whereas the apparent transmissibility constant β severely reduces t_hi. Eq. 14 can be used at the initial stages of the pandemics and requires only a valid estimate for β. This will certainly provide valuable information for authorities, if coupled with estimates of the mortality rate and deaths, prior to a decision for a herd-immunity policy.³³ Caution should be exercised with the use of Eq. 14, since it can be applied only under the strict assumption of herd kinetics operating throughout the entire period of the disease spreading. The example of Sweden (Figure 6) shows that societies can exhibit self-organization and move to a fractal kinetics’ mode.

Figure 8.

A contour plot based on Eq. 14 showing the time required to reach herd immunity level I (t_hi) = 0.6 for various values of parameter β and population size N.

Deviation from the herd kinetic profile after the imposition of lockdown

Cumulative data of infected people from nine countries (Austria, Belgium, Denmark, France, Germany, Italy, Spain, Switzerland, United Kingdom) were gathered and analyzed under two different prisms. Analysis was broken down into two parts, before and after imposition of strict preventive measures (lockdown) (Figure 9). For the first period, the herd kinetic motif where h = 1 (Eq. 6) was found to be adequate, whereas after lockdown clearly fails. The latter period was also analyzed using the fractal kinetic motif of h > 1 with very persuasive goodness of fit (Figure 9). In all cases, R² was greater than 0.98. This pictorial divergence shows that after implementing mobility restrictions the evolution of the pandemic could not be captured by a power law expression, but rather by a fractal kinetic one (Eq. 4) which eventually leads to a plateau of cumulative cases.

Figure 9. I(t) versus time plots for Austria, Belgium, Denmark, France, Germany, Italy, Spain, Switzerland, United Kingdom.³⁰

The blue dots represent cumulative infected cases up to lockdown datum points.¹⁶ The orange lines depict the power fit to these data. Purple dots represent data after lockdown imposition whereas the purple lines are their superimposed fractal fits. Red lines depict the hypothetical power fit to the aforementioned data points in the event that Covid-19 propagation followed a power law pattern. R² values for all nine countries were measured higher than 0.98.

Model predictions

According to Jewell et al.,¹ the ability of current models to predict is very poor. Our work demonstrates that the herd kinetics’ period is described by Eq. 6, while the kinetic motif “herd-fuzzy-fractal” should be taken into account in the modeling work. Apparently, these approaches have not been implemented so far. Roughly, predictions during the herd kinetics’ period can be based on a valid estimate for β, Eq. 6. Under preventive measures, valid estimates for the parameters of the model (Eq. 4, h>1) can be derived and used for predictive purposes provided that data beyond the point of inflection are available (see Table 1).