Countless research contributions reflect two major concepts for modeling the spread of the COVID-19 pandemic: (i) ordinary differential equations for population compartments, such as infected or deceased persons (these approaches often exhibit limited predictive capabilities); and (ii) rules applied to digitally realized agents in the populations (these approaches often lack reliable input data and may become computationally overly expensive). As a remedy, we here introduce and discuss convolutional integrodifferential equations adapted from Boltzmann's hereditary mechanics, so as to predict COVID-19 fatality trends from the evolutions of newly infected persons. Replacing the classical statistical reasoning by deliberations arising from the notion of “virus loads” and the corresponding compliance of the infected population to these loads, model errors with respect to data recorded in 102 countries, territories, or US states can be drastically reduced, namely, up to 98% when compared to the traditional kinetics equation of Kermack and McKendrick. The coefficients of determination between model predictions and recorded data range from 94% to 100%, a precision hitherto unachieved in equation-based epidemic modeling.

With the advent of the COVID-19 pandemic, mathematical modeling of disease spreading has experienced an unprecedented boost, along with a lively debate on its potentials and limitations.1 Broadly speaking, two classes of models have seen massive use in this context: (i) compartment models, also referred to as “susceptible–infected–removed (SIR)” models, based on ordinary differential equations2–8 as originally proposed in the landmark paper of Kermack and McKendrick,9 and (ii) agent-based models based on computationally implemented rules governing the interaction of very many entities.10–12 The main argument put forward in favor of the second type of models is the limitation of ordinary differential equations when it comes to represent complex disease spreading events.13 The classically employed ordinary differential equations express that the temporal derivative at a particular time instant, of each of the three population compartments (S, I, and R), solely depends on the size of the compartments S and I at the very same time instant. In order to overcome this limitation, roughly four types of improvements (and combinations thereof) have been proposed:

  1. introducing additional compartments, including the exposed (E), the hospitalized (H), or the quarantined (Q);14–18 

  2. replacing the ordinary by partial differential equations where the compartments are resolved into spatially distributed population densities of persons undergoing different health and illness stages, and where the temporal derivatives of these population densities depend not only on their local magnitude, but also on the spatial gradients of the latter;19 

  3. replacing the classical ordinary by delay differential equations where the rate of one compartment size at a particular time instant depends on one or several compartment sizes at one or several earlier time instants;20–24 

  4. including statistical deliberations in terms of integrodifferential equations, where either the aforementioned single time instants are extended to a distribution of infinitely many such earlier time instants;25,26 or where the compartments themselves are further resolved with respect to additional, statistically distributed attributes related to social heterogeneity in terms of contact or age.27–29 

Typically, the aforementioned extensions of the SIR models aim at an improved representation of the person-to-person virus transmission dynamics for which statistical deliberations appear as very appropriate. Much less discussed are the limitations of the SIR models (and their extensions) when it comes to modeling the effect of the virus within the infected persons, that is, to more deeply exploring the relation between the infected and the removed. This is the focus of the present study, where we propose a fundamentally different approach to the problem: rather than resting on statistical deliberations having their origin in transmission dynamics, we resort to the very first physical problem which was tackled by means of integrodifferential equations: it was the famous Austrian physicist Ludwig Boltzmann,30–32 who, as early as 1874, introduced integrodifferential equations in the context of hereditary mechanics and the so-called “elastic aftereffect,” based on pioneering experimental work in material mechanics.33–42 These integrodifferential equations express that the response of a system does not only depend on its current state, but also on the entire “loading” history it underwent before.

In the realm of continuum mechanics, the key integrodifferential equation is often referred to as the “Boltzmann superposition principle,” which mathematically reads as43–46 

ε(t)=tJ(tτ):σ̇(τ)dτ,
(1)

where τ denotes the time instant where the rates of the second-order stress tensor σ(τ) have been applied to the investigated material, (·)̇ denotes the temporal derivatives of quantity (·),J(tτ) denotes the fourth-order creep function tensor describing the time-dependent behavior of the investigated material loaded at time τ, J(tτ)0 for t<τ, the symbol “:” is the second-order tensor contraction operator, and t denotes the time when the second-order strain tensor ε is recorded. Equation (1) expresses that the mechanical strain a viscoelastic material is experiencing at a certain point in time does not only depend on the mechanical stress at this time, but also on the stress history it has experienced hitherto. Equation (1) is also applied in cases where temporal stress discontinuities (“load steps” or finite stress increments Δσi) occur; at corresponding time point τi, we have

σ̇(τi)=δ(ττi)×Δσi,
(2)

with δ standing for the Dirac delta function, being zero everywhere except for the origin, where its value is infinite, with the integral over this infinity amounting to one. For Ni of such load steps, and any other temporal variations of the stresses being absent, insertion of Eq. (2) into Eq. (1) yields

ε(t)=i=1NiJ(tτi):Δσ(τi).
(3)

Equations (1) and (3) take a central position in the rheology of hard and soft solids, including cementitious materials,47,48 polymers,49 and various biological materials.50,51 The focus of the present paper is to transfer Boltzmann's “hereditary concept” from continuum mechanics to epidemiology, thereby introducing the notion of hereditary epidemiology: accordingly, we replace the piece of material with a human population exposed to an epidemic, the stress rate with an infection rate Ċ (“number of confirmed cases per day”), and the stress history with the fatality trend F(t). This gives rise to a new epidemiologic quantity, the fatality function JF(tτ): it quantifies the number of fatalities over time, within all the individuals who were infected at time τ. Mathematically, our proposition reads as

FHER(t)=tJF(tτ)×Ċ(τ)dτ,
(4)

where the subscript “her” refers to “hereditary,” in order to distinguish the new modeling approach from reference models to which we will compare its predictive capabilities. In this sense, the remainder of the present paper is organized as follows: first, the novel epidemiologic integrodifferential equations are outlined in Sec. II, together with the transition from the logistic to Heaviside fatality functions and with a traditional ordinary differential equation as a reference model. This is followed by a collection of data recorded during the COVID-19 pandemic, as a basis for the model validation strategy described in Sec. III. Corresponding results are documented in Sec. IV and discussed in Sec. V, which also concludes the paper.

For the definition of JF(tτ), we recall that in many cases biological processes can be reasonably well described by means of so-called logistic functions.52–54 In particular, in order to consider the growth of death-inducing viruses within that portion of the population, which has been infected at time instant τ, we adopt the following mathematical format for JF:

JF(tτ)=fF,HER1+exp[sHER×(tτTF,HER)],
(5)

where fF,HER is the hereditary model-related fatality fraction, TF,HER is the hereditary model-related characteristic time of fatal illness, and sHER is the shape parameter of the logistic function. Inserting Eq. (5) into Eq. (4) yields

FHER(t)=tfF,HER×Ċ(τ)1+exp[sHER×(tτTF,HER)]dτ.
(6)

Notably, the model defined by Eq. (6) requires, as input data, the temporal derivatives of the total number C of infected persons (also referred to as “confirmed cases”). However, these rates are typically accessible in incremental format, as detailed in Sec. III. Hence, it makes sense to reformulate Eq. (6) in terms of Dirac-type derivatives as seen in Eq. (2). In more detail, let us consider one particular increment of the total number of confirmed cases, termed ΔC(τi)—the newly confirmed cases at time τi. Then, the corresponding fatality evolution follows for time instants tτi as

FHER[t,ΔC(τi)]=fF,HER×ΔC(τi)1+exp[sHER×(tτiTF,HER)],
(7)

whereas FHER(t,τi)=0 if t<τi. The overall fatality trend is eventually obtained through summing up all fatality evolutions due to single increments ΔC(τi). Thus,

FHER(t)=i=1NiFHER[t,ΔC(τi)],
(8)

with Ni now standing for the number of time instants at which new increments ΔC(τi) were recorded. Figure 1 shows schematically how the model defined by Eqs. (7) and (8) yields the overall fatality trend based on three increments of confirmed cases; namely, ΔC(τ1),ΔC(τ2), and ΔC(τ3). The fatality fraction fF,HER needs additional attention: it quantifies how many of the people infected at time τi will eventually die. Hence, it is strongly influenced by medical and healthcare progress made during the investigated pandemic; and the latter progress might influence the characteristic time of fatal illness and the shape parameter as well. In the present paper, we focus on COVID-19 data recorded until the end of the year 2020, where most of the countries documenting the spread of the disease have experienced two infection waves of quite distinct nature: in most of these countries, the first waves (observed in the first half of 2020) were intercepted by comparably strict measures, whereas the second waves (starting sometime in the third quarter of 2020) struck with much higher infection numbers. At the same time, the disease characteristics differed significantly between the first and second waves; on the one hand, due to the above-mentioned progress in medical treatment options, and, on the other hand, due to changed infection distributions among the populations. In order to take these effects into account, we divide the total time domain into two time intervals, the first time interval corresponding to the first wave, and the second time interval corresponding to the second wave. Within each time interval, the increments of confirmed cases translate into corresponding fatalities based on Eqs. (7) and (8). Hence, denoting the time instant at which the first wave ends and the second wave begins by twtr (with subscript “wtr” standing for wave transition), Eq. (7) needs to be extended as follows:

forτitwtr:FHERI[t,ΔC(τi)]=fF,HERI×ΔC(τi)1+exp[sHERI×(tτiTF,HERI)],
(9)

and

forτi>twtr:FHERII[t,ΔC(τi)]=fF,HERII×ΔC(τi)1+exp[sHERII×(tτiTF,HERII)],
(10)

with parameters fF,HERI,sHERI,TF,HERI,fF,HERII,sHERII, and TF,HERII being independent of each other. The overall fatality trend eventually follows from summing up the fatality trends associated with increments of confirmed cases occurring both during the first and the second wave

FHER2(t)=i=1NiIFHERI[t,ΔC(τi)]+i=NiI+1NiFHERII[t,ΔC(τi)],
(11)

with NiI referring to the number of time instants at which new case increments were recorded during the first wave. As concerns the definition of twtr, we consider the local minimum of the 30-day moving average of newly confirmed infections between the two infection peaks.

FIG. 1.

Boltzmann superposition principle in the context of hereditary epidemiology: (a) examples for increments of confirmed cases, ΔC(τ1),ΔC(τ2),ΔC(τ3), and the corresponding contributions to the overall fatality trend, F(τ1),F(τ2),F(τ3); the latter follow from Eq. (7), and are based on the same fatality fraction fF,HER, the same shape parameters sHER, and the same characteristic times of fatal illness TF,HER,; (b) the sum of the individual fatality trends results in the overall fatality trend FHER(t), according to Eq. (8).

FIG. 1.

Boltzmann superposition principle in the context of hereditary epidemiology: (a) examples for increments of confirmed cases, ΔC(τ1),ΔC(τ2),ΔC(τ3), and the corresponding contributions to the overall fatality trend, F(τ1),F(τ2),F(τ3); the latter follow from Eq. (7), and are based on the same fatality fraction fF,HER, the same shape parameters sHER, and the same characteristic times of fatal illness TF,HER,; (b) the sum of the individual fatality trends results in the overall fatality trend FHER(t), according to Eq. (8).

Close modal

Next, we are interested in the limit case where all fatally infected people die at exactly the same time, actually after a time period TDEL after the infection, with “del” standing for delay. This limit case is associated with the shape parameter s going to infinity, and considering s in Eq. (5) yields a fatality function of the format

JF,DEL(tτ)=fF,DEL×H(tτTF,DEL),
(12)

whereby H stands for the Heaviside function, reading as

H(x)={0forx<0,12forx=0,1forx>0.
(13)

Further simplification may be reached by omitting the convolution integral in Eq. (4) and replacing the confirmed case increments by the total number of confirmed cases, leading to the very simple expression

FDEL(t)=fF,DEL×C(tTF,DEL),
(14)

a format which we have recently described and discussed in greater detail;55 thereby, fF,DEL is the delay model-related fatality fraction, and TF,DEL is the delay model-related characteristic time of fatal illness.

As a further model reference, we consider the equation standardly used in SIR models for relating the increase in fatalities at time instant t to the number of currently infected people at the very same time instant. Mathematically, this death kinetics law reads typically as follows:9,56

dFKIN(t)dt=βF×I(t),
(15)

where FKIN is the kinetics model-predicted number of fatalities, I is the number of currently infected people, and βF,KIN is the death rate parameter,6 also referred to as mortality rate.16 Alternatively, Eq. (15) can be formulated in incremental manner, yielding

ΔFKIN(tj)=FKIN(tj)FKIN(tj1)=βF×I(tj1).
(16)

Ever since COVID-19 was declared a pandemic, the basic disease-characterizing data have been made available on various, publicly accessible platforms. For the mathematical model presented in Sec. II, the following data are particularly relevant: the total (cumulated) cases of people infected with COVID-19 until the respective dates, denoted by C; the active cases of (currently) infected people at the respective dates, denoted by I; the total (cumulated) deaths until the respective dates, denoted by F; the total (cumulated) recoveries until the respective date, denoted as R; and the corresponding daily changes ΔC,ΔI,ΔF, and ΔR. In this paper, we focus on countries where the reported numbers of fatalities are large enough to be statistically significant, yielding a sufficiently smooth time course of F. As related (quantitative) criterion, we have considered only countries with fatality numbers >80. As of December 31, 2020, this applies to the following 102 countries, territories, or US states (given in alphabetical order): Afghanistan, Albania, Andorra, Arizona, Arkansas, Armenia, Australia, Austria, Azerbaijan, Bahrain, Bangladesh, Belarus, Belgium, Bolivia, Bosnia and Herzegovina, Brazil, Bulgaria, California, Canada, Chile, China, Colorado, Connecticut, Costa Rica, Cote d'Ivoire, Croatia, Czechia, Delaware, Democratic Republic of the Congo, Denmark, Dominican Republic, Ecuador, Egypt, El Salvador, Estonia, Eswatini, Finland, France, Germany, Ghana, Greece, Haiti, Hawaii, Hungary, Indonesia, Ireland, Israel, Italy, Kansas, Kazakhstan, Kenya, Kosovo, Malawi, Malaysia, Massachusetts, Mauritania, Mexico, Michigan, Minnesota, Montenegro, Morocco, Namibia, Nebraska, Nepal, New Hampshire, New Jersey, New Mexico, New York, Nigeria, North Macedonia, Norway, Ohio, Oman, Panama, Pakistan, Paraguay, Pennsylvania, Peru, Poland, Portugal, Qatar, Rhode Island, Romania, Russia, Senegal, Somalia, South Africa, South Dakota, State of Palestine, Sudan, Suriname, Switzerland, Tennessee, Texas, Turkey, United Arab Emirates, Utah, Uzbekistan, Virginia, Wisconsin, Zambia, and Zimbabwe. The data related to Kosovo were collected from a Kosovo-specific website,57 while the data related to all other countries or territories were collected from the reference website Worldometer58 (which does not provide Kosovo-related data). For the purpose of reproducibility, all raw data used in this study are included in the supplementary material attached to this paper.

These data are employed for the determination of optimized model parameters, namely, fF,HERopt,TF,HERopt, and sHERopt entering Eqs. (7) and (8), fF,HERI,opt,TF,HERI,opt,sHERI,opt,fF,HERII,opt,TF,HERII,opt,sHERII,opt entering Eqs. (9)–(11), fF,DELopt,TF,DELopt entering Eq. (14), and βFopt entering Eq. (16). To that end, we considered the temporal average over the absolute errors between modeled and recorded fatalities. At time point ti, the aforementioned absolute errors read as

Em,i=|Fm(ti)F(ti)|,
(17)

if m=HER,DEL,HER2,DEL2, with Fm(ti) and F(ti) standing for the model-predicted and recorded fatality numbers, respectively, and the absolute errors read as

Em,i=|ΔFm(ti)ΔF(ti)|,
(18)

if m=KIN,KIN2, with ΔFm(ti) and ΔF(ti) standing for the increments of model-predicted and recorded fatality numbers, respectively. The temporal average reads as

Em=1Nii=1NiEm,i,
(19)

with m=HER,DEL,KIN,HER2,DEL2,KIN2. In more detail, the following optimization problems were solved for each and every of the chosen 102 countries, territories, and US states:

  • minimizing the time-averaged absolute error of the kinetics model with one death rate parameter covering the entirety of the investigated pandemic time period, according to
    minimizeEKIN(βF)βFopt,
    (20)

    providing 102 country-, territory-, and US-state-specific optimized parameters βFopt;

  • minimizing the time-averaged absolute error associated with two, wave-specific, death rate parameters, according to
    minimizeEKIN2(βFI,βFII)βFI,opt,βFII,opt,
    (21)

    providing 102 country-, territory-, and US-state-specific optimized parameters βFI,opt, and as many optimized parameters βFII,opt;

  • minimizing the time-averaged absolute error of the delay model with one pair of parameters (fF,DEL,TF,DEL) covering the entirety of the investigated pandemic time period, according to
    minimizeEDEL(fF,DEL,TF,DEL)fF,DELopt,TF,DELopt,
    (22)

    providing 102 country-, territory-, and US-state-specific optimized parameter pairs (fF,DELopt,TF,DELopt);

  • minimizing the time-averaged absolute error of the delay model with two, wave-specific, pairs of parameters, (fF,DELI,TF,DELI) and (fF,DELII,TF,DELII), according to
    minimizeEDEL2(fF,DELI,TF,DELI,fF,DELII,TF,DELII)fF,DELI,opt,TF,DELI,opt,fF,DELII,opt,TF,DELII,opt,
    (23)

    providing 102 country-, territory-, and US-state-specific optimized parameter pairs (fF,DELI,opt,TF,DELI,opt), and as many optimized parameter pairs (fF,DELII,opt,TF,DELII,opt);

  • minimizing the time-averaged absolute error of the hereditary model with one triple of parameters (fF,HER,TF,HER,sHER) covering the entirety of the investigated pandemic time period, according to
    minimizeEHER(fF,HER,TF,HER,sHER)fF,HERopt,TF,HERopt,sHERopt,
    (24)

    providing 102 country-, territory-, and US-state-specific optimized parameter triples (fF,HERopt,TF,HERopt,sHERopt);

  • minimizing the time-averaged absolute error of the hereditary model with two, wave-specific, triples of parameters (fF,HERI,TF,HERI,sHERI) and (fF,HERII,TF,HERII,sHERII), according to
    minimizeEHER2(fF,HERI,TF,HERI,sHERI,fF,HERII,TF,HERII,sHERII)fF,HERI,opt,TF,HERI,opt,sHERI,opt,fF,HERII,opt,TF,HERII,opt,sHERII,opt,
    (25)

    providing 102 country-, territory-, and US-state-specific optimized parameter triples (fF,HERI,opt,TF,HERI,opt,sHERI,opt), and as many optimized parameter triples (fF,HERII,opt,TF,HERII,opt,sHERII,opt).

Equations (20)–(25) are actually solved by an interior point algorithm,59–61 which is readily available via the fmincon-function implemented in the Optimization Toolbox of the commercial mathematics software Matlab.62 Thereby, we used version 2020b, run on a Linux operating system (Fedora 32—Workstation Edition). Considering earlier experience with the kinetics and the infection-to-death delay model,55 the following bounds were imposed onto the parameters to be optimized: 0<βFk<0.003withk=I,II;0<fF,mk<0.6withm=DEL,HERandk=I,II;5d<TF,mk<60dwithm=DEL,HERandk=I,II;andsmk>0withm=HERandk=I,II.

We also compare the model performances among each other, by computing relative differences between model prediction errors as

ΔEmn=EmEnEn,
(26)

with m,n=HER,HER2,DEL,DEL2,KIN,KIN2. In this context, Emn refers to the case when the optimized model parameters cover the entire time period studied, and an additional subscript “2” indicates the use of wave-specifically optimized model parameters.

Finally, the exactness of the model predictions with the optimized parameters is assessed through the coefficient of determination R2, according to

Rm2=1(iNi(F(ti)Fm(ti))2iNi(F(ti)1NiiNiF(ti))2),
(27)

with m,n=HER,HER2,DEL,DEL2,KIN,KIN2; and through normalized time-averaged errors (“relative errors”),

Rm=EmFmax,
(28)

with m,n=HER,HER2,DEL,DEL2,KIN,KIN2, whereby Fmax stands for the maximum number of fatalities, that is, the total numbers recorded until December 31, 2020.

For the epidemic model parameters referring to the entirety of the investigated pandemic time period, which spans from February to December 2020, our newly introduced hereditary model outperforms the traditional kinetics model in terms of fatality trend predictions for 101 out of 102 countries, territories, or US states, see Table I, as well as the third column of Table II for model error reductions encountered when comparing the kinetics model with the hereditary model. These reductions reach values as high as 96.4% (in the case of Ghana), and on average, they amount to 64.5%. The superiority of the new modeling approach is even more pronounced in the case of wave-specific parameter optimization, see the sixth column of Table II; for actually all of the investigated countries, territories, and US states, the hereditary model outperforms the kinetics model, with a maximum error reduction of 98.3% (in the case of New Jersey), and the average over all 102 country-, territory-, and US state-specific error reductions amounting to 83.9%. This is consistent with the hereditary model capturing much better the significantly changing epidemic characteristics between the first and the second wave, which can be clearly seen from Fig. 2 and in particular, from Fig. 3. Let us first go into details regarding the Austrian case depicted in Fig. 2. Comparing Figs. 2(c) and 2(e), it is evident that all of the three investigated modeling approaches perform (significantly) better when dividing the studied time domain into a first and a second infection wave. While the deviations between the model-predicted and recorded fatalities are partly significant when considering only one set of model parameters for the whole time domain, see Fig. 2(c), these deviations are clearly reduced when considering infection wave-specific model parameters, and this is particularly true for the hereditary model, as impressively illustrated in Fig. 2(e). From a qualitative point of view, the same findings can be reported for the state of New York, see Fig. 3. However, in this case, the benefits of considering infection wave-specific model parameters are even more pronounced, and particularly striking for the hereditary model. While evaluating the hereditary model with infection wave-specific model parameters yields an extremely good agreement between model-predicted and recorded fatalities over the whole time domain, see Fig. 3(e), evaluating it with model parameters valid for the whole time domain leads to significant deviations, especially for t > 210 days. Obviously, this deficit can be attributed to the fact that one set of model parameters cannot satisfyingly represent the relation between confirmed infections and correspondingly arising fatalities over a long period of time (irrespective of the chosen modeling approach), at least when this period comprises different pandemic waves. As a rule, the fatality fraction decreases with progression from the first to the second wave (this statement holds for 69 out of the 102 investigated countries, territories, and US states, see columns eleven and fourteen of Table III), and the characteristic time of fatal illness is, as a rule, larger in the second wave when compared to the first wave (this statement holds for 62 out of the 102 investigated countries, territories, and US states, see columns twelve and fifteen of Table III). The superior performance of the hereditary model is also underlined by very low relative predictions errors (see the last column of Table IV). The minimum amounts to 0.11% (in the case of Russia), the maximum does not exceed 5.3% (in the case of Zambia), and the relative average prediction error amounts to 1.2%. Similarly impressive are the coefficients of determination characterizing the fitting procedures (see the seventh column of Table V): the minimum still amounting to 93.5% (in the case of Somalia), the maximum reaching 100.0% (in the case of Russia), and the average over all 102 countries, territories, and US states amounting to 99.3%.

TABLE I.

Optimal parameters covering the entirety of the investigated pandemic time period, for the death kinetics law (βFopt), for the infection-to-death delay model (fF,DELopt and TF,DELopt), and for the hereditary model (sHERopt,fF,HERopt, and TF,HERopt) as well as the corresponding time-averaged absolute errors (EKIN,EDEL,EHER).

CountryβFopt (10−4 d−1)EKIN (—)fF,DELopt (—)TF,DELopt (d)EDEL (—)sHERopt (d−1)fF,HERopt (—)TF,HERopt (d)EHER (—)
Afghanistan 8.16 162.51 0.04 19.40 60.53 0.09 0.04 19.07 46.69 
Albania 6.16 115.54 0.04 24.80 33.17 0.14 0.04 27.23 32.70 
Andorra 0.00 50.79 0.06 58.53 24.33 8.03 0.06 60.00 23.75 
Arizona 1.38 2143.82 0.03 15.10 338.52 3.63 0.03 16.03 337.77 
Arkansas 15.57 62.08 0.02 26.10 44.64 0.06 0.02 26.77 35.50 
Armenia 8.33 67.57 0.02 7.50 53.65 0.07 0.02 10.86 42.45 
Australia 11.77 107.42 0.03 26.30 47.30 1.05 0.03 25.82 47.09 
Austria 12.93 253.65 0.02 18.60 227.92 0.07 0.04 46.09 162.17 
Azerbaijan 8.04 95.70 0.02 8.70 31.56 0.06 0.02 8.89 21.25 
Bahrain 3.19 26.04 0.00 12.20 3.94 1.46 0.00 12.55 3.88 
Bangladesh 3.83 600.08 0.01 10.10 77.08 0.11 0.01 5.00 64.60 
Belarus 3.66 110.76 0.01 18.50 66.79 0.03 0.01 36.77 25.03 
Belgium 2.20 7389.85 0.14 58.64 3676.68 >10 0.14 60.00 3526.52 
Bolivia 13.39 1004.58 0.06 10.20 55.73 0.11 0.06 6.96 43.98 
Bosnia and Herzegovina 14.74 51.74 0.04 14.40 45.88 0.61 0.04 14.44 45.34 
Brazil 13.50 27 244.64 0.03 6.00 10 622.69 1.17 0.03 5.00 10 476.22 
Bulgaria 14.34 61.52 0.04 14.90 58.65 0.15 0.05 18.55 38.06 
California 2.04 5265.39 0.02 12.60 1518.19 >10 0.02 14.02 1512.63 
Canada 15.14 3233.87 0.08 49.10 1807.43 3.51 0.08 56.43 1782.96 
Chile 24.59 1395.10 0.03 13.70 72.88 >10 0.03 12.93 67.32 
China 18.20 1263.72 0.05 12.32 251.68 0.05 0.06 11.37 135.14 
Colorado 1.94 1138.63 0.04 48.10 468.02 9.81 0.04 45.99 468.02 
Connecticut 2.94 2921.95 0.09 45.70 836.92 >10 0.09 38.15 826.60 
Costa Rica 3.78 73.87 0.01 11.50 14.87 2.23 0.01 8.86 11.86 
Cote d'Ivoire 0.00 80.44 0.01 6.50 6.66 1.51 0.01 5.00 6.48 
Croatia 25.16 115.15 0.02 13.90 32.41 0.15 0.02 18.08 28.93 
Czechia 17.98 330.37 0.02 13.00 125.18 0.14 0.02 12.52 108.53 
Delaware 1.33 403.38 0.03 18.60 109.12 1.34 0.03 25.51 107.75 
Dem. Rep. Congo 5.11 90.86 0.03 6.00 16.45 0.34 0.03 5.00 16.09 
Denmark 3.92 433.59 0.04 58.59 214.49 9.36 0.04 60.00 211.65 
Dominican Republic 2.89 714.78 0.02 5.99 145.00 0.69 0.02 5.00 143.42 
Ecuador 24.41 3118.53 0.08 6.00 1569.36 0.85 0.08 5.00 1556.14 
Egypt 12.99 584.30 0.06 12.20 131.77 0.05 0.06 5.00 61.63 
El Salvador 9.95 102.44 0.03 6.00 10.88 0.28 0.03 5.00 10.09 
Estonia 4.22 27.16 0.03 44.50 11.72 0.28 0.03 45.41 11.62 
Eswatini 7.79 17.93 0.02 6.46 2.74 2.28 0.02 5.02 2.70 
Finland 2.46 233.24 0.05 58.51 71.99 8.94 0.05 60.00 71.78 
France 7.76 21 315.15 0.03 6.01 14 766.54 0.30 0.03 5.00 14 682.89 
Germany 10.48 4094.74 0.05 45.80 1408.29 0.17 0.05 46.30 1376.55 
Ghana 0.00 179.90 0.01 12.49 7.63 0.10 0.01 7.18 6.45 
Greece 24.06 150.02 0.04 21.00 70.25 1.32 0.04 20.23 69.75 
Haiti 0.01 142.51 0.03 15.40 8.32 0.06 0.03 5.13 5.90 
Hawaii 2.25 54.24 0.02 17.00 7.31 0.08 0.02 20.53 5.68 
Hungary 9.27 393.18 0.03 9.20 273.25 0.03 0.08 60.00 204.67 
Indonesia 18.00 843.63 0.04 5.99 996.76 1.26 0.03 5.00 988.69 
Ireland 1.17 1396.31 0.07 58.53 446.55 >10 0.07 60.00 443.74 
Israel 5.30 120.02 0.01 12.90 79.81 0.10 0.01 11.30 59.75 
Italy 8.90 23 343.57 0.15 56.67 7298.56 2.39 0.15 56.24 7298.84 
Kansas 1.60 445.87 0.01 6.00 64.06 0.12 0.01 16.22 59.58 
Kazakhstan 8.70 109.36 0.02 22.50 64.25 0.09 0.02 21.97 58.44 
Kenya 5.22 141.41 0.02 5.99 35.73 0.35 0.02 5.00 33.80 
Kosovo 10.61 163.86 0.04 20.40 59.02 0.08 0.04 29.12 50.89 
Malawi 0.00 106.38 0.03 7.42 4.47 >10 0.03 7.00 4.48 
Malaysia 2.79 85.91 0.01 44.00 43.33 0.04 0.02 59.98 36.73 
Massachusetts 12.89 3860.40 0.08 24.02 1070.64 >10 0.08 27.96 1065.22 
Mauritania 9.03 33.54 0.02 6.50 14.33 9.76 0.02 5.00 13.99 
Mexico 30.00 25 788.29 0.10 6.00 3956.11 0.00 0.20 5.09 3272.22 
Michigan 4.11 4601.48 0.07 56.60 1761.48 >10 0.07 53.07 1755.66 
Minnesota 13.60 819.50 0.03 45.50 432.21 7.36 0.03 46.91 430.26 
Montenegro 6.57 50.51 0.02 6.00 12.80 0.46 0.02 5.00 12.32 
Morocco 16.65 171.88 0.02 6.03 104.91 0.55 0.02 5.00 98.66 
Namibia 4.18 13.35 0.01 8.70 3.39 >10 0.01 6.00 3.21 
Nebraska 2.59 143.59 0.01 23.60 43.91 0.14 0.01 23.54 41.88 
Nepal 7.13 101.66 0.01 8.70 47.69 0.03 0.01 30.38 29.68 
New Hampshire 10.67 206.44 0.06 45.30 68.12 >10 0.06 45.10 68.19 
New Jersey 4.69 9089.18 0.07 14.50 2277.78 5.46 0.08 21.48 2251.66 
New Mexico 4.22 268.90 0.04 34.50 125.46 0.66 0.04 35.42 124.76 
New York 2.65 22 331.89 0.07 6.48 4467.88 >10 0.07 5.89 4474.14 
Nigeria 3.91 302.88 0.02 5.99 78.11 8.66 0.02 5.00 77.03 
North Macedonia 12.98 180.69 0.03 6.49 93.27 0.05 0.05 35.47 73.19 
Norway 1.25 174.39 0.03 58.56 56.34 9.88 0.03 60.00 56.11 
Ohio 10.07 1714.77 0.04 29.70 823.38 0.15 0.04 34.01 811.88 
Oman 4.46 151.06 0.01 31.50 53.46 0.03 0.01 54.21 10.94 
Panama 8.05 181.41 0.02 6.00 126.81 9.77 0.02 5.00 125.95 
Pakistan 10.54 239.55 0.02 6.01 174.43 >10 0.02 5.00 169.05 
Paraguay 7.56 188.68 0.02 6.50 25.58 1.14 0.02 5.00 25.30 
Pennsylvania 9.76 2857.13 0.06 41.10 1461.34 >10 0.06 41.00 1461.39 
Peru 14.56 7639.74 0.04 6.00 3173.60 1.09 0.04 5.00 3127.06 
Poland 12.64 499.77 0.02 10.70 605.27 0.04 0.04 37.92 380.11 
Portugal 8.50 397.55 0.03 37.20 352.23 0.06 0.03 44.01 315.79 
Qatar 0.57 62.53 0.00 30.80 5.51 0.11 0.00 23.22 2.32 
Rhode Island 1.77 605.04 0.05 49.00 189.67 7.48 0.05 45.79 189.27 
Romania 14.79 600.12 0.03 6.47 850.67 0.00 0.05 33.18 764.87 
Russia 8.47 2638.08 0.02 6.00 664.97 1.12 0.02 5.10 663.64 
Senegal 5.90 53.34 0.02 9.50 7.19 0.81 0.02 8.48 7.17 
Somalia 0.00 80.48 0.03 6.42 8.73 0.94 0.03 5.00 8.62 
South Africa 16.62 498.81 0.03 15.80 775.22 0.03 0.03 7.01 231.02 
South Dakota 9.01 31.70 0.02 21.70 12.84 >10 0.02 21.39 12.85 
State of Palestine 7.53 83.06 0.01 7.60 17.32 0.37 0.01 5.08 16.79 
Sudan 3.96 676.19 0.08 6.48 60.03 2.56 0.08 5.01 59.97 
Suriname 10.49 24.03 0.02 10.40 1.82 0.34 0.02 10.01 1.78 
Switzerland 8.99 1008.01 0.06 57.70 548.10 0.07 0.06 60.00 469.10 
Tennessee 8.41 265.40 0.01 6.50 98.78 0.04 0.02 15.97 61.07 
Texas 7.24 1325.58 0.02 14.50 578.44 7.52 0.02 13.86 578.32 
Turkey 13.11 488.97 0.01 6.00 373.10 0.28 0.01 5.00 355.55 
United Arab Emirates 1.51 141.56 0.00 6.47 75.71 0.58 0.00 5.00 75.43 
Utah 2.03 122.11 0.01 48.00 61.54 4.24 0.01 49.40 61.33 
Uzbekistan 6.78 58.92 0.01 14.20 12.51 0.88 0.01 12.28 12.19 
Virginia 1.19 1353.94 0.02 5.99 377.81 7.95 0.02 5.00 377.83 
Wisconsin 8.23 235.32 0.01 5.50 268.78 0.03 0.01 20.74 253.41 
Zambia 7.08 140.03 0.02 6.47 22.41 1.58 0.02 5.00 22.04 
Zimbabwe 6.64 70.91 0.03 9.49 5.51 1.12 0.03 7.84 5.45 
CountryβFopt (10−4 d−1)EKIN (—)fF,DELopt (—)TF,DELopt (d)EDEL (—)sHERopt (d−1)fF,HERopt (—)TF,HERopt (d)EHER (—)
Afghanistan 8.16 162.51 0.04 19.40 60.53 0.09 0.04 19.07 46.69 
Albania 6.16 115.54 0.04 24.80 33.17 0.14 0.04 27.23 32.70 
Andorra 0.00 50.79 0.06 58.53 24.33 8.03 0.06 60.00 23.75 
Arizona 1.38 2143.82 0.03 15.10 338.52 3.63 0.03 16.03 337.77 
Arkansas 15.57 62.08 0.02 26.10 44.64 0.06 0.02 26.77 35.50 
Armenia 8.33 67.57 0.02 7.50 53.65 0.07 0.02 10.86 42.45 
Australia 11.77 107.42 0.03 26.30 47.30 1.05 0.03 25.82 47.09 
Austria 12.93 253.65 0.02 18.60 227.92 0.07 0.04 46.09 162.17 
Azerbaijan 8.04 95.70 0.02 8.70 31.56 0.06 0.02 8.89 21.25 
Bahrain 3.19 26.04 0.00 12.20 3.94 1.46 0.00 12.55 3.88 
Bangladesh 3.83 600.08 0.01 10.10 77.08 0.11 0.01 5.00 64.60 
Belarus 3.66 110.76 0.01 18.50 66.79 0.03 0.01 36.77 25.03 
Belgium 2.20 7389.85 0.14 58.64 3676.68 >10 0.14 60.00 3526.52 
Bolivia 13.39 1004.58 0.06 10.20 55.73 0.11 0.06 6.96 43.98 
Bosnia and Herzegovina 14.74 51.74 0.04 14.40 45.88 0.61 0.04 14.44 45.34 
Brazil 13.50 27 244.64 0.03 6.00 10 622.69 1.17 0.03 5.00 10 476.22 
Bulgaria 14.34 61.52 0.04 14.90 58.65 0.15 0.05 18.55 38.06 
California 2.04 5265.39 0.02 12.60 1518.19 >10 0.02 14.02 1512.63 
Canada 15.14 3233.87 0.08 49.10 1807.43 3.51 0.08 56.43 1782.96 
Chile 24.59 1395.10 0.03 13.70 72.88 >10 0.03 12.93 67.32 
China 18.20 1263.72 0.05 12.32 251.68 0.05 0.06 11.37 135.14 
Colorado 1.94 1138.63 0.04 48.10 468.02 9.81 0.04 45.99 468.02 
Connecticut 2.94 2921.95 0.09 45.70 836.92 >10 0.09 38.15 826.60 
Costa Rica 3.78 73.87 0.01 11.50 14.87 2.23 0.01 8.86 11.86 
Cote d'Ivoire 0.00 80.44 0.01 6.50 6.66 1.51 0.01 5.00 6.48 
Croatia 25.16 115.15 0.02 13.90 32.41 0.15 0.02 18.08 28.93 
Czechia 17.98 330.37 0.02 13.00 125.18 0.14 0.02 12.52 108.53 
Delaware 1.33 403.38 0.03 18.60 109.12 1.34 0.03 25.51 107.75 
Dem. Rep. Congo 5.11 90.86 0.03 6.00 16.45 0.34 0.03 5.00 16.09 
Denmark 3.92 433.59 0.04 58.59 214.49 9.36 0.04 60.00 211.65 
Dominican Republic 2.89 714.78 0.02 5.99 145.00 0.69 0.02 5.00 143.42 
Ecuador 24.41 3118.53 0.08 6.00 1569.36 0.85 0.08 5.00 1556.14 
Egypt 12.99 584.30 0.06 12.20 131.77 0.05 0.06 5.00 61.63 
El Salvador 9.95 102.44 0.03 6.00 10.88 0.28 0.03 5.00 10.09 
Estonia 4.22 27.16 0.03 44.50 11.72 0.28 0.03 45.41 11.62 
Eswatini 7.79 17.93 0.02 6.46 2.74 2.28 0.02 5.02 2.70 
Finland 2.46 233.24 0.05 58.51 71.99 8.94 0.05 60.00 71.78 
France 7.76 21 315.15 0.03 6.01 14 766.54 0.30 0.03 5.00 14 682.89 
Germany 10.48 4094.74 0.05 45.80 1408.29 0.17 0.05 46.30 1376.55 
Ghana 0.00 179.90 0.01 12.49 7.63 0.10 0.01 7.18 6.45 
Greece 24.06 150.02 0.04 21.00 70.25 1.32 0.04 20.23 69.75 
Haiti 0.01 142.51 0.03 15.40 8.32 0.06 0.03 5.13 5.90 
Hawaii 2.25 54.24 0.02 17.00 7.31 0.08 0.02 20.53 5.68 
Hungary 9.27 393.18 0.03 9.20 273.25 0.03 0.08 60.00 204.67 
Indonesia 18.00 843.63 0.04 5.99 996.76 1.26 0.03 5.00 988.69 
Ireland 1.17 1396.31 0.07 58.53 446.55 >10 0.07 60.00 443.74 
Israel 5.30 120.02 0.01 12.90 79.81 0.10 0.01 11.30 59.75 
Italy 8.90 23 343.57 0.15 56.67 7298.56 2.39 0.15 56.24 7298.84 
Kansas 1.60 445.87 0.01 6.00 64.06 0.12 0.01 16.22 59.58 
Kazakhstan 8.70 109.36 0.02 22.50 64.25 0.09 0.02 21.97 58.44 
Kenya 5.22 141.41 0.02 5.99 35.73 0.35 0.02 5.00 33.80 
Kosovo 10.61 163.86 0.04 20.40 59.02 0.08 0.04 29.12 50.89 
Malawi 0.00 106.38 0.03 7.42 4.47 >10 0.03 7.00 4.48 
Malaysia 2.79 85.91 0.01 44.00 43.33 0.04 0.02 59.98 36.73 
Massachusetts 12.89 3860.40 0.08 24.02 1070.64 >10 0.08 27.96 1065.22 
Mauritania 9.03 33.54 0.02 6.50 14.33 9.76 0.02 5.00 13.99 
Mexico 30.00 25 788.29 0.10 6.00 3956.11 0.00 0.20 5.09 3272.22 
Michigan 4.11 4601.48 0.07 56.60 1761.48 >10 0.07 53.07 1755.66 
Minnesota 13.60 819.50 0.03 45.50 432.21 7.36 0.03 46.91 430.26 
Montenegro 6.57 50.51 0.02 6.00 12.80 0.46 0.02 5.00 12.32 
Morocco 16.65 171.88 0.02 6.03 104.91 0.55 0.02 5.00 98.66 
Namibia 4.18 13.35 0.01 8.70 3.39 >10 0.01 6.00 3.21 
Nebraska 2.59 143.59 0.01 23.60 43.91 0.14 0.01 23.54 41.88 
Nepal 7.13 101.66 0.01 8.70 47.69 0.03 0.01 30.38 29.68 
New Hampshire 10.67 206.44 0.06 45.30 68.12 >10 0.06 45.10 68.19 
New Jersey 4.69 9089.18 0.07 14.50 2277.78 5.46 0.08 21.48 2251.66 
New Mexico 4.22 268.90 0.04 34.50 125.46 0.66 0.04 35.42 124.76 
New York 2.65 22 331.89 0.07 6.48 4467.88 >10 0.07 5.89 4474.14 
Nigeria 3.91 302.88 0.02 5.99 78.11 8.66 0.02 5.00 77.03 
North Macedonia 12.98 180.69 0.03 6.49 93.27 0.05 0.05 35.47 73.19 
Norway 1.25 174.39 0.03 58.56 56.34 9.88 0.03 60.00 56.11 
Ohio 10.07 1714.77 0.04 29.70 823.38 0.15 0.04 34.01 811.88 
Oman 4.46 151.06 0.01 31.50 53.46 0.03 0.01 54.21 10.94 
Panama 8.05 181.41 0.02 6.00 126.81 9.77 0.02 5.00 125.95 
Pakistan 10.54 239.55 0.02 6.01 174.43 >10 0.02 5.00 169.05 
Paraguay 7.56 188.68 0.02 6.50 25.58 1.14 0.02 5.00 25.30 
Pennsylvania 9.76 2857.13 0.06 41.10 1461.34 >10 0.06 41.00 1461.39 
Peru 14.56 7639.74 0.04 6.00 3173.60 1.09 0.04 5.00 3127.06 
Poland 12.64 499.77 0.02 10.70 605.27 0.04 0.04 37.92 380.11 
Portugal 8.50 397.55 0.03 37.20 352.23 0.06 0.03 44.01 315.79 
Qatar 0.57 62.53 0.00 30.80 5.51 0.11 0.00 23.22 2.32 
Rhode Island 1.77 605.04 0.05 49.00 189.67 7.48 0.05 45.79 189.27 
Romania 14.79 600.12 0.03 6.47 850.67 0.00 0.05 33.18 764.87 
Russia 8.47 2638.08 0.02 6.00 664.97 1.12 0.02 5.10 663.64 
Senegal 5.90 53.34 0.02 9.50 7.19 0.81 0.02 8.48 7.17 
Somalia 0.00 80.48 0.03 6.42 8.73 0.94 0.03 5.00 8.62 
South Africa 16.62 498.81 0.03 15.80 775.22 0.03 0.03 7.01 231.02 
South Dakota 9.01 31.70 0.02 21.70 12.84 >10 0.02 21.39 12.85 
State of Palestine 7.53 83.06 0.01 7.60 17.32 0.37 0.01 5.08 16.79 
Sudan 3.96 676.19 0.08 6.48 60.03 2.56 0.08 5.01 59.97 
Suriname 10.49 24.03 0.02 10.40 1.82 0.34 0.02 10.01 1.78 
Switzerland 8.99 1008.01 0.06 57.70 548.10 0.07 0.06 60.00 469.10 
Tennessee 8.41 265.40 0.01 6.50 98.78 0.04 0.02 15.97 61.07 
Texas 7.24 1325.58 0.02 14.50 578.44 7.52 0.02 13.86 578.32 
Turkey 13.11 488.97 0.01 6.00 373.10 0.28 0.01 5.00 355.55 
United Arab Emirates 1.51 141.56 0.00 6.47 75.71 0.58 0.00 5.00 75.43 
Utah 2.03 122.11 0.01 48.00 61.54 4.24 0.01 49.40 61.33 
Uzbekistan 6.78 58.92 0.01 14.20 12.51 0.88 0.01 12.28 12.19 
Virginia 1.19 1353.94 0.02 5.99 377.81 7.95 0.02 5.00 377.83 
Wisconsin 8.23 235.32 0.01 5.50 268.78 0.03 0.01 20.74 253.41 
Zambia 7.08 140.03 0.02 6.47 22.41 1.58 0.02 5.00 22.04 
Zimbabwe 6.64 70.91 0.03 9.49 5.51 1.12 0.03 7.84 5.45 
TABLE II.

Relative differences between model-specific prediction errors according to Eq. (26), for parameters optimized over the entire investigated pandemic time period, and for wave-specific parameters (indicated by subscript 2).

CountryΔEDELKIN (—)ΔEHERKIN (—)ΔEHERDEL (—)ΔEDEL2KIN2 (—)ΔEHER2KIN2 (—)ΔEHER2DEL2 (—)
Afghanistan −0.6275 −0.7127 −0.2287 −0.7389 −0.8938 −0.5934 
Albania −0.7129 −0.7170 −0.0142 −0.7436 −0.8749 −0.5122 
Andorra −0.5211 −0.5324 −0.0237 0.1773 −0.9039 −0.9184 
Arizona −0.8421 −0.8424 −0.0022 −0.8933 −0.9247 −0.2942 
Arkansas −0.2809 −0.4281 −0.2047 −0.7375 −0.7365 0.0036 
Armenia −0.2060 −0.3717 −0.2086 −0.2124 −0.8608 −0.8233 
Australia −0.5597 −0.5616 −0.0045 −0.6175 −0.9182 −0.7861 
Austria −0.1014 −0.3606 −0.2885 0.0197 −0.6599 −0.6665 
Azerbaijan −0.6703 −0.7780 −0.3267 −0.4034 −0.8469 −0.7435 
Bahrain −0.8488 −0.8509 −0.0141 −0.8504 −0.8545 −0.0275 
Bangladesh −0.8716 −0.8923 −0.1619 −0.8971 −0.9017 −0.0439 
Belarus −0.3970 −0.7741 −0.6253 −0.7525 −0.9509 −0.8014 
Belgium −0.5025 −0.5228 −0.0408 −0.6541 −0.9637 −0.8950 
Bolivia −0.9445 −0.9562 −0.2109 −0.9398 −0.9591 −0.3205 
Bosnia and Herzegovina −0.1132 −0.1236 −0.0118 −0.5737 −0.7271 −0.3599 
Brazil −0.6101 −0.6155 −0.0138 −0.5854 −0.6269 −0.1001 
Bulgaria −0.0465 −0.3812 −0.3510 0.1871 −0.3675 −0.4672 
California −0.7117 −0.7127 −0.0037 −0.7154 −0.6363 0.2779 
Canada −0.4411 −0.4487 −0.0135 −0.1926 −0.9125 −0.8916 
Chile −0.9478 −0.9517 −0.0763 −0.9620 −0.9661 −0.1072 
China −0.8008 −0.8931 −0.4631 −0.9020 −0.9315 −0.3011 
Colorado −0.5890 −0.5890 0.0000 −0.5518 −0.9555 −0.9007 
Connecticut −0.7136 −0.7171 −0.0123 −0.9040 −0.9725 −0.7130 
Costa Rica −0.7988 −0.8395 −0.2025 −0.8101 −0.8593 −0.2590 
Cote d'Ivoire −0.9172 −0.9195 −0.0269 −0.8282 −0.8309 −0.0158 
Croatia −0.7185 −0.7487 −0.1074 −0.7259 −0.8387 −0.4116 
Czechia −0.6211 −0.6715 −0.1330 −0.6675 −0.7607 −0.2802 
Delaware −0.7295 −0.7329 −0.0125 −0.5363 −0.9404 −0.8714 
Dem. Rep. Congo −0.8190 −0.8229 −0.0217 −0.8351 −0.8446 −0.0574 
Denmark −0.5053 −0.5119 −0.0132 0.8680 −0.6938 −0.8361 
Dominican Republic −0.7971 −0.7993 −0.0109 −0.7453 −0.7755 −0.1185 
Ecuador −0.4968 −0.5010 −0.0084 −0.5079 −0.9605 −0.9196 
Egypt −0.7745 −0.8945 −0.5323 −0.9048 −0.9330 −0.2959 
El Salvador −0.8938 −0.9015 −0.0725 −0.8901 −0.9373 −0.4293 
Estonia −0.5684 −0.5720 −0.0083 −0.9086 −0.9302 −0.2366 
Eswatini −0.8472 −0.8495 −0.0149 −0.8388 −0.8902 −0.3187 
Finland −0.6913 −0.6922 −0.0029 −0.6570 −0.9046 −0.7218 
France −0.3072 −0.3112 −0.0057 −0.4326 −0.9661 −0.9402 
Germany −0.6561 −0.6638 −0.0225 −0.3079 −0.6724 −0.5267 
Ghana −0.9576 −0.9641 −0.1544 −0.9648 −0.9738 −0.2573 
Greece −0.5318 −0.5350 −0.0070 −0.5869 −0.6371 −0.1213 
Haiti −0.9416 −0.9586 −0.2908 −0.8877 −0.9284 −0.3624 
Hawaii −0.8652 −0.8953 −0.2237 −0.8622 −0.9459 −0.6075 
Hungary −0.3050 −0.4794 −0.2510 −0.1328 −0.7747 −0.7403 
Indonesia 0.1815 0.1719 −0.0081 −0.0916 −0.1498 −0.0640 
Ireland −0.6802 −0.6822 −0.0063 −0.6131 −0.9686 −0.9187 
Israel −0.3350 −0.5022 −0.2514 −0.4314 −0.8981 −0.8207 
Italy −0.6873 −0.6873 0.0000 −0.8487 −0.9806 −0.8716 
Kansas −0.8563 −0.8664 −0.0700 −0.8860 −0.9086 −0.1985 
Kazakhstan −0.4125 −0.4656 −0.0904 −0.5086 −0.6280 −0.2430 
Kenya −0.7473 −0.7610 −0.0539 −0.7893 −0.8673 −0.3704 
Kosovo −0.6398 −0.6894 −0.1377 1.3136 −0.3554 −0.7214 
Malawi −0.9580 −0.9579 0.0020 −0.9618 −0.9607 0.0303 
Malaysia −0.4956 −0.5724 −0.1523 −0.4420 −0.8200 −0.6774 
Massachusetts −0.7227 −0.7241 −0.0051 −0.7295 −0.8083 −0.2913 
Mauritania −0.5728 −0.5830 −0.0238 −0.8151 −0.8315 −0.0886 
Mexico −0.8466 −0.8731 −0.1729 −0.9200 −0.9634 −0.5428 
Michigan −0.6172 −0.6185 −0.0033 0.0155 −0.9134 −0.9147 
Minnesota −0.4726 −0.4750 −0.0045 −0.5130 −0.9364 −0.8694 
Montenegro −0.7466 −0.7560 −0.0371 −0.7668 −0.9359 −0.7250 
Morocco −0.3897 −0.4260 −0.0595 −0.5955 −0.6649 −0.1716 
Namibia −0.7459 −0.7595 −0.0537 −0.7513 −0.7925 −0.1658 
Nebraska −0.6942 −0.7083 −0.0461 −0.7355 −0.9048 −0.6400 
Nepal −0.5309 −0.7080 −0.3776 −0.2002 −0.7372 −0.6714 
New Hampshire −0.6700 −0.6697 0.0010 −0.3781 −0.8370 −0.7379 
New Jersey −0.7494 −0.7523 −0.0115 −0.8683 −0.9831 −0.8720 
New Mexico −0.5334 −0.5360 −0.0056 −0.7241 −0.8284 −0.3781 
New York −0.7999 −0.7997 0.0014 −0.9104 −0.9805 −0.7819 
Nigeria −0.7421 −0.7457 −0.0138 −0.8234 −0.8167 0.0384 
North Macedonia −0.4838 −0.5950 −0.2153 −0.5260 −0.8567 −0.6977 
Norway −0.6769 −0.6783 −0.0041 −0.8498 −0.9542 −0.6948 
Ohio −0.5198 −0.5265 −0.0140 −0.1078 −0.9288 −0.9202 
Oman −0.6461 −0.9275 −0.7953 −0.6889 −0.9574 −0.8632 
Panama −0.3010 −0.3057 −0.0068 −0.7505 −0.7992 −0.1952 
Pakistan −0.2718 −0.2943 −0.0309 −0.8555 −0.8769 −0.1481 
Paraguay −0.8644 −0.8659 −0.0108 −0.5704 −0.8187 −0.5779 
Pennsylvania −0.4885 −0.4885 0.0000 1.8014 −0.7414 −0.9077 
Peru −0.5846 −0.5907 −0.0147 −0.4176 −0.9231 −0.8680 
Poland 0.2111 −0.2394 −0.3720 −0.0705 −0.8477 −0.8361 
Portugal −0.1140 −0.2057 −0.1035 −0.3837 −0.9205 −0.8710 
Qatar −0.9119 −0.9628 −0.5781 −0.9185 −0.9704 −0.6372 
Rhode Island −0.6865 −0.6872 −0.0021 −0.7136 −0.9552 −0.8435 
Romania 0.4175 0.2745 −0.1009 1.4254 −0.7277 −0.8877 
Russia −0.7479 −0.7484 −0.0020 −0.7220 −0.9369 −0.7731 
Senegal −0.8653 −0.8656 −0.0028 −0.8579 −0.8890 −0.2189 
Somalia −0.8916 −0.8929 −0.0119 −0.9305 −0.9316 −0.0164 
South Africa 0.5541 −0.5369 −0.7020 0.1877 −0.6204 −0.6804 
South Dakota −0.5948 −0.5945 0.0007 −0.8607 −0.8598 0.0068 
State of Palestine −0.7915 −0.7978 −0.0306 −0.4876 −0.7797 −0.5701 
Sudan −0.9112 −0.9113 −0.0010 −0.9362 −0.9630 −0.4201 
Suriname −0.9242 −0.9258 −0.0210 −0.9230 −0.9260 −0.0390 
Switzerland −0.4563 −0.5346 −0.1441 0.0261 −0.6698 −0.6782 
Tennessee −0.6278 −0.7699 −0.3817 −0.7386 −0.8066 −0.2602 
Texas −0.5636 −0.5637 −0.0002 −0.4128 −0.5862 −0.2954 
Turkey −0.2370 −0.2729 −0.0471 −0.4293 −0.7325 −0.5312 
United Arab Emirates −0.4652 −0.4671 −0.0037 −0.7112 −0.8757 −0.5696 
Utah −0.4960 −0.4978 −0.0035 −0.6203 −0.7828 −0.4278 
Uzbekistan −0.7877 −0.7931 −0.0256 −0.8168 −0.8527 −0.1960 
Virginia −0.7210 −0.7209 0.0001 −0.6689 −0.9234 −0.7686 
Wisconsin 0.1422 0.0769 −0.0572 1.1242 −0.5304 −0.7789 
Zambia −0.8399 −0.8426 −0.0165 −0.8774 −0.8663 0.0903 
Zimbabwe −0.9223 −0.9231 −0.0114 −0.9243 −0.9279 −0.0472 
CountryΔEDELKIN (—)ΔEHERKIN (—)ΔEHERDEL (—)ΔEDEL2KIN2 (—)ΔEHER2KIN2 (—)ΔEHER2DEL2 (—)
Afghanistan −0.6275 −0.7127 −0.2287 −0.7389 −0.8938 −0.5934 
Albania −0.7129 −0.7170 −0.0142 −0.7436 −0.8749 −0.5122 
Andorra −0.5211 −0.5324 −0.0237 0.1773 −0.9039 −0.9184 
Arizona −0.8421 −0.8424 −0.0022 −0.8933 −0.9247 −0.2942 
Arkansas −0.2809 −0.4281 −0.2047 −0.7375 −0.7365 0.0036 
Armenia −0.2060 −0.3717 −0.2086 −0.2124 −0.8608 −0.8233 
Australia −0.5597 −0.5616 −0.0045 −0.6175 −0.9182 −0.7861 
Austria −0.1014 −0.3606 −0.2885 0.0197 −0.6599 −0.6665 
Azerbaijan −0.6703 −0.7780 −0.3267 −0.4034 −0.8469 −0.7435 
Bahrain −0.8488 −0.8509 −0.0141 −0.8504 −0.8545 −0.0275 
Bangladesh −0.8716 −0.8923 −0.1619 −0.8971 −0.9017 −0.0439 
Belarus −0.3970 −0.7741 −0.6253 −0.7525 −0.9509 −0.8014 
Belgium −0.5025 −0.5228 −0.0408 −0.6541 −0.9637 −0.8950 
Bolivia −0.9445 −0.9562 −0.2109 −0.9398 −0.9591 −0.3205 
Bosnia and Herzegovina −0.1132 −0.1236 −0.0118 −0.5737 −0.7271 −0.3599 
Brazil −0.6101 −0.6155 −0.0138 −0.5854 −0.6269 −0.1001 
Bulgaria −0.0465 −0.3812 −0.3510 0.1871 −0.3675 −0.4672 
California −0.7117 −0.7127 −0.0037 −0.7154 −0.6363 0.2779 
Canada −0.4411 −0.4487 −0.0135 −0.1926 −0.9125 −0.8916 
Chile −0.9478 −0.9517 −0.0763 −0.9620 −0.9661 −0.1072 
China −0.8008 −0.8931 −0.4631 −0.9020 −0.9315 −0.3011 
Colorado −0.5890 −0.5890 0.0000 −0.5518 −0.9555 −0.9007 
Connecticut −0.7136 −0.7171 −0.0123 −0.9040 −0.9725 −0.7130 
Costa Rica −0.7988 −0.8395 −0.2025 −0.8101 −0.8593 −0.2590 
Cote d'Ivoire −0.9172 −0.9195 −0.0269 −0.8282 −0.8309 −0.0158 
Croatia −0.7185 −0.7487 −0.1074 −0.7259 −0.8387 −0.4116 
Czechia −0.6211 −0.6715 −0.1330 −0.6675 −0.7607 −0.2802 
Delaware −0.7295 −0.7329 −0.0125 −0.5363 −0.9404 −0.8714 
Dem. Rep. Congo −0.8190 −0.8229 −0.0217 −0.8351 −0.8446 −0.0574 
Denmark −0.5053 −0.5119 −0.0132 0.8680 −0.6938 −0.8361 
Dominican Republic −0.7971 −0.7993 −0.0109 −0.7453 −0.7755 −0.1185 
Ecuador −0.4968 −0.5010 −0.0084 −0.5079 −0.9605 −0.9196 
Egypt −0.7745 −0.8945 −0.5323 −0.9048 −0.9330 −0.2959 
El Salvador −0.8938 −0.9015 −0.0725 −0.8901 −0.9373 −0.4293 
Estonia −0.5684 −0.5720 −0.0083 −0.9086 −0.9302 −0.2366 
Eswatini −0.8472 −0.8495 −0.0149 −0.8388 −0.8902 −0.3187 
Finland −0.6913 −0.6922 −0.0029 −0.6570 −0.9046 −0.7218 
France −0.3072 −0.3112 −0.0057 −0.4326 −0.9661 −0.9402 
Germany −0.6561 −0.6638 −0.0225 −0.3079 −0.6724 −0.5267 
Ghana −0.9576 −0.9641 −0.1544 −0.9648 −0.9738 −0.2573 
Greece −0.5318 −0.5350 −0.0070 −0.5869 −0.6371 −0.1213 
Haiti −0.9416 −0.9586 −0.2908 −0.8877 −0.9284 −0.3624 
Hawaii −0.8652 −0.8953 −0.2237 −0.8622 −0.9459 −0.6075 
Hungary −0.3050 −0.4794 −0.2510 −0.1328 −0.7747 −0.7403 
Indonesia 0.1815 0.1719 −0.0081 −0.0916 −0.1498 −0.0640 
Ireland −0.6802 −0.6822 −0.0063 −0.6131 −0.9686 −0.9187 
Israel −0.3350 −0.5022 −0.2514 −0.4314 −0.8981 −0.8207 
Italy −0.6873 −0.6873 0.0000 −0.8487 −0.9806 −0.8716 
Kansas −0.8563 −0.8664 −0.0700 −0.8860 −0.9086 −0.1985 
Kazakhstan −0.4125 −0.4656 −0.0904 −0.5086 −0.6280 −0.2430 
Kenya −0.7473 −0.7610 −0.0539 −0.7893 −0.8673 −0.3704 
Kosovo −0.6398 −0.6894 −0.1377 1.3136 −0.3554 −0.7214 
Malawi −0.9580 −0.9579 0.0020 −0.9618 −0.9607 0.0303 
Malaysia −0.4956 −0.5724 −0.1523 −0.4420 −0.8200 −0.6774 
Massachusetts −0.7227 −0.7241 −0.0051 −0.7295 −0.8083 −0.2913 
Mauritania −0.5728 −0.5830 −0.0238 −0.8151 −0.8315 −0.0886 
Mexico −0.8466 −0.8731 −0.1729 −0.9200 −0.9634 −0.5428 
Michigan −0.6172 −0.6185 −0.0033 0.0155 −0.9134 −0.9147 
Minnesota −0.4726 −0.4750 −0.0045 −0.5130 −0.9364 −0.8694 
Montenegro −0.7466 −0.7560 −0.0371 −0.7668 −0.9359 −0.7250 
Morocco −0.3897 −0.4260 −0.0595 −0.5955 −0.6649 −0.1716 
Namibia −0.7459 −0.7595 −0.0537 −0.7513 −0.7925 −0.1658 
Nebraska −0.6942 −0.7083 −0.0461 −0.7355 −0.9048 −0.6400 
Nepal −0.5309 −0.7080 −0.3776 −0.2002 −0.7372 −0.6714 
New Hampshire −0.6700 −0.6697 0.0010 −0.3781 −0.8370 −0.7379 
New Jersey −0.7494 −0.7523 −0.0115 −0.8683 −0.9831 −0.8720 
New Mexico −0.5334 −0.5360 −0.0056 −0.7241 −0.8284 −0.3781 
New York −0.7999 −0.7997 0.0014 −0.9104 −0.9805 −0.7819 
Nigeria −0.7421 −0.7457 −0.0138 −0.8234 −0.8167 0.0384 
North Macedonia −0.4838 −0.5950 −0.2153 −0.5260 −0.8567 −0.6977 
Norway −0.6769 −0.6783 −0.0041 −0.8498 −0.9542 −0.6948 
Ohio −0.5198 −0.5265 −0.0140 −0.1078 −0.9288 −0.9202 
Oman −0.6461 −0.9275 −0.7953 −0.6889 −0.9574 −0.8632 
Panama −0.3010 −0.3057 −0.0068 −0.7505 −0.7992 −0.1952 
Pakistan −0.2718 −0.2943 −0.0309 −0.8555 −0.8769 −0.1481 
Paraguay −0.8644 −0.8659 −0.0108 −0.5704 −0.8187 −0.5779 
Pennsylvania −0.4885 −0.4885 0.0000 1.8014 −0.7414 −0.9077 
Peru −0.5846 −0.5907 −0.0147 −0.4176 −0.9231 −0.8680 
Poland 0.2111 −0.2394 −0.3720 −0.0705 −0.8477 −0.8361 
Portugal −0.1140 −0.2057 −0.1035 −0.3837 −0.9205 −0.8710 
Qatar −0.9119 −0.9628 −0.5781 −0.9185 −0.9704 −0.6372 
Rhode Island −0.6865 −0.6872 −0.0021 −0.7136 −0.9552 −0.8435 
Romania 0.4175 0.2745 −0.1009 1.4254 −0.7277 −0.8877 
Russia −0.7479 −0.7484 −0.0020 −0.7220 −0.9369 −0.7731 
Senegal −0.8653 −0.8656 −0.0028 −0.8579 −0.8890 −0.2189 
Somalia −0.8916 −0.8929 −0.0119 −0.9305 −0.9316 −0.0164 
South Africa 0.5541 −0.5369 −0.7020 0.1877 −0.6204 −0.6804 
South Dakota −0.5948 −0.5945 0.0007 −0.8607 −0.8598 0.0068 
State of Palestine −0.7915 −0.7978 −0.0306 −0.4876 −0.7797 −0.5701 
Sudan −0.9112 −0.9113 −0.0010 −0.9362 −0.9630 −0.4201 
Suriname −0.9242 −0.9258 −0.0210 −0.9230 −0.9260 −0.0390 
Switzerland −0.4563 −0.5346 −0.1441 0.0261 −0.6698 −0.6782 
Tennessee −0.6278 −0.7699 −0.3817 −0.7386 −0.8066 −0.2602 
Texas −0.5636 −0.5637 −0.0002 −0.4128 −0.5862 −0.2954 
Turkey −0.2370 −0.2729 −0.0471 −0.4293 −0.7325 −0.5312 
United Arab Emirates −0.4652 −0.4671 −0.0037 −0.7112 −0.8757 −0.5696 
Utah −0.4960 −0.4978 −0.0035 −0.6203 −0.7828 −0.4278 
Uzbekistan −0.7877 −0.7931 −0.0256 −0.8168 −0.8527 −0.1960 
Virginia −0.7210 −0.7209 0.0001 −0.6689 −0.9234 −0.7686 
Wisconsin 0.1422 0.0769 −0.0572 1.1242 −0.5304 −0.7789 
Zambia −0.8399 −0.8426 −0.0165 −0.8774 −0.8663 0.0903 
Zimbabwe −0.9223 −0.9231 −0.0114 −0.9243 −0.9279 −0.0472 
FIG. 2.

Data related to Austria, comprising (a) recorded data; (b) daily increments of confirmed cases, ΔC, and their 30 days average; (c) comparison between recorded fatalities and corresponding kinetics, delay, and hereditary model predictions based on parameters optimized for the entirety of the investigated pandemic time period; (d) absolute model errors associated with parameters optimized for the entirety of the investigated pandemic time period; (e) comparison between recorded fatalities and corresponding kinetics, delay, and hereditary model predictions based on wave-specifically optimized parameters, with the wave transition time amounting to twtr=105d; (f) absolute model errors associated with wave-specifically optimized parameters.

FIG. 2.

Data related to Austria, comprising (a) recorded data; (b) daily increments of confirmed cases, ΔC, and their 30 days average; (c) comparison between recorded fatalities and corresponding kinetics, delay, and hereditary model predictions based on parameters optimized for the entirety of the investigated pandemic time period; (d) absolute model errors associated with parameters optimized for the entirety of the investigated pandemic time period; (e) comparison between recorded fatalities and corresponding kinetics, delay, and hereditary model predictions based on wave-specifically optimized parameters, with the wave transition time amounting to twtr=105d; (f) absolute model errors associated with wave-specifically optimized parameters.

Close modal
FIG. 3.

Data related to New York, comprising (a) recorded data; (b) daily increments of confirmed cases, ΔC, and their 30 days average; (c) comparison between recorded fatalities and corresponding kinetics, delay, and hereditary model predictions based on parameters optimized for the entirety of the investigated pandemic time period; (d) absolute model errors associated with parameters optimized for the entirety of the investigated pandemic time period; (e) comparison between recorded fatalities and corresponding kinetics, delay, and hereditary model predictions based on wave-specifically optimized parameters, with the wave transition time amounting to twtr=110d; (f) absolute model errors associated with wave-specifically optimized parameters.

FIG. 3.

Data related to New York, comprising (a) recorded data; (b) daily increments of confirmed cases, ΔC, and their 30 days average; (c) comparison between recorded fatalities and corresponding kinetics, delay, and hereditary model predictions based on parameters optimized for the entirety of the investigated pandemic time period; (d) absolute model errors associated with parameters optimized for the entirety of the investigated pandemic time period; (e) comparison between recorded fatalities and corresponding kinetics, delay, and hereditary model predictions based on wave-specifically optimized parameters, with the wave transition time amounting to twtr=110d; (f) absolute model errors associated with wave-specifically optimized parameters.

Close modal
TABLE III.

Wave-specific optimal parameters for the death kinetics law (βFI,opt and βFII,opt), for the infection-to-death delay model (fF,DELI,opt,fF,DELII,opt,TF,DELI,opt, and TF,DELII,opt), and for the hereditary model (sHERI,opt,sHERII,opt,fF,HERI,opt,fF,HERII,opt,TF,HERI,opt, and TF,HERII,opt) as well as the corresponding time-averaged absolute errors (EKIN2,EDEL2,EHER2).

CountryβFI,opt (10−4 d−1)βFII,opt (10−4 d−1)EKIN2 (—)fF,DELI,opt (—)TF,DELI,opt (d)fF,DELII,opt (—)TF,DELII,opt (d)EDEL2 (—)sHERI,opt (d−1)fF,HERI,opt (—)TF,HERI,opt (d)sHERII,opt (d−1)fF,HERII,opt (—)TF,HERII,opt (d)EHER2 (—)
Afghanistan 7.91 9.65 169.79 0.04 20.23 0.04 6.00 44.34 0.11 0.04 18.35 0.08 0.10 29.16 18.03 
Albania 0.01 6.16 121.44 0.05 6.49 0.04 24.80 31.14 0.03 0.14 59.85 0.09 0.02 14.48 15.19 
Andorra 22.62 0.00 15.95 0.06 8.50 0.06 58.50 18.78 >10 0.07 10.00 9.14 0.00 5.07 1.53 
Arizona 2.27 1.38 1555.91 0.03 9.30 0.03 34.20 165.95 0.06 0.03 5.42 0.14 0.02 22.79 117.13 
Arkansas 10.79 16.16 120.54 0.01 6.00 0.02 30.20 31.64 0.06 0.02 25.78 0.22 0.02 18.41 31.76 
Armenia 9.05 8.04 40.27 0.02 6.48 0.02 9.90 31.71 >10 0.02 7.00 0.09 0.02 12.85 5.60 
Australia 6.38 13.64 85.24 0.01 12.50 0.03 26.40 32.61 0.16 0.01 11.51 1.99 0.04 20.11 6.98 
Austria 22.54 11.06 160.33 0.04 12.70 0.02 19.10 163.49 0.25 0.04 12.26 0.20 0.02 24.25 54.53 
Azerbaijan 9.88 7.74 44.02 0.02 6.46 0.02 10.50 26.26 0.31 0.02 5.43 0.02 0.02 8.52 6.74 
Bahrain 2.90 3.82 25.04 0.00 13.49 0.00 13.10 3.75 >10 0.00 11.03 >10 0.00 14.00 3.64 
Bangladesh 3.91 3.13 610.18 0.01 6.00 0.01 10.20 62.76 0.11 0.01 5.00 0.02 0.03 59.13 60.00 
Belarus 2.63 3.98 187.86 0.01 10.41 0.01 45.40 46.49 0.02 0.01 55.25 0.01 0.01 33.45 9.23 
Belgium 9.78 2.19 5976.70 0.17 6.48 0.14 58.55 2067.41 >10 0.16 5.00 0.14 0.02 20.93 217.02 
Bolivia 14.27 4.14 829.27 0.06 10.20 0.06 30.80 49.94 0.09 0.06 5.14 0.00 0.02 33.76 33.94 
Bosnia and Herzegovina 12.80 15.03 92.16 0.03 6.43 0.04 15.80 39.29 0.01 0.06 39.89 0.15 0.04 16.53 25.15 
Brazil 15.65 10.74 19 821.18 0.03 6.00 0.03 51.20 8218.72 0.87 0.03 5.00 >10 0.01 6.35 7396.08 
Bulgaria 16.12 14.16 42.20 0.04 6.46 0.04 15.00 50.09 0.06 0.05 10.19 0.23 0.04 16.32 26.69 
California 3.56 1.65 2682.75 0.02 6.00 0.02 30.90 763.62 >10 0.02 14.96 >10 0.01 14.87 975.79 
Canada 29.72 13.32 855.03 0.09 10.30 0.08 59.01 690.36 7.99 0.09 8.04 0.83 0.02 20.64 74.84 
Chile 22.76 30.00 1789.07 0.03 13.50 0.03 29.60 67.90 >10 0.03 12.15 >10 0.03 12.87 60.62 
China 18.29 0.00 1384.12 0.04 6.50 0.06 40.57 135.60 0.05 0.06 13.49 >10 0.00 6.00 94.78 
Colorado 7.70 1.94 598.74 0.06 6.48 0.04 51.70 268.34 >10 0.06 5.00 1.00 0.01 15.88 26.66 
Connecticut 8.36 2.44 1948.87 0.09 5.49 0.09 58.51 187.05 >10 0.09 5.87 >10 0.03 38.14 53.68 
Costa Rica 0.00 3.78 78.17 0.02 23.31 0.01 11.50 14.85 >10 0.01 7.88 1.68 0.01 8.56 11.00 
Cote d'Ivoire 1.70 0.00 37.77 0.01 6.50 0.01 6.47 6.49 7.00 0.01 5.00 >10 0.00 11.00 6.39 
Croatia 16.19 25.16 95.93 0.05 18.50 0.02 13.90 26.30 0.23 0.05 16.88 0.30 0.02 16.32 15.47 
Czechia 19.24 17.84 331.05 0.04 10.40 0.02 13.10 110.06 >10 0.03 8.00 0.14 0.02 13.16 79.22 
Delaware 14.75 1.18 169.97 0.04 6.40 0.04 59.00 78.81 0.07 0.05 7.19 >10 0.02 31.00 10.14 
Dem. Rep. Congo 5.11 10.45 87.08 0.03 6.50 0.03 6.50 14.36 0.41 0.03 5.01 >10 0.06 12.00 13.53 
Denmark 30.00 3.87 56.64 0.05 6.46 0.04 58.52 105.80 8.96 0.05 5.01 0.04 0.01 59.92 17.34 
Dominican Republic 4.89 0.88 413.56 0.02 6.00 0.02 32.71 105.34 0.51 0.02 5.00 >10 0.00 22.63 92.86 
Ecuador 30.00 23.01 3029.74 0.14 8.50 0.08 6.00 1490.89 0.03 0.26 21.53 0.11 0.03 5.00 119.80 
Egypt 11.61 27.53 844.86 0.05 10.10 0.06 8.60 80.44 0.06 0.06 5.00 0.00 0.09 32.12 56.64 
El Salvador 7.76 12.15 158.50 0.03 6.46 0.04 45.70 17.42 0.25 0.03 5.00 0.27 0.03 5.01 9.94 
Estonia 0.00 4.22 51.43 0.03 11.20 0.03 42.70 4.70 >10 0.03 9.00 0.06 0.02 41.18 3.59 
Eswatini 7.10 16.92 20.11 0.02 7.49 0.02 58.50 3.24 2.31 0.02 5.39 >10 0.12 31.95 2.21 
Finland 19.53 0.00 78.92 0.05 6.20 0.04 58.51 27.07 >10 0.05 5.50 8.94 0.02 60.00 7.53 
France 30.00 6.83 17 462.61 0.21 6.50 0.10 59.00 9908.80 >10 0.20 5.00 0.13 0.02 26.07 592.16 
Germany 28.27 9.16 887.36 0.05 13.82 0.05 45.60 614.16 0.31 0.05 12.89 7.02 0.03 33.74 290.67 
Ghana 0.00 0.00 179.91 0.01 12.38 0.01 48.12 6.34 0.07 0.01 5.11 0.01 0.00 32.51 4.71 
Greece 30.00 24.06 145.40 0.06 6.70 0.04 21.10 60.06 >10 0.06 8.00 1.62 0.04 20.62 52.77 
Haiti 3.13 0.01 69.26 0.03 12.50 0.03 52.80 7.78 0.06 0.03 5.13 >10 0.34 53.97 4.96 
Hawaii 3.15 0.00 50.14 0.01 15.30 0.02 30.49 6.91 >10 0.03 9.93 0.09 0.02 23.87 2.71 
Hungary 30.00 9.14 213.06 0.14 8.49 0.03 10.70 184.77 0.00 0.27 32.86 0.07 0.04 26.81 47.99 
Indonesia 18.46 17.11 748.48 0.04 5.99 0.05 43.20 679.91 0.63 0.04 5.00 >10 0.03 14.03 636.37 
Ireland 30.00 1.10 474.73 0.07 7.00 0.07 58.51 183.69 >10 0.07 7.08 0.04 0.01 34.13 14.93 
Israel 6.78 5.17 118.23 0.02 9.50 0.01 13.20 67.22 >10 0.02 10.99 0.11 0.01 14.91 12.05 
Italy 30.00 8.75 12 006.31 0.15 6.43 0.15 58.51 1816.28 0.71 0.15 5.00 0.21 0.02 16.49 233.24 
Kansas 5.68 1.12 431.26 0.03 6.48 0.02 20.80 49.18 8.92 0.02 5.00 0.13 0.01 20.25 39.42 
Kazakhstan 9.08 6.46 132.27 0.02 19.80 0.02 33.70 64.99 0.19 0.02 22.35 0.00 0.02 34.41 49.20 
Kenya 4.47 5.59 165.44 0.02 6.48 0.02 6.48 34.86 0.01 0.03 5.21 0.78 0.01 5.00 21.95 
Kosovo 22.16 10.01 16.03 0.04 14.47 0.05 33.51 37.08 >10 0.04 11.00 0.20 0.02 6.46 10.33 
Malawi 0.00 0.00 106.38 0.03 6.42 0.03 20.20 4.06 1.98 0.03 5.52 >10 0.00 6.00 4.18 
Malaysia 5.67 2.62 53.05 0.02 6.50 0.01 42.90 29.60 0.00 0.03 32.05 0.00 0.01 32.72 9.55 
Massachusetts 26.53 8.08 1049.05 0.08 10.90 0.08 59.00 283.80 >10 0.08 13.49 >10 0.06 50.09 201.13 
Mauritania 3.83 20.29 74.44 0.02 6.48 0.02 6.48 13.76 9.76 0.02 5.00 9.94 0.03 7.29 12.55 
Mexico 30.00 30.00 25 788.21 0.12 6.47 0.12 31.49 2062.82 0.00 0.21 33.67 0.00 0.13 29.07 943.14 
Michigan 30.00 4.00 841.48 0.10 6.20 0.07 59.01 854.56 >10 0.09 5.00 0.52 0.02 14.65 72.86 
Minnesota 30.00 12.22 601.83 0.05 6.01 0.04 55.50 293.12 8.01 0.05 5.00 0.12 0.01 14.57 38.29 
Montenegro 6.40 6.63 50.37 0.02 6.50 0.02 6.48 11.74 0.04 0.03 18.76 0.06 0.02 5.02 3.23 
Morocco 6.13 16.76 237.54 0.03 6.48 0.02 6.00 96.08 0.66 0.03 5.00 0.68 0.02 5.00 79.59 
Namibia 4.18 5.24 12.67 0.01 6.45 0.01 54.51 3.15 >10 0.01 5.99 >10 0.01 9.00 2.63 
Nebraska 3.55 2.51 113.55 0.01 6.49 0.01 28.70 30.04 0.14 0.02 5.01 1.10 0.01 15.14 10.82 
Nepal 0.00 7.32 51.49 0.00 6.29 0.01 8.60 41.18 0.09 0.02 60.00 0.07 0.01 18.26 13.53 
New Hampshire 30.00 8.92 43.79 0.07 13.60 0.07 57.55 27.23 >10 0.07 17.00 >10 0.03 42.04 7.14 
New Jersey 10.97 2.87 4895.44 0.08 8.60 0.08 58.97 644.88 0.21 0.08 8.72 0.25 0.02 23.14 82.54 
New Mexico 4.95 4.12 231.03 0.03 6.00 0.04 42.30 63.74 0.67 0.03 5.00 0.09 0.02 16.13 39.64 
New York 4.70 2.02 19 330.62 0.08 6.45 0.07 50.50 1732.21 >10 0.08 5.00 >10 0.01 11.29 377.77 
Nigeria 3.83 4.41 309.09 0.02 6.50 0.02 39.70 54.57 8.66 0.02 5.00 8.73 0.00 7.62 56.67 
North Macedonia 16.58 12.83 119.33 0.05 6.47 0.03 6.00 56.56 0.72 0.05 5.00 0.23 0.03 9.18 17.10 
Norway 3.02 0.00 129.09 0.03 13.30 0.03 59.01 19.39 >10 0.03 12.91 9.89 0.02 59.87 5.92 
Ohio 30.00 9.77 687.98 0.07 7.50 0.04 38.90 613.82 0.09 0.07 5.00 0.06 0.02 5.02 49.01 
Oman 4.10 6.35 125.72 0.01 14.70 0.01 43.80 39.11 0.03 0.01 52.13 >10 0.01 20.01 5.35 
Panama 7.89 8.43 192.07 0.02 5.99 0.02 28.00 47.92 9.77 0.02 5.00 0.56 0.02 15.23 38.57 
Pakistan 9.24 13.38 634.27 0.02 6.00 0.02 11.70 91.63 1.03 0.02 5.00 0.06 0.07 59.70 78.06 
Paraguay 10.68 6.37 80.61 0.02 7.20 0.03 50.90 34.63 >10 0.02 6.00 0.32 0.02 8.33 14.61 
Pennsylvania 24.66 8.34 293.55 0.08 12.00 0.06 48.60 822.34 >10 0.08 10.67 >10 0.02 16.96 75.92 
Peru 29.26 12.18 4171.68 0.07 6.01 0.04 5.99 2429.67 0.05 0.08 5.00 1.00 0.02 5.00 320.60 
Poland 13.02 12.64 502.35 0.05 6.50 0.02 11.60 466.93 0.93 0.04 5.00 0.10 0.02 14.44 76.52 
Portugal 7.76 8.53 381.36 0.04 6.60 0.04 40.80 235.03 0.15 0.04 6.18 0.23 0.02 14.55 30.32 
Qatar 0.59 0.00 61.92 0.00 30.30 0.00 30.40 5.05 0.12 0.00 23.35 0.01 0.00 22.87 1.83 
Rhode Island 8.37 1.61 219.94 0.06 7.70 0.06 56.55 63.00 >10 0.06 9.00 0.13 0.02 27.25 9.86 
Romania 30.00 14.75 238.61 0.07 6.50 0.08 59.00 578.73 0.02 0.17 60.00 0.05 0.03 8.34 64.98 
Russia 6.76 9.55 1030.70 0.02 18.30 0.02 6.00 286.55 0.08 0.02 14.46 0.12 0.02 13.04 65.03 
Senegal 5.63 30.00 56.75 0.02 9.20 0.02 51.80 8.07 1.50 0.02 8.99 2.18 0.04 12.24 6.30 
Somalia 0.00 0.00 80.48 0.03 6.48 0.03 52.58 5.60 >10 0.03 6.00 0.09 0.03 51.18 5.50 
South Africa 14.84 21.89 359.86 0.02 12.50 0.03 6.00 427.41 0.06 0.03 12.98 0.33 0.04 8.95 136.62 
South Dakota 0.00 9.01 85.72 0.01 14.30 0.02 21.70 11.94 >10 0.02 15.95 >10 0.02 21.39 12.02 
State of Palestine 4.23 8.44 18.49 0.01 6.50 0.01 7.10 9.47 0.33 0.01 15.17 0.16 0.01 14.53 4.07 
Sudan 3.42 3.96 687.79 0.09 6.49 0.09 56.50 43.86 1.11 0.08 5.00 2.10 0.03 5.06 25.44 
Suriname 10.72 0.00 23.50 0.02 10.30 0.02 33.90 1.81 0.21 0.02 9.75 0.17 0.26 34.05 1.74 
Switzerland 30.00 7.98 270.56 0.06 9.00 0.06 57.60 277.62 0.00 0.12 44.01 0.12 0.02 23.28 89.35 
Tennessee 5.77 12.92 306.75 0.01 6.00 0.01 9.70 80.18 0.03 0.02 14.68 0.07 0.01 5.01 59.32 
Texas 8.56 6.49 560.22 0.02 14.70 0.02 34.70 328.95 0.16 0.02 12.70 0.15 0.01 5.00 231.79 
Turkey 14.65 11.35 590.40 0.01 6.00 0.01 6.50 336.94 7.84 0.01 5.00 0.02 0.02 60.00 157.94 
United Arab Emirates 1.51 1.46 143.25 0.01 6.50 0.01 59.00 41.36 8.23 0.01 5.00 8.60 0.00 15.79 17.80 
Utah 3.24 1.89 62.25 0.01 5.99 0.01 55.50 23.63 0.06 0.01 5.01 >10 0.01 24.01 13.52 
Uzbekistan 6.38 9.15 49.11 0.01 14.30 0.01 5.99 9.00 5.93 0.01 13.87 0.90 0.01 5.01 7.23 
Virginia 4.13 1.19 932.89 0.03 6.49 0.03 38.00 308.87 0.08 0.04 5.00 7.20 0.02 28.88 71.46 
Wisconsin 18.53 7.89 105.98 0.04 6.50 0.03 59.01 225.11 6.38 0.03 5.00 0.23 0.01 19.10 49.77 
Zambia 5.02 8.06 152.56 0.02 6.44 0.02 34.40 18.70 0.86 0.02 5.00 1.45 0.03 28.34 20.39 
Zimbabwe 6.49 9.30 69.50 0.03 8.46 0.03 11.42 5.26 1.06 0.03 7.71 0.13 0.03 5.21 5.01 
CountryβFI,opt (10−4 d−1)βFII,opt (10−4 d−1)EKIN2 (—)fF,DELI,opt (—)TF,DELI,opt (d)fF,DELII,opt (—)TF,DELII,opt (d)EDEL2 (—)sHERI,opt (d−1)fF,HERI,opt (—)TF,HERI,opt (d)sHERII,opt (d−1)fF,HERII,opt (—)TF,HERII,opt (d)EHER2 (—)
Afghanistan 7.91 9.65 169.79 0.04 20.23 0.04 6.00 44.34 0.11 0.04 18.35 0.08 0.10 29.16 18.03 
Albania 0.01 6.16 121.44 0.05 6.49 0.04 24.80 31.14 0.03 0.14 59.85 0.09 0.02 14.48 15.19 
Andorra 22.62 0.00 15.95 0.06 8.50 0.06 58.50 18.78 >10 0.07 10.00 9.14 0.00 5.07 1.53 
Arizona 2.27 1.38 1555.91 0.03 9.30 0.03 34.20 165.95 0.06 0.03 5.42 0.14 0.02 22.79 117.13 
Arkansas 10.79 16.16 120.54 0.01 6.00 0.02 30.20 31.64 0.06 0.02 25.78 0.22 0.02 18.41 31.76 
Armenia 9.05 8.04 40.27 0.02 6.48 0.02 9.90 31.71 >10 0.02 7.00 0.09 0.02 12.85 5.60 
Australia 6.38 13.64 85.24 0.01 12.50 0.03 26.40 32.61 0.16 0.01 11.51 1.99 0.04 20.11 6.98 
Austria 22.54 11.06 160.33 0.04 12.70 0.02 19.10 163.49 0.25 0.04 12.26 0.20 0.02 24.25 54.53 
Azerbaijan 9.88 7.74 44.02 0.02 6.46 0.02 10.50 26.26 0.31 0.02 5.43 0.02 0.02 8.52 6.74 
Bahrain 2.90 3.82 25.04 0.00 13.49 0.00 13.10 3.75 >10 0.00 11.03 >10 0.00 14.00 3.64 
Bangladesh 3.91 3.13 610.18 0.01 6.00 0.01 10.20 62.76 0.11 0.01 5.00 0.02 0.03 59.13 60.00 
Belarus 2.63 3.98 187.86 0.01 10.41 0.01 45.40 46.49 0.02 0.01 55.25 0.01 0.01 33.45 9.23 
Belgium 9.78 2.19 5976.70 0.17 6.48 0.14 58.55 2067.41 >10 0.16 5.00 0.14 0.02 20.93 217.02 
Bolivia 14.27 4.14 829.27 0.06 10.20 0.06 30.80 49.94 0.09 0.06 5.14 0.00 0.02 33.76 33.94 
Bosnia and Herzegovina 12.80 15.03 92.16 0.03 6.43 0.04 15.80 39.29 0.01 0.06 39.89 0.15 0.04 16.53 25.15 
Brazil 15.65 10.74 19 821.18 0.03 6.00 0.03 51.20 8218.72 0.87 0.03 5.00 >10 0.01 6.35 7396.08 
Bulgaria 16.12 14.16 42.20 0.04 6.46 0.04 15.00 50.09 0.06 0.05 10.19 0.23 0.04 16.32 26.69 
California 3.56 1.65 2682.75 0.02 6.00 0.02 30.90 763.62 >10 0.02 14.96 >10 0.01 14.87 975.79 
Canada 29.72 13.32 855.03 0.09 10.30 0.08 59.01 690.36 7.99 0.09 8.04 0.83 0.02 20.64 74.84 
Chile 22.76 30.00 1789.07 0.03 13.50 0.03 29.60 67.90 >10 0.03 12.15 >10 0.03 12.87 60.62 
China 18.29 0.00 1384.12 0.04 6.50 0.06 40.57 135.60 0.05 0.06 13.49 >10 0.00 6.00 94.78 
Colorado 7.70 1.94 598.74 0.06 6.48 0.04 51.70 268.34 >10 0.06 5.00 1.00 0.01 15.88 26.66 
Connecticut 8.36 2.44 1948.87 0.09 5.49 0.09 58.51 187.05 >10 0.09 5.87 >10 0.03 38.14 53.68 
Costa Rica 0.00 3.78 78.17 0.02 23.31 0.01 11.50 14.85 >10 0.01 7.88 1.68 0.01 8.56 11.00 
Cote d'Ivoire 1.70 0.00 37.77 0.01 6.50 0.01 6.47 6.49 7.00 0.01 5.00 >10 0.00 11.00 6.39 
Croatia 16.19 25.16 95.93 0.05 18.50 0.02 13.90 26.30 0.23 0.05 16.88 0.30 0.02 16.32 15.47 
Czechia 19.24 17.84 331.05 0.04 10.40 0.02 13.10 110.06 >10 0.03 8.00 0.14 0.02 13.16 79.22 
Delaware 14.75 1.18 169.97 0.04 6.40 0.04 59.00 78.81 0.07 0.05 7.19 >10 0.02 31.00 10.14 
Dem. Rep. Congo 5.11 10.45 87.08 0.03 6.50 0.03 6.50 14.36 0.41 0.03 5.01 >10 0.06 12.00 13.53 
Denmark 30.00 3.87 56.64 0.05 6.46 0.04 58.52 105.80 8.96 0.05 5.01 0.04 0.01 59.92 17.34 
Dominican Republic 4.89 0.88 413.56 0.02 6.00 0.02 32.71 105.34 0.51 0.02 5.00 >10 0.00 22.63 92.86 
Ecuador 30.00 23.01 3029.74 0.14 8.50 0.08 6.00 1490.89 0.03 0.26 21.53 0.11 0.03 5.00 119.80 
Egypt 11.61 27.53 844.86 0.05 10.10 0.06 8.60 80.44 0.06 0.06 5.00 0.00 0.09 32.12 56.64 
El Salvador 7.76 12.15 158.50 0.03 6.46 0.04 45.70 17.42 0.25 0.03 5.00 0.27 0.03 5.01 9.94 
Estonia 0.00 4.22 51.43 0.03 11.20 0.03 42.70 4.70 >10 0.03 9.00 0.06 0.02 41.18 3.59 
Eswatini 7.10 16.92 20.11 0.02 7.49 0.02 58.50 3.24 2.31 0.02 5.39 >10 0.12 31.95 2.21 
Finland 19.53 0.00 78.92 0.05 6.20 0.04 58.51 27.07 >10 0.05 5.50 8.94 0.02 60.00 7.53 
France 30.00 6.83 17 462.61 0.21 6.50 0.10 59.00 9908.80 >10 0.20 5.00 0.13 0.02 26.07 592.16 
Germany 28.27 9.16 887.36 0.05 13.82 0.05 45.60 614.16 0.31 0.05 12.89 7.02 0.03 33.74 290.67 
Ghana 0.00 0.00 179.91 0.01 12.38 0.01 48.12 6.34 0.07 0.01 5.11 0.01 0.00 32.51 4.71 
Greece 30.00 24.06 145.40 0.06 6.70 0.04 21.10 60.06 >10 0.06 8.00 1.62 0.04 20.62 52.77 
Haiti 3.13 0.01 69.26 0.03 12.50 0.03 52.80 7.78 0.06 0.03 5.13 >10 0.34 53.97 4.96 
Hawaii 3.15 0.00 50.14 0.01 15.30 0.02 30.49 6.91 >10 0.03 9.93 0.09 0.02 23.87 2.71 
Hungary 30.00 9.14 213.06 0.14 8.49 0.03 10.70 184.77 0.00 0.27 32.86 0.07 0.04 26.81 47.99 
Indonesia 18.46 17.11 748.48 0.04 5.99 0.05 43.20 679.91 0.63 0.04 5.00 >10 0.03 14.03 636.37 
Ireland 30.00 1.10 474.73 0.07 7.00 0.07 58.51 183.69 >10 0.07 7.08 0.04 0.01 34.13 14.93 
Israel 6.78 5.17 118.23 0.02 9.50 0.01 13.20 67.22 >10 0.02 10.99 0.11 0.01 14.91 12.05 
Italy 30.00 8.75 12 006.31 0.15 6.43 0.15 58.51 1816.28 0.71 0.15 5.00 0.21 0.02 16.49 233.24 
Kansas 5.68 1.12 431.26 0.03 6.48 0.02 20.80 49.18 8.92 0.02 5.00 0.13 0.01 20.25 39.42 
Kazakhstan 9.08 6.46 132.27 0.02 19.80 0.02 33.70 64.99 0.19 0.02 22.35 0.00 0.02 34.41 49.20 
Kenya 4.47 5.59 165.44 0.02 6.48 0.02 6.48 34.86 0.01 0.03 5.21 0.78 0.01 5.00 21.95 
Kosovo 22.16 10.01 16.03 0.04 14.47 0.05 33.51 37.08 >10 0.04 11.00 0.20 0.02 6.46 10.33 
Malawi 0.00 0.00 106.38 0.03 6.42 0.03 20.20 4.06 1.98 0.03 5.52 >10 0.00 6.00 4.18 
Malaysia 5.67 2.62 53.05 0.02 6.50 0.01 42.90 29.60 0.00 0.03 32.05 0.00 0.01 32.72 9.55 
Massachusetts 26.53 8.08 1049.05 0.08 10.90 0.08 59.00 283.80 >10 0.08 13.49 >10 0.06 50.09 201.13 
Mauritania 3.83 20.29 74.44 0.02 6.48 0.02 6.48 13.76 9.76 0.02 5.00 9.94 0.03 7.29 12.55 
Mexico 30.00 30.00 25 788.21 0.12 6.47 0.12 31.49 2062.82 0.00 0.21 33.67 0.00 0.13 29.07 943.14 
Michigan 30.00 4.00 841.48 0.10 6.20 0.07 59.01 854.56 >10 0.09 5.00 0.52 0.02 14.65 72.86 
Minnesota 30.00 12.22 601.83 0.05 6.01 0.04 55.50 293.12 8.01 0.05 5.00 0.12 0.01 14.57 38.29 
Montenegro 6.40 6.63 50.37 0.02 6.50 0.02 6.48 11.74 0.04 0.03 18.76 0.06 0.02 5.02 3.23 
Morocco 6.13 16.76 237.54 0.03 6.48 0.02 6.00 96.08 0.66 0.03 5.00 0.68 0.02 5.00 79.59 
Namibia 4.18 5.24 12.67 0.01 6.45 0.01 54.51 3.15 >10 0.01 5.99 >10 0.01 9.00 2.63 
Nebraska 3.55 2.51 113.55 0.01 6.49 0.01 28.70 30.04 0.14 0.02 5.01 1.10 0.01 15.14 10.82 
Nepal 0.00 7.32 51.49 0.00 6.29 0.01 8.60 41.18 0.09 0.02 60.00 0.07 0.01 18.26 13.53 
New Hampshire 30.00 8.92 43.79 0.07 13.60 0.07 57.55 27.23 >10 0.07 17.00 >10 0.03 42.04 7.14 
New Jersey 10.97 2.87 4895.44 0.08 8.60 0.08 58.97 644.88 0.21 0.08 8.72 0.25 0.02 23.14 82.54 
New Mexico 4.95 4.12 231.03 0.03 6.00 0.04 42.30 63.74 0.67 0.03 5.00 0.09 0.02 16.13 39.64 
New York 4.70 2.02 19 330.62 0.08 6.45 0.07 50.50 1732.21 >10 0.08 5.00 >10 0.01 11.29 377.77 
Nigeria 3.83 4.41 309.09 0.02 6.50 0.02 39.70 54.57 8.66 0.02 5.00 8.73 0.00 7.62 56.67 
North Macedonia 16.58 12.83 119.33 0.05 6.47 0.03 6.00 56.56 0.72 0.05 5.00 0.23 0.03 9.18 17.10 
Norway 3.02 0.00 129.09 0.03 13.30 0.03 59.01 19.39 >10 0.03 12.91 9.89 0.02 59.87 5.92 
Ohio 30.00 9.77 687.98 0.07 7.50 0.04 38.90 613.82 0.09 0.07 5.00 0.06 0.02 5.02 49.01 
Oman 4.10 6.35 125.72 0.01 14.70 0.01 43.80 39.11 0.03 0.01 52.13 >10 0.01 20.01 5.35 
Panama 7.89 8.43 192.07 0.02 5.99 0.02 28.00 47.92 9.77 0.02 5.00 0.56 0.02 15.23 38.57 
Pakistan 9.24 13.38 634.27 0.02 6.00 0.02 11.70 91.63 1.03 0.02 5.00 0.06 0.07 59.70 78.06 
Paraguay 10.68 6.37 80.61 0.02 7.20 0.03 50.90 34.63 >10 0.02 6.00 0.32 0.02 8.33 14.61 
Pennsylvania 24.66 8.34 293.55 0.08 12.00 0.06 48.60 822.34 >10 0.08 10.67 >10 0.02 16.96 75.92 
Peru 29.26 12.18 4171.68 0.07 6.01 0.04 5.99 2429.67 0.05 0.08 5.00 1.00 0.02 5.00 320.60 
Poland 13.02 12.64 502.35 0.05 6.50 0.02 11.60 466.93 0.93 0.04 5.00 0.10 0.02 14.44 76.52 
Portugal 7.76 8.53 381.36 0.04 6.60 0.04 40.80 235.03 0.15 0.04 6.18 0.23 0.02 14.55 30.32 
Qatar 0.59 0.00 61.92 0.00 30.30 0.00 30.40 5.05 0.12 0.00 23.35 0.01 0.00 22.87 1.83 
Rhode Island 8.37 1.61 219.94 0.06 7.70 0.06 56.55 63.00 >10 0.06 9.00 0.13 0.02 27.25 9.86 
Romania 30.00 14.75 238.61 0.07 6.50 0.08 59.00 578.73 0.02 0.17 60.00 0.05 0.03 8.34 64.98 
Russia 6.76 9.55 1030.70 0.02 18.30 0.02 6.00 286.55 0.08 0.02 14.46 0.12 0.02 13.04 65.03 
Senegal 5.63 30.00 56.75 0.02 9.20 0.02 51.80 8.07 1.50 0.02 8.99 2.18 0.04 12.24 6.30 
Somalia 0.00 0.00 80.48 0.03 6.48 0.03 52.58 5.60 >10 0.03 6.00 0.09 0.03 51.18 5.50 
South Africa 14.84 21.89 359.86 0.02 12.50 0.03 6.00 427.41 0.06 0.03 12.98 0.33 0.04 8.95 136.62 
South Dakota 0.00 9.01 85.72 0.01 14.30 0.02 21.70 11.94 >10 0.02 15.95 >10 0.02 21.39 12.02 
State of Palestine 4.23 8.44 18.49 0.01 6.50 0.01 7.10 9.47 0.33 0.01 15.17 0.16 0.01 14.53 4.07 
Sudan 3.42 3.96 687.79 0.09 6.49 0.09 56.50 43.86 1.11 0.08 5.00 2.10 0.03 5.06 25.44 
Suriname 10.72 0.00 23.50 0.02 10.30 0.02 33.90 1.81 0.21 0.02 9.75 0.17 0.26 34.05 1.74 
Switzerland 30.00 7.98 270.56 0.06 9.00 0.06 57.60 277.62 0.00 0.12 44.01 0.12 0.02 23.28 89.35 
Tennessee 5.77 12.92 306.75 0.01 6.00 0.01 9.70 80.18 0.03 0.02 14.68 0.07 0.01 5.01 59.32 
Texas 8.56 6.49 560.22 0.02 14.70 0.02 34.70 328.95 0.16 0.02 12.70 0.15 0.01 5.00 231.79 
Turkey 14.65 11.35 590.40 0.01 6.00 0.01 6.50 336.94 7.84 0.01 5.00 0.02 0.02 60.00 157.94 
United Arab Emirates 1.51 1.46 143.25 0.01 6.50 0.01 59.00 41.36 8.23 0.01 5.00 8.60 0.00 15.79 17.80 
Utah 3.24 1.89 62.25 0.01 5.99 0.01 55.50 23.63 0.06 0.01 5.01 >10 0.01 24.01 13.52 
Uzbekistan 6.38 9.15 49.11 0.01 14.30 0.01 5.99 9.00 5.93 0.01 13.87 0.90 0.01 5.01 7.23 
Virginia 4.13 1.19 932.89 0.03 6.49 0.03 38.00 308.87 0.08 0.04 5.00 7.20 0.02 28.88 71.46 
Wisconsin 18.53 7.89 105.98 0.04 6.50 0.03 59.01 225.11 6.38 0.03 5.00 0.23 0.01 19.10 49.77 
Zambia 5.02 8.06 152.56 0.02 6.44 0.02 34.40 18.70 0.86 0.02 5.00 1.45 0.03 28.34 20.39 
Zimbabwe 6.49 9.30 69.50 0.03 8.46 0.03 11.42 5.26 1.06 0.03 7.71 0.13 0.03 5.21 5.01 
TABLE IV.

Model-specific time-averaged absolute and normalized error measures according to Eq. (19) and Eq. (28).

CountryEKIN (—)EDEL (—)EHER (—)EKIN2 (—)EDEL2 (—)EHER2 (—)Fmax (—)RKIN (—)RDEL (—)RHER (—)RKIN2 (—)RDEL2 (—)RHER2 (—)
Afghanistan 162.51 60.53 46.69 169.79 44.34 18.03 2201 0.0738 0.0275 0.0212 0.0771 0.0201 0.0082 
Albania 115.54 33.17 32.70 121.44 31.14 15.19 1181 0.0978 0.0281 0.0277 0.1028 0.0264 0.0129 
Andorra 50.79 24.33 23.75 15.95 18.78 1.53 84 0.6047 0.2896 0.2828 0.1899 0.2236 0.0182 
Arizona 2143.82 338.52 337.77 1555.91 165.95 117.13 8864 0.2419 0.0382 0.0381 0.1755 0.0187 0.0132 
Arkansas 62.08 44.64 35.50 120.54 31.64 31.76 3570 0.0174 0.0125 0.0099 0.0338 0.0089 0.0089 
Armenia 67.57 53.65 42.45 40.27 31.71 5.60 2823 0.0239 0.0190 0.0150 0.0143 0.0112 0.0020 
Australia 107.42 47.30 47.09 85.24 32.61 6.98 909 0.1182 0.0520 0.0518 0.0938 0.0359 0.0077 
Austria 253.65 227.92 162.17 160.33 163.49 54.53 6222 0.0408 0.0366 0.0261 0.0258 0.0263 0.0088 
Azerbaijan 95.70 31.56 21.25 44.02 26.26 6.74 2641 0.0362 0.0119 0.0080 0.0167 0.0099 0.0026 
Bahrain 26.04 3.94 3.88 25.04 3.75 3.64 352 0.0740 0.0112 0.0110 0.0711 0.0106 0.0103 
Bangladesh 600.08 77.08 64.60 610.18 62.76 60.00 7559 0.0794 0.0102 0.0085 0.0807 0.0083 0.0079 
Belarus 110.76 66.79 25.03 187.86 46.49 9.23 1424 0.0778 0.0469 0.0176 0.1319 0.0326 0.0065 
Belgium 7389.85 3676.68 3526.52 5976.70 2067.41 217.02 194 41 0.3801 0.1891 0.1814 0.3074 0.1063 0.0112 
Bolivia 1004.58 55.73 43.98 829.27 49.94 33.94 9149 0.1098 0.0061 0.0048 0.0906 0.0055 0.0037 
Bosnia and Herzegovina 51.74 45.88 45.34 92.16 39.29 25.15 4050 0.0128 0.0113 0.0112 0.0228 0.0097 0.0062 
Brazil 27 244.64 10 622.69 10 476.22 19 821.18 8218.72 7396.08 194 976 0.1397 0.0545 0.0537 0.1017 0.0422 0.0379 
Bulgaria 61.52 58.65 38.06 42.20 50.09 26.69 7576 0.0081 0.0077 0.0050 0.0056 0.0066 0.0035 
California 5265.39 1518.19 1512.63 2682.75 763.62 975.79 26 343 0.1999 0.0576 0.0574 0.1018 0.0290 0.0370 
Canada 3233.87 1807.43 1782.96 855.03 690.36 74.84 15 606 0.2072 0.1158 0.1142 0.0548 0.0442 0.0048 
Chile 1395.10 72.88 67.32 1789.07 67.90 60.62 16 608 0.0840 0.0044 0.0041 0.1077 0.0041 0.0037 
China 1263.72 251.68 135.14 1384.12 135.60 94.78 4634 0.2727 0.0543 0.0292 0.2987 0.0293 0.0205 
Colorado 1138.63 468.02 468.02 598.74 268.34 26.66 4814 0.2365 0.0972 0.0972 0.1244 0.0557 0.0055 
Connecticut 2921.95 836.92 826.60 1948.87 187.05 53.68 5995 0.4874 0.1396 0.1379 0.3251 0.0312 0.0090 
Costa Rica 73.87 14.87 11.86 78.17 14.85 11.00 2185 0.0338 0.0068 0.0054 0.0358 0.0068 0.0050 
Cote d'Ivoire 80.44 6.66 6.48 37.77 6.49 6.39 137 0.5872 0.0486 0.0473 0.2757 0.0474 0.0466 
Croatia 115.15 32.41 28.93 95.93 26.30 15.47 3920 0.0294 0.0083 0.0074 0.0245 0.0067 0.0039 
Czechia 330.37 125.18 108.53 331.05 110.06 79.22 11 813 0.0280 0.0106 0.0092 0.0280 0.0093 0.0067 
Delaware 403.38 109.12 107.75 169.97 78.81 10.14 926 0.4356 0.1178 0.1164 0.1836 0.0851 0.0109 
Dem. Rep. Congo 90.86 16.45 16.09 87.08 14.36 13.53 591 0.1537 0.0278 0.0272 0.1473 0.0243 0.0229 
Denmark 433.59 214.49 211.65 56.64 105.80 17.34 1298 0.3340 0.1652 0.1631 0.0436 0.0815 0.0134 
Dominican Republic 714.78 145.00 143.42 413.56 105.34 92.86 2414 0.2961 0.0601 0.0594 0.1713 0.0436 0.0385 
Ecuador 3118.53 1569.36 1556.14 3029.74 1490.89 119.80 14 034 0.2222 0.1118 0.1109 0.2159 0.1062 0.0085 
Egypt 584.30 131.77 61.63 844.86 80.44 56.64 7631 0.0766 0.0173 0.0081 0.1107 0.0105 0.0074 
El Salvador 102.44 10.88 10.09 158.50 17.42 9.94 1336 0.0767 0.0081 0.0076 0.1186 0.0130 0.0074 
Estonia 27.16 11.72 11.62 51.43 4.70 3.59 229 0.1186 0.0512 0.0508 0.2246 0.0205 0.0157 
Eswatini 17.93 2.74 2.70 20.11 3.24 2.21 205 0.0875 0.0134 0.0132 0.0981 0.0158 0.0108 
Finland 233.24 71.99 71.78 78.92 27.07 7.53 561 0.4158 0.1283 0.1280 0.1407 0.0483 0.0134 
France 21 315.15 14 766.54 14 682.89 17 462.61 9908.80 592.16 64 780 0.3290 0.2279 0.2267 0.2696 0.1530 0.0091 
Germany 4094.74 1408.29 1376.55 887.36 614.16 290.67 34 194 0.1198 0.0412 0.0403 0.0260 0.0180 0.0085 
Ghana 179.90 7.63 6.45 179.91 6.34 4.71 335 0.5370 0.0228 0.0193 0.5370 0.0189 0.0141 
Greece 150.02 70.25 69.75 145.40 60.06 52.77 4838 0.0310 0.0145 0.0144 0.0301 0.0124 0.0109 
Haiti 142.51 8.32 5.90 69.26 7.78 4.96 236 0.6038 0.0353 0.0250 0.2935 0.0330 0.0210 
Hawaii 54.24 7.31 5.68 50.14 6.91 2.71 288 0.1883 0.0254 0.0197 0.1741 0.0240 0.0094 
Hungary 393.18 273.25 204.67 213.06 184.77 47.99 9537 0.0412 0.0287 0.0215 0.0223 0.0194 0.0050 
Indonesia 843.63 996.76 988.69 748.48 679.91 636.37 22 138 0.0381 0.0450 0.0447 0.0338 0.0307 0.0287 
Ireland 1396.31 446.55 443.74 474.73 183.69 14.93 2237 0.6242 0.1996 0.1984 0.2122 0.0821 0.0067 
Israel 120.02 79.81 59.75 118.23 67.22 12.05 3325 0.0361 0.0240 0.0180 0.0356 0.0202 0.0036 
Italy 23 343.57 7298.56 7298.84 12 006.31 1816.28 233.24 74 159 0.3148 0.0984 0.0984 0.1619 0.0245 0.0031 
Kansas 445.87 64.06 59.58 431.26 49.18 39.42 2788 0.1599 0.0230 0.0214 0.1547 0.0176 0.0141 
Kazakhstan 109.36 64.25 58.44 132.27 64.99 49.20 2286 0.0478 0.0281 0.0256 0.0579 0.0284 0.0215 
Kenya 141.41 35.73 33.80 165.44 34.86 21.95 1670 0.0847 0.0214 0.0202 0.0991 0.0209 0.0131 
Kosovo 163.86 59.02 50.89 16.03 37.08 10.33 1336 0.1227 0.0442 0.0381 0.0120 0.0278 0.0077 
Malawi 106.38 4.47 4.48 106.38 4.06 4.18 189 0.5628 0.0236 0.0237 0.5628 0.0215 0.0221 
Malaysia 85.91 43.33 36.73 53.05 29.60 9.55 471 0.1824 0.0920 0.0780 0.1126 0.0629 0.0203 
Massachusetts 3860.40 1070.64 1065.22 1049.05 283.80 201.13 12 423 0.3107 0.0862 0.0857 0.0844 0.0228 0.0162 
Mauritania 33.54 14.33 13.99 74.44 13.76 12.55 347 0.0967 0.0413 0.0403 0.2145 0.0397 0.0362 
Mexico 25 788.29 3956.11 3272.22 25 788.21 2062.82 943.14 124 897 0.2065 0.0317 0.0262 0.2065 0.0165 0.0076 
Michigan 4601.48 1761.48 1755.66 841.48 854.56 72.86 13 018 0.3535 0.1353 0.1349 0.0646 0.0656 0.0056 
Minnesota 819.50 432.21 430.26 601.83 293.12 38.29 5463 0.1500 0.0791 0.0788 0.1102 0.0537 0.0070 
Montenegro 50.51 12.80 12.32 50.37 11.74 3.23 682 0.0741 0.0188 0.0181 0.0738 0.0172 0.0047 
Morocco 171.88 104.91 98.66 237.54 96.08 79.59 7388 0.0233 0.0142 0.0134 0.0322 0.0130 0.0108 
Namibia 13.35 3.39 3.21 12.67 3.15 2.63 205 0.0651 0.0166 0.0157 0.0618 0.0154 0.0128 
Nebraska 143.59 43.91 41.88 113.55 30.04 10.82 1651 0.0870 0.0266 0.0254 0.0688 0.0182 0.0066 
Nepal 101.66 47.69 29.68 51.49 41.18 13.53 2703 0.0376 0.0176 0.0110 0.0190 0.0152 0.0050 
New Hampshire 206.44 68.12 68.19 43.79 27.23 7.14 759 0.2720 0.0898 0.0898 0.0577 0.0359 0.0094 
New Jersey 9089.18 2277.78 2251.66 4895.44 644.88 82.54 19 184 0.4738 0.1187 0.1174 0.2552 0.0336 0.0043 
New Mexico 268.90 125.46 124.76 231.03 63.74 39.64 2477 0.1086 0.0507 0.0504 0.0933 0.0257 0.0160 
New York 22 331.89 4467.88 4474.14 19 330.62 1732.21 377.77 38 007 0.5876 0.1176 0.1177 0.5086 0.0456 0.0099 
Nigeria 302.88 78.11 77.03 309.09 54.57 56.67 1289 0.2350 0.0606 0.0598 0.2398 0.0423 0.0440 
North Macedonia 180.69 93.27 73.19 119.33 56.56 17.10 2503 0.0722 0.0373 0.0292 0.0477 0.0226 0.0068 
Norway 174.39 56.34 56.11 129.09 19.39 5.92 436 0.4000 0.1292 0.1287 0.2961 0.0445 0.0136 
Ohio 1714.77 823.38 811.88 687.98 613.82 49.01 13 237 0.1295 0.0622 0.0613 0.0520 0.0464 0.0037 
Oman 151.06 53.46 10.94 125.72 39.11 5.35 1499 0.1008 0.0357 0.0073 0.0839 0.0261 0.0036 
Panama 181.41 126.81 125.95 192.07 47.92 38.57 4022 0.0451 0.0315 0.0313 0.0478 0.0119 0.0096 
Pakistan 239.55 174.43 169.05 634.27 91.63 78.06 10 105 0.0237 0.0173 0.0167 0.0628 0.0091 0.0077 
Paraguay 188.68 25.58 25.30 80.61 34.63 14.61 2262 0.0834 0.0113 0.0112 0.0356 0.0153 0.0065 
Pennsylvania 2857.13 1461.34 1461.39 293.55 822.34 75.92 16 073 0.1778 0.0909 0.0909 0.0183 0.0512 0.0047 
Peru 7639.74 3173.60 3127.06 4171.68 2429.67 320.60 37 680 0.2028 0.0842 0.0830 0.1107 0.0645 0.0085 
Poland 499.77 605.27 380.11 502.35 466.93 76.52 28 556 0.0175 0.0212 0.0133 0.0176 0.0164 0.0027 
Portugal 397.55 352.23 315.79 381.36 235.03 30.32 6906 0.0576 0.0510 0.0457 0.0552 0.0340 0.0044 
Qatar 62.53 5.51 2.32 61.92 5.05 1.83 245 0.2552 0.0225 0.0095 0.2527 0.0206 0.0075 
Rhode Island 605.04 189.67 189.27 219.94 63.00 9.86 1911 0.3166 0.0992 0.0990 0.1151 0.0330 0.0052 
Romania 600.12 850.67 764.87 238.61 578.73 64.98 15 767 0.0381 0.0540 0.0485 0.0151 0.0367 0.0041 
Russia 2638.08 664.97 663.64 1030.70 286.55 65.03 57 019 0.0463 0.0117 0.0116 0.0181 0.0050 0.0011 
Senegal 53.34 7.19 7.17 56.75 8.07 6.30 410 0.1301 0.0175 0.0175 0.1384 0.0197 0.0154 
Somalia 80.48 8.73 8.62 80.48 5.60 5.50 130 0.6191 0.0671 0.0663 0.6191 0.0430 0.0423 
South Africa 498.81 775.22 231.02 359.86 427.41 136.62 28 921 0.0172 0.0268 0.0080 0.0124 0.0148 0.0047 
South Dakota 31.70 12.84 12.85 85.72 11.94 12.02 1488 0.0213 0.0086 0.0086 0.0576 0.0080 0.0081 
State of Palestine 83.06 17.32 16.79 18.49 9.47 4.07 1400 0.0593 0.0124 0.0120 0.0132 0.0068 0.0029 
Sudan 676.19 60.03 59.97 687.79 43.86 25.44 1468 0.4606 0.0409 0.0409 0.4685 0.0299 0.0173 
Suriname 24.03 1.82 1.78 23.50 1.81 1.74 122 0.1970 0.0149 0.0146 0.1926 0.0148 0.0143 
Switzerland 1008.01 548.10 469.10 270.56 277.62 89.35 7645 0.1319 0.0717 0.0614 0.0354 0.0363 0.0117 
Tennessee 265.40 98.78 61.07 306.75 80.18 59.32 6907 0.0384 0.0143 0.0088 0.0444 0.0116 0.0086 
Texas 1325.58 578.44 578.32 560.22 328.95 231.79 28 227 0.0470 0.0205 0.0205 0.0198 0.0117 0.0082 
Turkey 488.97 373.10 355.55 590.40 336.94 157.94 20 881 0.0234 0.0179 0.0170 0.0283 0.0161 0.0076 
United Arab Emirates 141.56 75.71 75.43 143.25 41.36 17.80 669 0.2116 0.1132 0.1128 0.2141 0.0618 0.0266 
Utah 122.11 61.54 61.33 62.25 23.63 13.52 1269 0.0962 0.0485 0.0483 0.0491 0.0186 0.0107 
Uzbekistan 58.92 12.51 12.19 49.11 9.00 7.23 614 0.0960 0.0204 0.0199 0.0800 0.0147 0.0118 
Virginia 1353.94 377.81 377.83 932.89 308.87 71.46 5032 0.2691 0.0751 0.0751 0.1854 0.0614 0.0142 
Wisconsin 235.32 268.78 253.41 105.98 225.11 49.77 4859 0.0484 0.0553 0.0522 0.0218 0.0463 0.0102 
Zambia 140.03 22.41 22.04 152.56 18.70 20.39 388 0.3609 0.0578 0.0568 0.3932 0.0482 0.0526 
Zimbabwe 70.91 5.51 5.45 69.50 5.26 5.01 363 0.1954 0.0152 0.0150 0.1915 0.0145 0.0138 
CountryEKIN (—)EDEL (—)EHER (—)EKIN2 (—)EDEL2 (—)EHER2 (—)Fmax (—)RKIN (—)RDEL (—)RHER (—)RKIN2 (—)RDEL2 (—)RHER2 (—)
Afghanistan 162.51 60.53 46.69 169.79 44.34 18.03 2201 0.0738 0.0275 0.0212 0.0771 0.0201 0.0082 
Albania 115.54 33.17 32.70 121.44 31.14 15.19 1181 0.0978 0.0281 0.0277 0.1028 0.0264 0.0129 
Andorra 50.79 24.33 23.75 15.95 18.78 1.53 84 0.6047 0.2896 0.2828 0.1899 0.2236 0.0182 
Arizona 2143.82 338.52 337.77 1555.91 165.95 117.13 8864 0.2419 0.0382 0.0381 0.1755 0.0187 0.0132 
Arkansas 62.08 44.64 35.50 120.54 31.64 31.76 3570 0.0174 0.0125 0.0099 0.0338 0.0089 0.0089 
Armenia 67.57 53.65 42.45 40.27 31.71 5.60 2823 0.0239 0.0190 0.0150 0.0143 0.0112 0.0020 
Australia 107.42 47.30 47.09 85.24 32.61 6.98 909 0.1182 0.0520 0.0518 0.0938 0.0359 0.0077 
Austria 253.65 227.92 162.17 160.33 163.49 54.53 6222 0.0408 0.0366 0.0261 0.0258 0.0263 0.0088 
Azerbaijan 95.70 31.56 21.25 44.02 26.26 6.74 2641 0.0362 0.0119 0.0080 0.0167 0.0099 0.0026 
Bahrain 26.04 3.94 3.88 25.04 3.75 3.64 352 0.0740 0.0112 0.0110 0.0711 0.0106 0.0103 
Bangladesh 600.08 77.08 64.60 610.18 62.76 60.00 7559 0.0794 0.0102 0.0085 0.0807 0.0083 0.0079 
Belarus 110.76 66.79 25.03 187.86 46.49 9.23 1424 0.0778 0.0469 0.0176 0.1319 0.0326 0.0065 
Belgium 7389.85 3676.68 3526.52 5976.70 2067.41 217.02 194 41 0.3801 0.1891 0.1814 0.3074 0.1063 0.0112 
Bolivia 1004.58 55.73 43.98 829.27 49.94 33.94 9149 0.1098 0.0061 0.0048 0.0906 0.0055 0.0037 
Bosnia and Herzegovina 51.74 45.88 45.34 92.16 39.29 25.15 4050 0.0128 0.0113 0.0112 0.0228 0.0097 0.0062 
Brazil 27 244.64 10 622.69 10 476.22 19 821.18 8218.72 7396.08 194 976 0.1397 0.0545 0.0537 0.1017 0.0422 0.0379 
Bulgaria 61.52 58.65 38.06 42.20 50.09 26.69 7576 0.0081 0.0077 0.0050 0.0056 0.0066 0.0035 
California 5265.39 1518.19 1512.63 2682.75 763.62 975.79 26 343 0.1999 0.0576 0.0574 0.1018 0.0290 0.0370 
Canada 3233.87 1807.43 1782.96 855.03 690.36 74.84 15 606 0.2072 0.1158 0.1142 0.0548 0.0442 0.0048 
Chile 1395.10 72.88 67.32 1789.07 67.90 60.62 16 608 0.0840 0.0044 0.0041 0.1077 0.0041 0.0037 
China 1263.72 251.68 135.14 1384.12 135.60 94.78 4634 0.2727 0.0543 0.0292 0.2987 0.0293 0.0205 
Colorado 1138.63 468.02 468.02 598.74 268.34 26.66 4814 0.2365 0.0972 0.0972 0.1244 0.0557 0.0055 
Connecticut 2921.95 836.92 826.60 1948.87 187.05 53.68 5995 0.4874 0.1396 0.1379 0.3251 0.0312 0.0090 
Costa Rica 73.87 14.87 11.86 78.17 14.85 11.00 2185 0.0338 0.0068 0.0054 0.0358 0.0068 0.0050 
Cote d'Ivoire 80.44 6.66 6.48 37.77 6.49 6.39 137 0.5872 0.0486 0.0473 0.2757 0.0474 0.0466 
Croatia 115.15 32.41 28.93 95.93 26.30 15.47 3920 0.0294 0.0083 0.0074 0.0245 0.0067 0.0039 
Czechia 330.37 125.18 108.53 331.05 110.06 79.22 11 813 0.0280 0.0106 0.0092 0.0280 0.0093 0.0067 
Delaware 403.38 109.12 107.75 169.97 78.81 10.14 926 0.4356 0.1178 0.1164 0.1836 0.0851 0.0109 
Dem. Rep. Congo 90.86 16.45 16.09 87.08 14.36 13.53 591 0.1537 0.0278 0.0272 0.1473 0.0243 0.0229 
Denmark 433.59 214.49 211.65 56.64 105.80 17.34 1298 0.3340 0.1652 0.1631 0.0436 0.0815 0.0134 
Dominican Republic 714.78 145.00 143.42 413.56 105.34 92.86 2414 0.2961 0.0601 0.0594 0.1713 0.0436 0.0385 
Ecuador 3118.53 1569.36 1556.14 3029.74 1490.89 119.80 14 034 0.2222 0.1118 0.1109 0.2159 0.1062 0.0085 
Egypt 584.30 131.77 61.63 844.86 80.44 56.64 7631 0.0766 0.0173 0.0081 0.1107 0.0105 0.0074 
El Salvador 102.44 10.88 10.09 158.50 17.42 9.94 1336 0.0767 0.0081 0.0076 0.1186 0.0130 0.0074 
Estonia 27.16 11.72 11.62 51.43 4.70 3.59 229 0.1186 0.0512 0.0508 0.2246 0.0205 0.0157 
Eswatini 17.93 2.74 2.70 20.11 3.24 2.21 205 0.0875 0.0134 0.0132 0.0981 0.0158 0.0108 
Finland 233.24 71.99 71.78 78.92 27.07 7.53 561 0.4158 0.1283 0.1280 0.1407 0.0483 0.0134 
France 21 315.15 14 766.54 14 682.89 17 462.61 9908.80 592.16 64 780 0.3290 0.2279 0.2267 0.2696 0.1530 0.0091 
Germany 4094.74 1408.29 1376.55 887.36 614.16 290.67 34 194 0.1198 0.0412 0.0403 0.0260 0.0180 0.0085 
Ghana 179.90 7.63 6.45 179.91 6.34 4.71 335 0.5370 0.0228 0.0193 0.5370 0.0189 0.0141 
Greece 150.02 70.25 69.75 145.40 60.06 52.77 4838 0.0310 0.0145 0.0144 0.0301 0.0124 0.0109 
Haiti 142.51 8.32 5.90 69.26 7.78 4.96 236 0.6038 0.0353 0.0250 0.2935 0.0330 0.0210 
Hawaii 54.24 7.31 5.68 50.14 6.91 2.71 288 0.1883 0.0254 0.0197 0.1741 0.0240 0.0094 
Hungary 393.18 273.25 204.67 213.06 184.77 47.99 9537 0.0412 0.0287 0.0215 0.0223 0.0194 0.0050 
Indonesia 843.63 996.76 988.69 748.48 679.91 636.37 22 138 0.0381 0.0450 0.0447 0.0338 0.0307 0.0287 
Ireland 1396.31 446.55 443.74 474.73 183.69 14.93 2237 0.6242 0.1996 0.1984 0.2122 0.0821 0.0067 
Israel 120.02 79.81 59.75 118.23 67.22 12.05 3325 0.0361 0.0240 0.0180 0.0356 0.0202 0.0036 
Italy 23 343.57 7298.56 7298.84 12 006.31 1816.28 233.24 74 159 0.3148 0.0984 0.0984 0.1619 0.0245 0.0031 
Kansas 445.87 64.06 59.58 431.26 49.18 39.42 2788 0.1599 0.0230 0.0214 0.1547 0.0176 0.0141 
Kazakhstan 109.36 64.25 58.44 132.27 64.99 49.20 2286 0.0478 0.0281 0.0256 0.0579 0.0284 0.0215 
Kenya 141.41 35.73 33.80 165.44 34.86 21.95 1670 0.0847 0.0214 0.0202 0.0991 0.0209 0.0131 
Kosovo 163.86 59.02 50.89 16.03 37.08 10.33 1336 0.1227 0.0442 0.0381 0.0120 0.0278 0.0077 
Malawi 106.38 4.47 4.48 106.38 4.06 4.18 189 0.5628 0.0236 0.0237 0.5628 0.0215 0.0221 
Malaysia 85.91 43.33 36.73 53.05 29.60 9.55 471 0.1824 0.0920 0.0780 0.1126 0.0629 0.0203 
Massachusetts 3860.40 1070.64 1065.22 1049.05 283.80 201.13 12 423 0.3107 0.0862 0.0857 0.0844 0.0228 0.0162 
Mauritania 33.54 14.33 13.99 74.44 13.76 12.55 347 0.0967 0.0413 0.0403 0.2145 0.0397 0.0362 
Mexico 25 788.29 3956.11 3272.22 25 788.21 2062.82 943.14 124 897 0.2065 0.0317 0.0262 0.2065 0.0165 0.0076 
Michigan 4601.48 1761.48 1755.66 841.48 854.56 72.86 13 018 0.3535 0.1353 0.1349 0.0646 0.0656 0.0056 
Minnesota 819.50 432.21 430.26 601.83 293.12 38.29 5463 0.1500 0.0791 0.0788 0.1102 0.0537 0.0070 
Montenegro 50.51 12.80 12.32 50.37 11.74 3.23 682 0.0741 0.0188 0.0181 0.0738 0.0172 0.0047 
Morocco 171.88 104.91 98.66 237.54 96.08 79.59 7388 0.0233 0.0142 0.0134 0.0322 0.0130 0.0108 
Namibia 13.35 3.39 3.21 12.67 3.15 2.63 205 0.0651 0.0166 0.0157 0.0618 0.0154 0.0128 
Nebraska 143.59 43.91 41.88 113.55 30.04 10.82 1651 0.0870 0.0266 0.0254 0.0688 0.0182 0.0066 
Nepal 101.66 47.69 29.68 51.49 41.18 13.53 2703 0.0376 0.0176 0.0110 0.0190 0.0152 0.0050 
New Hampshire 206.44 68.12 68.19 43.79 27.23 7.14 759 0.2720 0.0898 0.0898 0.0577 0.0359 0.0094 
New Jersey 9089.18 2277.78 2251.66 4895.44 644.88 82.54 19 184 0.4738 0.1187 0.1174 0.2552 0.0336 0.0043 
New Mexico 268.90 125.46 124.76 231.03 63.74 39.64 2477 0.1086 0.0507 0.0504 0.0933 0.0257 0.0160 
New York 22 331.89 4467.88 4474.14 19 330.62 1732.21 377.77 38 007 0.5876 0.1176 0.1177 0.5086 0.0456 0.0099 
Nigeria 302.88 78.11 77.03 309.09 54.57 56.67 1289 0.2350 0.0606 0.0598 0.2398 0.0423 0.0440 
North Macedonia 180.69 93.27 73.19 119.33 56.56 17.10 2503 0.0722 0.0373 0.0292 0.0477 0.0226 0.0068 
Norway 174.39 56.34 56.11 129.09 19.39 5.92 436 0.4000 0.1292 0.1287 0.2961 0.0445 0.0136 
Ohio 1714.77 823.38 811.88 687.98 613.82 49.01 13 237 0.1295 0.0622 0.0613 0.0520 0.0464 0.0037 
Oman 151.06 53.46 10.94 125.72 39.11 5.35 1499 0.1008 0.0357 0.0073 0.0839 0.0261 0.0036 
Panama 181.41 126.81 125.95 192.07 47.92 38.57 4022 0.0451 0.0315 0.0313 0.0478 0.0119 0.0096 
Pakistan 239.55 174.43 169.05 634.27 91.63 78.06 10 105 0.0237 0.0173 0.0167 0.0628 0.0091 0.0077 
Paraguay 188.68 25.58 25.30 80.61 34.63 14.61 2262 0.0834 0.0113 0.0112 0.0356 0.0153 0.0065 
Pennsylvania 2857.13 1461.34 1461.39 293.55 822.34 75.92 16 073 0.1778 0.0909 0.0909 0.0183 0.0512 0.0047 
Peru 7639.74 3173.60 3127.06 4171.68 2429.67 320.60 37 680 0.2028 0.0842 0.0830 0.1107 0.0645 0.0085 
Poland 499.77 605.27 380.11 502.35 466.93 76.52 28 556 0.0175 0.0212 0.0133 0.0176 0.0164 0.0027 
Portugal 397.55 352.23 315.79 381.36 235.03 30.32 6906 0.0576 0.0510 0.0457 0.0552 0.0340 0.0044 
Qatar 62.53 5.51 2.32 61.92 5.05 1.83 245 0.2552 0.0225 0.0095 0.2527 0.0206 0.0075 
Rhode Island 605.04 189.67 189.27 219.94 63.00 9.86 1911 0.3166 0.0992 0.0990 0.1151 0.0330 0.0052 
Romania 600.12 850.67 764.87 238.61 578.73 64.98 15 767 0.0381 0.0540 0.0485 0.0151 0.0367 0.0041 
Russia 2638.08 664.97 663.64 1030.70 286.55 65.03 57 019 0.0463 0.0117 0.0116 0.0181 0.0050 0.0011 
Senegal 53.34 7.19 7.17 56.75 8.07 6.30 410 0.1301 0.0175 0.0175 0.1384 0.0197 0.0154 
Somalia 80.48 8.73 8.62 80.48 5.60 5.50 130 0.6191 0.0671 0.0663 0.6191 0.0430 0.0423 
South Africa 498.81 775.22 231.02 359.86 427.41 136.62 28 921 0.0172 0.0268 0.0080 0.0124 0.0148 0.0047 
South Dakota 31.70 12.84 12.85 85.72 11.94 12.02 1488 0.0213 0.0086 0.0086 0.0576 0.0080 0.0081 
State of Palestine 83.06 17.32 16.79 18.49 9.47 4.07 1400 0.0593 0.0124 0.0120 0.0132 0.0068 0.0029 
Sudan 676.19 60.03 59.97 687.79 43.86 25.44 1468 0.4606 0.0409 0.0409 0.4685 0.0299 0.0173 
Suriname 24.03 1.82 1.78 23.50 1.81 1.74 122 0.1970 0.0149 0.0146 0.1926 0.0148 0.0143 
Switzerland 1008.01 548.10 469.10 270.56 277.62 89.35 7645 0.1319 0.0717 0.0614 0.0354 0.0363 0.0117 
Tennessee 265.40 98.78 61.07 306.75 80.18 59.32 6907 0.0384 0.0143 0.0088 0.0444 0.0116 0.0086 
Texas 1325.58 578.44 578.32 560.22 328.95 231.79 28 227 0.0470 0.0205 0.0205 0.0198 0.0117 0.0082 
Turkey 488.97 373.10 355.55 590.40 336.94 157.94 20 881 0.0234 0.0179 0.0170 0.0283 0.0161 0.0076 
United Arab Emirates 141.56 75.71 75.43 143.25 41.36 17.80 669 0.2116 0.1132 0.1128 0.2141 0.0618 0.0266 
Utah 122.11 61.54 61.33 62.25 23.63 13.52 1269 0.0962 0.0485 0.0483 0.0491 0.0186 0.0107 
Uzbekistan 58.92 12.51 12.19 49.11 9.00 7.23 614 0.0960 0.0204 0.0199 0.0800 0.0147 0.0118 
Virginia 1353.94 377.81 377.83 932.89 308.87 71.46 5032 0.2691 0.0751 0.0751 0.1854 0.0614 0.0142 
Wisconsin 235.32 268.78 253.41 105.98 225.11 49.77 4859 0.0484 0.0553 0.0522 0.0218 0.0463 0.0102 
Zambia 140.03 22.41 22.04 152.56 18.70 20.39 388 0.3609 0.0578 0.0568 0.3932 0.0482 0.0526 
Zimbabwe 70.91 5.51 5.45 69.50 5.26 5.01 363 0.1954 0.0152 0.0150 0.1915 0.0145 0.0138 
TABLE V.

Model-specific coefficients of determination according to Eq. (27).

CountryRKIN2 (—)RDEL2 (—)RHER2 (—)RKIN22 (—)RDEL22 (—)RHER22 (—)
Afghanistan 0.9191 0.9787 0.9846 0.9128 0.9909 0.9986 
Albania 0.7991 0.9678 0.9740 0.7839 0.9683 0.9956 
Andorra −5.6649 −3.1722 −2.9544 0.1408 −2.6712 0.9869 
Arizona 0.1019 0.9599 0.9634 0.5432 0.9921 0.9944 
Arkansas 0.9925 0.9969 0.9980 0.9734 0.9984 0.9983 
Armenia 0.9886 0.9925 0.9939 0.9946 0.9959 0.9999 
Australia 0.8701 0.9674 0.9676 0.8883 0.9778 0.9992 
Austria 0.9456 0.9617 0.9769 0.9513 0.9731 0.9955 
Azerbaijan 0.9570 0.9773 0.9975 0.9852 0.9863 0.9995 
Bahrain 0.9289 0.9985 0.9986 0.9410 0.9986 0.9986 
Bangladesh 0.9250 0.9980 0.9989 0.9250 0.9989 0.9989 
Belarus 0.8739 0.9630 0.9914 0.6525 0.9734 0.9991 
Belgium −1.6718 −0.9789 −0.7575 −0.7391 −0.4383 0.9953 
Bolivia 0.8863 0.9996 0.9997 0.9230 0.9997 0.9998 
Bosnia and Herzegovina 0.9965 0.9971 0.9974 0.9905 0.9977 0.9989 
Brazil 0.7781 0.9577 0.9588 0.8814 0.9732 0.9735 
Bulgaria 0.9980 0.9985 0.9990 0.9987 0.9987 0.9996 
California 0.3479 0.9044 0.9166 0.8342 0.9768 0.9612 
Canada 0.2828 0.6493 0.6208 0.9357 0.9210 0.9993 
Chile 0.9177 0.9996 0.9996 0.8626 0.9996 0.9996 
China −0.3581 0.8493 0.9668 −0.6412 0.9517 0.9657 
Colorado −0.3366 0.6672 0.6648 0.6290 0.8521 0.9983 
Connecticut −2.9238 0.3884 0.4484 −0.7358 0.9629 0.9979 
Costa Rica 0.9747 0.9991 0.9994 0.9729 0.9991 0.9995 
Cote d'Ivoire −2.6992 0.9642 0.9662 0.2541 0.9669 0.9674 
Croatia 0.9591 0.9979 0.9984 0.9690 0.9983 0.9990 
Czechia 0.9723 0.9977 0.9984 0.9729 0.9979 0.9984 
Delaware −1.9764 0.6974 0.7355 0.4522 0.7779 0.9967 
Dem. Rep. Congo 0.2871 0.9340 0.9356 0.3730 0.9491 0.9732 
Denmark −2.1232 −0.1720 −0.1375 0.9350 0.6172 0.9881 
Dominican Republic 0.1084 0.9551 0.9558 0.7308 0.9779 0.9795 
Ecuador 0.4651 0.8395 0.8415 0.4873 0.8453 0.9982 
Egypt 0.9283 0.9957 0.9990 0.8640 0.9982 0.9991 
El Salvador 0.9134 0.9989 0.9991 0.8093 0.9935 0.9991 
Estonia 0.5106 0.8217 0.8202 −0.7391 0.9699 0.9836 
Eswatini 0.7921 0.9936 0.9940 0.7537 0.9829 0.9964 
Finland −2.2283 0.3989 0.3913 0.5456 0.9031 0.9922 
France −1.1195 −0.2893 −0.2848 −0.4202 0.0877 0.9973 
Germany 0.5025 0.8912 0.8962 0.9674 0.9820 0.9948 
Ghana −1.9060 0.9940 0.9956 −1.9062 0.9949 0.9971 
Greece 0.9509 0.9935 0.9934 0.9514 0.9940 0.9923 
Haiti −2.3813 0.9843 0.9937 0.2589 0.9853 0.9952 
Hawaii 0.4136 0.9916 0.9952 0.4351 0.9930 0.9980 
Hungary 0.9613 0.9792 0.9864 0.9869 0.9857 0.9990 
Indonesia 0.9789 0.9659 0.9664 0.9838 0.9788 0.9841 
Ireland −4.2866 −0.1687 −0.1652 0.3819 0.5837 0.9988 
Israel 0.9775 0.9920 0.9944 0.9757 0.9931 0.9998 
Italy −1.2471 0.4979 0.4947 0.4150 0.9295 0.9996 
Kansas 0.0836 0.9853 0.9876 0.0063 0.9913 0.9935 
Kazakhstan 0.9718 0.9903 0.9901 0.9624 0.9895 0.9921 
Kenya 0.8904 0.9934 0.9941 0.8552 0.9933 0.9973 
Kosovo 0.7110 0.9022 0.9547 0.9977 0.9657 0.9988 
Malawi −1.6772 0.9934 0.9935 −1.6773 0.9934 0.9933 
Malaysia 0.3015 0.6813 0.8056 0.6804 0.7543 0.9814 
Massachusetts −0.4859 0.7348 0.7711 0.8776 0.9827 0.9937 
Mauritania 0.7131 0.9285 0.9324 −0.0961 0.9248 0.9359 
Mexico 0.4127 0.9831 0.9865 0.4127 0.9952 0.9989 
Michigan −1.8219 0.3489 0.3645 0.8966 0.8063 0.9987 
Minnesota 0.5305 0.8083 0.8118 0.7477 0.8916 0.9973 
Montenegro 0.8609 0.9924 0.9931 0.8634 0.9927 0.9993 
Morocco 0.9890 0.9965 0.9969 0.9797 0.9967 0.9981 
Namibia 0.9128 0.9936 0.9946 0.9227 0.9935 0.9965 
Nebraska 0.8277 0.9798 0.9836 0.8786 0.9872 0.9987 
Nepal 0.9735 0.9934 0.9972 0.9851 0.9942 0.9994 
New Hampshire −0.4577 0.7806 0.7778 0.9256 0.9321 0.9977 
New Jersey −2.2400 0.4818 0.6009 0.0776 0.9667 0.9995 
New Mexico 0.7457 0.9001 0.9103 0.8113 0.9808 0.9887 
New York −4.7500 0.4773 0.4794 −3.2930 0.9451 0.9941 
Nigeria 0.4475 0.9513 0.9525 0.4237 0.9718 0.9705 
North Macedonia 0.8884 0.9695 0.9765 0.9492 0.9817 0.9987 
Norway −2.3801 0.3435 0.3327 −0.9966 0.9259 0.9933 
Ohio 0.6335 0.8827 0.8913 0.9349 0.9226 0.9994 
Oman 0.8216 0.9713 0.9989 0.8830 0.9834 0.9998 
Panama 0.9598 0.9762 0.9757 0.9548 0.9976 0.9982 
Pakistan 0.9882 0.9949 0.9954 0.9474 0.9982 0.9985 
Paraguay 0.8434 0.9978 0.9979 0.9747 0.9902 0.9992 
Pennsylvania 0.2677 0.7266 0.7247 0.9908 0.8847 0.9990 
Peru 0.5930 0.9042 0.9066 0.8828 0.9250 0.9982 
Poland 0.9860 0.9909 0.9948 0.9856 0.9926 0.9998 
Portugal 0.8984 0.8922 0.9266 0.9110 0.9412 0.9994 
Qatar 0.2893 0.9919 0.9990 0.2935 0.9935 0.9992 
Rhode Island −0.9539 0.6499 0.6721 0.7188 0.9506 0.9991 
Romania 0.9737 0.9436 0.9499 0.9953 0.9540 0.9996 
Russia 0.9492 0.9961 0.9961 0.9883 0.9991 1.0000 
Senegal 0.7756 0.9941 0.9941 0.7506 0.9914 0.9948 
Somalia −4.6607 0.8979 0.9018 −4.6609 0.9407 0.9345 
South Africa 0.9950 0.9845 0.9981 0.9971 0.9958 0.9995 
South Dakota 0.9710 0.9976 0.9977 0.9166 0.9977 0.9981 
State of Palestine 0.8847 0.9929 0.9922 0.9926 0.9966 0.9996 
Sudan −1.6530 0.9511 0.9500 −1.7482 0.9797 0.9957 
Suriname 0.5469 0.9971 0.9972 0.5653 0.9971 0.9973 
Switzerland 0.5459 0.7286 0.8194 0.9647 0.8658 0.9921 
Tennessee 0.9514 0.9947 0.9980 0.9582 0.9964 0.9981 
Texas 0.9655 0.9886 0.9887 0.9945 0.9974 0.9983 
Turkey 0.9801 0.9904 0.9916 0.9599 0.9903 0.9979 
United Arab Emirates 0.3365 0.7903 0.7911 0.3179 0.9013 0.9729 
Utah 0.8245 0.9386 0.9377 0.9562 0.9839 0.9967 
Uzbekistan 0.8977 0.9941 0.9950 0.9302 0.9972 0.9984 
Virginia −0.1140 0.8548 0.8530 0.4800 0.9208 0.9948 
Wisconsin 0.9499 0.9326 0.9320 0.9864 0.9345 0.9972 
Zambia −0.3864 0.9566 0.9586 −0.6414 0.9654 0.9611 
Zimbabwe 0.3476 0.9956 0.9958 0.3820 0.9964 0.9965 
CountryRKIN2 (—)RDEL2 (—)RHER2 (—)RKIN22 (—)RDEL22 (—)RHER22 (—)
Afghanistan 0.9191 0.9787 0.9846 0.9128 0.9909 0.9986 
Albania 0.7991 0.9678 0.9740 0.7839 0.9683 0.9956 
Andorra −5.6649 −3.1722 −2.9544 0.1408 −2.6712 0.9869 
Arizona 0.1019 0.9599 0.9634 0.5432 0.9921 0.9944 
Arkansas 0.9925 0.9969 0.9980 0.9734 0.9984 0.9983 
Armenia 0.9886 0.9925 0.9939 0.9946 0.9959 0.9999 
Australia 0.8701 0.9674 0.9676 0.8883 0.9778 0.9992 
Austria 0.9456 0.9617 0.9769 0.9513 0.9731 0.9955 
Azerbaijan 0.9570 0.9773 0.9975 0.9852 0.9863 0.9995 
Bahrain 0.9289 0.9985 0.9986 0.9410 0.9986 0.9986 
Bangladesh 0.9250 0.9980 0.9989 0.9250 0.9989 0.9989 
Belarus 0.8739 0.9630 0.9914 0.6525 0.9734 0.9991 
Belgium −1.6718 −0.9789 −0.7575 −0.7391 −0.4383 0.9953 
Bolivia 0.8863 0.9996 0.9997 0.9230 0.9997 0.9998 
Bosnia and Herzegovina 0.9965 0.9971 0.9974 0.9905 0.9977 0.9989 
Brazil 0.7781 0.9577 0.9588 0.8814 0.9732 0.9735 
Bulgaria 0.9980 0.9985 0.9990 0.9987 0.9987 0.9996 
California 0.3479 0.9044 0.9166 0.8342 0.9768 0.9612 
Canada 0.2828 0.6493 0.6208 0.9357 0.9210 0.9993 
Chile 0.9177 0.9996 0.9996 0.8626 0.9996 0.9996 
China −0.3581 0.8493 0.9668 −0.6412 0.9517 0.9657 
Colorado −0.3366 0.6672 0.6648 0.6290 0.8521 0.9983 
Connecticut −2.9238 0.3884 0.4484 −0.7358 0.9629 0.9979 
Costa Rica 0.9747 0.9991 0.9994 0.9729 0.9991 0.9995 
Cote d'Ivoire −2.6992 0.9642 0.9662 0.2541 0.9669 0.9674 
Croatia 0.9591 0.9979 0.9984 0.9690 0.9983 0.9990 
Czechia 0.9723 0.9977 0.9984 0.9729 0.9979 0.9984 
Delaware −1.9764 0.6974 0.7355 0.4522 0.7779 0.9967 
Dem. Rep. Congo 0.2871 0.9340 0.9356 0.3730 0.9491 0.9732 
Denmark −2.1232 −0.1720 −0.1375 0.9350 0.6172 0.9881 
Dominican Republic 0.1084 0.9551 0.9558 0.7308 0.9779 0.9795 
Ecuador 0.4651 0.8395 0.8415 0.4873 0.8453 0.9982 
Egypt 0.9283 0.9957 0.9990 0.8640 0.9982 0.9991 
El Salvador 0.9134 0.9989 0.9991 0.8093 0.9935 0.9991 
Estonia 0.5106 0.8217 0.8202 −0.7391 0.9699 0.9836 
Eswatini 0.7921 0.9936 0.9940 0.7537 0.9829 0.9964 
Finland −2.2283 0.3989 0.3913 0.5456 0.9031 0.9922 
France −1.1195 −0.2893 −0.2848 −0.4202 0.0877 0.9973 
Germany 0.5025 0.8912 0.8962 0.9674 0.9820 0.9948 
Ghana −1.9060 0.9940 0.9956 −1.9062 0.9949 0.9971 
Greece 0.9509 0.9935 0.9934 0.9514 0.9940 0.9923 
Haiti −2.3813 0.9843 0.9937 0.2589 0.9853 0.9952 
Hawaii 0.4136 0.9916 0.9952 0.4351 0.9930 0.9980 
Hungary 0.9613 0.9792 0.9864 0.9869 0.9857 0.9990 
Indonesia 0.9789 0.9659 0.9664 0.9838 0.9788 0.9841 
Ireland −4.2866 −0.1687 −0.1652 0.3819 0.5837 0.9988 
Israel 0.9775 0.9920 0.9944 0.9757 0.9931 0.9998 
Italy −1.2471 0.4979 0.4947 0.4150 0.9295 0.9996 
Kansas 0.0836 0.9853 0.9876 0.0063 0.9913 0.9935 
Kazakhstan 0.9718 0.9903 0.9901 0.9624 0.9895 0.9921 
Kenya 0.8904 0.9934 0.9941 0.8552 0.9933 0.9973 
Kosovo 0.7110 0.9022 0.9547 0.9977 0.9657 0.9988 
Malawi −1.6772 0.9934 0.9935 −1.6773 0.9934 0.9933 
Malaysia 0.3015 0.6813 0.8056 0.6804 0.7543 0.9814 
Massachusetts −0.4859 0.7348 0.7711 0.8776 0.9827 0.9937 
Mauritania 0.7131 0.9285 0.9324 −0.0961 0.9248 0.9359 
Mexico 0.4127 0.9831 0.9865 0.4127 0.9952 0.9989 
Michigan −1.8219 0.3489 0.3645 0.8966 0.8063 0.9987 
Minnesota 0.5305 0.8083 0.8118 0.7477 0.8916 0.9973 
Montenegro 0.8609 0.9924 0.9931 0.8634 0.9927 0.9993 
Morocco 0.9890 0.9965 0.9969 0.9797 0.9967 0.9981 
Namibia 0.9128 0.9936 0.9946 0.9227 0.9935 0.9965 
Nebraska 0.8277 0.9798 0.9836 0.8786 0.9872 0.9987 
Nepal 0.9735 0.9934 0.9972 0.9851 0.9942 0.9994 
New Hampshire −0.4577 0.7806 0.7778 0.9256 0.9321 0.9977 
New Jersey −2.2400 0.4818 0.6009 0.0776 0.9667 0.9995 
New Mexico 0.7457 0.9001 0.9103 0.8113 0.9808 0.9887 
New York −4.7500 0.4773 0.4794 −3.2930 0.9451 0.9941 
Nigeria 0.4475 0.9513 0.9525 0.4237 0.9718 0.9705 
North Macedonia 0.8884 0.9695 0.9765 0.9492 0.9817 0.9987 
Norway −2.3801 0.3435 0.3327 −0.9966 0.9259 0.9933 
Ohio 0.6335 0.8827 0.8913 0.9349 0.9226 0.9994 
Oman 0.8216 0.9713 0.9989 0.8830 0.9834 0.9998 
Panama 0.9598 0.9762 0.9757 0.9548 0.9976 0.9982 
Pakistan 0.9882 0.9949 0.9954 0.9474 0.9982 0.9985 
Paraguay 0.8434 0.9978 0.9979 0.9747 0.9902 0.9992 
Pennsylvania 0.2677 0.7266 0.7247 0.9908 0.8847 0.9990 
Peru 0.5930 0.9042 0.9066 0.8828 0.9250 0.9982 
Poland 0.9860 0.9909 0.9948 0.9856 0.9926 0.9998 
Portugal 0.8984 0.8922 0.9266 0.9110 0.9412 0.9994 
Qatar 0.2893 0.9919 0.9990 0.2935 0.9935 0.9992 
Rhode Island −0.9539 0.6499 0.6721 0.7188 0.9506 0.9991 
Romania 0.9737 0.9436 0.9499 0.9953 0.9540 0.9996 
Russia 0.9492 0.9961 0.9961 0.9883 0.9991 1.0000 
Senegal 0.7756 0.9941 0.9941 0.7506 0.9914 0.9948 
Somalia −4.6607 0.8979 0.9018 −4.6609 0.9407 0.9345 
South Africa 0.9950 0.9845 0.9981 0.9971 0.9958 0.9995 
South Dakota 0.9710 0.9976 0.9977 0.9166 0.9977 0.9981 
State of Palestine 0.8847 0.9929 0.9922 0.9926 0.9966 0.9996 
Sudan −1.6530 0.9511 0.9500 −1.7482 0.9797 0.9957 
Suriname 0.5469 0.9971 0.9972 0.5653 0.9971 0.9973 
Switzerland 0.5459 0.7286 0.8194 0.9647 0.8658 0.9921 
Tennessee 0.9514 0.9947 0.9980 0.9582 0.9964 0.9981 
Texas 0.9655 0.9886 0.9887 0.9945 0.9974 0.9983 
Turkey 0.9801 0.9904 0.9916 0.9599 0.9903 0.9979 
United Arab Emirates 0.3365 0.7903 0.7911 0.3179 0.9013 0.9729 
Utah 0.8245 0.9386 0.9377 0.9562 0.9839 0.9967 
Uzbekistan 0.8977 0.9941 0.9950 0.9302 0.9972 0.9984 
Virginia −0.1140 0.8548 0.8530 0.4800 0.9208 0.9948 
Wisconsin 0.9499 0.9326 0.9320 0.9864 0.9345 0.9972 
Zambia −0.3864 0.9566 0.9586 −0.6414 0.9654 0.9611 
Zimbabwe 0.3476 0.9956 0.9958 0.3820 0.9964 0.9965 

The superiority of the hereditary model with respect to the delay model is less pronounced when determining optimized model parameters for the entirety of the investigated time period, with virtually no such superiority observed for the following countries, territories, or US states (see fourth column of Table II): Arizona, Australia, California, Colorado, Finland, Italy, Malawi, Michigan, Minnesota, New Hampshire, New York, Norway, Pennsylvania, Rhode Island, Russia, Senegal, Somalia, South Dakota, Sudan, Texas, United Arab Emirates, Utah, and Virginia. The latter observation is evident in Fig. 3(c) where the graph representing the delay model (applied to the state of New York) virtually falls together with the graph representing the hereditary model. However, these cases are, on average, characterized by normalized time-averaged errors, which are still as high as 7.13% for the delay model and 7.11% for the hereditary model; and these errors are significantly reduced through splitting the investigated pandemic time period into two, wave-specific, portions, namely, to 3.57% for the delay model, and further down to 1.31% for the hereditary model. This again underlines the superiority of the hereditary model, while the total-number-of-cases-dependent delay model exhibits unrealistic kinks at the wave transition point twtr (see, e.g., Figs. 2(e) and 3(e)). The hereditary model with two triples of optimized parameters is particularly suitable for countries or US states with two distinctively developed infection waves, irrespective of whether these waves are of very different size (as is the case with Austria, see Fig. 2) or of similar size (as is the case with New York, see Fig. 3). Resting on the increment of confirmed cases, it is also very stable with respect to data inaccuracies, as observed, for example, with the number of recoveries in the case of Norway, which was heavily corrected 86 days after the onset of COVID-19 infections, see the supplementary material. Under these conditions, the traditional kinetics model, building on the number of active infections (which depends on the recorded recoveries) cannot reach a Norway-specific mean relative model error below some 40%, see the ninth column of Table IV. By comparison, the wave-specific use of the hereditary model in the context of Norway yields a normalized time-averaged model error, which is as low as 1.4%, see the fourteenth column of Table IV.

We also take note of the cases where the recorded data virtually turn the general fatality function given by Eq. (5) into the Heaviside format, see Eq. (13), that is, when the fitting procedure yields very high values of sm, with m=HER,HER2. This is true for the following countries, territories, or US states (see Tables I and III): Andorra, Bahrain, Canada, China, Cote d'Ivoire, Delaware, Democratic Republic of Congo, Estonia, Eswatini, Finland, Greece, Hawaii, Kosovo, Namibia, New Hampshire, Paraguay, Rhode Island, Somalia, South Dakota, and Utah. Still, the differences between the corresponding errors of the hereditary model, see Eqs. (4) and (6)–(11), and errors of the delay model, see Eq. (14), remain, as a rule, significant (see Table II), as the incremental format of the hereditary model (4) differs fundamentally from the simplified total-number-based format of the delay model (14).

While the drawbacks of traditional compartmental SIR modeling have normally motivated “microscopic” agent-based approaches,10–12 the present contribution followed a different path: not abolishing, but modifying the governing equations for infected, recovered, or deceased persons in an epidemic population. This quest for “new” equations was met by resorting to a mathematical concept, which was very successful in the context of continuum mechanics: integrodifferential equations proposed by Boltzmann for the “deformation aftereffect” in 1874.63,64 This adds a new conceptual dimension to disease modeling, regarding the infection-to-death transition not so much as a statistical process, but as a “pseudo-mechanical phenomenon.” In this context, the “pseudo-mechanical system” (i.e., the population undergoing the pandemic) is subjected to virus loads developing over time (in terms of the new infections spreading within this population), with these loads leading, in a delayed and convolution-type fashion, to “mechanical strains” (in terms of virus-induced fatalities). In other words, the system partially complies, concedes, or gives up with respect to the virus load, and it is this type of concession or compliance, which drives the fatality trends—quantified here through a logistic function reflecting the corresponding “mechanical creep”; that is, the way the system reacts to the “mechanical” virus load imposed at some past time instant. This particular choice allows for very precisely capturing the fatality trends in 102 countries, territories, and US states, as underlined by coefficients of determination between model predictions and recorded fatalities ranging from 94% to 100%. To the best knowledge of the authors, this marks a new level of precision in the equation-based description of pandemic-related fatality trends.

Hence, our novel approach adds an original aspect to the few recent research endeavors which employ convolution integrals in the context of COVID-19 modeling, so as to relate, in quite different ways, the number of infected people over time to numbers or also to rates of newly infected people. In this context, three main paths were followed:

  • Convolution integrals were employed to spatial variables in the context of social heterogeneity.27 

  • The number of previously infected persons was linked to the evolution of the newly infected people, with the corresponding kernels being Gaussian or Weibull probability density functions associated with the incubation period, which varies from patient to patient.65 

  • The number of previously infected people was linked to the rate of newly confirmed cases, with the corresponding kernels being normal distributions, delta-functions, or Weibull distributions,66 and similar approaches have been applied to the number of susceptibles25,26 with calling the respective kernel the “evolution of infectiousness” and “infectivity probability,” respectively, and choosing a Gamma-distribution for the latter.

As a common characteristic, all these recent endeavors, which somehow mark the advent of hereditary epidemiology as well, rely upon a statistical interpretation of the kernels occurring in the integrodifferential equations, with classical statistical choices such as Gaussian, Weibull, Gamma, or delta functions. As stated above, we here took a distinctively different perspective, by adopting a mechanical rather than statistical motivation for proposing integrodifferential equations relating infections to fatality trends, and we supported the novel formulation by comparison with very many, world-widely recorded data.

In this context, it is appropriate to make a few comments on the nature of the experimental (or rather observational) data against which we have tested the three modeling strategies studied in this paper: the confirmed cases typically relate to persons who have either felt symptoms of COVID-19, or have undergone polymerase chain reaction-testing (commonly known as PCR-testing) or antigen-testing. Symptomatic persons are, as a rule, already infectious,67 and the share of asymptotically or presymptomically tested persons who are already infectious is typically very small.68 Hence, within the precision limits of mathematical models based on compartmentalization of the population, it is safe to say that the confirmed cases recorded in publicly available data bases (such as Worldometer58) refer to infectious, rather than solely exposed persons. Hence, the choice of a benchmark model in the SEIR format, with the letter E standing for the population of exposed persons (while the letters S, I, R stand for the populations of susceptible, infectious, and recovered persons), is rather not advisable. In the same context, the question may arise whether variable I, occurring, for example, in Eq. (15), should represent “infectious” rather than “infected” persons. However, before full recovery (and hence, before leaving the compartment represented by the letter I), those persons are not infectious any more.68 Thus, infected appears as an appropriate term for the compartment referred to by variable I.

It is also interesting to discuss different modifications, extensions, and further application ranges of our novel approach: currently, all model parameters are evaluated country-, US state-, or territory-specifically. This appears as a natural choice when considering that healthcare systems, the main drivers of fatality trends, are typically defined at this geographical level, and the wide agreement on this reasoning is reflected by the very existence of the comprehensive Worldometer database,58 the data fundament of the present study. Still, also data associated with larger geographical entities, such as continents,69,70 or smaller entities such as regions, provinces, countries, or districts,25,28,29 could, in principle, enter the optimization process of Sec. III of the present paper. Regional data may be particularly interesting for countries with diverse settlement structures such as Sweden or Switzerland. At the same time, the partitioning of the global population hit by the pandemic need not necessarily be driven by geographical categories alone, with three additional factors being of eminent interest in the context of COVID-19 fatality trends: age, immune level, and virus mutations.71–77 Accordingly, a geography-specific population may be further subdivided into different groups discriminated by age (e.g., resolved into decades), immune level (e.g., not-infected–not-vaccinated, partially vaccinated, fully vaccinated, recovered), or virus clade (e.g., wildtype, alpha variant, delta variant). Correspondingly optimized model parameters may provide valuable quantitative insight into the dynamics of mortality in these groups, and hence in the effectiveness of different strategies, such as reducing the contacts of the elderly, or vaccination. Still, we have to be aware that age-, vaccination-, and mutation-specific data are only sporadically available at the level of countries,78 so that—for the time being—the subdivision of all 102 population sets investigated here into several age and/or immune level groups is simply impossible.

It is advisable to link our novel modeling approach to the standardly used epidemiological terminology, so as to highlight similarities as well as differences of the epidemiological model expressed by Eq. (4) with respect to earlier investigations, such as the work of Kaniadakis et al.79 There, the logistic function was shown to very satisfactorily fit the overall evolution of the fatalities, referred to as F(t) in the present paper, and denoted as “cumulative distribution function” by Kaniadakis et al.79 We note that such an approach does not establish any link between infected and dead persons, as it was our goal when establishing Eq. (4). Still, the aforementioned good experience with the logistic function corroborated our choice of a logistic function for defining the fatality function JF occurring in Eq. (4). However, the conceptual idea of Eq. (4) goes beyond the particular choice of JF and is open toward any other mathematical formal appearing as appropriate for the representation of actual epidemic data. In this context, generalized logistic functions80 may play a beneficial role in potential future applications.

An obvious practical application of our model concerns the forecast of future fatalities based on currently available datasets. In a corresponding preliminary study, we have calibrated the three models (her, kin, del) for two infection waves, but considered the time domain to end two weeks earlier than for the numerical studies presented above, that is, on December 17, 2020. The parameters of this time domain were optimized again via the fmincon-function of Matlab, by minimizing the temporal average of the absolute errors between model-predicted fatality increments and recorded fatality increments, according to

Em=1Nii=1Ni|ΔFm(ti)ΔF(ti)|,
(29)

with m=HER2,DEL2,KIN2, and with ΔFm(ti) and ΔF(ti) standing for the model-predicted and recorded fatality increments, respectively. Thereafter, we have used the correspondingly optimized model parameters in order to forecast the fatalities for the time period from December 18 to 31, 2020. The quality of such forecasts was assessed by the coefficient of determination, R2, between the forecast fatalities, and those actually encountered (see Fig. 4). For both Austria and New York, our novel hereditary approach performed very well, as indicated by an R2>0.95, while the traditional kinetics model is associated with very low R2-values. The delay model provided a good forecast in the case of New York, but performed badly in the case of Austria. This again underlines the validity of our novel approach.

FIG. 4.

Fatalities forecasts for a time period of two weeks, starting on December 18, 2021, based on the hereditary model (FHER), the delay model (FDEL), and the kinetics model (FKIN); for (a) Austria (RKIN2=1.8986; RDEL2=0.1268; RHER2=0.9618), and (b) New York (RKIN2=0.0463; RDEL2=0.9941; RHER2=0.9710); the coefficients of determination relate to the recorded fatalities F.

FIG. 4.

Fatalities forecasts for a time period of two weeks, starting on December 18, 2021, based on the hereditary model (FHER), the delay model (FDEL), and the kinetics model (FKIN); for (a) Austria (RKIN2=1.8986; RDEL2=0.1268; RHER2=0.9618), and (b) New York (RKIN2=0.0463; RDEL2=0.9941; RHER2=0.9710); the coefficients of determination relate to the recorded fatalities F.

Close modal

Finally, we want to emphasize that the concept of hereditary epidemiology as proposed in this paper is applicable to many different phenomena taking place within the subpopulations already struck by an epidemic (in contrast to the transmission process between infected and susceptible individuals). In this sense, further application ranges for models similar to Eq. (4) may concern the transition from exposed to infectious persons, from infectious to hospitalized persons, or from hospitalized persons to patients requiring intensive care.

See the supplementary material for the raw data related to the each of the investigated countries, territories, and US states are provided. Furthermore, the optimized parameters obtained for each country are provided as well as graphical illustrations of the resulting fatality trends.

The authors gratefully acknowledge the financial support provided by the TU Wien Bibliothek through its Open Access Funding Programme.

All authors declare that they have no conflict of interest.

The data that support the findings of this study are available within the article and its supplementary material.

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Supplementary Material