Long Tails of Mean-Field COVID-19 Epidemic Curves: Implications of a Hidden Metapopulational Dynamics

Abstract

This chapter addresses the problem of a possible complex origin of the observable country-wide epidemic data detected for a number of epidemic curves of COVID-19. The key idea is the actual existence of decomposition into several approximate solutions within the mean-field compartmental approach of mathematical epidemiology and the direct summation of outbreaks localized within weakly connected socio-geographic areas. It is argued that the type of a multiscale distribution of such components can be estimated using the interrelations between the analytical invariant-based solution of the susceptible–infected–recovered/removed (SIR) model and the continuous wavelet transform based on the Derivative of Gaussian wavelet. The analysis is illustrated and verified by the comparison between the principal SIR-based mode decompositions and geographically spread outbreaks, which made their input in the full epidemic curve for countries, where COVID-19 dynamics demonstrated a long tail in the time course of registered cases.

Appendices

Appendix A: practical computation of the CWT with the Gaussian wavelet

The practical computing the continuous wavelet transform of the equispaced sample of the daily registered data on the number of infected persons can be carried out using the FFT-based algorithm based on the convolution theorem as a discretization of the explicit formula (34). For the practical implementation in this work, the program code was written in MATLAB language using its special possibility of the parallel element-by-element multiplication and functions’ computations discussed in details in [52]. The function, which calculates the CWT with the wavelet (24) reads as

Program Code

The processed data were taken from the aggregation and analytic system “Our World in Data” [48]; the program code used for data downloading is provided in https://github.com/postnicov/owid_loglet_interface!.

Appendix B: Simulation of Kendall’s wave

The data used to plot Fig. 5 were obtained by the numerical solutions of the system of partial differential equations (45)–(47) via the specialised software FlexPDE 5.0.22 (https://www.pdesolutions.com/) using the following program code (it is practically self-explanatory because of the closeness of the FlexPDE syntax to the common mathematical notation):

Program Code

The exported data were imported to MATLAB for plotting using the function fpreade from the special interface package [53].

Appendix C: global analytical properties of the SIR-X system

Recently, B.F. Maier and D. Brockmann within the context of COVID-19 modelling proposed [11] a modification of the classical SIR system, which is called SIR-X model. It answers two valuable challenges of the respective mathematical modelling: (i) in fact, nobody knows the real number of infected persons, the epidemic curves deals with the reported cases, which are only the part of the total infected population; (ii) the outbreak of COVID-19 in Mainland China demonstrated a subexponential (power-law in time) increase of confirmed cases during the initial stage of epidemics (instead of the conventional exponential time dependence, which follows from the SIR model).

The new variant of a compartment model introduced by Maier&Brockmann reads as

$$\begin{aligned} \frac{dS}{dt}= & {} -\alpha S I-\kappa _{0} S, \end{aligned}$$

(50)

$$\begin{aligned} \frac{dI}{dt}= & {} \alpha S I-\beta I-\kappa _{0} I-\kappa I \end{aligned}$$

(51)

$$\begin{aligned} \frac{dR}{dt}= & {} \beta I+\kappa _{0} S \end{aligned}$$

(52)

$$\begin{aligned} \frac{dX}{dt}= & {} \left( \kappa +\kappa _{0}\right) I, \end{aligned}$$

(53)

where $\alpha$ and $\beta$ are the conventional transmission and recovery rates, respectively, but $\kappa _0$ and $\kappa$ are the coefficients, which characterise the containment rate and removal of symptomatic infected persons. In addition, new variable X proportional to the empirically confirmed and reported cases, is introduced. The analysis carried out by the authors of this model is based on the linearisation under the condition of large $k_0$ and numerical fitting the obtained solution to the medical records. It demonstrates that the model (50)–(53) can catch the desired dynamical behaviour.

The key idea, which leads to revealing principal global features of the classic SIR system is reducing dimensionality of the system due to existence of algebraic invariants, which simplify the system considered in (S, I) phase plane in the same way as it is discussed above for the SIR model and its spatio-temporal extension. As it will be shown below, it is possible to follow this way for the Eqs. (50)–(53) due to the specific introduction of the containment and removal terms.

First of all, the considered system is supplied by its construction with the conservation law for the full population N:

$$\begin{aligned} S+I+R+X=N. \end{aligned}$$

(54)

Note that it is convenient to put $N=1$ as denoting the full accessible population; in this case, all variables will have the meaning of the respective relative part of the full population.

Further, the right-hand sides of the both Eqs. (50) and (51) has a common multiplier (S and I, respectively), which can be taken out, and both sides of the equations can be divided by these multipliers. Applying the standard formula of the derivative of a natural logarithm, Eqs.  (50) and (51) can be rewritten as

$$\begin{aligned} \frac{d\ln (S)}{dt}= & {} -\alpha \exp (\ln (I))-\kappa _{0}, \end{aligned}$$

(55)

$$\begin{aligned} \frac{d\ln (I)}{dt}= & {} \alpha \exp (\ln (S)) -(\beta +\kappa _{0} +\kappa ), \end{aligned}$$

(56)

where the remaining variables at right-hand sides are expressed as $I=\exp (\ln (I))$ and $S=\exp (\ln (S))$ to get the uniform variable alteration.

Considered together, Eqs.  (55) and (56) result in the equation

$$\begin{aligned} \frac{d\ln (S)}{d\ln (I)}= \frac{\alpha \exp (\ln (I))+\kappa _{0}}{(\beta +\kappa _{0} +\kappa )-\alpha \exp (\ln (S))}, \end{aligned}$$

(57)

which can be easily solved analytically by the separation of variables.

However, it is important to take into account the principal difference between the time course of S and I. While S is a monotonically decaying function that is obvious from Eq. (50) since its right-hand side is always negative, the variable I has one maximum corresponding to the peak of epidemics. This value $I_{max}$ is determined by the condition $dI/dt=0$ that gives the equality

$$\begin{aligned} S_m=\frac{(\beta +\kappa _{0} +\kappa )}{\alpha }\equiv \mathbb {R}^{-1}_{0,eff}, \end{aligned}$$

(58)

where $S=S_m$ when $I=I_{max}$, and $\mathbb {R}_{0,eff}$ is the effective basic reproduction number of the SIR-X model.

Thus, within the intervals $I\in [I_0,\,I_{max}]$, $S\in [S_0,\,S_m]$ both functions, I and S are monotonous and the integration from the initial to the current values gives the algebraic expression, which defines a unique curve on the (S, I) phase plane:

$$ (I-I_0)+\frac{\kappa _0}{\alpha }\ln \left( \frac{I}{I_0}\right) = S_m\ln \left( \frac{S}{S_0}\right) -(S-S_0). $$

It is convenient to rewrite this equality in the form

$$ -\frac{S_0}{S_m}\frac{S}{S_0}e^{-\frac{S_0}{S_m}\frac{S}{S_0}}= -\frac{S_0}{S_m}\exp \left[ \frac{1}{S_m}(I-I_0)+\frac{\kappa _0}{\alpha S_m}\ln \left( \frac{I}{I_0}\right) -\frac{S_0}{S_m} \right] , $$

which indicates that the number of susceptible persons on the time interval of growing I can be exactly analytically expressed via the Lambert W-function:

$$\begin{aligned} S=-S_m W\left( -\frac{S_0}{S_m}e^{-\frac{S_0}{S_m}}\times \left( \frac{I}{I_0}\right) ^{\frac{\kappa _0}{\alpha S_m}}e^{\frac{(I-I_0)}{S_m}}\right) . \end{aligned}$$

(59)

For $I=I_0$, the argument of the Lambert function in Eq. (59) is its reciprocal, $-S_m$ cancel each other and $S=S_0$. On the other hand, this form allows the exact determining the maximal number of infected persons since in this case the Lambert function should be equal to $-1=W(-e^{-1})$, i.e.

$$\begin{aligned} \frac{S_0}{S_m}e^{-\frac{S_0}{S_m}}\times \left( \frac{I_{max}}{I_0}\right) ^{\frac{\kappa _0}{\alpha S_m}}e^{\frac{(I_{max}-I_0)}{S_m}}=\frac{1}{e}. \end{aligned}$$

(60)

This algebraic equation again is exactly solvable:

$$\begin{aligned} I_{max}=\frac{\kappa _0}{\alpha } W\left( \frac{\alpha I_0}{\kappa _0}\left( \frac{S_0}{S_m}\right) ^{-\frac{\alpha S_m}{\kappa _0}} e^{-\frac{\alpha }{\kappa _0}(S_0+I_0-S_m)}\right) . \end{aligned}$$

(61)

Subsequently, both functions I(t) and S(t) are also monotonous when $I\in [I_{max},\,0]$, $S\in [S_m,\,S_{max}]$; the former decays to zero while the latter grows to the saturation. Integrating Eq. (57) within these intervals but starting now from $I=I_{max}$, $S=S_m$ and repeating similar procedures, it is possible to get the expression for the second stage of the outbreak again as the Lambert W-function:

$$\begin{aligned} S=-S_m W\left( -\frac{1}{e} \left( \frac{I}{I_{max}}\right) ^{\frac{\kappa _0}{\alpha S_m}}e^{\frac{(I-I_{max})}{S_m}}\right) . \end{aligned}$$

(62)

Therefore, substituting either (59) or (62) into Eq. (51), it is possible to obtain the unique solution for the number of infected persons as an implicit quadrature expression

$$\begin{aligned} t=\int \limits _0^I\frac{dI}{\alpha I\left( -S_m W\left( -\frac{S_0}{S_m}e^{-\frac{S_0}{S_m}}\times \left( \frac{I}{I_0}\right) ^{\frac{\kappa _0}{\alpha S_m}}e^{\frac{(I-I_0)}{S_m}}\right) -S_m \right) } \end{aligned}$$

(63)

for $t\in [0,\,t_m]$, where $t_m$ is the time moment corresponding to $I(t_m)=I_{max}$, and

$$\begin{aligned} t=t_m+\int \limits _{I_m}^I\frac{dI}{\alpha I\left( -S_m W\left( -\frac{1}{e} \left( \frac{I}{I_{max}}\right) ^{\frac{\kappa _0}{\alpha S_m}}e^{\frac{(I-I_{max})}{S_m}}\right) -S_m \right) } \end{aligned}$$

(64)

for $t\in [t_m,\,\infty ]$.

Respectively, having I(t) found by this way, the time evolution of the number of susceptible persons will be found via the algebraic equations (59) and (62), and the rest of variables—by the simple integration of the obtained two functions of time substituted into Eqs.  (52) and Eqs.  (53) with the lower bounds of integration equal to $R(0)=X(0)=0$ when starting from the very beginning of the outbreak.

Note that all singularities in (63) and (64) are weak and the integrals do not diverge. For $t\rightarrow 0$ and $t\rightarrow \infty$ the denominators $\propto I$, i.e. the integration gives logarithm leading to the standard exponential growth and decay of the number of infected in the vicinity of the initial and asymptotically large times. In the vicinity of the maximum point, using the known asymptotic expansion [54]

$$ W(\xi )+1\approx \sqrt{2(1+e\xi )}, $$

and the equality (60), the denominator tends to zero as the power-law function $\propto (1-I/I_{max})^{\kappa _0/(2\alpha S_m)}$, i.e. also sufficiently slowly to avoid a divergence of the respective integral.

Note also that the complete procedure can also be carried out using S instead of I due to the symmetry of Eq. (57) respectively to both variables; in this case, I(S) will be expressed via the Lambert W-function too. However, the form of solution given above is more convenient from the point of view that the main X(t) can be obtained by the direct integration of I(t) that makes possible to reveal the origin of the replacement of an exponential growth immanent for the classic SIR model by the power-law growth, for which the system (50)–(53) was invented in [11].

Now let us consider the growth’s properties at the first stage of an outbreak. First of all, it should be pointed out that all derivations and equations for I and S are valid for the standard SIR model too if simply put $\kappa _0=0$ that eliminates the second term in Eq. (50). The functional form of Eq. (51) will not be affected, the only difference is in the value of $S_m$. The solution in quadratures (63) takes the form

$$\begin{aligned} t=\int \limits _0^I\frac{dI}{\alpha I\left( -S_m W\left( -\frac{S_0}{S_m}e^{-\frac{S_0}{S_m}}\times e^{\frac{(I-I_0)}{S_m}}\right) -S_m \right) }. \end{aligned}$$

(65)

For small $(I-I_0)/S_m$, when the exponential function in the Lambert W function’s argument is close to unity, it naturally reduces to

$$ t=\int \limits _0^I\frac{dI}{ I\alpha \left( S_0-S_m \right) } $$

that gives the standard exponential growth of the epidemic beginning within the SIR model:

$$\begin{aligned} I(t)=I_0e^{\left[ \alpha \left( S_0-S_m \right) t\right] }. \end{aligned}$$

(66)

On the contrary, one can not neglect the factor $(I/I_0)^{\kappa _0/(\alpha S_m)}$ in Eq. (63) even for small $(I-I_0)/S_m$, which eliminates the exponential factor, since the growth of the former affects the integral directly, not as a small addition to unity. The respective influence far from the outbreak’s peak, when the absolute values of an argument of Lambert W function are sufficiently smaller than 1/e, can be estimated using Taylor’s expansion taking into account the equality

$$ W(z)=\frac{1}{z}\frac{W(z)}{1+W(z)} $$

as

$$ W\left( -\frac{S_0}{S_m}e^{-\frac{S_0}{S_m}}\times \left( \frac{I}{I_0}\right) ^{\frac{\kappa _0}{\alpha S_m}}\right) \approx -\frac{S_0}{S_m}+ \frac{\left( -\frac{S_0}{S_m}\right) }{1-\frac{S_0}{S_m}} \left[ \left( \frac{I}{I_0}\right) ^{\frac{\kappa _0}{\alpha S_m}}-1\right] . $$

Respectively, the integral (63) takes the form

$$ t=\int \limits _0^I\frac{dI}{I\alpha \left( \left[ S_0-S_m\right] - \frac{S_0}{\frac{S_0}{S_m}-1} \left[ \left( \frac{I}{I_0}\right) ^{\frac{\kappa _0}{\alpha S_m}}-1\right] \right) }, $$

which can be taken analytically and the resulting solution is

$$\begin{aligned} I(t)=\frac{I_0\left[ S_0-S_m+\frac{S_0}{\frac{S_0}{S_m}-1}\right] ^{\frac{\alpha S_m}{\kappa _0}}e^{\alpha \left[ S_0-S_m+\frac{S_0}{\frac{S_0}{S_m}-1}\right] t}}{\left[ S_0-S_m+\frac{S_0}{\frac{S_0}{S_m}-1}e^{\alpha \left[ S_0-S_m+\frac{S_0}{\frac{S_0}{S_m}-1}\right] \frac{\kappa _0}{\alpha S_m}t}\right] ^{\frac{\alpha S_m}{\kappa _0}}}. \end{aligned}$$

(67)

Thus, it is seen that the growth of infected persons under the same conditions, when the standard SIR model gives pure exponential growth, is slower. Except for the very first moment, when the time-dependent part of the denominator in Eq. (67) (that corresponds to the initial conditions in Eq. (51)) vanishes, the exponential function is divided by relatively fast-growing time-dependent denominator. Interestingly, the solution (67) has a form of the power-law function of the solution of the logistic equation, which slows down from the exponential to linear growth (the further tending to saturation will be out of the considered time interval corresponding to the active growth stage only, where the SIR model exhibit the exponential growth). The rate of this slowing depends directly on the value of $\kappa _0$, which is the main controlling factor. In the limit $\kappa _0\rightarrow 0$, Eq. (67) will be reduced to the pure exponential solution, Eq. (66) that can be easily demonstrated by the expansion of the exponential function in the denominator up to the linear function of time with the subsequent transition to the exponential function (due to the exponent $1/k_0$), which eliminates $\exp \left[ \alpha \frac{S_0}{\frac{S_0}{S_m}-1}\right] t$ from the nominator as well as the amplitude multiplier additional to $I_0$.

Now, the last step: calculating the dynamics of symptomatic, quarantined infected persons X(t) from Eqs.  (53) and (67). Fortunately, Eq. (67) can be integrated in an analytic form that results in the desired explicit formula

$$\begin{aligned} X(t)= & {} \frac{\left( \kappa +\kappa _{0}\right) I_0 \left[ S_0-S_m+\frac{S_0}{\frac{S_0}{S_m}-1}\right] ^{\frac{\alpha S_m}{\kappa _0}-1}}{\alpha \left[ S_0-S_m\right] } \left\{ e^{\alpha \left[ S_0-S_m+\frac{S_0}{\frac{S_0}{S_m}-1}\right] t}\right. \times \nonumber \\&\left[ S_0-S_m+\frac{S_0}{\frac{S_0}{S_m}-1}e^{\alpha \left[ S_0-S_m+\frac{S_0}{\frac{S_0}{S_m}-1}\right] \frac{\kappa _0}{\alpha S_m}t}\right] ^{1-{\frac{\alpha S_m}{\kappa _0}}}\times \\&\left. {}_2F_1\left( 1,1;1+{\frac{\alpha S_m}{\kappa _0}},-\frac{S_0}{S_m}e^{\alpha \left[ S_0-S_m+\frac{S_0}{\frac{S_0}{S_m}-1}\right] \frac{\kappa _0}{\alpha S_m}t} \right) -\right. \nonumber \\&\left. \left[ S_0-S_m+\frac{S_0}{\frac{S_0}{S_m}-1}\right] ^{1-{\frac{\alpha S_m}{\kappa _0}}} \right\} \nonumber \end{aligned}$$

(68)

Since three terms in the curly braces are combined multiplicatively, it is obvious that two of them, being time-dependent, heavily affect the dynamics and lead to its deviation from the exponential growth provided by the first multiplier. Note finally, that linearisation route described in the work [11] also showed the deviation of X(t) dynamics from the exponential growth expressed as emergence of an additional multiplicative time-dependent factor. But in the case of Eq. (68) such influence is derived within the non-linear route based on the exact (S, I) phase space representation.