Determination of a Key Pandemic Parameter of the SIR-Epidemic Model from Past Covid-19 Mutant Waves and Its Variation for the Validity of the Gaussian Evolution

1. Introduction

The susceptible-infected-recovered/removed (SIR) pandemics model developed originally by Kermack and McKendrick [1] and refined by Kendall [2] is the simplest, but still realistic compartment model where persons from a considered population are assigned to the three compartments S (susceptible), I (infectious) and R (recovered/removed). The infection ($a\left(t\right)$) and recovery ($\mu \left(t\right)$) rates then regulate the transition probability between the compartment fractions. Later refinements of the SIR-model such as the SEIR [3,4,5,6,7,8,9,10,11,12], SVEIR [13,14], SEIRD [15], SIRD [16,17,18], SIRS [19,20] and SIRV [21,22,23,24] have introduced additional compartments (for reviews see refs. [25,26,27,28,29,30,31]). The SIR-epidemic model provides a good explanation for the temporal evolution of Covid-19 waves from different mutants [32,33,34].

An important key parameter of the SIR pandemics model is the ratio $k\left(t\right)=\mu \left(t\right)/a\left(t\right)$ of the recovery to infection rate. Existing analytical solutions to the SIR equations in the literature have adopted originally stationary values of $\mu \left(t\right)={\mu }_0$ and $a\left(t\right)=a_0$ leading to the Kermack and McKendrick integral solution [1]. Recently [35,36] this integral solution has been generalized to arbitrary time-dependent infection rates $a\left(t\right)$ assuming a stationary value of the ratio $k\left(t\right)=k_0$ so that the recovery rate has the same time dependence as the infection rate. A further generalization to slowly time-dependent ratios $k\left(t\right)$ is also possible [37].

It is the purpose of the present manuscript to investigate for the first time a different approach. Instead of adopting different choices of the time dependence of the key parameter $k\left(t\right)$ and then solve the SIR equations as before we follow a different line of argument: in Section 2 we express the ratio $k\left(t\right)$ in terms of the observable rate of new infections $\dot{J}\left(t\right)$ and its corresponding cumulative fraction $J\left(t\right)$. By adopting the earlier considered Gaussian evolution function [38,39,40,41] for the rate of new infections $\dot{J}\left(t\right)$ we then calculate in Section 3 the corresponding time dependence of the ratio $k\left(t\right)$. If this required dependence of $k\left(t\right)$ agrees with the determination of the ratio $k\left(t\right)$ from monitored infection rates in different pandemic waves (Section 4) it would prove the validity of the Gauss model for the time evolution of pandemic waves.

2. SIR model

2.1. Starting equations

The original SIR-equations read

$$\frac{dS}{dt}=-a\left(t\right)SI,$$

(1a)

$$\frac{dI}{dt}=a\left(t\right)SI-\mu \left(t\right)I,$$

(1b)

$$\frac{dR}{dt}=\mu \left(t\right)I,$$

(1c)

obeying the sum constraint

$$S\left(t\right)+I\left(t\right)+R\left(t\right)=1$$

(2)

at all times $t\ge t_0$ after the start of the wave at time $t_0$ with the initial consitions [36]

$$I\left(t_0\right)=\eta ,S\left(t_0\right)=1-\eta ,R\left(t_0\right)=0.$$

(3)

where $\eta$ is positive and usually very small, $\eta \ll 1$. Very accurate analytical approximations [42] of the solutions of the SIR-equations (1)–(2) have been derived recently assuming a stationary value of the ratio $k\left(t\right)=a\left(t\right)/\mu \left(t\right)=k_0$. In Section 4 we also use the monitored data on the temporal evolution of pandemic waves in different countries to test the validity of the stationarity of $k\left(t\right)$.

2.2. Key parameter

In terms of the reduced time

$$\tau ={\int }_{t_0}^{t}d\xi a\left(\xi \right)$$

(4)

the SIR equations (1) read

$$\frac{dS}{d\tau }=-SI,$$

(5a)

$$\frac{dI}{d\tau }=SI-k\left(\tau \right)I,$$

(5b)

$$\frac{dR}{d\tau }=k\left(\tau \right)I,$$

(5c)

as with the time-dependent ratio

$$k\left(\tau \left(t\right)\right)=\frac{\mu \left(t\right)}{a\left(t\right)}.$$

(6)

Equation (5a) provides

$$I\left(\tau \right)=-\frac{dS\left(\tau \right)/d\tau }{S\left(\tau \right)}=-\frac{dlnS\left(\tau \right)}{d\tau }=\frac{d}{d\tau }ln{(1-J\left(\tau \right))}^{-1}=\frac{j\left(\tau \right)}{1-J\left(\tau \right)}$$

(7)

in terms of the rate of new infections $j\left(\tau \right)=SI=dJ\left(\tau \right)/d\tau$ and the cumulative number of new infections $J={\int }_0^{\tau }d\xi j\left(\xi \right)$. In deriving Equation (7) we have used that $S\left(\tau \right)=1-J\left(\tau \right)$. Likewise Equation (5b) yields

$$k\left(\tau \right)=S-\frac{dlnI}{d\tau }=1-J\left(\tau \right)-\frac{d}{d\tau }ln\left[\frac{d}{d\tau }ln{(1-J\left(\tau \right))}^{-1}\right],$$

(8)

where we inserted Equation (7).

As an aside we note that the same relation (8) results if we use Eqs. (5c)–(7) and the sum constraint (2), i.e.

$$k\left(\tau \right)=\frac{dR/d\tau }{I}=\frac{\frac{d}{d\tau }(1-S-I)}{I}=-\left(\frac{dlnI}{d\tau }+\frac{\frac{dS}{d\tau }}{I}\right)=-\frac{dlnI}{d\tau }+S,$$

(9)

where in the last step we inserted Equation (7).

In terms of the real time $\dot{J}\left(t\right)=a\left(t\right)j\left(\tau \right)$ and $J\left(t\right)=J\left(\tau \right)$ Equation (8) finally reads

$$k\left(t\right)=\frac{\mu \left(t\right)}{a\left(t\right)}=1-J\left(t\right)-\frac{1}{a\left(t\right)}\frac{d}{dt}ln\left[\frac{1}{a\left(t\right)}\frac{d}{dt}{(1-J\left(t\right))}^{-1}\right]$$

(10)

in terms of the monitored cumulative rate of new infections $J\left(t\right)$ and the time-dependent infection rate $a\left(t\right)$.

Multiplying Equation (10) with $a\left(t\right)$ then provides

$$\mu \left(t\right)=\left[1-J\left(t\right)\right]a\left(t\right)-\frac{d}{dt}ln\left[\frac{\dot{J}\left(t\right)}{a\left(t\right)(1-J(t\left)\right)}\right],$$

(11)

and for a stationary infection rate $a_0$, Eqs. (10)–(11) simplify to

$$\begin{array}{ccc}\hfill k\left(t\right)& =& \frac{\mu \left(t\right)}{a_0}=1-J\left(t\right)-\frac{1}{a_0}\frac{d}{dt}ln\left[\frac{\dot{J}\left(t\right)}{a_0(1-J\left(t\right))}\right]\hfill \\ & =& 1-J\left(t\right)-\frac{1}{a_0}\frac{d}{dt}ln\left[\frac{1}{a_0}\frac{d}{dt}ln{(1-J\left(t\right))}^{-1}\right]\hfill \\ & =& 1-J\left(t\right)-\frac{1}{a_0}\left[\frac{\ddot{J}\left(t\right)}{\dot{J}\left(t\right)}+\frac{\dot{J}\left(t\right)}{1-J\left(t\right)}\right]\hfill \end{array}$$

(12)

and

$$\mu \left(t\right)=[1-J\left(t\right)]a_0-\frac{d}{dt}ln\left[\frac{1}{a_0}\frac{d}{dt}ln{(1-J\left(t\right))}^{-1}\right],$$

(13)

respectively. In the case of stationary infection rate the entire real time dependence of the ratio $k\left(t\right)$ is attributed to a time-dependent recovery rate $\mu \left(t\right)$.

We emphasize that Eqs. (9)–(13) hold for all reduced ($\tau$) and real (t) times.

2.3. Limiting case $J\left(t\right)\ll 1$

In practically all Covid-19 mutant waves the final cumulative fraction of infected persons is much less than unity, i.e. $J_{\infty }\ll 1$. In the limit $J\left(t\right)\le J_{\infty }\ll 1$ we use the approximations

$$1-J\left(t\right)\simeq 1,ln{(1-J\left(t\right))}^{-1}\simeq J\left(t\right),$$

(14)

so that Eqs. (10)–(11) reduce to

$$\begin{array}{ccc}\hfill k\left(t\right)& \simeq & 1-J\left(t\right)-\frac{1}{a\left(t\right)}\frac{d}{dt}ln\frac{\dot{J}\left(t\right)}{a\left(t\right)},\hfill \\ \hfill \mu \left(t\right)& \simeq & [1-J\left(t\right)]a\left(t\right)-\frac{d}{dt}ln\frac{\dot{J}\left(t\right)}{a\left(t\right)},\hfill \end{array}$$

(15)

and in case of stationary infection rates

$$\begin{array}{ccc}\hfill k\left(t\right)& \simeq & 1-J\left(t\right)-\frac{1}{a_0}\frac{d}{dt}ln\frac{\dot{J}\left(t\right)}{a_0}=1-J\left(t\right)-\frac{\ddot{J}\left(t\right)}{a_0\dot{J}\left(t\right)}\hfill \\ & =& 1-J\left(t\right)-\frac{1}{a_0}\frac{dln\dot{J}\left(t\right)}{dt},\hfill \\ \hfill \mu \left(t\right)& \simeq & a_0-\frac{d}{dt}ln\dot{J}\left(t\right).\hfill \end{array}$$

(16)

The first Equation (16) indicates that a small value of the stationary infection rate $a_0$ provides a smaller value of the ratio $k\left(t\right)$ close to zero.

3. Condition for the validity of the Gaussian evolution

In this Section we adopt the Gaussian evolution for the rate of new infections [40,41]

$$\dot{J}\left(t\right)=\frac{J_{\infty }}{\sqrt{\pi }\Delta }exp\left[-\frac{{(t-t_{\max })}^2}{{\Delta }^2}\right].$$

(17)

The corresponding accumulative fraction of infections at any time then is given by

$$J\left(t\right)={\int }_{-\infty }^{t}d{t}^{\prime }{e}^{-\frac{{(t^{\prime }-t_{\max })}^2}{{\Delta }^2}}=\frac{J_{\infty }}{2}\left[1+\mathrm{erf}\left(\frac{t-t_{\max }}{\Delta }\right)\right]$$

(18)

in terms of the error function. $J_{\infty }$ denotes the final cumulative fraction of infections after infinite time $t\to \infty$.

The Gaussian evolution (17) implies

$$\ddot{J}\left(t\right)=-\frac{2(t-t_{\max })}{{\Delta }^2}\dot{J}\left(t\right).$$

(19)

In the general case of arbitrarily large $J_{\infty }$ the full Equation (12) yields for the Gauss evolution

$$\begin{array}{ccc}\hfill k\left(t\right)& =& 1-\frac{J_{\infty }}{2}\left[1+\mathrm{erf}\left(\frac{t-t_{\max }}{\Delta }\right)\right]+\frac{2(t-t_{\max })}{a_0{\Delta }^2}\hfill \\ & & -\frac{J_{\infty }{e}^{-\frac{{(t-t_{\max })}^2}{{\Delta }^2}}}{\sqrt{\pi }{a}_0\Delta \left\{1-\frac{J_{\infty }}{2}\left[1+\mathrm{erf}\left(\frac{t-t_{\max }}{\Delta }\right)\right]\right\}}.\hfill \end{array}$$

(20)

In the limit $J_{\infty }\ll 1$, as also indicated by the first Equation (16), the last term in Equation (20) is negligibly small providing in this limit

$$k\left(t\right)\simeq 1-\frac{J_{\infty }}{2}\left[1+\mathrm{erf}\left(\frac{t-t_{\max }}{\Delta }\right)\right]+\frac{2(t-t_{\max })}{a_0{\Delta }^2}.$$

(21)

The full ratio (20) and its approximation (21) are shown in Figure 1. One notices that the approximation (21) agrees very well with the exact time variation (20). Obviously Figure 1 indicates three different regimes in time for the Gauss ratio:

 (i)

At early times $t\ll t_{\max }$ the Gauss ratio increases linearly starting from ratio values less than unity.

 (ii)

At times near maximum, i.e. close to $t_{\max }$ near the maximum of $\dot{J}\left(t\right)$, the Gauss ratio exhibits a dip which is more pronounced for smaller values of $a_0$ which is also indicated by Equation (21) as the third linear term is inversely proportional to $a_0$.

(iii)

At late times beyond $t_{\max }$ the Gauss ratio resumes its linear increase with time.

We conclude that for the time dependency (20) of the ratio $k\left(t\right)$ the Gaussian evolution (17) is an exact analytical solution of the SIR equations. Likewise, if the monitored differential and cumulative infection rates $\dot{J}\left(t\right)$ and $J\left(t\right)$ inserted in Equation (12) are well approximated by the Gaussian ratios (20) and (21) in all three distinct time regimes we can justify the use the Gaussian evolution (17) as good approximations of the solutions of the SIR equations.

4. Determination of the ratio (12) from monitored infection rates of Covid-19 waves

In Figure 2 we show in the left panels the differential infection rate $\dot{J}\left(t\right)$ in the five countries Germany, Switzerland, The Netherlands, United States, and Sweden inferred from the reported death rates adopting a fatality rate of 0.005. In each country the raw data (in grey) have been smoothed (black curves) in order to infer the second derivative $\ddot{J}\left(t\right)$. More details on the considered Corona waves are given in the caption of Figure 2. The respective right panels show the derived ratio $k\left(t\right)$ calculated from Equation (16) for different values of the stationary infection rate $a_0$.

The monitored infection rates exhibit one clear maximum during the selected time spans capturing a single wave. The corresponding inferred ratios $k\left(t\right)$ in these countries remarkably show more or less similar behaviors. In all cases, except Sweden, one notices a linear increase at early times below the first maximum of $\dot{J}\left(t\right)$ before the ratios approach a nearly constant value close to unity at the time of the first maximum with small amplitude oscillations at later times. A resumed linear increase of the ratio at late times is not visible. Consequently, we may conclude that the Gaussian evolution (17) provides a good approximation of the solutions of the SIR equations during early and peak times. It however fails at late times beyond the first maximum of the considered past corona waves in these five countries. This good agreement of the Gauss modeling at times prior the maximum in the monitored differential infection rates justifies a posteriori the earlier approaches [38,39,40,41]. Moreover, the almost constant values of the ratio $k\left(t\right)$ at times after the maximum time indicates that the analytical SIR-solutions [35,36] based on a constant ratio are well applicable at these times.

Common to all five examples shown in Figure 1 is that only one clear maximum in the differential rate of new infections is visible. In Figure 3 we show the differential rate of new infections on a longer time scales in the United States. Here at least three maxima are exhibited so that the ratio $k\left(t\right)$ at late times deviates from its nearly constant value and decreases substantially with time. This clearly demonstrates that a decreasing ratio $k\left(t\right)$ to values considerably less than unity is responsible for leading to new maxima in the rate of new infections of the same mutant.

5. Summary and conclusions

The SIR-epidemic model provides a good explanation for the temporal evolution of Covid-19 waves from different mutants. An important key parameter of the SIR model is the ratio $k\left(t\right)=\mu \left(t\right)/a\left(t\right)$ of the recovery to infection rate. Here, apparently for the first time, monitored differential infection rates of past Corona waves have been used to infer a posteriori the real time variation of this key parameter. By attributing its time dependence formally to a time dependent recovery rate the temporal evolution of the ratio $k\left(t\right)$ is inferred for different values of the stationary infection rate $a_0$.

For past corona waves in five different countries it is found that the ratio $k\left(t\right)$ exhibits a linear increase at early times below the first maximum of the differential infection rate before the ratio approaches a nearly constant value close to unity at the time of the first maximum with small amplitude oscillations at later times.

The observed time dependencies at early times and at times near the first maximum agree favorably well with the behavior of the calculated ratio $k\left(t\right)$ for the Gaussian evolution for the rate of new infections, although the predicted linear increase of the Gauss ratio at late times is not observed.

Likewise, the near constancy of the ratio $k\left(t\right)$ at times after the maximum time indicates that earlier analytical solutions of the SIR equations for constant ratios are well justified.

On longer time intervals more than one maximum in the differential rate of new infections indicate the presence of several maxima which can be explained by a decrease of the ratio $k\left(t\right)$ at times after the first maximum to values considerably less than unity. This decrease then is responsible for causing new maxima in the rate of new infections of the same mutant.