Estimation of the Instantaneous Reproduction Number and Its Confidence Interval for Modeling the COVID-19 Pandemic

Abstract

In this paper, we derive an optimal model for calculating the instantaneous reproduction number, which is an important metric to help in controlling the evolution of epidemics. Our approach, within a frequentist framework, gave us the opportunity to calculate a more realistic confidence interval, a fundamental tool for a safe interpretation of the instantaneous reproduction number value, so that health and governmental people pay more attention to it. Our reasoning begins by decoupling the incidence data in mean and Gaussian noise by using practical series analysis techniques; then, we continue with a likely relationship between the present and past incidence data. Monte Carlo simulations and numerical integrations were conducted to complement the analytical proofs, and illustrations are provided for each stage of analysis to validate the analytical results. Finally, a real case study is discussed with the incidence data of the Republic of Panama regarding the COVID-19 pandemic. We have shown that, for the calculation of the confidence interval of the instantaneous reproduction number, it is essential to include all sources of variability, not only the Poissonian processes of the incidences. This proposal is delivered with analysis tools developed with Microsoft Excel.

1. Introduction

Throughout history, epidemics have been a threat to humanity due to their rapid ability to spread locally, regionally, and globally. Today, humanity is living one of these epidemics, COVID-19, which was classified as a pandemic by the World Health Organization on 11 March 2020, becoming one of the most devastating pandemic phenomena for human health in the last century. COVID-19 challenged health systems all over the world. To study the reproduction of this virus, health experts designed a series of strategies [1]. Consequently, recent investigations have been developed to model the COVID-19 pandemic [2,3,4].

The number of infections that occur in an epidemic is associated with the number of cases in the past [5]. This observation has led to mathematical models to explain the evolution of epidemics [6]. Usually, these models propose methods to calculate a frequently used epidemiological indicator, the instantaneous reproduction number $R_t$, which is the average number of secondary cases caused by an infected individual [1,7]. As this indicator comes from incidence (cases per day) data with a lot of variability, we should not take decisions based on the $R_t$ value alone, because it will be affected by this instability. The statisticians dealt with this problem by conceiving the confidence interval, which is a range of values where it is probable to find the true mean (the $R_t$ in our case). Therefore, to take an informed decision we should take care of both the $R_t$ (a sample mean) and its confidence interval. To date, the current methods to calculate the $R_t$ had consistent results between them [8]. Thus, our main concern is the confidence interval because, when it is calculated by current methods, there is an accepted practice of considering only the Poisson process of the incidences, discarding other kinds of variability (registry errors, delayed tasks, counting errors, etc.). Unfortunately, this practice produces too narrow confidence intervals for the $R_t$. A narrow confidence interval could be interpreted as positive at first glance; however, in this case, the people responsible for making the decisions are not going to have the complete information available. This is the problem that we are proposing to solve.

In summary, the main objective of this paper is to propose a new model to calculate the $R_t$ confidence interval, addressing not only the Poissonian random process but also considering all sources of variability. We provide a valid link to the repository (Available at https://github.com/publiodariocortes/seguiepi, accessed on 30 October 2021) that implements the estimation method described and allows reproduction of the results.

The paper is organized into the following sections. Section 2 comprises the theoretical foundations and the literature review. In Section 3, a model is proposed to simplify the interpretation of the series of incidence, decoupling it into a mean and a Gaussian component of pure noise. The contagion model is described, and a special noisy reproduction number is deduced, obtaining an explicit expression for the estimation of the instantaneous reproduction number and the confidence intervals. Section 4 provides graphical demonstrations that support the assumptions made throughout this paper. At the end of this section, we present a real case with incidence data of the COVID-19 pandemic of the Republic of Panama. Finally, in Section 5, we present our conclusions.

2. Literature Review

Although the modeling of epidemics had its beginning in 1760 with a study about the spread of smallpox [9], it was not until the beginning of the 20th century that modern modeling began with Kermack and MacKendrick [10], who formulated the first compartmentalized deterministic model (SIR). These models captured important features of disease transmission, but their discrete stage structuring was in contradiction with the complex medical realities of disease progression [11]. For this reason, as an alternative more adjusted to reality, different models have been proposed that recognize the importance of the time-since-infection, the so-called TSI models. Within this category of models, different methods have been devised to calculate the effective reproduction number. This is an indicator that has become very relevant to monitor the chain of infections during epidemics, which precisely uses the time-since-infection to calculate it. The distribution of this time is the base to calculate a series of $w_s$ factors, whose values are used to adjust the importance of past incidence to cause current incidence, considering the elapsed time from the instant of infection to the instant when the symptoms appear. According to Fraser [12], the effective reproduction number has two modalities. The first, called the case (or cohort) reproduction number, was proposed by Wallinga and Teunis [13]. In this modality, this indicator was calculated dividing the incidence that will occur in the future caused by infected individuals (incidence) at the present instant t, by the incidence at instant t. The second modality, called the instantaneous reproduction number, was proposed by Fraser himself. In this modality, this indicator was calculated dividing the incidence at time t by an estimate of the incidence before t (past), which could have contributed to the contagion of individuals counted as incidence at time t. In both modalities, it is necessary to relate the infected individuals with those who infected them. Since this relationship is very difficult to establish, it has been necessary to make estimates based on the TSI model.

Many have contributed to calculating the effective reproduction number in the modalities identified by Fraser [12], and we briefly review them here. First, as a departure study, we have the work of Fine [14]. He completed a detailed study of the time and place of the infection events, including the time elapsed between infection events of successive individuals, in a chain of transmission, the transmission interval. Wallinga and Teunis [13] developed a method to calculate the case reproduction number. Svensson [15] proposed a mathematical model that characterized the serial interval and generation times, clarifying the differences between these two concepts. The generation times were measured from the time individual A was infected, until the moment that an individual B was infected by individual A. On the other hand, the serial interval is measured from the time individual A presented symptoms, until the time that an individual B, infected by individual A, exhibited symptoms. Fraser [12] deduced a method to calculate the instantaneous reproduction number, mentioned above. Cori et al. [16] proposed another method to calculate the instantaneous reproduction number ($R_t$), by assuming that the distribution of infectivity (or serial interval) was independent of the calendar time [16], considering that the incidence in the observation time t follows a Poissonian distribution, with a mean given by $R_t{{\displaystyle \sum }}_{s=1}^t{I}_{t-s}{w}_s.$ With this, following a Bayesian analysis framework, the $R_t$ and the corresponding confidence interval were calculated. Thomson et al. [17] developed a proposal that moved away from the assumption of an invariant serial interval with the time, and from the assumption of local transmission of the infections, maintaining the assumption of the Poissonian distribution of the incidences. In this context, the Bayesian model has been adopted by several researchers [18,19].

Finally, below is shown a comparison of the investigations regarding epidemic models (

Table 1. Comparative table of investigations regarding to epidemics process models. The blank cells mean the criterion was not applied.

Year	Author	Subject	Deterministic (O)/Stochastic (X)	Stochastic Framework	Variability Considered	Method for Confidence Interval Calculation	Potential to Control the Sanitary and Governmental Processes of the Epidemics from the Perspective of Statistical Control Process
1927	Kermack & MacKendrick	SIR	O
2003	Fine	Transmission interval	X
2004	Wallinga & Teunis	Effective reproduction (case)	X	Bayesian	Poisson	Not declared	No
2007	Svensson	Generation time and serial interval	X
2007	Fraser	Instantaneous reproduction number and case (cohort) reproduction number	X	Frequentist	All	Experimental (Bootstrapping)	Could be difficult with bootstrapping techniques
2013	Cori et al.	Instantaneous reproduction number	X	Bayesian	Poisson	Analytically (Gamma distribution)	No
2019	Thompson et al.	Instantaneous reproduction number	X	Bayesian	Poisson	Bayesian method (Credible interval)	No
2020	Peterson & Adhikari	Time since infection models	O
2020	Zhao et al.	Serial interval	X
2021	Cortés-Carvajal, P.D.; Cubilla Montilla, M.; González-Cortés, D.R.	Instantaneous reproduction number	X	Frequentist	All	Analytically (Normal distribution, using the central limit theorem)	Yes

). It should be noted that only two investigations considered all variability (Fraser [12] and ours). The analytical method for calculating the confidence interval we are using includes all variability, a basic requirement for Statistical Process Control.

3. Materials and Methods

This section provides details about the data simulation, the model used to simplify the interpretation of the series of incidence, the contagion model, the estimation of the instantaneous reproduction number and the confidence intervals.

3.1. Data

As the $R_t$ calculation is based on the $I_t$ incidence serial data and considering we aspire to include all sources of variability, we must have a practical statistical model that describes $I_t$ incidence serial data. This model should facilitate the statistical calculations of statistical concepts such as means, standard deviations, distribution models, and confidence intervals. Taking care of our objective by observing the graphs of some pandemic data (Figure 1) it is common to find an aleatory pattern around the mean, with a bandwidth growing following the mean growing.

It is unrealistic to think that $I_t$ will have the same type of distribution in all observing instants. However, for simplicity, we need to select only one type, which should be something in the middle (symmetric and mesokurtic). The best candidate for this purpose is normal distribution, which has mathematical properties that simplify the analysis and the results interpretations. For example, the normal distribution is the core of the Central Limit Theorem. The normal distribution is useful modeling the distribution of large integer values, as it is the case of the incidences we can expect in an epidemic. Consequently, it should be tolerable that a few incidences of low integer values have a somewhat distorted statistical representation.

Therefore, we are proposing a model that assumes the existence of a “smooth” curve $I_t^{\mu }$ to which a random sequence of noise $e_t$ has been added. This noise will be a composition of two independent types of noise: counting error and Poissonian variation. Counting error is assumed to follow a normal distribution, and Poissonian variation is considered to be approximated with a normal distribution also. Therefore, the mix of these noises produce a single noise, $e_t$, which follows a unified normal distribution, $N\left(0,{\sigma }_t^2\right)$, at each measurement time t. Furthermore, to achieve a more realistic model, the noise, $e_t$, is assigned a standard deviation, ${\sigma }_t$, that varies according to a function $f\left\{I_t^{\mu }\right\}$, which is all time positive and increasing with $I_t^{\mu }$.

$$I_t=I_t^{\mu }+e_t\begin{array}{cc},& e_t\end{array}\sim N\left(0,{\sigma }_t^2\right),{\sigma }_t=f\left\{I_t^{\mu }\right\}$$

(1)

To simplify the presentation and demonstration of the theory that we will be developing in this paper, we will use simulated data constructed as follows:

$$I_t^{\mu }=120e^{{\left(\frac{t-20}{16}\right)}^2}+480e^{{\left(\frac{t-60}{20}\right)}^2}+246e^{{\left(\frac{t-80}{10}\right)}^2}+120e^{{\left(\frac{t-100}{12}\right)}^2},\begin{array}{c}10<t<120\end{array}$$

(2)

The variability of the noise, $e_t$, is arbitrarily modeled:

$${\sigma }_t=f\left\{I_t^{\mu }\right\}=\sqrt{10I_t^{\mu }}\Rightarrow e_t\sim N\left(0,10I_t^{\mu }\right)$$

(3)

With the previous definitions, it is possible to construct a simulated time series. The simulation strategy will allow us to determine the effectiveness of the applied analysis in extracting the statistical parameters of a time series, which hypothetically behaves like the proposed model (1).

The values of $I_t^{\mu }$ can be easily generated in a spreadsheet. The values of $e_t$, which arises from a random process, require the application of a simulation process, which can also be implemented in a spreadsheet (Figure 2):

In this way, it is possible to construct an infinity of graphs of $I_t$. Figure 3 shows some examples of them:

Appendix A develops the theory we propose for the estimation of the model parameters (1) from real incidence data series. The estimation of the population mean, $I_t^{\mu }$, will be $I{m}_t$, and the estimation of the standard deviation, ${\sigma }_t$, will be ${\widehat{\sigma }}_t$. Hence, from now on we will assume that $I{m}_t$ and ${\widehat{\sigma }}_t$ are known values for all incidences.

3.2. Bayesian Framework

Today the most popular method to calculate the instantaneous reproduction number $R_t$ is based on Cori et al. [16], which assumes that $I_t$ follows a Poissonian distribution, with a mean $I{m}_t$ given by:

$$I{m}_t=R_t{\displaystyle \sum }_{s=1}^t{w}_s{I}_{t-s}$$

(4)

The mean depends on the known values of incidences $I_{t-s}$, on the infectivity profiles given by the factors $w_s$ calculated based on the serial interval distribution (explained below), and on an $R_t$ value to be discovered. Using a Bayesian framework, Cori et al. [16] deduced a method to calculate $R_t$, which follows a Gamma distribution. Now we propose a new method following an alternative approach within the Frequentist framework, without the restriction of a Poissonian distribution for the $I_t$ values.

3.3. Serial Interval

As we will be using the serial interval in our proposal, we will briefly review this concept. The $w_s$ values based on the serial interval are obtained in epidemiological studies for the current epidemic and are independent of the calendar time [12]. The values of $w_s$ are usually modeled with a distribution chosen by the analyst based on the most accepted information. Typically, it is modeled with Gamma (Figure 4), Lognormal, or Weibull distributions [20], facilitating its parametric generation by defining the parameters of the mean ${\mu }_s$ and the standard deviation ${\sigma }_s$.

The mean ${\mu }_s$ can be interpreted as the average time elapsed measured from the time a symptomatic individual A infects a susceptible individual B, until the time that individual B manifests symptoms. Therefore, ${\sigma }_s$ is the standard deviation of this time.

In the analysis we will be developing, it may be known that a symptomatic individual A infected a susceptible individual B, but usually it will not be known exactly when individual B became infected after individual A became symptomatic. Thus, to simplify this lack of information, we will assume that individual B was infected just after individual A became symptomatic.

Taking care of the above assumption, the $w_s$ factors are calculated based on the distribution of the serial interval $f\left(s\right)$. In this example, the $w_s$ values are calculated by the rectangular rule of integration ($f\left(s\right)\times b$). The main parameter values (k and λ) of the distribution can be defined by calculus based on a provided mean (${\mu }_s$) and a standard deviation (${\sigma }_s$), which should be obtained by experimentation.

The $w_s$ factors have a special meaning regarding one susceptible individual infected in s = 0. An infected individual in s = 0 will manifest symptoms on any instant after s = 0, and the probability that the symptoms appear in a specific instant s = e in the future is $w_e$. In Figure 4, for example, we can see that the probability that an infected individual manifests symptoms in s = 10, is $w_{10}$ = 0.08. Now, suppose that at s = 0 we have 100 susceptible individuals just infected, then the expected number of new individuals (infected at s = 0) with symptoms at s = 10 will be $0.08\times 100=8$.

Not all the susceptible individuals will be infected at s = 0. The actual number of susceptible individuals infected at s = 0 will depend on the number of infectious individuals at s = 0, the number of susceptible individuals just before s = 0, and many other factors (population size, health conditions, environment, use of face masks, social distancing, population immunity, and so forth).

3.4. The Contagion Model

Let us consider a likely relation based on the fact that contagion in an epidemic comes from an infected individual:

$$I_t={\displaystyle \sum }_{s=1}^t{r}_{t,s}{I}_{t-s}$$

(5)

The incidence $I_t$ of an instant t results from the individual contributions $\left(r_{t,s}{I}_{t-s}\right)$ to the infections of all the $I_{t-s}$ incidences of the past, where the factors $r_{t,s}$ are the specific rates of these contributions.

Equation (5) is too simple, we present it here just as didactical tool to emphasize the exclusive relationship of $I_t$ (present) with the past values of $I_{t-s}$ $\left(t-s=1,2,\dots ,t-1\right)$, but in fact, the situation is more complex. So, to have a more realistic analysis, we will redefine the $r_{t,s}$ factors as follow:

$$r_{t,s}=R_{t,s}^{*}{w}_s$$

(6)

In this way, we still have a factor that contains “the specific rate” of the contribution of all the $I_{t-s}$ values to the value of $I_t$, but now we have split $r_{t,s}$ into two factors, $R_{t,s}^{*}$ and $w_s$, to which we will be assigning important meanings.

The $w_s$ factors are known from the serial interval, representing the effects that have occurred since the elapsed time from the infection in the instant (t − s) to the instant t of symptoms manifestation, considering we are constructing a TSI model. It can be said, these factors represent the natural law that governs the relationship between the infectious agent and its host. Thus, by definition, these factors must be statistically constants (Figure 4) and independent of t.

The $R_{t,s}^{*}$ mathematical interpretation will be deduced below. In the meantime, assume it is a convenient factor necessary to make (5) valid.

Replacing $r_{t,s}$ from (6) in (5):

$$I_t{\displaystyle \sum }_{s=1}^t{R}_{t,s}^{*}{w}_s{I}_{t-s}={\displaystyle \sum }_{s=1}^t{w}_s\left(R_{t,s}^{*}{I}_{t-s}\right)$$

(7)

We can still say the relation presented in (5) is accomplished, but now, with the specific meaning of $w_s$, it is possible to deepen the mathematical interpretation of (7). Considering the example presented at the end of Section 3.3 and Equation (7), the product $\left(R_{t,s}^{*}{I}_{t-s}\right)$ could be interpreted as the total number of susceptible individuals just infected in the instant $\left(t-s\right)$. Therefore, the product $w_s\left(R_{t,s}^{*}{I}_{t-s}\right)$ is the number of individuals infected in $\left(t-s\right)$ who will become symptomatic in t. The sum of all $w_s\left(R_{t,s}^{*}{I}_{t-s}\right)$ products from s = 1 to s = t will result precisely in the incidence $I_t$ (all the symptomatic individuals counted in t).

The above is consistent with the example of Section 3.3; our model still works, but we do not have a specific mathematical interpretation of $R_{t,s}^{*}$ yet. Now, we take one step forward rewriting (7) in two possible forms:

(a)

Keeping $R_{t,s}^{*}$ with the original dependency on t and s:

$$I_t={\displaystyle \sum }_{s=1}^t{R}_{t,s}^{*}\left(w_s{I}_{t-s}\right)$$

(8)

(b)

Declaring $R_{t,s}^{*}$ as independent of s, and keeping the dependency on t:

$$I_t=R_t^{*}{\displaystyle \sum }_{s=1}^t\left(w_s{I}_{t-s}\right)$$

(9)

Both Equations (8) and (9) are valid and maintain the relation proposed in (5). There is no mathematical reason to prefer one or the other. Both Equations (8) and (9) let us know that not all the incidence $I_{t-s}$ will have an effective responsibility in the resulting incidence $I_t$, because of the elapsed time effect. This condition is observed in the product $\left(w_s{I}_{t-s}\right)$. Thus, we are free to select one of the Equations (8) or (9) by appealing any other reason instead of a mathematical one. Therefore, we will think of practical and useful reasons that help us to control the evolution of the epidemics. Equation (8) has one $R_{t,s}^{*}$ per combination of t and s indexes. Equation (8) is useless because there are infinite $R_{t,s}^{*}$ factors for each t that can be solutions for Equation (8). So, we will discard Equation (8) as impractical. In Equation (9), in each instant t we can find a specific $R_t^{*}$. Solving from Equation (9):

$$R_t^{*}=\frac{I_t}{{{\displaystyle \sum }}_{s=1}^t{w}_s{I}_{t-s}}$$

(10)

There is only one solution for $R_t^{*}$, which has an expression (10) that provides an easy and practical interpretation of $R_t^{*}$. As the summation ${{\displaystyle \sum }}_{s=1}^t{w}_s{I}_{t-s}$ in (9) can be interpreted as the summation of the effective incidences $I_{t-s}$ ($s=1,2,\dots ,t$), now the practical (and mathematical) interpretation for $R_t^{*}$ is:

$R_t^{*}$ is the ratio of $I_t$ with respect to the effective number of infectious individuals in the past. In simple words, it means $R_t^{*}$ is the number of secondary cases caused by an infected individual. This is a very important concept. As the summation ${{\displaystyle \sum }}_{s=1}^t{w}_s{I}_{t-s}$ in (9) is something we cannot handle, because all the incidences $I_{t-s}$ are in the past, and the $w_s$ factors represent a natural law, we should act on the $R_t^{*}$ factor, which represents some elements that can be controlled by us (for example, population education, sanitary and governmental process, quarantines, and so forth), even if there are some other elements that cannot be handled directly (such as weather conditions). Thus, through our actions, we should try to reduce $R_t^{*}$ as much as possible; to reduce the number of incidences, $R_t^{*}$ must be less than 1.

3.5. Frequentist Framework

Since the denominator ${{\displaystyle \sum }}_{s=1}^t{w}_s{I}_{t-s}$ (10) behaves as a weighted average (low variability), the variability of $R_t^{*}$ is mainly caused by the variability of $I_t$. Therefore, as $R_t^{*}$ is a kind of reproduction number, we will name it the noisy reproduction number (just a name to distinguish it from other reproduction numbers). To stabilize $R_t^{*}$ for useful purposes, we propose to obtain the mean by the standard Frequentist method:

$$E\left[R_t^{*}\right]={{\displaystyle \int }}_{-\infty }^{\infty }{R}_t^{*}f\left(R_t^{*}\right)d{R}_t^{*}$$

(11)

where $f\left(R_t^{*}\right)$ is the probability density function of $R_t^{*}$. The deduction of the $f\left(R_t^{*}\right)$ equation is a little complex and extensive, so it is explained in Appendix B. Next, assuming we know $f\left(R_t^{*}\right)$, it is possible to solve Equation (11) numerically, but it is not possible to obtain an explicit expression for $R_t^{*}$ mean. Therefore, we will try using expected values operators and Taylor approximations. Replacing $I_t$ and $I_{t-s}$ with their model components (1) in (10):

$$R_t^{*}=\frac{I_t^{\mu }+e_t}{{{\displaystyle \sum }}_{s=1}^t{w}_s\left(I_{t-s}^{\mu }+e_{t-s}\right)}=\frac{I_t^{\mu }+e_t}{{{\displaystyle \sum }}_{s=1}^t{w}_s{I}_{t-s}^{\mu }+{{\displaystyle \sum }}_{s=1}^t{w}_s{e}_{t-s}}$$

(12)

Defining auxiliary variables:

$$c_t={\displaystyle \sum }_{s=1}^t{w}_s{I}_{t-s}^{\mu }, {\epsilon }_t={\displaystyle \sum }_{s=1}^t{w}_s{e}_{t-s}, g_t\left({\epsilon }_t\right)=\frac{1}{c_t+{\epsilon }_t}$$

(13)

To convert $g_t\left({\epsilon }_t\right)$ in a more manageable form, we will replace it with its Taylor approximation, if ${\epsilon }_t$ has small values around zero (which is true in general):

$$g_t\left({\epsilon }_t\right)=\frac{1}{c_t+{\epsilon }_t}\approx \frac{1}{c_t}-\left(\frac{1}{c_t^2}\right){\epsilon }_t$$

(14)

Applying (14) in (12):

$$R_t^{*}\approx \left(I_t^{\mu }+e_t\right)\left[\frac{1}{c_t}-\left(\frac{1}{c_t^2}\right){\epsilon }_t\right]=\frac{I_t^{\mu }}{c_t}-\left(\frac{I_t^{\mu }}{c_t^2}\right){\epsilon }_t+\frac{e_t}{c_t}-\left(\frac{e_t}{c_t^2}\right){\epsilon }_t=\frac{I_t^{\mu }}{c_t}+\frac{e_t}{c_t}-\left(\frac{I_t^{\mu }}{c_t^2}\right){\epsilon }_t-\left(\frac{1}{c_t^2}\right)e_t{\epsilon }_t$$

(15)

Applying expected value operators:

$$E\left[R_t^{*}\right]\approx \frac{I_t^{\mu }}{c_t}+\frac{E\left[e_t\right]}{c_t}-\left(\frac{I_t^{\mu }}{c_t^2}\right)E\left[{\epsilon }_t\right]-\left(\frac{1}{c_t^2}\right)E\left[e_t{\epsilon }_t\right]$$

(16)

In (13) it is implicit that ${\epsilon }_t$ and $e_t$ are independent normal distributed variables, because $e_t$ is not included in the weighted average of errors; therefore, there is no correlation between them, and consequently $E\left[e_t{\epsilon }_t\right]=E\left[e_t\right]E\left[{\epsilon }_t\right]$. Moreover, it is noted that $E\left[e_t\right]=0$, and $E\left[{\epsilon }_t\right]=0$. Thus:

$$E\left[R_t^{*}\right]\approx \frac{I_t^{\mu }}{c_t}+\frac{E\left[e_t\right]}{c_t}-\left(\frac{I_t^{\mu }}{c_t^2}\right)E\left[{\epsilon }_t\right]-\left(\frac{1}{c_t^2}\right)E\left[e_t\right]E\left[{\epsilon }_t\right]=\frac{I_t^{\mu }}{c_t}$$

(17)

Replacing $c_t$ from (13) in (17):

$$E\left[R_t^{*}\right]=R_t^{\mu }\approx \frac{I_t^{\mu }}{{{\displaystyle \sum }}_{s=1}^t{w}_s{I}_{t-s}^{\mu }}$$

(18)

Thus, considering that the estimated value for $I_t^{\mu }$ is $I{m}_t$, the estimated value of $R_t^{\mu }$ is:

$$R_t^m=\frac{I{m}_t}{{{\displaystyle \sum }}_{s=1}^t{w}_{s}I{m}_{t-s}}$$

(19)

This result we will be tested later in Section 4.2.

This equation is similar to the equation that Fraser [12] proposed, but he used a different approach by starting from a continuous model.

As he called this ratio the instantaneous reproduction number, we named it the same. Thus, this paper mainly proposes some steps forward to find its probability density function and the confidence interval.

3.6. The Probability Density Function of $R_t^m$

Now we will proceed to calculate the probability density function of the estimator $R_t^m$ (19). To begin with, it is noted that the numerator and denominator (19) are constructed with moving averages of the incidence series $I_t$ (Appendix A), which is a procedure that involves the overlap of “crude” incidence $I_t$ in the calculations of the means $I{m}_t$ and of $I{m}_{t-s}$. Therefore, it cannot be said that the numerator and the denominator are independent. The same occurs between the means $I{m}_{t-s}$ of the denominator ${{\displaystyle \sum }}_{s=1}^t{w}_{s}I{m}_{t-s}$ (from now on denoted as $\Lambda m_t$). As this lack of independence makes the analytical calculation of a formula for the probability density function of $R_t^m$ impossible, the problem was approached by applying the experimental method of Monte Carlo simulations, for which we worked with the generator of a series of incidence (Figure 2).

With the generator, 320 series of $I_t$ of 128 days were obtained, each of which was subjected to a calculation of moving averages, to finally obtain the curves of $R_t^m$, using Equation (19). Figure 5 shows a superposition of 10 of these curves.

For each instant of observation, we obtained a sample of 320 $R_t^m$ values coming from the curves (you can imagine 320 curves in Figure 5) and calculated the mean and the standard deviation of $R_t^m$ from these values. Furthermore, at 10-day intervals, the corresponding histograms were constructed. Figure 6 shows the histogram of 320 $R_t^m$ values, for t = 100.

For practical purposes, this histogram can be said to follow a normal distribution. Thus:

$$R_t^m\sim N\left(R_t^{\mu },{\sigma }_{R_t^m}^2\right)$$

(20)

This experimental finding suggests an analytical way to calculate the probability density of $R_t^m$, even if it is by way of approximation.

We can discompose $I{m}_{t-s}$ into its model components ($I{m}_{t-s}=I_{t-s}^{\mu }+e_{t-s}^m$), as we performed with $I_{t-s}$ (1). Thus, let us replace $I{m}_{t-s}$ in the denominator of Equation (19):

$$R_t^m=\frac{I{m}_t}{{{\displaystyle \sum }}_{s=1}^t{w}_{t,s}\left(I_{t-s}^{\mu }+e_{t-s}^m\right)}=\frac{I{m}_t}{{{\displaystyle \sum }}_{s=1}^t{w}_s{I}_{t-s}^{\mu }+{{\displaystyle \sum }}_{s=1}^t{w}_s{e}_{t-s}^m}$$

(21)

where the $e_{t-s}^m$ values are error components which follow a normal distribution $N\left(0,\frac{{\sigma }_{t-s}^2}{n_t}\right)$, with $n_t$ as the number of incidences defined to calculate the incidence mean in the moving average procedure (Appendix A). This means the time series values of $e_{t-s}^m$ delimit a compressed bandwidth compared with the bandwidth of $e_{t-s}$. Thus, it gives us confidence to assume that ${\displaystyle \sum }_{s=1}^t{w}_{t,s}{I}_{t-s}^{\mu }\gg {\displaystyle \sum }_{s=1}^t{w}_{t,s}{e}_{t-s}^m$ (which is true in general):

$$R_t^m\approx \frac{I{m}_t}{{{\displaystyle \sum }}_{s=1}^t{w}_s{I}_{t-s}^{\mu }}$$

(22)

In summary, considering the denominator ${\displaystyle \sum }_{s=1}^t{w}_s{I}_{t-s}^{\mu }$ is a constant, an alternative function for the probability density of $R_t^m$, could be:

$$\begin{gathered} g\left(R_t^m\right)=\frac{1}{\sqrt{2\pi }{\sigma }_{R_t^m}}{e}^{\frac{-1}{2}{\left(\frac{R_t^m-R_t^{\mu }}{{\sigma }_{R_t^m}}\right)}^2}\Rightarrow R_t^m\sim N\left(R_t^{\mu },{\sigma }_{R_t^m}^2\right) \\ {R}_t^{\mu }=\frac{I_t^{\mu }}{{\Lambda }_t^{\mu }}\begin{array}{cc},& {\sigma }_{R_t^m}\end{array}=\frac{{\sigma }_t}{{\Lambda }_t^{\mu }\sqrt{n_t}}\begin{array}{cc},& {\Lambda }_t^{\mu }={\displaystyle {\displaystyle \sum }_{s=1}^t}{w}_s{I}_{t-s}^{\mu }\end{array} \end{gathered}$$

(23)

Consecutively, the estimators for the parameters of Equation (23) are:

$$R_t^m=\frac{I{m}_t}{{\Lambda }_t^m}\begin{array}{cc},& {\widehat{\sigma }}_{R_t^m}\end{array}=\frac{{\widehat{\sigma }}_t}{{\Lambda }_t^m\sqrt{n_t}}\begin{array}{cc},& {\Lambda }_t^m={\displaystyle {\displaystyle \sum }_{s=1}^t}{w}_{s}I{m}_{t-s}\end{array}$$

(24)

3.7. The Confidence Interval of $R_t^m$

The confidence interval is straightforwardly calculated from Equation (24):

$$R_t^m-t_{\frac{\alpha }{2},n_t-2}{\widehat{\sigma }}_{R_t^m}<R_t^{\mu }<R_t^m+t_{\frac{\alpha }{2},n_t-2}{\widehat{\sigma }}_{R_t^m}$$

(25)

This result will be tested later in Section 4.3.

We use a t-Student distribution to compensate for any effect that could cause the number $n_t$ used in the moving average calculation of $I{m}_t$. The degree of freedom ($n_t-2$) used is a consequence of $I{m}_t$ (22), because it can be interpreted as the middle point of a line through $n_t$ points (see Appendix A, we use the centered moving average method), the same reasoning as if it were a case of simple regression analysis.

Equation (25) specifically is the instantaneous reproduction number confidence interval without uncertainty in the serial interval, because there is another type of confidence interval which considers serial interval uncertainty [16]. This means, although the serial interval is still considered to have stable values in theory, it could be impossible to obtain serial interval data to calculate the $w_s$ factors. This could happen, for example, at the beginning of an epidemic. This condition can be handled by modeling the uncertainty in the serial interval parameters [16], which are the mean (μ_S) and the standard deviation (σ_S). It is assumed that these parameters are random independent normal distributed variables, which they really are not. We follow the same procedure of the Monte Carlo simulation, but instead of changing $I_t$, we changed μ_S and σ_S simultaneously. To emphasize that μ_S and σ_S will be considered as random variables, they will be replaced by ms and ss, respectively. Thus:

$$ms\sim N\left(m{s}_{\mu },m{s}_{\sigma }^2\right),$$

(26)

where $m{s}_{\mu }$ and $m{s}_{\sigma }^2$ are the mean and the variance of ms.

Moreover:

$$ss\sim N\left(s{s}_{\mu },s{s}_{\sigma }^2\right)$$

(27)

where $s{s}_{\mu }$ and $s{s}_{\sigma }^2$ are the mean and the variance of ss.

To identify the effect of a specific random pair j of each Monte Carlo iteration, superscripts will be applied:

$$\left(m{s}^{\left(j\right)},s{s}^{\left(j\right)}\right)\to w_1^{\left(j\right)},w_2^{\left(j\right)},w_3^{\left(j\right)}\dots$$

(28)

Consider Equation (19), again:

$$R_t^m=\frac{I{m}_t}{{{\displaystyle \sum }}_{s=1}^t{w}_{s}I{m}_{t-s}}$$

(29)

For each observation instant t, we propose a Monte Carlo simulation process as follows:

The denominator of Equation (29):

1.

Generate N random pairs $\left(m{s}^{\left(j\right)},s{s}^{\left(j\right)}\right)$.

2.

Calculate the weights $w_s^{\left(j\right)}$ for each pair j.

3.

Calculate the denominator ${\displaystyle \sum }_{s=1}^t{w}_s^{\left(j\right)}I{m}_{t-s}$ for each pair j.

Apply each pair $\left(m{s}^{\left(j\right)},s{s}^{\left(j\right)}\right)$ in the calculation of $R_t^m$:

4.

Calculate the values of the means and standard deviations corresponding to each pair:

$$R_{t,j}^m=\frac{I{m}_t}{{\Lambda }_{t,j}^m}\begin{array}{cc},& {\widehat{\sigma }}_{R_t^m}^{\left(j\right)}\end{array}=\frac{{\widehat{\sigma }}_t}{{\Lambda }_{t,j}^m\sqrt{n_t}}\begin{array}{cc},& {\Lambda }_{t,j}^m={\displaystyle {\displaystyle \sum }_{s=1}^t}{w}_s^{\left(j\right)}I{m}_{t-s}\end{array}$$

(30)

5.

Calculate the general mean of all means $R_{t,j}^m$:

$$R_t^m=\frac{1}{N}{\displaystyle \sum }_j^N{R}_{t,j}^m$$

(31)

6.

Calculate the general standard deviation by using the following formula:

$${\widehat{\sigma }}_{R_t^m}=\sqrt{\frac{1}{N}{\displaystyle \sum }_j^N\left[{\left({\widehat{\sigma }}_{R_t^m}^j\right)}^2+{\left(R_{t,j}^m-R_t^m\right)}^2\right]}$$

(32)

7.

Given the mean (31), the standard deviation (32), and a significance level α, assuming a Gamma distribution, calculate the lower and upper limits of the confidence interval for the population mean of $R_t^m$, for each observation instant t.

The derivation and proof of this procedure are in Appendix C.

It should be noticed that this procedure used a Gamma distribution in step 7, when this distribution is in fact a subrogated function of a summatory of normal distributions (Appendix C). To take care of the effect caused by time of moving average ($n_t$), we should have used Student’s t-test distributions instead to obtain a more conservative confidence interval, but unfortunately it cannot be handled with a Gamma distribution as a subrogated density function. Therefore, steps 5, 6, and 7 produce a confidence interval a little narrower than the conservative confidence interval to which we could aspire. It is a tradeoff to speed up the computation process.

4. Results

This section provides graphical demonstrations which support the assumptions completed through Section 3. At the end, a case with real incidence data of a pandemic will be presented.

4.1. Simulated Incidence Data Series

Figure 7 shows a superposition of different types of incidence series. Crude incidence data $I_t$ was generated by simulation in the spreadsheet (Figure 2), adding aleatory noise $e_t$ to the ideal model of $I_t^{\mu }$, following the model Equations (2) and (3). Moving average incidence series ($I{m}_t$) was generated using 15 crude incidence data per calculation ($n_k$), as described in Appendix A.

It is important to note there is no appreciable delay between these graphs (Figure 7), especially the moving average incidence series. We used a centered moving average method.

4.2. Process Calculation of $R_t^m$

The $I{m}_t$ series was generated with the moving average of $I_t$; then, ${\Lambda }_t^m$ was generated with the denominator of the Equation (19). With this, for each t instant, we can calculate $R_t^m$ (19). In fact, in Figure 8, it is easy to say when the $R_t^m$ curve will be greater than 1, or less than 1. Note the smoothness of the ${\Lambda }_t^m$ curve. As there is no appreciable randomness in the ${\Lambda }_t^m$ curve, we can assume it is statistically constant in each instant t. This condition simplifies the deduction of the probability density function of $R_t^m$ (23) and its confidence interval (25).

In Figure 9, we have the $R_t^{*}$ calculated with Equation (10), the graph of $R_t^m$-approximate calculated with Equation (19), and the graph of $R_t^m$-exact calculated by the numeric integration of Equation (11). There is no visible difference between both types of $R_t^m$ graph; therefore, we can say the Taylor approximation used to obtain Equation (19) was good enough. The numerical integration was performed, for each instant t, with the standard method of the trapezoidal rule.

4.3. Process Calculation of the $R_t^m$ Confidence Interval

Table 2. Partial table of 320 Monte Carlo iterations of 128 days $R_t^m$ series.

		$R_t^m$ iterations
Day	Rtm 314	Rtm 315	Rtm 316	Rtm 317	Rtm 318	Rtm 319	Rtm 320	SD MCS
11	2.17001	2.26392	2.47387	1.97609	2.00207	2.25481	2.07686	0.542613		Standard
12	2.06127	2.11066	2.41122	1.87018	1.79859	2.13629	1.93210	0.589714		deviation
13	1.91099	2.06799	2.32853	1.74570	1.75767	2.05797	1.89229	0.640457
14	1.89529	1.86216	2.25943	1.59517	1.69615	1.94120	1.83866	0.69466
15	1.85282	1.79917	2.14870	1.47665	1.70210	1.86042	1.77167	0.751537
16	1.86321	1.68052	1.92846	1.42572	1.68435	1.76157	1.60419	0.808612
17	1.75908	1.62979	1.83304	1.35254	1.57679	1.67131	1.57936	0.866715
18	1.69122	1.59028	1.75054	1.38187	1.50763	1.55256	1.51021	0.924422
19	1.59302	1.53044	1.68361	1.35205	1.43605	1.54926	1.40478	0.982537
20	1.53617	1.46193	1.61977	1.28381	1.42267	1.43287	1.31839	1.040908
21	1.52046	1.38184	1.61057	1.25251	1.45462	1.36895	1.22546	1.099110
22	1.45647	1.30902	1.54020	1.23288	1.42035	1.35385	1.13442	1.157020

, on the right side, has a column of the standard deviation of all the 320 $R_t^m$ iterations for each t instant (days), as described in Section 3.6. This large quantity of values gives us confidence that the standard deviation calculated was almost the standard deviation of the population, which we denote as ${\sigma }_{R_t^m}^{MCS}$ to emphasize it coming from a Monte Carlo simulation. Therefore, considering $R_t^m$ follows a normal distribution (20), we can use the values of ${\sigma }_{R_t^m}^{MCS}$ to construct an exact confidence interval to test the confidence interval we proposed in Equation (25). Figure 10 shows the superposition of both types of confidence intervals.

Although it is a qualitative graphic test (Figure 10), the approximated method (25) is good enough for practical purposes. The confidence interval width of the approximated method is the widest, because it takes care of the sample size $n_t$ used for calculating the moving average of the incidences.

4.4. Comparison between Bayesian Method and Frequentist Method for Calculation of the Instantaneous Reproduction Number ($R_t$ vs. $R_t^m$)

All the methods in Figure 11 share the same $I_t$ series and each graph was calculated with the serial interval parameters shown in the figure. The “Length of time steps” (same as time for moving average, $n_k$) was 15 days. The calculus of the graph data of the Bayesian framework was obtained with the Excel software EpiEstim, developed by Cori et al. [16]. This procedure requires one additional special parameter, the posterior coefficient of variation (CV) = 0.3. Both Bayesian framework graphs, Figure 11a,c, delay the Frequentist framework graphs, Figure 11b,d, by 7 days, which is a number rounded down from the length of time steps divided by 2. It is not a relationship by chance. This happens because we are using a centered moving average (Appendix A) method to calculate the mean of the incidences $I_t$. We selected this method because, in this way, the mean series is synchronized with the crude incidence series of $I_t$ (Figure 7). Thus, the Frequentist framework method produces $R_t^m$ values synchronized with the steps (days) of observations, which is a great advantage.

Most of the time, the Frequentist framework graphs, Figure 11b,d, produce wider confidence interval graphs than the Bayesian framework graphs, Figure 11a,c. This has happened because the Frequentist framework method is open to accept any aleatory variation, not only Poissonian variation, as is the case of the Bayesian framework method. Usually, we would like to have a thin confidence interval but, in this case, “thin” means we are not aware of the real variability of the data; therefore, the Frequentist framework had a better performance than the Bayesian framework, because they show us the complete panorama.

Figure 12 shows an interesting relationship between the Bayesian and Frequentist framework methods; when the Frequentist framework method is 7 days delayed, both graphs are almost the same. This happens because if we make appropriate approximations, the equations of both methods are also nearly the same.

4.5. A Real Case Application: Pandemic COVID-19 in Panama

The data collection of the pandemic of COVID-19 in Panama began in March 2021, with some isolated cases (Figure 13).

Both instantaneous reproduction number graphs were constructed by using $n_k$ time (Frequentist framework) and the “length of time steps” (Bayesian framework) equal to 15 days. As expected, it found a delay of 7 days; moreover, the confidence interval is wider in the Frequentist framework than in the Bayesian framework.

In Panama, the health authorities used the Bayesian framework method to calculate the $R_t$. The health authorities and scientists knew the pandemic could begin in our country at any moment (by knowing the world epidemic reports from the beginning in China, at the end of 2019), but there was no way to know when, where, and how, and the best procedure to follow. The incidence growth rate was high during the first days, doubling its value every three days, which caused very large $R_t^m$ (from day 1 to 20, Figure 13a). As soon as the first pandemic procedures were implemented, and the population mobility was controlled (around day 50), the pandemic process arrived at a stage the epidemiologists named community transmission. This means the contagion was between individuals of the community, locally, following a natural behavior, with the infectious agents moving through the “windows” not closed by the epidemic procedures. On day 85, the new procedures began to show effectiveness, because the $R_t^m$ (Figure 13b) diminished, but the analysts did not notice it until seven days later, because they were using $R_t$ (Figure 13c). In any case, this condition could be difficult to detect by observing incidence data alone (Figure 13a). The ideal condition to control an epidemic, according to epidemiologists, is to obtain a reproductive number below 1. The first time it seemed to be happening was day 60 (Figure 13c), but now, with our $R_t^m$ curve (Figure 13b), we noticed there was too much variability (wide confidence interval) to have enough confidence. For this reason, we must be cautious when the instantaneous reproduction number seems to be diminishing and use the upper limit of its confident interval to take appropriate decisions. We had four moments when the $R_t$ (Figure 13c) seemed to be less than 1 (days 60, 140, 223, and 312, Figure 13c), considering its thin confidence interval. Retrospectively thinking, it should be noted that day 223 (18 October 2021) was a very risky day to take decisions, considering $R_t<1$ (Figure 13c), because something happened a few days later when the incidence began to grow, resulting in the largest incidence peak of Panama. Of course, the health system used many other indicators to take decisions, and what was important was the proximity of the end of the year festivities. If we had used our frequentist method (Figure 13b), the first moment in which the $R_t^m$ would seem to have decreased would have been on day 310 (13 January 2021), using the criterion of the upper confidence limit.

4.6. Computational Aspects

The computational experiments were carried out on a computer with the following hardware characteristics: (i) OS: Windows 10 for 64 bits; (ii) RAM: 4 Gigabytes; and (iii) processor: Intel Core i3 (7th Gen) i3-7020U Dual-core, 2.30 GHz-4 GB DDR4 SDRAM. We used Microsoft Excel, from Office 2019, as our software platform. Here we implemented the theory of this paper with our software SeguiEpi, programmed with standard Excel instructions, including macros. This software is available at: https://github.com/publiodariocortes/seguiepi/ Size: 11,349 KB (accessed on 29 November 2021).

After downloading, the user must read the instructions in the worksheet “User manual”. As the software is an application of the theory for $R_t$ calculation, there are two special modes, with their corresponding run times:

Rt without serial interval uncertainty < 5 s.

Rt with serial interval uncertainty < 60 s.

5. Conclusions

The results of this study are relevant, since epidemiological models are important to monitor the contagion of diseases and evaluate the effectiveness of health policies, and the measures adopted by governments to guarantee adequate hospital care. Estimating the instantaneous reproduction number is a key factor in detecting changes in disease transmission over time.

The method of calculating the confidence interval of the instantaneous reproduction number developed in this article is a significant change with respect to other proposed methods. The explicit formula for calculating $R_t$ (Bayesian framework) produced a curve delayed with respect to the curve of $R_t^m$ (our Frequentist framework). The delayed time was about a half of the time of moving average time ($n_k$). Nevertheless, it was not a large issue, because we can adjust our interpretation of the $R_t$, considering this delay. The situation differed with the confidence interval because we included all sources of variability. Therefore, the main contribution of this work is the proposition of an expression for the confidence interval of the instantaneous reproduction number, considering all sources of variability of the incidences. We defend our approach affirming it produced a 95% confidence interval much more appropriate for using it as a decision tool in controlling the evolution of a pandemic event, such as COVID-19. We hope the software (SeguiEpi) we are providing with the implementation of this theory can help in controlling the evolution of epidemics.

Although our results are encouraging, more tests are still needed with data from other countries and different infectious agents. In any case, it will be necessary to conduct comparatives studies between the Bayesian frequentist method and our Frequentist method, such the comparation we made with the case of Panama, including the measures taken by the authorities, the population behavior, and other relevant situations, to find if there are connections with the evolution of the epidemics and for recommending improvements in the decision processes. Moreover, we must continue the investigation of time series analysis for decoupling the incidences in their model components and, in relation with this matter, it is necessary to conduct sensitivity analysis to determine the robustness of the $R_t^m$ and its confidence interval, regarding the parameters defined for decoupling the incidences. This sensitivity analysis should be performed with simulated and actual incidence data, by using the software we already have. We have shown how to calculate an optimum confidence interval considering serial interval uncertainty, but we still need to find a way to develop a software to perform these tasks, so that we can replace our solution through the subrogated Gamma function. The solution for this matter is already conceptually completed with what we obtained with the summation of normal distribution at Appendix C; we only need to replace the summation of normal distribution with a summation of the Student’s t-test distribution. The only important problem to solve is the implementation of a software that automatically performs all the manual tasks we have in our current method. It will be a great advantage if we could automatically actualize the serial interval parameters by incorporating the new information collected in the local area; moreover, it will be necessary to consider the cases of infected people that come from other regions. This problem has already been solved from a point of view of the Bayesian framework, so we should begin by deeply understanding this approach. In parallel with the proposed investigations, we should perform an important revolution to the method of how to control the epidemics with the implementation of the Statistical Process Control (SPC). It is unquestionable that the controls applied to combat epidemics are a collection of processes. We implemented a special tool in the software we are proposing, a Control Chart designed to monitor the transformed error (Control Panel 3, SeguiEpi Excel software). In this direction, many changes can be performed, including the critical thinking and organization of the Six Sigma Methodology.