Predicting the number of total COVID-19 cases and deaths in Brazil by the Gompertz model

Abstract

In this work, we estimate the total number of infected and deaths by COVID-19 in Brazil and two Brazilian States (Rio de Janeiro and Sao Paulo). To obtain the unknown data, we use an iterative method in the Gompertz model, whose formulation is well known in the field of biology. Based on data collected from the Ministry of Health from February 26, 2020, to July 2, 2020, we predict, from July 3 to 9 and at the end of the epidemic, the number of infected and killed for the whole country and for the Brazilian states of Sao Paulo and Rio de Janeiro. We estimate, until July 9, 2020, a total of 1,709,755 cases and 65,384 deaths in Brazil, 331,718 cases and 15,621 deaths in Sao Paulo, 134,454 cases and 11,574 deaths in Rio de Janeiro. We also estimate the basic reproduction number $R_0$ for Brazil and its two states. The estimated values $(R_0)$ were 1.3, 1.3, and 1.4 for Brazil, Sao Paulo, and Rio de Janeiro, respectively. The results show a good fit between the observed data and those obtained by the Gompertz. The proposed methodology can also be applied to other countries and Brazilian states, and we provide an executable as well as the source code for a straightforward application of the method on such data.

Introduction

The World Health Organization declared the coronavirus disease 2019 a pandemic on March 11, pointing to the over 118,000 cases in over 110 countries and territories around the world at that time [18]. The COVID-19 pandemic in Brazil began on February 26, 2020, when a man from Sao Paulo who returned from Italy tested positive for the virus. The first case of COVID-19 in Rio de Janeiro was confirmed on March 5, 2020. Nowadays, Brazil is considered the epidemic center of Latin America, occupying the second place in the total number of cases and, more recently, in the total number of deaths. Currently (July 9, 2020), Sao Paulo and Rio de Janeiro are the states with the highest number of deaths by the new coronavirus in Brazil, according to data from the Brazilian Health Ministry.

In the past few months, a considerable number of studies related to the evolution of COVID-19 in the world have been submitted and published. In the following, we describe some of these works.

Ahmadi et al. [1] developed mathematical models to predict the number of COVID-19 cases in Iran from April 3, 2020, to May 13, 2020. The unknown parameters in these models were estimated by running the fminsearch, a MATLAB function, which is a least-squares algorithm. Torrealba et al. [22] analyzed the modeling and prediction of COVID-19 in Mexico, from an initial approximation, and using the Gauss–Newton algorithm, the authors estimated parameters in the Gompertz and Logistic models. Articles [12, 15, 16] estimated the number of total COVID-19 cases and deaths in world using the Gompertz model. In [11], they analyzed the number of deaths by COVID-19 with social distancing in Brazil, Sao Paulo. Using the SEIR model, the authors recommend temporary lockdowns, however, great the economic and social costs. In [2], the authors studied the impact on the evolution of cases in Rio de Janeiro, Brazil, considering a Susceptible-Infectious-Quarantined-Recovered (SIQR) model with containment.

In this work, we model the near-future trajectory of the cumulative number of infections and deaths by COVID-19 for Brazil and two Brazilian states, given the cumulative number of infected cases and deaths from February 26, 2020 to July 2, 2020. Data were obtained from the Brazilian Ministry of Health until July 2 (https://covid.saude.gov.br), and we considered the cumulative reports from the date on which the first case was notified in Brazil and in each analyzed state. We also estimate the basic reproduction number $(R_0)$, which represents the average number of secondary infections generated by each infected person. To estimate the unknown parameters, we use the Gompertz model and propose a gradient type iterative method (Minimal error method).

The Gompertz model is one of the particular cases of the Richards model, as well as Brody, negative exponential, logistic and von Bertalanffy models, see [20, 21].

The growth functions can be grouped in three main categories: those without inflection point (Brody and negative models), those with sigmoidal shape and a fixed inflection point (Gompertz, logistic, and von Bertalanffy models), and those with a flexible inflection point (Richards model). The logistic, Gompertz, and von Bertalanffy models exhibit inflection points at about 50, 37, and 30% of the upper asymptote, respectively, which means that the Gompertz and von Bertalanffy processes are asymmetric, whereas the logistic is a symmetric process, see [19].

The Gompertz and the logistic models are the most frequently used sigmoid functions, and the literature on these models is extensive [3, 21]. In general, the cumulative number of deaths and cases by COVID-19 presents an asymmetrical sigmoidal growth curve. Therefore, using an inappropriate growth curve can have a substantial impact on forecasting [3].

The Gompertz model [5] was proposed by Benjamin Gompertz in 1825. Since then, this exponential model has been used to describe growth in plants, animals, bacteria, and cancer cells [22]. The Gompertz differential equation has the following form,

$$\begin{aligned} \displaystyle N_t=r N\ln \Big (\frac{K }{N}\Big ), \;\text { with }\;\;\;N(t_0)=N^{1}. \end{aligned}$$

(1)

where t represents time; N(t) is the cumulative population size at time t; r the intrinsic growth rate of model; K is the maximum value of model (N) when t goes to infinity; $t_0$ represents the initial time; and $N^1$ is the initial population or condition.

The analytic solution from (1) is

$$\begin{aligned} N(t)=Ke^{\ln (N^1/K)\times e^{-r(t-t_0)}}, \end{aligned}$$

(2)

where $\lim \limits _{t\rightarrow \infty } N(t)=K$.

The turning point is the time at which the rate of accumulation changes from increasing to decreasing or vice versa and can be easily located by finding the inflection point of the epidemic curve, that is, the moment at which the trajectory begins to decline. Clearly, this quantity is of epidemiological importance, indicating either the beginning (i.e., the moment of acceleration after deceleration) or end (i.e., the moment of deceleration after acceleration) of a phase [6]. From Eq. (2), it is trivial to show that the inflection point is given by

$$\begin{aligned} t_{i.p}=t_0+\frac{\ln (\ln (K/N^1))}{r}. \end{aligned}$$

(3)

At the inflection point, the number of infected or killed is given by $N(t_{i.p})=K/e$.

In this paper, we consider the discrete function, from Eq. (2) and for $t_0=1$,

$$\begin{aligned} N^i= \displaystyle Ke^{\ln (N^1/K)\times e^{-r(i-1)}},\quad i =1,2, \ldots ,m; \end{aligned}$$

(4)

where i represents time in days and $N^i$ is the cumulative number of infected cases (or deaths) at day i.

Table 1 Estimates of parameters unknown

In this paper, we consider that equality (3) is the inflection point of Eq. (4).

We denote ${{\varvec{M}}}=\left( N^1,N^2,\ldots ,N^m\right) \in {\mathbb {R}}^m$. The inverse problem is to estimate ${{\varvec{x}}}=(K,r)$, from (4), given ${{\varvec{M}}}$.

Method: inverse problem to the Gompertz model

Here, we consider an application of the minimal error method (MEM) to the inverse problem at the Gompertz model. The MEM is a variant of the conjugate gradient method [4, 10, 17], and this approach was used to estimate unknown parameters in a computational neuroscience models [24, 25]. Also, [14] solves the Cauchy problem in linear elasticity with MEM.

Knowing the cumulative number of infections or deaths from COVID-19 $({{\varvec{M}}})$, we want to determine ${{\varvec{x}}}$ assuming that Eq. (4) holds.

The transpose of a vector ${{\varvec{y}}}$ is denoted by ${{\varvec{y}}}^T$. Let the nonlinear operator ${{\varvec{F}}}: {\mathbb {R}}^2\rightarrow {\mathbb {R}}^m,$ defined by

$$\begin{aligned}{{\varvec{F}}}({{\varvec{x}}})=\begin{bmatrix} f^1({{\varvec{x}}}) \\ f^2({{\varvec{x}}}) \\ \vdots \\ f^m({{\varvec{x}}}) \end{bmatrix}^T=\begin{bmatrix} N^1 \\ N^2 \\ \vdots \\ N^m \end{bmatrix}^T={{\varvec{M}}}, \end{aligned}$$

where $f^1,f^2,\cdots ,f^m:{\mathbb {R}}^2\rightarrow {\mathbb {R}}$ are multivariable functions with real value and ${{\varvec{M}}}$ solves (4).

To obtain an approximation for ${{\varvec{x}}}$, given ${{\varvec{M}}}$ and ${{\varvec{x}}}_1$, we used the minimal error iteration

$$\begin{aligned} {{\varvec{x}}}_{k+1}={{\varvec{x}}}_k+w_k{{\varvec{F}}}'({{\varvec{x}}}_k)^*({{\varvec{M}}}-F({{\varvec{x}}}_k)), \end{aligned}$$

(5)

where ${{\varvec{F}}}'({{\varvec{x}}}_k)$ is the directional derivative of ${{\varvec{F}}}$ computed at ${{\varvec{x}}}_k$, and ${{\varvec{F}}}'({{\varvec{x}}}_k)^*$ is its adjoint. We also define

$$\begin{aligned} w_k=\frac{{\Vert {{\varvec{M}}}-{{\varvec{F}}}({{\varvec{x}}}_k)\Vert }^2_{{\mathbb {R}}^m} }{{\Vert {{\varvec{F}}}'({{\varvec{x}}}_k)^*({{\varvec{M}}}-{{\varvec{F}}}({{\varvec{x}}}_k))\Vert }^2_{ {\mathbb {R}}^2 }}, \end{aligned}$$

where ${{\varvec{F}}}({{\varvec{x}}}_k)={{\varvec{M}}}_k$ solves (4), replacing ${{\varvec{x}}}=(K,r)$ by ${{\varvec{x}}}_k=(K_k,r_k)$. Note that ${{\varvec{M}}}_k=(N_k^1,N_k^2,\cdots ,N_k^m)$.

It is possible to show that, under certain conditions (we assume that is the case), ${{\varvec{x}}}_k$ converges to a solution of ${{\varvec{F}}}({{\varvec{x}}})={{\varvec{M}}}$; see [9, Theorem 3.21].

Table 2 Estimated number of cases and deaths by days in Brazil and its two states

Fig. 1

Plots for Brazil. Subplot a shows extrapolated infection trajectory from February 26 to July 9. Subplot b presents the number of COVID-19 deaths from March 17 to July 9. The blue curve shows the exact data. The red line represents the estimate of the blue curve with the Gompertz model. The green asterisk is the inflection point

From Eq. (5), we obtain an approximation ${{\varvec{x}}}_k$ for ${{\varvec{x}}}$, but the adjoint ${{\varvec{F}}}'({{\varvec{x}}}_k)$ is not known. In this paper, we obtain the unknown operator (see Appendix) together with the following iteration

$$\begin{aligned} K_{k+1}= & {} K_k+w_k\sum \nolimits _{i=1}^{m}(N^i-N^i_k)\frac{\partial f^i({{\varvec{x}}}_k)}{\partial K}, \end{aligned}$$

(6)

$$\begin{aligned} r_{k+1}= & {} r_k+w_k\sum \nolimits _{i=1}^m (N^i-N^i_k)\frac{\partial f^i({{\varvec{x}}}_k)}{\partial r}, \end{aligned}$$

(7)

where

$$\begin{aligned}&\frac{\partial f^i({{\varvec{x}}}_k)}{\partial K}=\frac{f^i({{\varvec{x}}}_k)}{K_k}\left( 1-e^{-r_k(i-1)}\right) ,\\&\frac{\partial f^i({{\varvec{x}}}_k)}{\partial r}=-f^i({{\varvec{x}}}_k)\left( N^{1}-\ln (K_k)\right) (i-1)e^{-r_k(t-1)},\\&w_k=\frac{{\Vert {{\varvec{M}}}-{{\varvec{M}}}_k\Vert }^2_{{\mathbb {R}}^m}}{{\Bigg \Vert \sum \nolimits _{i=1}^m\left( (N^i-N^i_k)\frac{\partial f^i({{\varvec{x}}}_k)}{\partial K},(N^i-N^i_k)\frac{\partial f^i({{\varvec{x}}}_k)}{\partial r}\right) \Bigg \Vert }^2_{ {\mathbb {R}}^2 }}. \end{aligned}$$

The regression coefficient $(R^2)$ is used to evaluate the fitting ability of various methods and can be obtained by the following equation,

$$\begin{aligned} R_k^2=1-\frac{\sum \nolimits _{i=1}^m \left( N^i-N^i_k\right) ^2 }{\sum \nolimits _{i=1}^m\left( N^i-{\bar{N}}\right) ^2 }, \end{aligned}$$

(8)

where

$$\begin{aligned} {\bar{N}}=\frac{\sum \nolimits _{i=1}^m N^i}{m} \end{aligned}$$

is the average of the cumulative confirmed COVID-19 cases (or deaths).

Fig. 2

Results for Sao Paulo. Subplots a and b are the cumulative number of infected individuals and deaths from February 26 to July 17 and March 17 to July 17, respectively. See Figure 1 for a description of each variable

Fig. 3

Plots for Rio de Janeiro. Subplots a and b show the cumulative number of infected individuals and deaths from March 5 to July 17 and March 19 to July 17, respectively. See Figure 1 for a description of each variable

Remark 1

From differential Eq. (1) and since ${{\varvec{M}}}$ is known, we consider the following initial approximation for $\varvec{x}$

$$\begin{aligned} {{\varvec{x}}}_1=\Bigg (1.2N^m\;,\;\frac{N^m-N^{m-1}}{N^m\ln (1.2N^m/N^{m-1})}\Bigg ). \end{aligned}$$

(9)

The numerical scheme for our method poposed is in Algorithm 1.

Remark 2

The basic reproduction number $R_0$ is the most important parameter to analyze any epidemic model for any disease. In a practical sense, $R_0$ is the average number of secondary infections caused by one infected individual during his/her entire infectious period. An epidemic occurs only if a single individual can spread its illness to more than one individual $(R_0>1)$. If $R_0 < 1$, on average an infectious individual infects less than one person and the contagion is expected to stop spreading.

Our result can be used to compute the basic reproduction number

$$\begin{aligned} {\mathcal {R}}_0 = \exp (r{\mathcal {T}}), \end{aligned}$$

(10)

see [6,7,8, 13], where r denotes the intrinsic growth rate in the Gompertz model and ${\mathcal {T}}$ is the generation time of disease transmission. In this paper, we consider ${\mathcal {T}}=14$.

Results

In this section, we will provide the results obtained for the whole country (Brazil) and for the Brazilian states of Sao Paulo and Rio de Janeiro. Table 1 describes the estimated values for the Gompertz model and also presents the regression coefficient, the inflection point, and the basic reproduction number.

The estimated daily values for the cumulative number of confirmed cases and deaths, from July 3 to 9, are illustrated in Table 2. Figures 1, 2, 3 illustrate the relationship between observed values and those predicted by the Gompertz model. The tables and figures show the efficiency of the proposed iterative method.

Conclusion

In this article, we propose an iterative method to estimate the unknown parameters in the Gompertz model. The results of the modeling show a good fit between the estimated and the observed data. We obtained an estimate of $99.9\% \;(R^2)$ for the cumulative number of infected and killed in Brazil. For Sao Paulo, we obtained an approximation of $99.7\%$ and $99.4\%$ for the cumulative number of infected and deaths, respectively. For Rio de Janeiro, the estimate was $99.8\%$ for the cumulative number of infected and deaths.

The basic reproduction number $R_0$ ( or the average number of infections caused by one typically infectious individual ) depends on the growth rate (r), which was estimated in this work, and the duration of infectiousness $({\mathcal {T}})$. Considering ${\mathcal {T}}=14$, the basic reproduction number was 1.3, 1.3, and 1.4 for Brazil, Sao Paulo, and Rio de Janeiro, respectively. Applying our proposed methodology for China, we obtained $R_0=3.2$, given the number of infected cases accumulated from February 2 to July 2, based on data collected from the world health organization. We did the same for Italy and obtained $R_0=1.8$. Many works obtained these values for China and Italy. Therefore, our estimates for the basic reproduction number are acceptable.

The inflection point of the curve provides vital information about changing trends in the epidemic and may possibly indicate changes in intervention and control. In this paper, the inflection points for the cumulative number of infected cases were July 9, July 4, and June 21 for Brazil, Sao Paulo, and Rio de Janeiro, respectively.

Regarding the methodology used, there are several methods for estimating or determining parameters in a mathematical model. We can divide these methods into two groups, iterative and non-iterative methods. In this paper, we used a gradient type iterative method. This method can recover unknown parameters with a non-uniform distribution (non-constant functions) given noisy data. On the other hand, like all iterative methods, it depends on an initial estimate. If this initial approximation is far from the solution, the iteration will diverge. Although it is difficult to determine an initial estimate for some problems, in this work we found an appropriate initial guess, see Remark 1.

Appendix

Theorem 1

Consider the iteration in Eq. (5). Then, Eqs. (6) and (7) holds.

Proof

Given ${{\varvec{x}}}_k=(K_k,r_k)\in {\mathbb {R}}^2$ and unit vector ${\varvec{\theta }}\in {\mathbb {R}}^2$, the directional derivative of F at ${{\varvec{x}}}_k$ in the direction ${\varvec{\theta }}$ is given by

$$\begin{aligned} {{\varvec{F}}}'({{\varvec{x}}}_k)({\varvec{\theta }})= & {} \lim _{\lambda \rightarrow 0}\frac{{{\varvec{F}}}({{\varvec{x}}}_k+\lambda {\varvec{\theta }})-{{\varvec{F}}}({{\varvec{x}}}_k)}{\lambda }\nonumber \\= & {} J_{{{\varvec{F}}}}({{\varvec{x}}}_k)\cdot {\varvec{\theta }}, \end{aligned}$$

(11)

where $J_{{{\varvec{F}}}}({{\varvec{x}}}_k)=(\nabla f^1({{\varvec{x}}}_k),\nabla f^2({{\varvec{x}}}_k),\cdots ,\nabla f^m({{\varvec{x}}}_k))^T$ $\in {\mathbb {R}}^{m\times 2}$ is the Jacobian matrix of ${{\varvec{F}}}$ at ${{\varvec{x}}}_k$, $\nabla f^i({{\varvec{x}}}_k)$ represents the gradient of function $f^i$ at ${{\varvec{x}}}_k$, and $J_{{{\varvec{F}}}}({{\varvec{x}}}_k)\cdot {\varvec{\theta }}$ denotes the multiplication between matrix $J_{{{\varvec{F}}}}({{\varvec{x}}}_k)$ and unit vector ${\varvec{\theta }}$.

From the minimal error iteration in Eq. (5), we gather that

$$\begin{aligned} {\big<{{\varvec{x}}}_{k+1}-{{\varvec{x}}}_k,{\varvec{\theta }}\big>}_{{\mathbb {R}}^2}= & {} w_k{\big< {{\varvec{F}}}'({{\varvec{x}}}_k)^*({{\varvec{M}}}-{{\varvec{F}}}({{\varvec{x}}}_k)),{\varvec{\theta }}\big>}_{{\mathbb {R}}^2},\\= & {} w_k\big < {{\varvec{F}}}'({{\varvec{x}}}_k)^*({{\varvec{M}}}-{{{\varvec{M}}}_k}),{\varvec{\theta }}\big >_{{\mathbb {R}}^2}. \end{aligned}$$

By the definition of adjoint operator,

$$\begin{aligned} {\big<{{\varvec{x}}}_{k+1}-{{\varvec{x}}}_k,{\varvec{\theta }}\big>}_{{\mathbb {R}}^2}= & {} w_k{\big< {{\varvec{M}}}}-{{\varvec{M}}}_k,{{\varvec{F}}}'({{\varvec{x}}}_k)({\varvec{\theta }})\big>_{ {\mathbb {R}}^m},\\= & {} w_k \big < {{\varvec{M}}}-{{\varvec{M}}}_k,J_{{\varvec{F}}}({{\varvec{x}}}_k)\cdot {\varvec{\theta }}\big >_{ {\mathbb {R}}^m}. \end{aligned}$$

From previous equation and by the definition of Euclidean inner product on ${\mathbb {R}}^m$, we obtain

$$\begin{aligned} {\big <{{\varvec{x}}}_{k+1}-{{\varvec{x}}}_k,{\varvec{\theta }}\big >}_{{\mathbb {R}}^2}=\sum _{i=1}^m \left( N^i-N^i_k\right) \left( \nabla f^i.{\varvec{\theta }}\right) , \end{aligned}$$

by definition of Euclidean inner product on ${\mathbb {R}}^2$, we have

$$\begin{aligned} {\big<{{\varvec{x}}}_{k+1}-{{\varvec{x}}}_k,{\varvec{\theta }}\big>}_{{\mathbb {R}}^2}= & {} w_k\Bigg<\sum _{i=1}^m(N^i-N^i_k)\nabla f^i({{\varvec{x}}}_k)\;,{\varvec{\theta }}\Bigg>_{{\mathbb {R}}^2},\\= & {} \Bigg <w_k\sum _{i=1}^m(N^i-N^i_k)\nabla f^i({{\varvec{x}}}_k)\; ,{\varvec{\theta }}\Bigg >_{{\mathbb {R}}^2}. \end{aligned}$$

Since ${\varvec{\theta }}\in {\mathbb {R}}^2$ is arbitrary, we gather that the following iteration holds:

$$\begin{aligned} {{\varvec{x}}}_{k+1}={{\varvec{x}}}_k+w_k\sum _{i=1}^m(N^i-N^i_k)\nabla f^i({{\varvec{x}}}_k), \end{aligned}$$

where

$$\begin{aligned} w_k=\frac{{\big \Vert {{\varvec{M}}}-{{\varvec{M}}}_k\big \Vert }^2_{{\mathbb {R}}^m}}{{\displaystyle \bigg \Vert \sum \nolimits _{i=1}^m(N^i-N^i_k)\nabla f^i({{\varvec{x}}}_k)\bigg \Vert }^2_{ {\mathbb {R}}^2 } }. \end{aligned}$$

$\square$