Richards Curve Implementation For Prediction of  Covid-19 Spread in Maluku Province


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(2) (2022), Pages 195-206 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: September 10, 2021 Reviewed: December 22, 2021 Accepted: January 05, 2022 
DOI: http://dx.doi.org/10.18860/ca.v7i1.13323   

   Richards Curve Implementation for Prediction of Covid-19 
Spread in Maluku Province 

Nanang Ondi, Francis Yunito Rumlawang, Yopi Andry Lesnussa*  

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Pattimura 
University, Indonesia 

 

*Corresponding Author  
Email: yopi_a_lesnussa@yahoo.com* 

rumlawang@yahoo.com, nanangondi21@gmail.com 

ABSTRACT  

The first case of COVID-19 in Maluku Province, Indonesia was reported at the end of March 2020 
as many as 1 case and the total cumulative cases reported were 3.884 cases on November 4, 2020. 
The purpose of this study is to predict the spread of COVID-19 cases in Maluku Province by 
estimating the Richards function parameters are I  is the population size, K  is carrying capacity, 

k  is the growth rate, a  is the scaling parameter and mt  is the turning point using the nonlinear 

least-squares (NLS) method. The method use in this research is Richards Curve method. The 
results of this research found the estimation results, with RMSE = 75,1057, the peak of the spread 
of COVID-19 cases in the Maluku province is predicted to occur on October 22, 2020, with a total 
of 3.623 cases and ends on May 25, 2023, with a total of 9.451 cases. This research can provide an 
overview of the results of predictions for the development of Covid-19 for the government, making 
it easier for the government to make decisions in the future. 

Keywords: Carrying Capacity; COVID-19; Prediction; Richards Curve; Turning Point 

INTRODUCTION 

Coronavirus is a group of viruses from the subfamily Orthocronavirinae in the 
Coronaviridae family and the order Nidovirales. This group of viruses can cause disease 
in birds and mammals, including humans [1]. In 2002, the SARS-CoV coronavirus (SARS 
Coronavirus)  caused Severe Acute Respiratory Syndrome (SARS) in Guangdong, China 
[2]. In 2012 the type of Coronavirus MERS-CoV (MERS Coronavirus) caused Middle 
Eastern Respiratory Syndrome (MERS) which occurred in Saudi Arabia and the Middle 
East [3]. 

In early 2020, WHO (World Health Organization) received a report from China that 
there were 44 patients with severe pneumonia in Wuhan City, Hubei Province, China [4]. 

Subsequent research showed a close relationship with the Coronavirus that caused SARS 
in 2002 [5]. On February 11, 2020, WHO inaugurated the term COVID-19 (Coronavirus 
Disease 2019) which is an infectious disease similar to influenza caused by Severe Acute 
Respiratory Syndrome 2 (SARS-CoV-2) [6], [7]. The first COVID-19 was reported in 
Indonesia on March 23, 2020, with two cases. Data on March 31, 2020, showed that there 
were 1,528 confirmed cases and 136 deaths.  

http://dx.doi.org/10.18860/ca.v7i1.13323
mailto:yopi_a_lesnussa@yahoo.com*
mailto:rumlawang@yahoo.com
mailto:nanangondi21@gmail.com


Richards Curve Implementation For Prediction of  Covid-19 Spread in Maluku Province 

Nanang Ondi 196 

In 1839 Verhulst introduced the Logistics Equation to model population growth 
which became known as the Verhulst equation and was rediscovered in 1912 [8], [9]. in 
1959 in research entitled: A Flexible Growth Function For Empirical Use, Richards 
modified the Verhulst Equation and became known as the Richards Curve [10] or 
Generalized Logistic Function [11] because it is an extension of the Logistic Model [12], 
[13] and in some literature, the Richards Curve is also called the Theta Logistic Model [14], 

[15] with parameters namely K  (carrying capacity), k  (growth rate), mt  (inflection point) 

and a  (scaling parameter) The shape of the Richards Curve resembles the shape of the 
Exponential Curve [16].  Richards Curve is a model of a population growth curve in 
conditions where growth is not symmetrical with inflexion points [17], [18]. 

In 2004 the Richards Curve was used to predict the spread of SARS in Singapore, 
Hong Kong and Beijing [19] After estimation with the Richards Curve, the results obtained 
are that the spread of SARS in Beijing is predicted to end on 27 June 2003 with a total of 
2.595 cases, in Hong Kong it is predicted to end on 29 June 2003 with a total of 1.748 cases 
and in Singapore it is predicted to end in May 28, 2003 with a total of 207 cases. The 
prediction results of the spread of SARS in Singapore, Hong Kong and Beijing using the 
Richards Curve were considered quite successful, because based on the data obtained, 
Singapore last reported cases of SARS on May 18, 2003 with a total of 206 cases, Hong 
Kong on June 11, 2003 with a total of cases of 1.755 cases and Beijing on June 11, 2003 
with a total of 2.631 cases. Besides that, the Richards Curve was widely used in other 
studies  [20]–[22] and in 2020, the Richards Curve was used to predict the spread of 
COVID-19 in the province of South Sulawesi, Indonesia, with the peak of the spread 
predicted to occur in mid-June 2020 - July 2020 with a total of 10,000-12,000 cases and 
the end of the spread is predicted to occur at the end of November 2020.    

Based on the above background, where the Richards Curve is considered quite good 
in predicting the spread of SARS in Singapore, Hong Kong and Beijing in 2002, therefore 
in this study the Richards Curve will be used to predict the spread of COVID-19 in Maluku 
province. 
 

METHODS 

In general, the differential form of the Richards Curve is : [10], [23] 

  1
a

dI I
I t rI

dt K

    
      
     

  (1) 

 
  Where I  is the population size,  K  is carrying capacity,  k  is the growth rate and 

a   is the scaling parameter. To find a solution to equation 1, the integration technique can 
be written as: 

     
a

a a

K
dI r dt

I K I

 
  
    








 

Or it can be written: 



Richards Curve Implementation For Prediction of  Covid-19 Spread in Maluku Province 

Nanang Ondi 197 

 

 

Based on the similarity of the two sides, the values of 1A   and 
1a

B I


  are obtained 
so as to obtain : 

     
a

a aa a

K A B
dI dI

I K II K I

   
           

 
    

                                                  

1
1

   
a

a a

I
dI dI

I K I

  
    

   


 
   

So we get: 

 
1

  ln ln
a

a
a a

a a

K
dI I K I

I K I

    
             




  

Since we get  r dt rt C  , we can write: :  

   
1

 ln ln  
a

a a
I K I rt C


 

    
   

Or it can be written : 

      

 

So we get: 

 
1

   

a
a a

rt C
K I

e
I



 

 


  
 
 
   

To simplify the above form, both sides can be raised to the power of a  so that we 
get: : 

 

  1

a
a

art aC

K
I

e e
 

 
 
 
 

 (2) 

   
 

   

a a

a a

A K I B I
dI r dt

I K I

  
  
 
 








 
1

ln   

a
a a

K I
rt C

I


 

 
 

 
 
 



Richards Curve Implementation For Prediction of  Covid-19 Spread in Maluku Province 

Nanang Ondi 198 

From equation 2, since ,a r and C  are constants, it is assumed that k  is the product 

of ar   and Q  is the product of 
 aC

e


, so it can be written as: 

 1  

a
a

kt

K
I

Q e


 
 
 
   

So we get: 

 

 
1

1  
kt a

K
I t

Q e


 
 

  
  
  

 (3) 

 

Since the inflection point of equation 3 is 

 
1

1 a

K

a

 
 
   

 [24] , let mt  be the parameter of the 

inflection point of equation 3 then it can be written as : [25] 

 
 

1

1  m
k t t a

K
I t

ae
 

 
 

  
   
  

 (4) 

Where  I  is the population size or the total number of cases that occurred at the 
time of  t ,  K  is the carrying capacity or total of the latest cases,  k  is the rate of growth 

of cases, mt  is the inflexion point or time of the peak of the spread of COVID-19 cases where 

 
   

1 1

1  1  m m
m

k t t a a

K K
I t

aae
 

 
 

 
   
          

. 

RESULTS AND DISCUSSION  

COVID-19 cases in Maluku province have continued to increase since it was first 
reported on March 23, 2020, and as of November 4, 2020, the total cumulative cases of 
COVID-19 in Maluku province were reported as many as 3,884 cases, including 551 
positive patient cases or with a percentage of 14.18%, 3,286 cases of patients cured or 
with a percentage of 84.6 and 47 cases of patients dying or with a percentage of 1.2%.  
Cumulative case developments and the addition of daily cases of COVID-19 in Maluku 
province from March 23, 2020 – November 4, 2020, can be described as follows: 

 
Table 1. Cumulative and daily case data of COVID-19 in Maluku province 

Date  Cumulative Cases Daily Cases  

March 23, 2020 1 0  

March 24, 2020 1 0  

March 25, 2020 1 0  

March 26, 2020 1 0  



Richards Curve Implementation For Prediction of  Covid-19 Spread in Maluku Province 

Nanang Ondi 199 

Date  Cumulative Cases Daily Cases  

March 27, 2020 1 0  

… … …  

October 30, 2020 3849 59  

October 31, 2020 3851 2  

November 1, 2020 3863 12  

November 2, 2020 3863 0  

November 3, 2020 3877 14  

November 4, 2020 3884 7  

 
The development of cumulative COVID-19 cases in Maluku province from 23 March 

– 4 November 2020 can be described as follows: 
 

 
Figure 1. Cumulative case development 

 

The first case of the spread of COVID-19 in Maluku province was reported on March 
23, 2020, as many as 1 case and up to July 5, 2020 the total cumulative cases reported 
were 794 cases or with a growth rate of 755, 23%.  On July 6 to October 22, 2020 the 

average daily addition of cases increased to 26 cases with the average growth rate 
increasing significantly as much as %1864, 953  from the previous one, which was 

2620,183%,  and from October 23 to November 4, 2020, the average increase in cases The 

daily rate of COVID-19 in Maluku province decreased by 18 cases. The graph of the daily 
increase in cases can be seen in Figure 2, where the Maluku province experienced the 
highest number of cases on October 2, 2020, which was 117 cases. 
  

 
Figure 2. Development of daily cases of COVID-19 in Maluku province 23 March – 4 November 2020  
 

 
 



Richards Curve Implementation For Prediction of  Covid-19 Spread in Maluku Province 

Nanang Ondi 200 

Parameter Estimation Results 
By using data on cumulative cases of COVID-19 in Maluku province from March 23 

– November 4, 2020, an estimate was made with the Richards Function parameter using 
the nonlinear least square method in Python with the following script: 
 
import scipy.optimize as optimize 

from scipy.optimize import curve_fit 

import numpy as np 

import pandas as pd 

def RichardsFunction(t,K,a,k,tm): 

        return K/(1 + a*(np.exp(-k*(t-tm))))**(1/a) 

df=pd.read_excel('CovidDate_Maluku.xlsx') 

data=df[0:227] 

y=data['CummulativeCases'] 

t=np.arange(1,228,1) 

popt,pcov=optimize.curve_fit(RichardsFunction,t,y,bounds=(0.01,np.inf)) 

 
The results obtained are: 

 
Table 2. Richards parameter estimation results (RMSE: 75,1057) 

K  a  k  mt  

9.451,245 0,085 0,01 213,918 

 

So by substituting the parameter values K , a , k  and mt    in equation (4) obtained 

the Richards equation, namely : 
 

 
 

1

20.01 0,08591 113, 8

9.451, 245

1 0, 085
t

I t

e
 

 
 

  
  
  

 

Then it can be illustrated that the comparison between the cumulative COVID- 19 
case data from the Richards Function parameter estimation results with the actual data 
for [1, 227]t   is as follows:  

 

 
Figure 3. Comparison of the results of predictions of cumulative cases of covid-19 in Maluku province with 

actual data  



Richards Curve Implementation For Prediction of  Covid-19 Spread in Maluku Province 

Nanang Ondi 201 

In Figure 3. The cumulative comparison between the actual data and the predicted 
data using the Richards function at the time of t = 1 to t = 227, we get RMSE = 75.1057 
while using the logistic function we get a larger RMSE value of 85.1813. The comparison 
of the error values between the predicted results and the actual data can be seen in the 
following Table 3: 
  

Table 3. The error value of the predicted data with the actual data 

t  Actual Predict Error 

1 1 16.65457518205870 15.65457518205870 

2 1 17.48844921400830 16.48844921400830 

3 1 18.35884011574600 17.35884011574600 

4 1 19.26706628665680 18.26706628665680 

5 1 20.21448005719650 19.21448005719650 

… … … … 

222 3849 3889.25714338151000 40.25714338151370 

223 3851 3922.50525096264000 71.50525096263660 

224 3863 3955.72658861666000 92.72658861666100 

225 3863 3988.91835825547000 125.91835825547500 

226 3877 4022.07779333937000 145.07779333936700 

227 3884 4055.20215931066000 171.20215931065600 

 
From Figure 3, the Richards Curve can be described from the estimation results as 

follows:  

 
Figure 4. Richards Curve of estimation results 

From Figure 4, suppose that  iI t  is the total cumulative cases on day i  and  1iI t   

is the total cumulative cases on day 1i  , then the total addition of daily cases can be 
formulated as follows : [26] 

     1 ; 1, 2, 3,...i i iJ t I t I t i        (5) 

So the comparison between the predicted data and actual data from daily COVID-19 
cases in Maluku province can be described as follows: 



Richards Curve Implementation For Prediction of  Covid-19 Spread in Maluku Province 

Nanang Ondi 202 

 
Figure 5. Comparison of the results of daily Covid-19 case predictions in Maluku province with actual data 

From Figure 5, it can be seen that the results of daily case predictions for COVID- 19 
in Maluku province are as follows: 

 
Figure 6. Daily cases of COVID-19 in Maluku province from estimated result 

 

Turning Point of Case Deployment 
From the results of Richards parameter estimation with data on COVID-19 cases in 

Maluku province, the parameter mt  value is 213.918, meaning that the time of the turning 

point for the spread of COVID-19 in Maluku province is predicted to occur on the 214th 
day, where the total cases on the 214th day are obtained. from the equation: 

 
 

1

0.01 214 0,085213,91 18

9.451, 245
214

1 0, 085

I

e
 

 
 

  
  
  

 

which is 3.622,654 means that the total cases at the inflection point are 3.623 cases 
or can be described as follows:  

 
Figure 7. Inflection Point 



Richards Curve Implementation For Prediction of  Covid-19 Spread in Maluku Province 

Nanang Ondi 203 

For the addition of daily cases, the total addition of cases can be obtained at the 

inflexion point or when 
m

t t  namely : 

   214 213I I 
 

   

 
1 1

0.01 214 0.01 2130,0851 0,081 5213,918 213,9 8 1

9.451, 245 9.451, 245
33, 358161

1 0, 085 1 0, 085e e
   

   
   

    
       




       
 

So the total addition of daily cases at the inflection point is 33 cases, so it can be 
concluded that the turning point of the COVID-19 case in Maluku province is based on the 

estimation results, namely     , 214, 33t I t   or can be described as follows :  
 

 
Figure 8. Turning Point 

In figure 7, the point      , 214, 33t I t 
 
which is the turning point of the curve is 

also the peak of the curve, namely when t = 214. 

 
End of Case Deployment 

From Richards parameter estimation results with data on COVID-19 cases in Maluku 
province, the parameter K  value is 9,451,245, meaning that the latest total cases for 

COVID-19 cases in Maluku province are predicted to be 9,451 cases. For example, if endt is 

the end time of COVID-19 cases in Maluku province, with a total of 9,450.5 cases or can be 

written as  endI t = 9.450,5 then the value of endt can be obtained from the equation: 

 . 213,918
1

0 01 0,0851

9.451, 245
9.450, 5

1 0, 085 end
t

e
 

 
 

  
   
    

That is endt  = 1.158,681 meaning that the time for the end of the COVID-19 case in 

Maluku province is predicted to occur on the 1.159th day.  

 



Richards Curve Implementation For Prediction of  Covid-19 Spread in Maluku Province 

Nanang Ondi 204 

 
Figure 9. Total case when 1.159t   

From Figure 9, when 1.159t   the population size will always be at number 1.159 
and will only move towards the value of  K  or carrying capacity. 

 

CONCLUSIONS 

From the estimation results of the Richards function parameter with the cumulative 
case data of COVID-19 in the Maluku province, the Richards equation is obtained to 
predict the spread of COVID-19 in the Maluku province, namely: 

 
 

1

20.01 0,08591 113, 8

9.451, 245

1 0, 085
t

I t

e
 

 
 

  
  
  

 

 
Where, the turning point or peak of the spread of COVID-19 in Maluku province is 

predicted to occur on October 22, 2020 with a total of 3.623 cases, while the time for the 
end of the spread of COVID-19 in Maluku province is predicted to occur on May 25, 2023 
with 9.451 cases. 

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