On the quick estimation of probability of recovery from COVID-19 during first wave of epidemic in India: a logistic regression approach

Abstract

The COVID-19 pandemic has recently become a threat all across the globe with the rising cases every day and many countries experiencing its outbreak. According to the WHO, the virus is capable of spreading at an exponential rate across countries, and India is now one of the worst-affected country in the world. Researchers all around the world are racing to come up with a cure or treatment for COVID-19, and this is creating extreme pressure on the policy makers and epidemiologists. However, in India the recovery rate has been far better than in other countries, and is steadily improving. Still in such a difficult situation with no effective medicine, it is essential to know if a patient with the COVID-19 is going to recover or die. To meet this end, a model has been developed in this article to estimate the probability of a recovery of a patient based on the demographic characteristics. The study used data published by the Ministry of Health and Family Welfare of India for the empirical analysis.

Full text

STATISTICS IN TRANSITION new series, June 2022
Vol. 23 No. 2, pp. 197–208, DOI 10.2478/stattrans-2022-0024
Received – 20.06.2020; accepted – 28.07.2021
On the quick estimation of probability of recovery from
COVID-19 during first wave of epidemic in India:
a logistic regression approach
Hemlata Joshi
1
,
S. Azarudheen
2
,
M. S. Nagaraja
3
,
Singh Chandraketu
4
ABSTRACT
The COVID-19 pandemic has recently become a threat all across the globe with the rising
cases every day and many countries experiencing its outbreak. According to the WHO,
the virus is capable of spreading at an exponential rate across countries, and India is now
one of the worst-affected country in the world. Researchers all around the world are racing
to come up with a cure or treatment for COVID-19, and this is creating extreme pressure
on the policy makers and epidemiologists. However, in India the recovery rate has been far
better than in other countries, and is steadily improving. Still in such a difficult situation
with no effective medicine, it is essential to know if a patient with the COVID-19 is going
to recover or die. To meet this end, a model has been developed in this article to estimate
the probability of a recovery of a patient based on the demographic characteristics.
The study used data published by the Ministry of Health and Family Welfare of India for
the empirical analysis.
Key words: COVID-19, epidemic, coronavirus disease, recovery estimation, logistic
regression, logit analysis.
1. Introduction
Coronaviruses are the group of related RNA viruses which has ribonucleic acid as
its genetic material. These viruses cause diseases in humans, other mammals and birds
and sickness may range from common cold to severe respiratory diseases. COVID-19
1
CHRIST (Deemed to be University), Bangalore, India. E-mail: hemlata.joshi28@gmail.com.
ORCID: https://orcid.org/0000-0002-4051-6330
2
CHRIST (Deemed to be University), Bangalore, India. E-mail: azarudheen.s@christuniversity.in.
ORCID: https://orcid.org/0000-0001-7568-4273.
3
CHRIST (Deemed to be University), Bangalore, India. E-mail: nagaraja.ms@christuniversity.in.
ORCID: https://orcid.org/0000-0002-6900-8436.
4
CHRIST (Deemed to be University), Bangalore, India. E-mail: chandraketu.lko@gmail.com.
ORCID: https://orcid.org/0000-0003-2367-5396.
© Hemlata Joshi, S. Azarudheen, M. S. Nagaraja, Singh Chandraketu. Article available under the CC BY-SA 4.0
licence

198 Hemlata Joshi et al.: On the quick estimation of probability of recovery…
is the most recent disease that has jumped off to humans. Initially the eruption of the
novel coronavirus was documented in China's Wuhan at the beginning of December
2019 and then circularized all across the world. Often during coughing or sneezing,
the infection of coronavirus disease disseminates from one human to others via
droplets raised from the respiratory system of the infected humans (WHO, 2020). The
COVID-19 symptoms generally include fever, dry cough, tiredness, and in severe
cases, infection can lead pneumonia, shortness of breath, chest pain, loss of speech or
movement, kidney failure, and even death (WHO, 2020), but approximately
20 percent of the cases have been deemed to be severe (Singh et al., 2020). The World
Health Organization (WHO) announced this COVID-19 a pandemic on 11 March
2020 and ingeminated the call for countries to take quick actions and scale up
response to treat, detect and reduce transmission to save people’s lives. The developed
countries such as the United States of America, Italy, Spain, France, UK, etc. are
struggling to overcome the disease spreading by novel coronavirus. According to
WHO, by the end of May 2020 it has spread in around 188 countries, the total
number of cases have exceeded 6 million and approximately 3.7 lakh deaths
worldwide. In India, the first case of coronavirus infection was observed in Kerela on
30 January 2020 and for the two months, the spread of the coronavirus disease was
extremely slow may be due to the strict nationwide lockdown. After that, the
Government of India gave the conditional relaxation in the nationwide lockdown and
during this period of lockdown, the coronavirus cases started increasing with the
exponential rate. Although the incubation period for the coronavirus disease has not
been confirmed yet, from the pooled analysis it is seen that the symptoms may appear
in 2 days to 14 days (Singhal, 2020) and the Government of India has declared
minimum 14 days quarantine period for the suspected cases. In the absence of any
efficacious medicine or vaccination, the social distancing has been consented as
a most efficient scheme for cutting the severity of this coronavirus disease all across
the globe (Ferguson et al., 2020; Singh et al., 2020).
As India is the second largest most populated country and majority of the
population live under the inadequate hygiene and with insufficient medical facilities
such as lack of testing kits, labs and health personnel, etc., and with the relaxation
in lockdown, the coronavirus disease may start spreading at community level. In the
middle of June, the total confirmed COVID-19 cases crossed 3.43 lakh with an
increase of more than ten thousand cases in a single day and the new cases was rising
at the record pace while the deaths have come up to 9900 with 380 fatalities. If the
same rate continues, India will reach the sixth position in the most affected countries
by COVID-19, and presently India is the 7th worst affected country after the USA,
Brazil, Russia, UK, Spain and Italy (WHO), and in terms of the fatality rate, India is at
the twelfth position while it is ranked 8th in terms of recovery rate from coronavirus
disease currently.

STATISTICS IN TRANSITION new series, June 2022
199
The Prime Minister of India Mr. Narendra Modi stated that currently India is
being listed amongst the countries with the least number of deaths due to coronavirus
and also said that the death rate can still be reduced if we all follow all the guidelines
suggested by WHO. PM Modi also said that the decision of nationwide lockdown on
time served better in controlling the speed of spreading of coronavirus disease
in India. According to the ICMR's serological survey, about 0.73% of the population
was exposed to the virus by the mid-June and India could have 200 million COVID
infected people by September. The Indian Council of Medical Research (ICMR) said
that India was not in community transmission yet but a large chunk of the population
is at risk and physical distancing and other similar measures need to continue. The
return of millions of migrants to villages in Bengal, UP, Bihar, Orissa, Chhattisgarh,
Jharkhand, etc. will lead to a surge of infections in these rural hinterlands.
As COVID-19 is a new pandemic, it has become a challenging task in front of the
scientists and researchers to fight with this coronavirus disease in the absence of
vaccine. Thus, to know its behaviour and nature a lot of research is being done all
across the globe, so that it could help the scientists or epidemiologists to possibly cure
humans from its infections. The published data on COVID-19 pandemic are analysed
by many researchers by using various mathematical modelling approaches (Rao et al.,
2020; Chen et al., 2020). Huang et al. (2020) worked on the clinical features of patients
infected with 2019 novel coronavirus in Wuhan, China. Modelling and forecasting of
the COVID-19 pandemic is done by Anastassopoulou et al. (2020), Corman et al.
(2020), Rothe et al. (2020) and Gamero, J. et al. (2020) and many interesting results
have been obtained using the principles of mathematical modelling. Nikolay et al.
(2020) used the coronavirus data and compared the Verhult model with the half-
logistic curve of growth with polynomail variable transfer model. Further, they have
compared the Verhulst growth model with Verhulst curve of growth with polynomail
variable transfer model on the Covid-19 data and also have studied the intrinsic
properties of some models of growth with polynomial variable transfer that give
a very good approximation of the specific data on the pandemics in Cuba. Zaliskyi et
al. (2020) built a mathematical model for COVID-19 data of European countries.
In this article, an effort has been made to estimate the probability of recovery from the
coronavirus disease using the indirect method of estimation. For this a logistic
regression techniques has been used and for the empirical analysis, the available
information about the demographic variables such as age and gender of the patients,
which was published by the Ministry of Health and Family Welfare, Government of
India, is utilized.

200 Hemlata Joshi et al.: On the quick estimation of probability of recovery…
2. The Model and Methodology
Here, the variable of the interest is the status of the patient whether the patient
recovered or deceased after the infection of COVID-19. The status of the patient can
take only two values – either 0 if the patient deceased due to COVID-19 or 1 if
recovered, and we want to estimate the probability of dying or survived after getting
the infection of COVID-19 as a function of the indicator variables such as gender
(male or female) and various age groups (020, 2140, 4160 and 60 and over).
Since the response variable is of a dichotomous type, the logistic regression modelling
technique is applied for the estimation of the probability whether the patient will die
or recover by using various age groups and gender of the patients.
Let

denote the probability of recovery from the corona disease of a patient for
the given values of
p
predictor variables and the relationship between the probability

and
p
predictors can be represented by the logistic model (see Chatterjee, S. and
Hadi, Ali S. (2006)), i.e.





p
X
p
XX
p
X
p
XX
pp
e
e
xXxXYPr
...,
22110
......,
22110
11
1
=
)=,...,=|1=(=
(1)
The function given in equation (1) is the logistic regression function. It is non-
linear in the regression coefficients p



..., 10 and it is linearised by the logit
transformation, i.e. if the probability of an event that the patient recovers from the
corona disease is

then the ratio


1 obtained is the odds for the recovery from
the coronavirus disease.
Since





p
x
p
xx
pp
e
xXxXYPr
......
22110
11
1
1
=
)=,...,=|0=(=1 (2)
Then,







p
x
p
xx
e......
22110
=
1
(3)
Taking natural log both sides, we get






pp xxx
loglogit
......=
)
1
(=)(
22110
(4)

STATISTICS IN TRANSITION new series, June 2022
201
Here, the function )(

logit in equation (4) is a linear function of explanatory
variables xi (i=1,…,p) in terms and it is called the logit function and the range of

in equation (1) is between 0 and 1 while the range of the values of )
1
(



log is
between  and , which makes the logits more appropriate linear regression
fitting, and the disturbance term

satisfies all the basic assumptions of ordinary least
squares.
Now, our predictor variables are categorical type so the dummy variables are
created for each of the categorical predictors. If the regression model contains an
intercept term, the number of dummies defined should always be one less than the
number of categories of that variable. Let G be the dummy variable for the gender of
the patient which have only two categories (male and female), i.e. 1=G if the patient
is male, 0 otherwise. Similarly, the dummy variables for the age having four age
groups is 1,2,3=; tAt and it can be defined as
Otherwise0;=
aboveand60groupagetheinliespatienttheIf1;=
ndOtherwisea0;=
6041groupagetheinliespatienttheIf1;=
Otherwise0;=
4021groupagetheinliespatienttheIf1;=
3
2
1
A
A
A


(5)
Here, the female category in the dummy variable G and the age group 020
in the t
A dummy variable are taken in the reference category and the logit model can
be written as






GAAA
loglogit
43322110
=
)
1
(=)( (6)
3. Empirical Analysis
For the estimation of the probability of recovery of a patient infected by
coronavirus disease in India, the data issued by the Ministry of Health and Family
Welfare (MoHFW, India) are utilized. In the analysis, 427 patients have been included
due to the lack of availability of data on all the patients and the data on the patients’
status from all over India are taken from between 30 January 2020 to 30 May 2020,
which is shown in Table 1. From the available data, an effort has been made to
estimate the probability of recovery from coronavirus disease in India. For this, the

202 Hemlata Joshi et al.: On the quick estimation of probability of recovery…
logistic regression technique is used and the developed model is shown in equation
(6), where age group and gender of the patients are the indicator variables and

is
the probability of recovery of a patient from coronavirus disease. The analysis is done
using 𝑅𝑆𝑡𝑢𝑑𝑖𝑜 (R Core Team (2020)) and the results obtained are shown in Table 2.
The estimated model is given as
GAAA
loglogit
0.10712.01011.59130.93460.0401=
)
1
(=)(
321 




(7)
Now, from Table 2, it can be seen that the age groups 4160 and 60 and over are
significant at 0.05 level of significance as their p-values are smaller than the 0.05and
the log odds of recovery from the corona disease are 1.5913 and 2.0101 for the age
group 4160 and 60 and over respectively. For a better understanding of the results,
the exponentiated terms of the regression coefficients has also been computed, which
is shown in Table 3. If we look at the exponentiated terms of these log odds of
significant variables, i.e. 0.20365=1.5913)(exp and 0.13397=2.0101)(exp , these
exponentiated terms show the odds of recovery from the coronavirus disease means
that recovery odds for the patients in the age group 4160 years is equal to 0.2036
times the recovery odds for the patients in the age group 020 years. Similarly, the
patients aged 60 and over have 0.13397 times the odds of being recovered from Covid-
19 disease compared to the patients in the age group 020 years on average, holding
all else constant. From these two odds ratios, it can also be discovered that the odds of
recovery from the corona disease is higher in the patients aged between 4160 than
the patients whose age is 60 and over. From Table 2, it can be assured that for the
patients in the age group 020 and who are male, the probability of recovery from
coronavirus disease is 0.6597 and the probability of recovery for the male patients
aged between 4160 is 0.6818. Also, the predicted recovery probability from
coronavirus disease of patients aged 60 and over is 0.6746, which is slightly lower than
the patients aged between 4160 and higher than the patients of aged between 020.
But on average, it can also be seen that the probability of recovery from coronavirus
disease during the first wave of pandemic is almost same in all the patients and lies
between the probability 0.65970.6818. If we look at the coefficient of gender (male)
in Table 2, which is also statistically insignificant, it means there is no strong evidence
for a gender difference in risk of dying due to coronavirus disease. This implies that
the probability of recovery from coronavirus disease is same in males and females,
keeping all else constant.
To test the goodness of fit of the model to the data, the log likelihood ratio 2
R
,
sometimes called McFadden R-squared, the C-Statistic (Concordance Statistic)

STATISTICS IN TRANSITION new series, June 2022
203
and Chi-Square goodness of fit test, has been used. The McFadden R-square is
defined as:
0
21= LL
LL
Rfull
MF  (8)
where full
LL is the full log likelihood model and 0
LL is the log likelihood function of
the model with the intercept only. Backhaus et al. (2000) suggested that a McFadden
2
R
value is in the range 0.20.4 indicates a good fit of the model and the obtained
value of the 2
MF
R is 1-384.12/482.96= 0.20465463 and shows the model is sufficiently
well fitted to the data and the C-statistics can be computed by considering all possible
pairs consisting of one patients who recovered from the coronavirus disease and one
patients who deceased. The obtained C-statistics is the proportion of such pairs
in which the patients who experienced a recovery from coronavirus disease had
a higher estimated probability of experiencing the recovery than the patients who did
not experience the recovery from the coronavirus disease. The value of C-statistics can
lie between 0.50 to 1.00 The closer the C-statistic is to 1, the better a model is able to
classify outcomes correctly. The value of C-statistics between 0.70 and 0.80 signals the
model is good fitted to the data and the value between 0.50 to 0.70 indicates poor
models (Hosmer & Lemeshow, 2000). Here, the obtained C-statistic is 0.7599994,
which also indicates that the model is good enough and is able to classify outcomes
correctly.
The Chi-square goodness of fit test is also used to test the goodness of fit of the
model. For this, the standardized residuals are calculated as
𝑟𝑦𝑦


𝑦󰇛1𝑦󰇜
And then the Chi-squared statistics is obtained as
𝜒𝑟



The 𝜒 statistics follows a 𝜒 distribution with n-(p+1) degree of freedom, where
p are the number of covariates. The obtained 𝜒 value is 427.228 with 422 degree of
freedom and the corresponding p-values is 0.4199021. This indicates that we cannot
reject the null hypothesis that the model is exactly correct and it shows that the model
fits the data well. From Figures (1 and 2), it can also be seen that the observed and
expected number of cases of recovered and deceased is almost same, which also
indicates that the model fits the data well.

204 Hemlata Joshi et al.: On the quick estimation of probability of recovery…
4. Conclusion
The coronavirus has wreaked havoc all across the world with the rising cases of
COVID-19 every day and with the absence of any effective treatment. In these
gravedigger circumstances, the Government of India adopted many preventive steps
such as lockdown, social distancing and urging people to live with extra cleanliness
and India benefited somewhat from the strict lockdown but this nationwide lockdown
cannot be continued for so long as it is not the solution for this pandemic, and it also
not good for the country’s economy. Hence, it is necessary to estimate the probability
of recovery from the coronavirus disease as most of the Indian population is living
in poor hygienic conditions. In this article, a probability model is developed using the
indirect method of estimation based on some demographic factors, and it is found
that the probability of recovery from coronavirus disease is statistically same in both
males and females. Also, the coronavirus patients in the age group 040 years have
almost equal probability of being recovered from this disease. In the patients aged
between 4160, the odds of recovery from the coronavirus disease is equal to 0.2036
times the recovery odds of the patients of the age group 020 years, while the patients
aged 60 and over have 0.13397 times the odds of recovery from coronavirus compared
to the patients of the age group 0-20 years on average. Also, the odds of recovery from
coronavirus is higher in the patients of the age group 4160 years than in the patients
aged 60 and over.
References
Anastassopoulou, C., Russo, L., Tsakris, A. and Siettos, C., (2020). Data-based
analysis, modelling and forecasting of the COVID-19 outbreak, PLOS ONE, 15(3),
e0230405. https://doi.org/10.1371/journal.pone.0230405.
Backhaus, K., Erichson, B., Plinke, W. and Weiber, R., (2000). Multivariate
analysenmethoden, Berlin: Springer.
Chen, Yi C., Lu, P. E., Chang, C. S. and Liu, T. H., (2020). A Time-dependent SIR
model for COVID-19 with Undetectable Infected Persons,
http://gibbs1.ee.nthu.edu.tw/A Time Dependent SIR Model For Covid 19.pdf.
Chatterjee, S. and Hadi, Ali S., (2006) Regression analysis by example. John Wiley &
Sons, Inc., Hoboken, New Jersey.
Corman, V. M., Landt, O., Kaiser, M., Molenkamp, R., Meijer, A., Chu, D. K.,
Bleicker, T., Brunink, S., Schneider, J. and Schmidt, M. L., (2020). Detection of
2019 novel coronavirus (2019-ncov) by realtime rt-pcr, Euro surveillance, 25(3),
2000045.

STATISTICS IN TRANSITION new series, June 2022
205
Ferguson, N. M., Laydon, D., Nedjati-Gilani, G., Imai, N., Ainslie, K., Baguelin, M.,
Bhatia, S., Boonyasiri, A., Cucunubã, Z., Cuomo-Dannenburg, G., Dighe, A.
Dorigatti, I., Fu, H., Gaythorpe, K., Green, W., Hamlet, A., Hinsley, W., Okell,
L. C., Elsland, S. V., Thompson, H., Verity, R., Volz, E., Wang H., Wang, Y.,
Walker, P. Gt., Walters, C., Winskill, P., Whittaker, C., Donnelly, C. A., Riley, S.
and Ghani, A. C., (2020). Report 9: Impact of non-pharmaceutical interventions
(NPIs) to reduce COVID-19 mortality and healthcare demand, Imperial College
COVID-19 Response Team.
Gamero, J., Tamayo, J. A. and Martinez-Roman J. A., (2020). Forecast of the evolution
of the contagious disease caused by novel corona virus (2019-ncov) in China, arXiv
preprint ar Xiv: 2002, 04739.
Huang, C., Wang, Y., Li, X., Ren, L., Zhao, J., Hu, Y., Zhang, L., Fan, G., Xu, J. and Gu,
X., (2020). Clinical features of patients infected with 2019 novel coronavirus
in Wuhan, China, The Lancet, 395(10223), pp. 497506.
Hosmer Dw, Lemeshow S., (2000). Applied Logistic Regression (2nd Edition), New
York, NY: John Wiley & Sons;.
Nikolay K., Anton I. and Asen R., (2020). On the half–logistic model with
”polynomial variable transfer”. Application to approximate the specific ”data
corona virus”. International Journal of Differential Equations and Applications,
19(1), pp. 4561.
Nikolay K., Anton I. and Asen R. (2020). On the Verhulst growth model with
polynomial variable transfer. Some applications. International Journal of
Differential Equations and Applications, 19(1), pp. 15-32.
Maksym Z., Roman O. B., Yuliia P., Maksim I. and Irakli P., (2020). Mathematical
model building for COVID-19 diseases data in European Countries. IDDM’2020:
3rd International Conference on Informatics & Data-Driven Medicine, November
19–21, 2020, Växjö, Sweden, Session 1: Artificial intelligence, CEUR Workshop
Proceedings.
Rao Srinivasa A., S. R., Krantz S., G., Kurien T. and Bhat R., (2020). Model based
retrospective estimates for COVID-19 or coronavirus in India: continued efforts
required to contain the virus spread. Current Science, 118(7), pp. 1023-1025.
R Core Team, (2020). R: A language and environment for statistical computing. R
Foundation for Statistical Computing, Vienna, Austria, URL https://www.R-
project.org/.

206 Hemlata Joshi et al.: On the quick estimation of probability of recovery…
Rothe, C., Schunk, M., Sothmann, P., Bretzel, G., Froeschl, G., Wallrauch, C., Zimmer,
T., Thiel, V., Janke C. and Guggemos, W., (2020). Transmission of 2019-ncov
infection from an asymptomatic contact in Germany. New England Journal of
Medicine, 382(10), pp. 970-971.
Singh, B. P., Singh, G., (2020). Modeling Tempo of COVID-19 Pandemic in India and
Significance of Lockdown, https://doi.org/10.1002/pa.2257.
Singh, B. P., (2020). Forecasting Novel Corona Positive Cases in India using Truncated
Information: A Mathematical Approach, medRxiv preprint, doi:
https://doi.org/10.1101/2020.04.29.20085175.
Singh, R., Adhikari, R., (2020). Age-structured impact of social distancing on the
COVID-19 epidemic in India, arXiv: 2003, 12055.
Singhal, T., (2020). A review of coronavirus disease-2019 (COVID-19). The Indian
Journal of Pediatrics, pp. 1-6.
World Health Organization, (2020). Updated WHO advice for international traffic
in relation to the outbreak of the novel coronavirus 2019-nCoV, Available at:
https://www.who.int/ith/2020-24-01-outbreak-of-Pneumonia-caused-by-new-
coronavirus/en/ (accessed January 2020).
World Health Organization, (2020). Coronavirus disease (COVID-19) Weekly
Epidemiological Update and Weekly Operational Update, Available at:
https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-
reports (accessed March 2020).

STATISTICS IN TRANSITION new series, June 2022
207
Appendix
Table 1. Number of patients deceased or recovered from the corona disease in India during 30
January 2020 to 30 May 2020
Age Group
Patient Status
Total
Decease
d
Recovere
d
Female Male Female Male
0-20 6 2 4 4 16
21-40 6 15 21 31 73
41-60 48 104 11 19 182
60 and over 51 87 6 12 156
Total 111 208 42 66 427
Table 2.

Coefficients showing the log odds ratios of recovery from the coronavirus disease
Deviance Residuals:
Min 1Q Median 3Q Max
-1.61 -0.59 -0.51 0.8 2.1
Coefficients:
Group Estimate Standard
Error z value Pr(>|z|)
Intercept 0.0401 0.5103 0.0790 0.9373
21-40 0.9346 0.5676 1.6470 0.0996
41-60 -1.5913 0.5441 -2.9250 0.0034*
60 and over -2.0101 0.5632 -3.5690 0.0003*
Gender(Male) -0.1071 0.2695 -0.3970 0.6913
Null Deviance: 482.96 on 426 degree of freedom
Residual
Deviance:
384.14 on 422 degree of freedom
AIC: 394.14
Number of Fisher scoring iterations: 4
The p-values denoted by * are significant at 0.05 level of significance
Table 3. Exponentiated estimated coefficients showing the odds ratios and their respective
confidence intervals
Group Estimates 95% Confidence Interval
Lower limit Upper limit
Intercept 1.04 0.38 2.89
21-40 2.55 0.83 7.89
41-60 0.20 0.07 0.60
60 and over 0.13 0.04 0.41
Gender (Male) 0.90 0.53 1.53

208 Hemlata Joshi et al.: On the quick estimation of probability of recovery…
Figure 1. Observed and expected number of cases recovered from the corona disease in groups
Figure 2. Observed and expected number of cases deceased from the corona disease in groups