Forecasting COVID-19 epidemic in India and high incidence states using SIR and logistic growth models.

BACKGROUND: Ever since the Coronavirus disease (COVID-19) outbreak emerged in China, there has been several attempts to predict the epidemic across the world with varying degrees of accuracy and reliability. This paper aims to carry out a short-term projection of new cases; forecast the maximum number of active cases for India and selected high-incidence states; and evaluate the impact of three weeks lock down period using different models.
METHODS: We used Logistic growth curve model for short term prediction; SIR models to forecast the maximum number of active cases and peak time; and Time Interrupted Regression model to evaluate the impact of lockdown and other interventions.
RESULTS: The predicted cumulative number of cases for India was 58,912 (95% CI: 57,960, 59,853) by May 08, 2020 and the observed number of cases was 59,695. The model predicts a cumulative number of 1,02,974 (95% CI: 1,01,987, 1,03,904) cases by May 22, 2020. As per SIR model, the maximum number of active cases is projected to be 57,449 on May 18, 2020. The time interrupted regression model indicates a decrease of about 149 daily new cases after the lock down period, which is statistically not significant.
CONCLUSION: The Logistic growth curve model predicts accurately the short-term scenario for India and high incidence states. The prediction through SIR model may be used for planning and prepare the health systems. The study also suggests that there is no evidence to conclude that there is a positive impact of lockdown in terms of reduction in new cases.

Background

Ever since a series of pneumonia cases of unknown cause emerged in Wuhan, China in December 2019 that was later confirmed and named as corona virus disease 2019 (COVID-19), it quickly spread around the planet within less than three months, infecting around 4.1 million cases and killing around 2,83,000 as of May 11, 2020.1, 2, 3 Countries have taken extreme steps such as total lock down to partial lock down coupled with social distancing, quarantine, and isolation that prevent human movement which could reduce the transmission. , India reported its first case of COVID-19 on January 30, 2020 which rose to 100th cases on March 14, 2020 and thereafter, the reported cases have increased steadily to reach around 70,000 cases, with 22,500 recoveries and 2300 deaths reported across the country as of May 11, 2020.

Since the beginning of the COVID-19 epidemic, there has been several mathematical and statistical modelling that have predicted the global and national epidemic with varying degrees of accuracy and reliability. , The accuracy of prediction and its uncertainty depend on the assumptions, availability and quality of data. The results can vary significantly if there is difference in the assumptions, and values of input parameters. During a pandemic like COVID-19, the availability and quality of data keep improving as the epidemic progress, which make predictions uncertain in the early stages and expected to improve in the later stages. Moreover, an epidemic may not always behave in the same manner as pathogens are likely to behave differently over time.

In terms of COVID-19, different models are used to estimate the key features of the disease such as the incubation period, transmissibility, asymptomaticity, severity, and the likely impact of different public health interventions. Among the models, Susceptible, Exposed, Infection and Recover (SEIR), Susceptible, Infection and Recover (SIR) models, Agent-based models and Curve-fitting, Logistic growth models due to the exponential nature of growth of the epidemic or extrapolation models, are commonly adopted using different biological and social processes. , 11, 12, 13, 14, 15, 16 Especially, the logistic growth curve model could be relevant for short term projection while SIR models could be useful to estimate the maximum number of active cases and the peak time of attaining it. Though the SIR model is used widely in COVID-19, there is not much information about how the SIR model performed in China and other countries where the pandemic is stabilizing or indicating a declining trajectory. Therefore, it is essential to validate the SIR model using the reported data from different countries and evolve a correction factor that is in relation to the maximum number of active cases to total number of cases reported. Few researchers have used the exponential growth model to predict the number of Cumulative cases. But, these models do not have upper bound and therefore does not get stabilised, rather likely to go on increasing. In this scenario, the logistic growth models are better preferred option. Choudhary (2020) has predicted the estimated cases very early till April 7, 2020, using time series models. However, it was found to be a gross underestimation. In spite of the limitations, considering the unprecedented nature of the pandemic, uncertainties about the disease and the need for urgent but appropriate social, economic and public health responses; accurate forecasting of the size, severity and duration of the epidemic is critical to inform policies, programme and strategies.

This paper aims to carry out short-term projection of new cases using the logistic growth curve model; forecast the maximum number of active cases for India and selected high-burden states using the SIR model with correction factor based on China, Italy and South Korea; and evaluate the impact of lockdown and other interventions on the incidence of daily cases.

Methods

Modified logistic growth model

Logistic Growth is characterized by an increasing growth in the beginning, but a decreasing growth at a later stage, as it approaches the maximum. In COVID-19, the maximum limit will be the total population and the growth will necessarily come down when a greater proportion of the population is sick. The reason for using logistic growth for modelling the Coronavirus outbreak is based on the evidence that the epidemic follows an exponential growth in the early stages and expected to come down during the later stages of the epidemic. The modified logistic growth model , is presented as follows,Where.

y(t) is the number of cases at any given time t.

C is the limiting value, the maximum capacity for y. a = (C/y0)–1. b is the rate of change.

the number of cases at the beginning, also called initial value is: C/(1+a)

the maximum growth rate is at t = ln(a)/b

When y is equal to C (that is, the population is at maximum size), y/C will be 1. Therefore, the (1- (y/C)) will be 0 and hence the growth will be 0. The optimum values of the parameters can be obtained by Non-linear least square method. The future prediction of covid-19 cases was done by Time series prophet model.

Susceptible, Infected and Recovered (SIR) Model

SIR model is a compartmental model in which individuals are separated into compartments based on their infectious status and track the corresponding population sizes through time. The model divides the population into three compartments that are Susceptible (S), Infectious (I) and Recovered (R). Susceptible is the group of people who are vulnerable to exposure with infectious people. Infected are those with the disease and can transmit it to the susceptible. Recovered are those who recovered from the disease, developed immunity and not susceptible to the same illness anymore. ,

β is a transmission parameter, which is the average number of individuals that one infected individual will infect per time unit. It is determined by the chance of contact and the probability of disease transmission. γ is the rate of recovery in a specific period.

D, the average time period during which an infected individual remains infectious which is derived from γ. .

The ratio, is the basic reproduction number. R is the average number of people infected by an infected individual over the disease infectivity period, in a totally susceptible population.

In order to fit a SIR model, the parameters were obtained by minimizing the residual sum of squares between the observed active cases and the predicted active cases. We have fixed R0 and as 2.5 and 7 days respectively. , Therefore, is 0.14 and the is 0.36. The data for India was taken from the crowd sourced database available on https://www.covid19india.org and the other countries data were taken from https://www.kaggle.com/sudalairajkumar/novel-corona-virus-2019-dataset. We used R software to estimate these parameters. In order to estimate the parameters in the logistic growth model, we used Python code in Google Colaboratory (Colab) platform.

Estimation of correction factor

Invariably, the SIR model overestimates the active number of cases. In order to compute the overestimation, the actual number of reported cases from China was obtained up to April 5, 2020 and used to estimate the maximum number of active cases in China. Subsequently, the ratio of maximum (peak) active cases projected by the model to the observed peak active cases was computed. The similar estimation was done for Italy and South Korea as well. In order to choose the best correction factor that is appropriate for India, we compared the age and gender distribution of population of these three countries with the age and gender distribution of population in India. China correction factor was applied to states such as Maharashtra, Rajasthan and Tamil Nadu. As the population size in Delhi is small which is about four to five times lower than the other states, SIR model was not done for Delhi. Data that were used in the modelling is presented in appendix.

Time interrupted regression analysis

Time interrupted regression analysis was done to assess the impact of 3 weeks’ lockdown on the incidence of new cases. Dummy variable was introduced at April 15, 2020. The hypothesis was that there will be a decline in the incidence of new cases after the lock down period, that is after April 14, 2020. That is, the regression coefficient will be significant and negative in direction. As there were only 3 cases reported from Jan 03 to March 01, 2020, we excluded these time points from the analysis.

Results

Table 1 presents the predicted cumulative number of cases with 95% CI for India and states such as Maharashtra, Tamil Nadu, Delhi and Rajasthan. The predicted cumulative number of cases in India was 58,912 (95% CI: 57,960, 59,853) by May 08, 2020 and the observed number of cases was 59,695. The predicted cumulative number of cases for Maharashtra was 18,857 and the observed number was 19,063. Similarly, the predicted cases for Tamil Nadu, Delhi and Rajasthan were 5,288, 6080 and 3634 respectively. According to the model, the expected number of cases in India by May 22, 2020 is 1,02,974 (95% CI: 1,01,987, 1,03,904). The predicted cases in Maharashtra, Tamil Nadu, Delhi and Rajasthan are 32,872, 15,258 10,332 and 4783 respectively. The diagrammatic representation of Table 1 is presented in Fig. 1a, Fig. 1b a & 1b.

Table 1

Projected cumulative number of cases for India and high incidence states.

Date	India	Maharashtra	Tamil Nadu	Delhi	Rajasthan
01-05-2020	38686 (37747, 39641)	11719 (11436, 12002)	3118 (2721, 3466)	4077 (3848, 4290)	2757 (2615, 2902)
02-05-2020	41264 (40345, 42264)	12625 (12347, 12894)	3246 (2894, 3617)	4332 (4134, 4551)	2893 (2756, 3027)
03-05-2020	43992 (43032, 45021)	13544 (13255, 13811)	3512 (3137, 3874)	4628 (4434, 4834)	3027 (2888, 3170)
04-05-2020	46842 (45861, 47853)	14600 (14308, 14894)	3814 (3416, 4194)	4924 (4710, 5134)	3155 (3019, 3300)
05-05-2020	49750 (48792, 50705)	15649 (15355, 15923)	4141 (3768, 4510)	5186 (4974, 5395)	3280 (3133, 3418)
06-05-2020	52722 (51793, 53640)	16687 (16395, 16981)	4502 (4162, 4899)	5470 (5260, 5673)	3404 (3257, 3540)
07-05-2020	55784 (54839, 56803)	17787 (17483, 18072)	4878 (4521, 5226)	5767 (5555, 5958)	3518 (3363, 3660)
08-05-2020	58912 (57960, 59853)	18857 (18563, 19129)	5288 (4930, 5693)	6080 (5874, 6292)	3634 (3502, 3777)
09-05-2020	62070 (61037, 63060)	19953 (19669, 20230)	5589 (5206, 5927)	6386 (6164, 6600)	3751 (3603, 3894)
10-05-2020	65316 (64300, 66290)	21028 (20746, 21296)	6041 (5692, 6445)	6725 (6510, 6931)	3863 (3719, 4011)
11-05-2020	68619 (67635, 69616)	22203 (21925, 22490)	6542 (6172, 6916)	7059 (6852, 7267)	3967 (3827, 4098)
12-05-2020	71904 (70911, 72883)	23331 (23063, 23610)	7086 (6728, 7469)	7352 (7142, 7548)	4065 (3932, 4193)
13-05-2020	75175 (74262, 76122)	24406 (24097, 24661)	7679 (7323, 8071)	7659 (7453, 7887)	4159 (4016, 4308)
14-05-2020	78453 (77521, 79401)	25502 (25238, 25784)	8306 (7911, 8674)	7972 (7749, 8174)	4242 (4102, 4388)
15-05-2020	81709 (80683, 82688)	26526 (26250, 26789)	8987 (8614, 9320)	8291 (8075, 8512)	4325 (4177, 4476)
16-05-2020	84906 (83940, 85866)	27535 (27244, 27804)	9579 (9186, 9951)	8596 (8389, 8821)	4410 (4273, 4542)
17-05-2020	88102 (87087, 89047)	28483 (28192, 28753)	10344 (9976, 10729)	8927 (8714, 9139)	4488 (4345, 4628)
18-05-2020	91266 (90297, 92172)	29495 (29219, 29784)	11184 (10845, 11564)	9243 (9023, 9436)	4557 (4412, 4699)
19-05-2020	94326 (93390, 95376)	30427 (30145, 30708)	12091 (11710, 12444)	9511 (9301, 9718)	4621 (4476, 4755)
20-05-2020	97288 (96335, 98269)	31276 (30999, 31565)	13076 (12716, 13466)	9785 (9572, 9984)	4682 (4542, 4818)
21-05-2020	100179 (99207, 101128)	32120 (31824, 32397)	14124 (13759, 14494)	10059 (9868, 10279)	4732 (4585, 4871)
22-05-2020	102974 (101987, 103904)	32872 (32584, 33148)	15258 (14881, 15653)	10332 (10103, 10531)	4783 (4652, 4929)

Note: Figures in parenthesis () represent 95% Confidence Intervals for the projected number of cumulative cases.

Fig. 1a

Cumulative number of predicted and actual COVID-19 cases for India.

Fig. 1b

Cumulative number of predicted and actual COVID-19 cases for high incidence States.

Goodness of fit statistics

Table 2 presents the goodness of fit statistics such as R, AIC, BIC and MSE for India and the states that are studied. The R2 statistic was 0.997 for India, suggesting that could be the best model. There is less scope for improvement as the unexplained variability is only 0.03%. The range of R2 ranged from 0.958 to 0.999, implying that these are very good fit.

Table 2

Goodness of fit criteria for Logistic Growth Model.

	India	Maharashtra	Rajasthan	Delhi	Tamil Nadu
R-Squared	0.997	0.998	0.990	0.991	0.958
AIC	1329.2	663.6	636.6	699.5	595.7
BIC	1337.1	669.9	643.2	706.2	601.5
MSE	575419.8	50549.3	12803.1	28110.1	89258.5

Akaike Information Criterion;– Bayesian Information Criterion;-Mean Square Error.

Based on the analysis of data from China, the ratio between the maximum number of active cases (as of April 5, 60,005) to the observed maximum number of active cases by SIR model was 5851. Similarly, the ratio between maximum numbers of active cases to the observed maximum number of active cases in Italy was 138; and for South Korea, the ratio was 1635. As the age and gender distribution of population of China are matching with Indian scenario, we choose the best possible correction factor of China.

Table 3 presents the estimates of maximum number of active cases and the time at which it could occur for India and other states from SIR model. The maximum number of active cases in India is projected to be 57,449 on May 18, 2020, while in Maharashtra, Rajasthan and Tamil Nadu, it will be 5,089, 3324 and 3221 respectively. The corresponding peak time was expected to be June 10, 2020, June 6, 2020 and June 21, 2020 respectively.

Table 3

Projected No. of active cases in peak day adjusted based on 3 Countries data modelling using SIR model.

State	Projected Population(as on 2020)	PeakDay	Projected Number of Maximum Active Cases Adjusted Based on 3 Countries
			China(CF-5851)	South Korea(CF-1635)	Italy(CF-138)
India	1376566797	18-May-2020	57449	–	–
Maharashtra	121924973	10-Jun-2020	5089	18215	215804
Rajasthan	79584255	6-Jun-2020	3324	11895	140930
Tamil Nadu	77177540	21-Jun-2020	3221	11528	136577

Note: CF=Correction Factor.

Impact of lockdown intervention in daily incidence cases

The diagrammatic representation of the trend is presented in Fig. 2 . The results of the interrupted time regression analyses are presented in Table 4 . The model indicates a decrease of about 149 daily new cases after April 14, 2020, 3 weeks after the lockdown which is not statistically significant.

Fig. 2

Time interrupted regression model with lock down intervention on April 14, 2020.

Table 4

Regression Coefficients, 95% CI and p value of time interrupted regression analysis.

Variables	Regression Coefficients (95% CI)	p value
Days	22.8 (17.0, 28.6)	<0.001
Lock down effect measured
April 15, 2020	−148.3 (−399.9, 103.5)	0.244
Time	88.6 (73.1, 104.1)	<0.001

Discussion

There have been several studies forecasting the incident cases of COVID-19 in various countries. However, there are a little peer reviewed articles about India. Forecasting COVID-19 through appropriate models can help us to understand the possible spread across the population so that appropriate measures can be taken to prevent further transmission and prepare the health systems for medical management of the disease. It is also essential to evaluate the effectiveness of interventions so that appropriate and timely programmatic changes can be made to mitigate the epidemic.

We forecasted the number of cumulative cases for India and four other high incidence states using logistic growth model which has projected the cumulative cases very closely to the observed cases. This model is based on the current trends of the cumulative cases in India and specific states. We have used the logistic growth model due to the exponential nature of growth of the epidemic which eventually get stabilised as against pure exponential model. , 11, 12, 13, 14, 15 A study by Ranjan (2020) who used the SIR model projected 13,000 active cases by the end of May 2020. However, the total number of cases had already crossed 20,000 by April 22, 2020, which was a gross underestimation.

The SIR model with correction factor predicted 57,450 cases which will be the maximum number of active cases by May 18, 2020. However, the peak time gets pushed to June in other states. When we performed the SIR model using the reported cases from China, South Korea and Italy, we found that the model predicted more number of active cases than what they observed up to a time point for which the data was analysed. In order to address the overestimation, we formulated a correction factor which is essential to predict the epidemic accurately. Besides, as suggested by Ranjan (2020), the SIR model depends heavily on the population who are susceptible. Therefore, it may overestimate the maximum cases when the epidemic is not generalized in the population. Therefore, this could be considered as a warning signal for preparing the health systems in terms of planning treatment facilities and other interventions.

In COVID-19 epidemic, assessing the effectiveness of lockdown is one of the key interest areas. India had a head start in imposing the lockdown relatively early, in addition to strong public health measures to mitigate the spread of the epidemic. It also raises an interesting question whether this lockdown has really impacted the incidence cases. Several studies have assessed the effectiveness of interventions with varying level of results. , We carried out interrupted time series analyses that suggested no significant decline in the number of daily cases immediately after the lock down. Ironically, there is an increase in the number of daily cases immediately after the 3 weeks of lockdown period. It indicates that the lockdown and other interventions did not have any impact on reducing the number of daily cases after a certain period. This may be due to the fact that the number of tests done over a period of time has increased significantly. However, we need to revise the model every week as and when the data gets accumulated.

Limitations: As in any other projection using models, the limitation is that each model would behave differently, not merely due to differences in underlying assumptions but differences in population density, existing capacity of the health systems, current level of interventions and socio-demographic and economic situation across and within the states and districts. Therefore, district level projections may be required, which would account the variations between the states and within the states. In Covid-19, there has been a higher level of uncertainly about the number of reported confirmed cases due to the issues in varying testing strategies, the proportion of asymptomatic cases and the effective transmission rate. Because of this, we may be missing a significant number of reported confirmed cases which may affect the accuracy of any models.

In conclusion, the short term projection predicts exactly well with the observed number of cases in India and in other states through the logistic growth model. The findings from SIR model may be used for planning the interventions and prepare the health systems for better clinical management of the infected in the country and respective states.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Ethical approval

Not required.

Declaration of competing interest

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Section I.

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Section II.

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Table: Data used for modelling COVID-19 cases of India and high incidence States (till May 08, 2020)

Days	Date	India	Maharashtra	Tamil Nadu	Delhi	Rajasthan
1	30-Jan-2020	1
2	31-Jan-2020	1
3	01-Feb-2020	1
4	02-Feb-2020	2
5	03-Feb-2020	3
6	04-Feb-2020	3
7	05-Feb-2020	3
8	06-Feb-2020	3
9	07-Feb-2020	3
10	08-Feb-2020	3
11	09-Feb-2020	3
12	10-Feb-2020	3
13	11-Feb-2020	3
14	12-Feb-2020	3
15	13-Feb-2020	3
16	14-Feb-2020	3
17	15-Feb-2020	3
18	16-Feb-2020	3
19	17-Feb-2020	3
20	18-Feb-2020	3
21	19-Feb-2020	3
22	20-Feb-2020	3
23	21-Feb-2020	3
24	22-Feb-2020	3
25	23-Feb-2020	3
26	24-Feb-2020	3
27	25-Feb-2020	3
28	26-Feb-2020	3
29	27-Feb-2020	3
30	28-Feb-2020	3
31	29-Feb-2020	3
32	01-Mar-2020	3
33	02-Mar-2020	5			1
34	03-Mar-2020	6			1	1
35	04-Mar-2020	28			1	2
36	05-Mar-2020	30			2	2
37	06-Mar-2020	31			3	2
38	07-Mar-2020	34			3	2
39	08-Mar-2020	39			3	2
40	09-Mar-2020	48	2		4	2
41	10-Mar-2020	63	5		4	3
42	11-Mar-2020	71	11		5	3
43	12-Mar-2020	81	14		6	3
44	13-Mar-2020	91	17		7	3
45	14-Mar-2020	102	26		7	4
46	15-Mar-2020	112	32		7	4
47	16-Mar-2020	126	39		7	4
48	17-Mar-2020	146	41		10	4
49	18-Mar-2020	171	45	1	11	7
50	19-Mar-2020	198	48	2	14	9
51	20-Mar-2020	256	52	2	20	17
52	21-Mar-2020	334	64	5	27	24
53	22-Mar-2020	403	74	8	28	29
54	23-Mar-2020	497	89	11	30	32
55	24-Mar-2020	571	107	17	30	32
56	25-Mar-2020	657	122	25	35	38
57	26-Mar-2020	730	125	28	39	43
58	27-Mar-2020	883	153	37	40	50
59	28-Mar-2020	1019	181	41	49	54
60	29-Mar-2020	1139	203	49	72	59
61	30-Mar-2020	1326	220	66	97	79
62	31-Mar-2020	1635	302	123	120	93
63	01-Apr-2020	2059	335	233	152	120
64	02-Apr-2020	2545	423	308	293	133
65	03-Apr-2020	3105	487	410	386	179
66	04-Apr-2020	3684	635	484	445	206
67	05-Apr-2020	4293	747	570	503	266
68	06-Apr-2020	4777	868	620	525	301
69	07-Apr-2020	5350	1018	689	576	343
70	08-Apr-2020	5915	1135	737	669	383
71	09-Apr-2020	6728	1364	833	720	463
72	10-Apr-2020	7599	1574	910	903	561
73	11-Apr-2020	8453	1761	968	1069	700
74	12-Apr-2020	9211	1982	1074	1154	804
75	13-Apr-2020	10454	2334	1172	1510	897
76	14-Apr-2020	11489	2684	1203	1561	1005
77	15-Apr-2020	12371	2916	1241	1578	1076
78	16-Apr-2020	13432	3201	1266	1640	1131
79	17-Apr-2020	14354	3321	1322	1707	1229
80	18-Apr-2020	15725	3648	1371	1893	1351
81	19-Apr-2020	17305	4200	1476	2003	1478
82	20-Apr-2020	18544	4666	1519	2081	1576
83	21-Apr-2020	20081	5218	1595	2156	1735
84	22-Apr-2020	21373	5649	1628	2248	1888
85	23-Apr-2020	23040	6427	1682	2376	1964
86	24-Apr-2020	24448	6817	1754	2514	2034
87	25-Apr-2020	26283	7628	1820	2625	2083
88	26-Apr-2020	27890	8068	1884	2918	2185
89	27-Apr-2020	29458	8590	1936	3108	2262
90	28-Apr-2020	31360	9318	2057	3314	2364
91	29-Apr-2020	33065	9915	2161	3439	2440
92	30-Apr-2020	34866	10498	2322	3515	2584
93	01-May-2020	37262	11506	2525	3738	2666
94	02-May-2020	39826	12296	2756	4122	2772
95	03-May-2020	42778	12974	3022	4549	2886
96	04-May-2020	46434	14541	3549	4898	3061
97	05-May-2020	49405	15525	4057	5104	3158
98	06-May-2020	53007	16758	4828	5532	3317
99	07-May-2020	56351	17974	5408	5980	3427
100	08-May-2020	59695	19063	6008	6318	3579