Prediction of Second Wave of COVID-19 in India using the Modifed SEIR Model
Changqing Sun Zhengzhou University Mingyang Zhao Zhengzhou University Zhuoyang Tian Zhengzhou University Wensen Zhang Zhengzhou University Hengzhen Zhang Zhengzhou University Wenqian He Zhengzhou University Rongrong Wang Zhengzhou University Ke Wu Zhengzhou University Biyao Wang Zhengzhou University Nan Sun University of Georgia Weihong Zhang Zhengzhou University Qiang Zhang ( [email protected] ) Zhengzhou University https://orcid.org/0000-0003-1566-1955
Research Article
Keywords: COVID-19, pandemic, SEIR model, the second wave, the Delta variant
Posted Date: August 23rd, 2021
DOI: https://doi.org/10.21203/rs.3.rs-800978/v1
Page 1/17 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License
Page 2/17 Abstract
Background: The second wave of the coronavirus disease 2019 (COVID-19) epidemic in India was caused by the COVID-19 Delta variant. However, the epidemiological characteristics and transmission mechanism of the Delta variant remain unclear. To explore whether the epidemic trend will change after effective isolation measures were taken and what is the minimum number of individuals who need to be vaccinated to end the epidemic.
Methods: We used actual data from March 5 to April 15, 2021, of daily updates confrmed cases and deaths, to estimate the parameters of the model and predict the severity of possible infection in the coming months. The classical Susceptible-Exposed-Infected-Removed (SEIR) model and extended models [Susceptible-Exposed-Infected-Removed-Quarantine (SERIQ) model and Susceptible-Exposed- Infected-Removed- medicine (SERIM) model] were developed to simulate the development of epidemic under the circumstances of without any measures, after effective isolation measures were taken and after being fully vaccinated.
Results: The result demonstrated good accuracy of the classic model. The SEIRQ model showed that after isolation measures were taken, the infections will decrease by 99.61% compared to the actual number of infections by April 15. And the SEIRQ model demonstrated that if the vaccine efcative rate was 90%, when the vaccination rate was 100%, the number of existing cases would reach a peak of 529,723 cases on the 52nd day.
Conclusion: Effective quarantine measures and COVID-19 vaccination from ofcial are critical prevention measures to help end the COVID-19 pandemic.
Introduction
The Coronavirus disease 2019 (COVID-19) is a new respiratory infectious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Since early December 2019, COVID-19 infections have occurred in many countries, and the number of COVID-19 cases has increased dramatically. On March 22, 2020, the World Health Organization declared a world pandemic [1]. Due to the lack of effective vaccines and therapeutic drug, governments have taken a series of measures to delay the transmission of COVID- 19, including case isolation and travel restrictions. These measures have helped several Asian countries represented by China achieve signifcant progress in the early stages of the epidemic [2].
As a vast and densely populated country, India is facing greater challenges in coping with the COVID-19 due to the inadequate and inconsistent federal public health infrastructure. In response to the COVID-19 pandemic, the Indian government implemented a nationwide quarantine measure. Due to the timely strictly implementation of quarantine measures by the government, the total number of early infections in India was lower than that in other countries [3]. However, there were still many problems in the isolation measure. For example, people's awareness of the seriousness of the disease was insufcient and the
Page 3/17 government supervision was not strict. Therefore, the situation in India further deteriorated after the lifting of the quarantine measures on May 3, 2020.
In early March 2021, India broke out the second COVID-19 epidemic, which was caused by the Delta variant of COVID-19 [4]. The infectivity and mortality of the Delta variant are higher than that of the common COVID-19 virus, and the transmission speed is faster at higher temperature [5]. Within two months, the continuous COVID-19 epidemic resulted in more than 60,000 deaths in India, and has plunged the society into chaos and panic. Currently, the epidemic characteristics and transmission mechanism of the mutant virus remained poorly understood, and how to deal with mutant virus is still an urgent problem to be solved.
The Susceptible-Infected-Removed (SIR) model is a classic mathematical modeling which was frequently used to simulate the dynamic mechanism of infectious diseases [7]. However, the classic SIR model only takes three compartments into consideration.
In this study, we used the extended SIR models, SEIR [8], SEIRQ and SEIRM, to simulate the second epidemic in India, which has taken isolation and vaccine factor into consideration, to explore whether the epidemic trend will change after effective isolation measures are taken and what is the minimum number of people who need different vaccines to end the epidemic.
Material And Methods Formulation of SEIR model
The SEIR model provides a practical quantitative research method for the analysis of the epidemiological characteristics of infectious diseases. The model was constructed based on the following assumptions. (1) The asymptomatic infected persons were not considered. (2) No consideration for reinfection. (3) The impacts of birth rate, death rate and immigration were not considered. (4) The model only considered the propagation dynamics in the natural state. In this model, the target population is divided into four compartments, including Susceptible (S), Exposed (E), Infected (I) and Removed (R). Individuals in SEIR move from one compartment to another based on basic parameters, simulating the spread of the disease through the population [9].
In the SEIR model (Fig. 1a), where N is the population size, β is the infection rate, γ is the removed rate, and σ is the incidence rate. Eq. 1 shows the system of ordinary different equations used to determine how much of the population is within each group at a specifc time for the model. Equation 1
Page 4/17 Formulation of SEIRQ model
In the absence of effective vaccines during the epidemic period, isolation measure was an effective measure to control the spread of infectious diseases. Therefore, the isolation factor was included in the model to analyze the trend of the epidemic in India under the implementation of effective isolation measures.
In the SEIRQ model (Fig. 1b), Q represents the population isolated after illness, α is the isolation rate, ω represents the probability that an isolated person will recover or die. Eq. 2 shows the system of ordinary different equations used to determine how much of the population is within each group at a specifc time for the model.
Equation 2
Page 5/17 Formulation of SEIRM model
Vaccination was the most effective means of prevention and control of the COVID-19 epidemic. The vaccine factor was included in the model to analyze what was the minimum number of people who need to be vaccinated with different vaccines to end the epidemic.
In the SEIRM model (Fig. 1c), M represents the population with effective antibodies after vaccination, λ is the average daily vaccination rate within 70 days of vaccination (According to reports, the number of single-day vaccination in China could reach up to 20 million. With reference to this vaccination rate, it was estimated that the full vaccination in India would be completed within 70 days), µ is the effective rate. Eq. 3 shows the system of ordinary different equations used to determine how much of the population is within each group at a specifc time for the model. Equation 3
Page 6/17 Main parameter settings and descriptions
Due to the lack of data on the second wave in India, we referred to the literature on the epidemic in India in 2020 and combined it with actual case data for parameter estimation. The main parameter settings and detailed descriptions of each parameter were shown in Table 1.
Page 7/17 Table 1 The main parameter settings Parameters Description Parameter Parameters values of the source
S (0) The initial susceptible population 1353713978 actual data 1
E (0) The initial exposed population 156008 data ftting
I (0) The initial infected population 181868 actual data 2
R (0) The initial removed population 0 actual data 3
N The total population 1354051854 actual data 4
Q (0) The initial isolated after illness population 0 model assumption
M (0) The initial population with antibodies after vaccination 0 model in a susceptible population assumption
β The probability that a susceptible person would 0.6537 data ftting become ill after coming into contact with an infected person
σ The probability that the latent patient developed 0.1294 data ftting symptoms and became the infected person
γ The probability that an infected person would recover 0.3485 data ftting or die
α The isolation rate 1; 0.5; 0.3; model 0.1 assumption
ω The probability that an isolated person would recover 0.3491 data ftting or die
λ The average daily vaccination rate within 70 days of 1/70; 1/100; model vaccination in the susceptible population 1/140 assumption
µ The effective rate of producing effective antibodies 0.9; 0.7; 0.5 model after vaccination assumption
Notes: 1、2、3: Data was collected from WHO daily updates;4༚Data was collected from World Bank Demographics.
Model analysis
Page 8/17 First, the SEIR model was formulated to simulate the number of daily infections and then compared with the actual number of infections. The average percentage error (APE) was used to evaluate the accuracy of the model, the smaller the APE value, the better the model ft. An APE value༜0.3 is generally considered a good ftting effect. After taking the isolation and vaccination measures into account, the SEIRQ and SEIRM models were developed to predict the future trend of the epidemic.
Data sources
We obtained the daily updates data from March 5 to April 15, 2021, of daily updates laboratory-confrmed cases, recovered, and deaths in India from World Health Organization (https://www.who.int/). All the analysis was performed in MATLAB 2021a.
Results SEIR model ftting
As shown in Fig. 2, the average percentage error was 0.12, indicating that the accuracy of the model was good. SEIRQ model ftting
The results are shown in Fig. 3. Due to the isolation management of cases, the unisolated infected population showed a trend of rapid decline in the early stage, and then, due to the difference of isolation intensity, the trend of the epidemic presented different trends. After isolating all infected people, the epidemic had been on a downward trend, with the number of existing cases standing at 6,167 on April 15, a 99.61% decrease compared to the actual number of cases. After isolating the infected by 50%, the epidemic had shown a downward trend, with the number of existing cases standing at 47,132 on April 15, a 96.98% reduction compared to the actual number of cases. After isolating the infected by 30%, the epidemic was on the rise, and the number of existing cases would be 152,079 on April 15, a decrease of 90.27% compared to the actual number of cases. With 10 percent of the infected isolated, the epidemic was on the rise, with 658,369 cases remaining on April 15, a 57.90% percent decrease compared to the actual number of cases.
SEIRM model ftting
If the vaccine efcacy rate was 90%, when the vaccination rate was 100%, the number of existing cases would reach the peak of 529,723 cases on the 52nd day. When the vaccination rate was 70%, the number of existing cases would peak at 1,041,780 cases on day 72. When the vaccination rate was 50%, the number of existing cases would peak on day 97 with the number of cases at 2,472,460 as shown in Fig. 4a
Page 9/17 If the vaccine efcacy rate was 70% and the vaccination rate was 100%, the number of existing cases would peak at 832,850 cases on day 65. When the vaccination rate was 70%, the number of existing cases would peak at 1,944,240 cases on day 90; When the vaccination rate was 50%, the number of existing cases would not peak within 100 days as shown in Fig. 4b.
If the vaccine efcacy rate was 50% and the vaccination rate was 100%, the number of existing cases would peak at 1,840,330 cases on day 89. When vaccination rates were 70% and 50%, the number of existing cases would not peak within 100 days as shown in Fig. 4c.
Discussion
In this study, three infection disease models were constructed using the actual number of infections of COVID-19 from March 5 to April 15 in India. The results showed that when effective isolation and vaccination measures were taken, the number of infections will be greatly reduced by 99%.
During the second wave of epidemic in India, the number of newly confrmed cases rose rapidly every day. As of May 7, there were 401,326 newly confrmed cases per day, and 3,931,673 existing cases. Due to the sharp increase in cases and the shortage of medical supplies in the India federation, the medical and health system is on the verge of collapse, and the society is facing huge panic [5].
During the epidemic, in the absence of effective vaccine, isolation is an effective measure to control the spread of infectious diseases. Therefore, we incorporated isolation measures into the model, and simulated the impact of the measures of different intensities on the number of existing cases. The results showed that when the effective isolation rate of the infected population reaches 0.5, the current infections will continue to decline, and as the effective isolation rate increases, the decline will be further increased. The prediction results of the model suggested that isolation measure can effectively control the epidemic. In addition, even if the isolation rate is low and the epidemic rebounds again, the current number of infections as of April 15 is far smaller than the actual number of cases. During the second wave of the epidemic, the India government claimed to have implemented isolation measure to control the epidemic [10], however based on the model ftting results, the measures had little effect. According to reports, the Indian government was unable to guarantee the normal supply for residents during the implementation of isolation measures, as a result that residents sneaked out of their homes at the risk of being infected [11], which has led to the ineffectiveness of the isolation measures. On April 14, millions of Indians poured into the Ganges to celebrate the festival, providing a breeding ground for the spread of disease. Formulating of measures does not mean achieving the ultimate goal. After the measures were formulated, how to implement them smoothly and how to achieve the expected results are the issues that those in power should consider.
Vaccination is the most effective prevention method to control COVID-19 epidemic. In the current study, we assumed that all patients will be vaccinated within 70 days, then we incorporated the vaccination into the extended SEIR model and simulated the future epidemic trend within 100 days from the start of
Page 10/17 vaccination on March 5 to one month after vaccination. Currently, India has approved three vaccines- Covishield, Covaxin and Russia's Sputnik V, and their effective rates are 70.42%, 60% and 95%, respectively [12]. Therefore, we simulated the development of the epidemic of COVID-19 with vaccine effective rate of 50%, 70%, and 90%, under different coverage rates. The results demonstrated that with the increases of effectiveness of the vaccine and the number of people vaccinated, the India epidemic will hit an early infection point.
The impact of vaccines on the trend of the epidemic has played an important role in guiding the prevention and control of the epidemic. The Indian government is currently expanding vaccine- manufacturing capacity and encouraging more individuals to be vaccinated. As of April 15th, more than 110 million people in India had been vaccinated, but judging from the actual number of infections, the current vaccination has not had a positive impact on the prevention and control of the epidemic. The possible reasons are as follows. First, In India, the priority population for vaccination is the upper class with relatively lower infection rate, however, people living in slums with high infection rate have a lower vaccination rate, which leads to most of the vulnerable population not being effectively protected [13]. Furthermore, the second wave of the epidemic in India was caused by a mutated virus strain, and the current vaccine maynot be effective as expected. Therefore, to better control the epidemic, the Indian government still needs to continue to expand the vaccination population and try to vaccinate more efcient vaccines.
There are several strengths of this study. First, additional factors were incorporated in the extended SEIR models, which makes the ftting effect of the model closer to the real situation. Second, this study provides guidance for early prevention and control of the epidemic under the circumstances that the underlying mechanisms of the delta coronavirus mutation remained unclear.
There are some limitations of this study. First, to better analyze the impact of isolation and vaccine factors on the epidemic, other parameters were combined and simplifed as much as possible, which might lead to overftting of the model. Second, the data used in this study was obtained from the WHO, and the authenticity of the data is subject to the ofcial announcement. However, during the second wave of epidemic in India, the upper limit of single-day testing of medical institutions might affect the authenticity of the data. Finally, the model tends to predict more optimistic result than the actual situation. The current study predicted the vaccination situation in India based on the highest single-day vaccination in China, however, the current single-day vaccination in India was less than 3 million [14].
Conclusion
The second wave of the epidemic in India was caused by the COVID-19 Delta variant. And the epidemiological characteristics and transmission mechanism of the Delta variant remain unclear. Fortunately, effective quarantine measures and COVID-19 vaccination from ofcial are critical prevention measures to help end the COVID-19 pandemic.
Page 11/17 Abbreviations
Abbreviation Full Name
COVID-19 coronavirus disease 2019
SARS-CoV-2 severe acute respiratory syndrome coronavirus 2
SIR Susceptible -Infected-Removed
SEIR Susceptible-Exposed-Infected-Removed
SEIRQ Susceptible-Exposed-Infected-Removed-Quarantine
SEIRM Susceptible-Exposed-Infected-Removed- medicine
APE average percentage error
Declarations Ethics approval and consent to participate
Not applicable
Consent for publication
Not applicable
Availability of data and materials
The datasets used and/or analyzed during the current study are publicly available from (https://www.who.int/).
Competing interests
The authors declared no conficts of interest.
Funding
Changqing Sun was partially supported by grants from National Social Science Fund Project [20BRK041]. Yibin Hao was partially supported by grants from Health Commission of Henan Province Fund Project [LHGJ20200681]. The funder had no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Page 12/17 Authors' contributions
MYZ performed data analysis and wrote the manuscript. ZYT, WSZ, HZZ, WQH, RRW, KW, BYW, NS collected and organized the data. WHZ gave suggestions while revision. CQS and QZ conceived and initiated this project, provided advice on experimental design, oversaw the implementation of the statistical method, and revised/fnalized the manuscript. All authors read and approved the fnal manuscript.
Acknowledgements
We would like to express our great attitude to WHO and World Bank Demographics for making the COVID- 19 data publicly available. We appreciate PhD Yibin Hao for his support. Finally, we appreciate all the medical workers for your sacrifces during the epidemic.
References
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Figures
Figure 1 a. The fowchart of the SEIR mode b. The fowchart of the SEIRQ model c. The fowchart of the SEIRM model
Page 14/17 Figure 2
The total number of infections based on the SEIR model against the real data in India between March 5 to April 15.
Page 15/17 Figure 3
The total number of infections based on the SEIRQ model in India between March 5 to April 15 for different isolation rates.
Page 16/17 Figure 4
The total number of infections based on the SEIRM model in India between March 5 to May 15 for different effective rate of vaccination, corresponding to effective rate of 0.9 (a), 0.7 (b) and 0.5 (c).
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