Epidemiological Models of SARS-CoV-2 (COVID-19) to Control the Transmission Based on Current Evidence: A Systematic Review

In Wuhan city of China, an episode of novel coronavirus (COVID-19) happened. during late December and it has quickly spread to all places in the world. Until May 29, 2020, cases were high in the USA with 1.7 Million, Russia with approximately 387 thousand, the UK with 271 thousand confirmed cases. Everybody on the planet is anxious to know when the coronavirus pandemic will end. In this scourge, most nations force extreme medication measures to contain the spread of COVID-19. Modeling has been utilized broadly by every national government and the World Health Organization in choosing the best procedures to seek after in relieving the impacts of COVID-19. Many epidemiological models are studied to understand the spread of the illness and its prediction to find maximum capacity for human-to-human transmission so that control techniques can be adopted. Also, arrangements for the medical facilities required such as hospital beds and medical supplies can be made in advance. Many models are used to anticipate the results keeping in view the present scenario. There is an urgent need to study the various models and their impacts. In this study, we present a systematic literature review on epidemiological models for the outbreak of novel coronavirus in India. The epidemiological dynamics of COVID-19 is also studied. Here, In addition, an attempt to take out the results from the exploration and comparing it with the real data. The study helps to choose the models that are progressive and dependable to predict and give legitimate methods for various strategies.


Introduction
Coronavirus is very infectious and can induce disorder in individuals with multiple complexities.
It can lead to Middle-East-Respiratory-Syndrome (MERS) and Severe-Acute-Respiratory-Syndrome (SARS). It has now tainted practically all pieces of the world as yet unfurling. The introductory cases were recorded in December 2019 and presumably spread by the contact through your eye, nose, and mouth. Signs differ from individual to individual after infection leading to fever, throbbing-painfulness, nasal-blockage, migraine, conjunctivitis, sore-throat, loose-bowels, loss-of-taste, and smell, or a rash on skin or staining of fingers or toes. These side-effects are typically gentle and start consistently while some become contaminated and take a serious adverse effect on health [1].  February, over 2,000 affirmed cases were accounted for, ascending to 3,150 on 29 February.
WHO has considered Europe the dynamic focal point of the pandemic starting on 13 March.
Cases by a nation over Europe had multiplied multiple times in commonly 3 days, with certain nations (for the most part those at prior phases of identification) indicating multiplying like clockwork [5]. On 31 January, The outbreak was confirmed to have spread to Italy. highest in states which includes the major industrial cities of the country. Figure 2c shows the changes in active cases, as well as deaths. From Figure 2d, we can conclude that the cases since the 100th case are high in India as compared to other countries [7].
Epidemiological models planned for understanding the human-to-human transmission and the control techniques to stop the transmission. The mathematical modeling studies may help to provide evidence and insights into the transmission through the environmental medium like air, and water. For India, there are different models used to decipher the information and predict the results of COVID-19. There are epidemiological models displaying the investigation of COVID-19 including the models like dynamic models, machine learning, and statistical models. Considering these examinations we are assessing the ends by recognizing qualities, confinements, and potential usage. In this paper, we are trying to take out the outcomes from the research and comparing the outcomes with the real-life scenario and evaluating their strengths, limitations, and potential implementations. This paper is divided into three sections as the introduction, mathematical models of COVID-19, and conclusion/discussions.

Epidemiological models of COVID-19
Mathematical models are being extensively used in the study of disease expansion and forecasting worldwide. Epidemiological-models are quite effective for foretelling the number of fresh cases or for identifying the best measures to diminish transmission. The researchers throughout the world have studied numerous mathematical-modeling and numerical-analysis on COVID-19 since its outbreak. This section will describe the past studies towards the growth of various mathematical-models of COVID-19 in India. Critical features of these models are a lifelike portrayal of the long-term physiological behavior, forecast of COVID-19 cases, and its prevention. Some of these forecasts also include environmental factors that are affected due to conditions like temperature, humidity, and transmittance through water and air.

SIR Models
Practically all numerical models of sicknesses start from a similar fundamental reason:

SIS Models
The SIS model is utilized for a given final population susceptible to a distinct disease and

SIRS and SIRD Models
In SIRS, the individual becomes susceptible again as resistance wanes, i.e. recovered individuals maintain only short-term resistance or no resistance. And as per the data shown by WHO and MoHFW, there are many cases of the recovered patient being infected again by COVID-19. Hence, this must be considered as one of the major models related to COVID-19.
The mathematical equations are given below. The Parameter denotes the degree of recovered individuals returns to the susceptible state being a deadly disease has a large number of people dying due to it. This model has the following system of differential equations. The new parameter represents the rate of mortality.

SEIR Models
For many diseases, there is a vital incubation phase through which somebody has been affected but is not yet contagious. During this phase, the individual is in Exposed (E) condition.
This model is important as many precautions are being taken to prevent exposure of infected persons like Lockdown, curfew, and home quarantining of people. The parameter α is the rate of being infected if exposed. This basic model has the following system of differential equations with Q (for quarantined cases -confirmed and infected) and mortality rate κ(t) = κ0 exp(−κ1t) are time-dependent and average quarantine time δ -1 . Predicting estimated total deaths by the end of the epidemic could be around 33000, which is around 4% of total estimated cases. For India, predicted value till May 19 is to reach 30,000. This model also failed, as on 19 May the numbers were even more than 100,000. Maji A. et al.
[33] used the same model on May 9 to predict the state-wise predictions on the major states of India. In comparison with real data, the predictions have a difference of around 25% of the real data. Sardar T. et al. [34] proposed two different models to be used with and without lockdown on April 7. Both models are based on SEIR with added asymptomatic A(t), hospitalized or notified C(t) population, and population in the lockdown compartment L(t) -SLEAICR. The model equations with lockdown are as follows where is recovery rate for asymptomatic infected, is the recovery rate for symptomatic γ 1 γ 2 infected, represents the recovery rate for hospitalized or notified individuals, is the γ 3 ρ reduction in COVID-19 transmission for asymptomatic infected, is a fraction of exposed κ population that become symptomatically infected, is incubation period for COVID-19, is /σ 1 δ the death rate of hospitalized or notified population, and 1/ are lockdown success rate and , ι ω lock-down period respectively. In this model, the reproduction number R 0 is defined as β κσ/((μ σ)(γ τ μ)) ρβ (1 κ)σ/((μ σ)(γ τ μ)) R 0 = 1 + 2 + 2 + + 1 − + 1 + 1 + The above model was implemented in three different states and overall India. According to the final results, they predicted that on May 7 the daily cases will go down to zero in Delhi and Tamil Nadu and will reach a peak of 1300 in Maharashtra and around 8000 in overall India.
Considering the current data, we had around 450, 580, 1200, and 3344 confirmed cases added on May 7. We found a close relation in the Maharashtra data but there is a huge difference in other states' data.  April 29. The SEQIR model with a quarantined (Q) population is given as The parameter values for simulation results are N = 1, 400, 000, 000, α 1 = 0.   [41] used two reproduction numbers R 0 = 1.5 and 4 for optimistic and pessimist, respectively. They also used the SEIQR model for four major states in India. They proposed multiple conclusions including optimistic/pessimist for with/without interventions. Pai C. et al. [42] used the SEIR model on April 28. The parameter β is optimized during the fitting procedure. The results predicted that the infected cases will be at a max of 43,000 in mid-May but current data shows cases crossing 100,000 around mid-May.
Hence, the prediction failed this may be due to the fact that the ratio in which people the interventions are different or the population density is very different in both countries.
Chowdhury R. et al. [43] used SEIPHURD with infected individuals (P), the hospitals (H), and patients who stay in ICU (U). The proposed model is given as The Therefore, predicting the max values of 2,200,000 at around mid-August. Ravinder R. et al. [49] an adaptive interacting cluster-based SEIR (AICSEIR) model and applied it to multiple countries. The India model was constructed with R 0 = 2 to 6. According to the predictions of the model, the current pandemic size in India must be around 50 million but we have just crossed 100,000. Marimuthu Y. et al. [50] used the SEIR model on May 12. They used their previous model for TB patients in Delhi and added COVID-19 to it. Predicting COVID-19 infected TB cases. They predicted that due to the interventions the number can be lower by 30% and also delaying the peak cases by 44 days. Gupta [51] used SEIR-QDPA with Qa (detected and quarantined asymptomatic), Qs (detected and quarantined symptomatic), Ru (undetected recovered asymptomatic), Ra (recovered detected asymptomatic), Rs (recovered detected symptomatic), D (dead) and P (protected; non susceptible). They have used R 0 = 2.083. Verma V.
R. et al. [52] used a dynamic model based on SEIR for predicting the number of hospital beds required. Concluding that augmented testing of 500,000 tests per day during peak (mid-July) under a moderate lockdown scenario would lead to more reported cases (5,500,000-6,000,000).
Gupta R. et al. [53] proposed a model on April 3. They have used the SEIR and regression models to predict the number of cases with R 0 = 2.01. Comparing their prediction with real data none of the SEIR or regression models matches real data. There is a difference of about 40% in the regression model and about 50% in the SEIR model. Chatterjee K. et al. [54] used the following SEIQRD model for the outbreak of COVID-19 Comparing the predictions with real data, the predictions seem to match with current data. Furthermore, the model predicts that the number will increase until July end and can reach up to 80 million. L. et al. [64] proposed an exponential model with Herd immunity. After calibrating the model, they predicted the active cases to reach 250,000 by April 30, But even the current data hasn't reached that number. Mishra P. et al. [65] proposed used an exponential regression model.

ARIMA and Exponential Models
Comparing the predictions with real data, the predictions matched real data for a long time then started to deviate. Currently, the model prediction lags within 10 days. Singh B. P. [66] proposed a model on May 5. They have used multiple growth models including linear growth, exponential growth and logistic growth models to predict future epidemic sizes. Comparing the prediction with real data, the predicted value comes at about 50% of the current data for all predicted values.

Machine Learning based Models
Sujatha R. [67] proposed a machine learning method based on feedforward artificial neural networks (FANN). The model is as follows Pasayat, A. K. et al. [68] proposed a model on May 20. They have applied machine learning algorithms to improve exponential growth and linear growth models. Compared with real data, the real data is following the exponential growth pattern given by the model. Tiwari U. et al. [69] proposed a machine learning-based model to predict the outcomes of COVID-19. The model's aim is to analyze whether the case of COVID-19 in India is going to be the same as in Italy or South Korea. The answer is yes. India might be going to face its worst days in the future if we look at the pattern of these countries and India too. Tiwari S. et al. [70] proposed a machine learning model. In the model, the outbreak in India has been predicted based on the pattern of China. Comparing the prediction with real data, the model predicts 68,978 cases by April 25, but the real values have reached around 24,000. This may be due to the fact that people are more aware of the disease now and are following interventions, similar was introduced by Punn N. S. et al. [71]. This system uses 11 symptoms like age, sex, fever, dry cough, breathing problem, flu and cold, medical history, travel history, and two recently identified symptoms by some infected patients anosmia (lost sense of smell), loss of hearing ability. In this study, they shown that if fever is yes, cough is yes then the probability of coronavirus infection seems to be positive, similarly, if fever is greater than 38 and cough is dry which comes in the category of yes and patient is facing difficulty in breathing and he has also a travel history to some infected countries than the

Other Models
There are many authors working on other than the above-mentioned models like arithmetic progression, polynomial based, network-based etc. Summary of these model is as

Discussion and Conclusions
About 95 significant contributions have been reviewed to explore the outbreak of COVID-19 in India by the mathematical approach. The demonstration for a progressing episode of COVID-19 is as yet a difficult errand at this phase of the pandemic. We locate that most existing models of COVID-19 are ordinarily founded on plague dynamic models instead of the factual models or AI. In this study, various authors have included the lockdown as a part in their model [8,20,24,37,52,74,83,84,93]. They have shown that the susceptibility rate decreases due to the containment of people and providing help in maintaining social-distancing.
Hence, we can conclude that the transmission rate has decreased due to lockdown implementation. The legislature carefully followed the initial two lockdowns. However, lockdown 3.0 -4.0 the administration gave more unwinding to people in general and expanded the number of testing. Numerous forecasts before lockdown 3.0 are excellent. After lockdown 3.0, the charts for various states are extremely dissimilar. Also, many authors had shown the linkage of COVID-19 with environmental factors. The summary is as follows • The prediction of required ICU beds for COVID-19 patients was given [43,52]. Being able to predict the number of ICU beds will help in the arrangement of proper facilities and providing better healthcare.
• The P-value of water and air for transmission is determined [5]. This study had shown that the P-value of water and air is enough that the transmit of medication through these mediums is possible and will reach the majority of the population.
• The study of temperature and humidity effect on COVID-19 spread in India was given [94]. They have shown that India's weather is unfavorable for people. Also, many states might have greater weather-based spread in the future.
• Also, most of the models have shown that there is a high death rate associated with COVID-19. Hence, the burying of these dead bodies may cause the soil-based transmission otherwise burning will cause various sorts of pollutants introduction in the environment.
There is as yet extensive vulnerability to evaluate the epidemiological attributes because of the novel idea of COVID-19 and the beginning period of the outbreak. Certain techniques present some inborn confinements because of deficient information and restricted information sources. Extra research is critically important to satisfy such research holes. We accept that as the pandemic arrives at the end and more information can be gathered, epidemiological models can be improved so as to introduce a genuine impression of the full picture. This survey will assume a significant job in future research to grow new scientific models for foreseeing and investigating the conduct of novel coronavirus (COVID-19).
The results of this review offer an overview of the current use of mathematical models in the context of the study of COVID-19 and indicate that direction of future research is needed. In the current work, we can without much of a stretch see which mathematical model has great outcomes in contrast with different models under Indian circumstances. The aftereffects of this survey offer an outline of the ebb and flow utilization of scientific models with regards to the investigation of COVID-19 and show that future research is required. In this investigation, we have attempted to sum up all the scientific models of COVID-19 in India which will help in giving an increasingly comprehensive structure to the analysts and specialists for picking up knowledge into the determinants of expectation and controlling the novel coronavirus.
Furthermore, these models can be related to further studies depicting the effect of lockdown on environmental factors such as pollution index, and temperature variation due to a decrease in industrial and vehicle activities. Also, these models will help in expanding more information about the example and epidemiological example of the COVID-19 which can help the governing body in making certain steps concerning prosperity gauges like lockdown, purification, movement, medicinal administrations, and to be set up for the cutting-edge conditions. This review will help in scientific and information-driven demonstrated proof and bits of knowledge into the transmission, seriousness, and identity of the malady, which could promptly advise the dynamic in battling the sickness.