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# Importance of Mathematical and Statistical models for COVID-19 Modeling

The SARS-CoV-2, which have been named as 2019 novel coronavirus disease (COVID-19) is a contagious disease. The virus that causes COVID-19 is mainly transferred through droplets caused when an infected person sneezes, or exhales and coughs. On March 11, 2020, the World Health Organization (WHO) on has declared this novel coronavirus (COVID-19) outbreak a global pandemic. Worldwide till date near about 3.2 million people got infected, out of which near about 1 million people recovered whereas at least 2, 28, 000 people has died due to COVID-19 so far.  In India nearly 38 thousand positive cases have registered, whereas more than 10,000 people return to their home after recovery whereas nearly 1,200 death cases have been recorded. Since still there is no vaccine is available for COVID-19 so far. Thus the absence of specific healing treatment or effective vaccine against it, scientist looking for other possibilities of the prevention and control measures. In this scenario modelling pursues to judge the likely impact of various non-pharmaceutical justification and dominance strategies on the spread of the virus and the number of deaths. Few questions may arises in almost every one’s mind that are: (i) will the COVID-19 disease infect millions in India? (ii) How many are likely to die due to this infection?

Results from mathematical and statistical models (studies) may not give accurate answers to these questions but may helpful to understand nature of spreading of  the COVID-19 virus and its impacts.  Models may also help to shed new light on this virus. By comparing predictions with field of statistics, scientists may detect those parameters that explain potential differences. They may be then deduce any information that would otherwise have escaped them, and refine their systems. Now as we all know the situation is changing day by day and so it is too important to understand the true magnitude and behavior of the pandemic is important in framing public policies. Scientists from different science fraternity are tried their best to find out the remedies from this pandemic situation. Mathematician as well as Statistician are also working hard to find the best out of way to model or forecast the nature of spreading and understand the impact of COVID-19. Here is some mathematical and statistical models which may help Scientist or researcher to predict or forecast the spreading COVID-19 virus and its influence.

Mathematical Models:

(i) SIR model:  This model consists of three compartments: the number of susceptible (S), the number of infectious (I), and the number of recovered (R) individuals. It is one of the simplest compartmental models and from this basic form many models can be derivative. The basic SIR model uses equations that take the assumptions and determine what percentage of S (the Susceptibles) are infected and move to (the Infectious), what percentage of I die and what percentage of I recover and move to R. These models often have additional parameters that govern how the infection rate, recovery rate, and death rate change at particular times.

(ii) SEIR model: In this model having four compartments such as susceptible (S), exposed (E), infected (I), and resistant (R). Each of those variables represents the number of people in those groups. Recently a mathematical model is developed based on susceptible-exposed-infectious-recovered (SEIR) model to describe the COVID-19 transmission dynamic in Korea.

(iii) SSqEqIHRM model: Recently this model has been used to study of mitigation spread of COVID-19 in the early phase of the outbreak in Wuhan, China (Weike Zhou et al., 2020). In this model, the population is divided into seven compartments, including susceptible (S), quarantined susceptible (Sq), infectious without symptoms (or exposed E), quarantined exposed (Eq), infectious with symptoms (or infected I), hospitalized (H) and recovered (R), M represents the cumulative density of awareness programs driven by the media reports.

Statistical models: The statistical models are supposed to be unsuccessful, often because of one or more key assumptions were not adjusted to reflect in the current data set. Therefore, one of the major exercise is to develop a dynamic model in which the process of ensuring an effective feedback loop to stress on test the assumptions, and revive the model with new reliable information. Here is the two models which may useful in modeling COVID-19 virus.

(i)  ARIMA model: Autoregressive integrated moving average (ARIMA) models can be treated as one of the simplest and direct tool to which may use to monitor national and regional level the health monitoring system. ARIMA model may be completely summarized by three numbers: p= the number of autoregressive terms, d = the number of non-seasonal differences and q = the number of moving-average terms. A standard notation is used of ARIMA (p, d, q) where the parameters are substituted with integer values to quickly indicate the specific ARIMA model being used. A value of 0 can be used for a parameter, which indicates to not use that element of the model. It may be used to forecast the epidemic trend over the period. An ARIMA-type models have proved to be useful to a very wide range of physical and human-generated dynamic systems in engineering, econometrics, Earth sciences, and other fields (Feigelson ED et al, 2018). One of the advantage of this model to surely forecasting approach and its ease of application and interpretation.

(ii) ARFIMA model: The autoregressive fractional integral moving average (ARFIMA) models are time series models that generalize ARIMA models by permitting fractional values of the differencing parameter (Granger C.W.J and Joyeux .R, 1980). In a simple manner, ARFIMA (p, d, q ) is a time series model where the autoregressive and moving average components treat many short-memory dependencies on recent past values, and the fractional integration component treats both many forms of trend and also a class of long memory models.

The mathematical and statistical models discussed above may be useful for the scientist to do modeling of spreading of COVID-19, if analysts would collect correct data from the medical authorities that distinguish between deaths caused by COVID-19 and deaths that would have happened anyway. The useful and more accurate forecasting may be provided these models provided scientists or researchers bring up to date data related to COVID-19 cases and applying the models to other countries as well.

References:

1. Weike Zhou, Aili Wang, Fan Xia, Yanni Xiao and Sanyi Tang, (2020) Effects of media reporting on mitigating spread of COVID-19 in the early phase of the outbreak, Mathematical Biosciences and Engineering Vol- 17(3), pp-2693–2707
2. Granger C.W.J and Joyeux .R, 1980. An introduction to long memory time series models and fractional differencing, Journal of Time series Analysis, 15-29.
3. Feigelson ED, Babu GJ and Caceres GA (2018) Autoregressive Times Series Methods for Time Domain Astronomy.  Phys. 6:80. doi: 10.3389/fphy.2018.00080
4. http://www.public.asu.edu/~hnesse/classes/seir.html.
5. https://eandt.theiet.org/content/articles/2020/04/predicting-the-pandemic-mathematical-modelling-tackles-covid-19/

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