key: cord-0059577-v25yims9
authors: Anitha, K.
title: Predictive Analysis of COVID-19 Transmission: Mathematical Modeling Study
date: 2021-02-16
journal: Emerging Technologies for Battling Covid-19
DOI: 10.1007/978-3-030-60039-6_15
sha: 5815e68e151e5a68d8e01e46ea169d522be72cab
doc_id: 59577
cord_uid: v25yims9

A novel COVID-19 caused by the virus SARS-CoV-2 was identified at China in December 2019, and it was declared as global pandemic on 11th March 2020 by the World Health Organization. In India on 30th January 2020, the first COVID-19 case was confirmed at Kerala. As of 6th June 2020, 6,702,662 cases have been reported in India even though stringent nationwide lockdown has been implemented. It is essential to identify the rate of disease transmission and its characterization. ICMR stated that 80% of COVID patients in India are without symptoms. In that case, we should focus on identifying the optimal attributes that will give the rate of transmission of this disease. Statistical machine learning is a powerful tool to fight against COVID-19 which gives the potential models for identifying the pattern of this disease, predictive analysis, and rate of transmission. This chapter aims to construct the predictive model to identify the severity of this disease transmission and also find the topographical spread of the contagion.

disease with the help of mathematical methods in epidemics. Daniel Bernoulli first gave the mathematical analysis for small pox in 1760. Mathematical models incorporated with computational tools bring the new insight to key issues like drug discovery, vaccination, transmission rate based on the various parameters, and treatment methods. The process of constructing the mathematical model that will improve the optimality in prediction is known as predictive analysis. Global search engine Google, Netflix, and financial sectors handling credit cards are widely using this predictive analysis. Always communicable diseases play vital role in human history. One of the best incidents is the 1918 Spanish flu which created more than 50,000,000 deaths. In 1980, more than 2,600,000 were occurred by measles, but it was drastically reduced in 2011 with the help of measles vaccine. The efficient understanding of the particular disease is more important to develop the drug. For this understanding, mathematical modeling helps the epidemiologist to identify the various factors involving the disease like age, rate of transmission, rate of mortality, demographic structure, etc. The first mathematical model was initiated by Daniel Bernoulli against smallpox in the year 1760. In his model, he focused the competitive risk about smallpox, and he focused on how to control the death rate. Another remarkable spatial model was introduced by John snow in the year 1855 for cholera (London). For constructing the mathematical model for communicable diseases, it is essential to maintain some basic assumptions about its rate of transmission. The transmission rate always depends on the number of susceptible and infective persons [10] . This idea is the building block for compartmental model. In 1911, Ross proposed the simple compartmental model for malaria transmission, and he stated that the population of mosquitoes below the critical level will minimize the infection. Kermack 

Pandemics are framed by bacteria and viruses. The COVID-19 pandemic is new to humans and it attacks the respiratory system; hence human body needs strong immune system. The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has started its impact in India on 30th January 2020 at Kerala. It is a singlestranded enveloped RNA Virus-1 which has a spherical shape with glycoprotein projections. It has many subtypes like alpha-, beta-, gamma-, and delta-type coronavirus. It has been found in bats, mice, cats, pigs, and some bird family. Figure 15 .1 describes the genome and structure of COVID-19.

As per the scientists point of view in India, COVID-19 may have plateaued and could fall after certain period of lockdown is lifted. So far, 6,702,662 positive cases, 393,210 deaths, and 3,045,695 recovered cases have been reported globally. In India, 226,770 positive cases, 6363 deaths, and 109,462 recovered cases have been reported as on first week of June 2020. The duration for peak will depend on the maintenance of social distancing even after the relaxation of lockdown. Once lockdown is lifted, the second wave of COVID-19 will start again. In that case, curve will reach the peak based on social distancing. Before that, vaccine or drug has to be initiated. More numbers of mathematical models are being developed to forecast and control COVID-19 worldwide.

The pathogen defined the significant challenges for health, planning about the pandemic, and the healthcare systems. Until the identification of vaccine, some of the non-pharmaceutical measures have to be considered. Infective individual's contact tracing and isolation will minimize the level of outbreak. Epidemiological models should reduce the morbidity and mortality.

The following pictures show the data of COVID-19 status in India up to 16th May 2020 ( Fig. 15 .2).

Let N(t) be the population at a time t based on the status of infection. The population is categorized as Susceptible (S), Infected (I), Recovered (R), and Death (D). The following flow chart exhibits the working method of this model (Fig. 15.3) .

Assume that the constant population of the state = ten million

Then, the rate of change of population is considered as where B = S + I + R, which is the exponential growth. The solution of the above equation is (Fig. 15.4 )

The differential equation of susceptible population is given by

where α 1 -quarantined individuals, α 2 -recovered individuals, α 3 -not confirmed, and d 1 -natural death. Similarly, the differential equations of infected and recovered population are defined by The SIRD model structure is locally asymptotically stable under the condition when R(0) < 1 and becomes unstable when R(0) < 1.

The Jacobian of the above system is given by 

The process of constructing the mathematical model or the particular model which generates the accurate prediction is called predictive modeling. Splitting of data, data prediction, performance analysis, and comparative study are the key roles of predictive modeling. The software R-has the package named applied predictive model package. For the implementation of above set of equations, we need to analyze and optimize with some set of data. Data splitting is the initial process in modeling. We have to check how well the derived model extrapolates for different population. Furthermore, it is essential to find how much data should be considered for testing and validating. Scaling and centering the predictor variables is the most significant process in data transformation. The predictor variable has zero mean in centering, while each variable is divided by its standard deviation in scaling process. These values increase the numerical stability of the calculation. For the transformation process, Box and Cox proposed the following transformation function: The value of θ can be identified using the training data. Principal component analysis (PCA) is one of the common methods to reduce the irrelevant information from the data. In this method, linear combination of predictor variables is considered as principal components. Mathematical formation of ith component of PC is written by

where P is the number of predictors. The technical term collinearity is the technical term which is being used when the pair of predictor variables have correlated between them. Multiple correlations are also available between the variables. In this process, the first step is formation of correlation matrix between the predictor variables. Next select the pair of highly correlated variables. Let the pair be C 1 and C 2 . Now calculate the average correlation between C 1 and other variables and similarly for C 2 and other variables. If C 1 has the larger correlated value, then remove the variable C 1 ; otherwise remove C 2 . Continue the process until no absolute correlation is found.

COVID-19 statewise status of India has been taken from https://www.mohfw.gov. in/ which displays the COVID-19 affected people from all the states of India up to June 6, 2020, and the values are displayed here. Figures 15.5 and 15.6 give the numerical values of number of active cases, number of death, and total number of affected people.

The predictive model has been verified with this data. For implementation process, Weka classifier is being used for analyzation. The results are being displayed here. Results (Fig. 15.7 

In this chapter, predictive epidemic model of COVID-19 has been framed. The confirmation cases are increasing rapidly for each and every day. The prediction factor of every infected individual is more important to avoid the transmission. The various parameters like age, health factors, respiratory syndrome level, and density of the population should be focused properly. It has been observed that when the value of R(0) < 1, then the system is locally stable. Based on the numerical results, it has been shown that there is the correlation between infected and susceptible individual. Hence, social distancing and invention of drug are more important. In future, the predict model can be extended to Bayesian network model which gives the result based on categorical responses. It is the acyclic graph which analyzes the various factors and constructs the network which will give the optimal result in decision-making. 

The final size of an epidemic and its relation to the basic reproduction number

Reflexions sur les avantges de l'inoculation

SIS and SIR epidemic models under virtual dispersal

Mathematical and statistical approaches to AIDS epidemiology

Mathematical epidemiology: past, present and future

No pharmaceutical measures for pandemic influenza in no healthcare settings-social distancing measures

The application of mathematics to epidemiology

Epidemic disease in England -the evidence of variability and of persistence

An introduction to compartmental MODELLING for the budding infectious disease modeler

A contribution to the mathematical theory of epidemics

The effect of control strategies that reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China

Contributions to the mathematical theory of epidemics, Part II

Contributions to the mathematical theory of epidemics, Part III

Laboratory surveillance for SARS-CoV-2 in India: performance of testing & descriptive epidemiology of detected COVID-19

The Prevention of Malaria (with Addendum)

Simulation of an SEIR infectious disease model on the dynamic contact network of conference attendees

Prediction of COVID-19 transmission dynamics using a mathematical model considering behavior changes in Korea

Using data on social contacts to estimate agespecific transmission parameters for respiratory-spread infectious agents

Clinical presentation and virological assessment of hospitalized cases of coronavirus disease 2019 in a travel-associated transmission cluster. medRxiv

Clinical characteristics of 138 hospitalized patients with 2019 novel coronavirus-infected pneumonia in

Mathematical MODELLING and epidemic prediction of COVID-19 and its significance to epidemic prevention and control measures

Patterns of human social contact and contact with animals in Shanghai