Covid-19 Pandemic in India Mathematical Model Study – Mathematics Assignment Help

Assignment Task

1- Abstract

The present novel coronavirus (SARSCoV-2) infection has created a global emergency situation by spreading all over the world in a large scale within very short time period. But there is no vaccine, anti-viral medicine for such infection. So at this moment, a major worldwide problem is that how we can control this pandemic. On the other hand, India is high population density country, where the coronavirus infection disease (COVID-19) has started from 1March 2020. Due to high population density, human to human social contact rate is very high in India. So controlling pandemic COVID-19 in early stage is very urgent and challenging problem of India.Mathematical models are employed to study the disease dynamics, identify the influential parameters and access the proper prevention strategies for reduction outbreak size. In this work, we have formulated a deterministic compartmental model to study the spreading of COVID-19 and estimated the model parameters by fitting the model with reported data of ongoing pandemic in India. Sensitivity analysis has been done to identify the influential model parameters. The basic reproduction number has been estimated from actual data and the effective basic reproduction number has been studied on the basis of reported cases. Some effective preventive measures and their impact S. K. Biswas Sripat Singh College, Murshidabad, West Bengal, India J. K. Ghosh · S. Sarkar · U. Ghosh (B) Department of Applied Mathematics, University of Calcutta, Kolkata, India e-mail: uttam_math@yahoo.co.in have also been studied. Prediction are given on the future trends of the virus transmission under some control measures. Finally, the positive measures to control the disease have been summarized in the conclusion section. Keywords Basic reproduction number.

2- Quarantine Sensitivity analysis

Prevention measure Mathematics Subject Classification 37N25 · 49J15 · 92D30 1 Introduction The novel coronavirus disease (COVID-19) is a worldwide infectious disease in the current time [1–5]. Including this year pandemic, world faces severe attack by coronavirus several times, and some of those are SARS-CoV [6], MERS-CoV [7–10] and SARS-CoV2 [11]. The symptom of all coronavirus patients is same; they suffer with respiratory problem, fever, dry cough, etc., but COVID-19 is more infectious compare to predecessors [11]. Most of the countries throughout the world are affected by this disease and its harmfulness is increasing day to day. The disease is spreading among different countries mainly through air-travel mode as large number of people is travelling from one country to another [12–14]. To control the disease spreading, WHO provided an advisory to all the counties regard123 S. K. Biswas et al. ing screening of people at both ends: exit and the entry of country [2,15]. After 5 April 2020, every countries have been suffering seriously due to the corona infection. A large number of people is confirmed as COVID-19 positive as well as a large number of people is in quarantine and also in asymptomatic stage. The exposed and the asymptomatic persons are more harmful as a result a large number of people is being infected every day. 

The exposed class are being infectious at any time as its incubation period is 2–14 days [16]. On the other hand, the asymptomatic class is most dangerous compared to any other class because the asymptomatic period is on an average three days [13], because the asymptomatic persons are not showing the symptoms of the disease as a result the people interacting with them are not taking any care about the disease so COVID-19 is spreading rapidly among the people. On the other hand due to large number of infected cases and limited medical capacity in maximum countries, the diagnosis test of the exposed, asymptomatic and quarantined classes for confirmation of COVID19 infection is low. This fact also promotes the number of the infected population [17]. The harmfulness of COVID-19 is so high that upto 11 May 2020, nearly forty two lakh people have been infected; among them, 2,87,131 are dead [16]. As the disease is spreading through interaction and no proper medicine is available till now, so minimizing the social distance and interaction among the people is only way to minimize the spreading of disease. To maintain social distancing, the China Government adopted the lockdown policy and is able to control the spreading of the disease [17,18]. Following this policy, every country is adopting this policy except some countries. In highly populated countries like India, Bangladesh, etc., a large number of people move from one place to another place due to job, also a large number of population came in these countries from the highly infected countries. 

So these countries have high chance of spreading this disease. To control and stop the movement of the population, the Government of India adopted lockdown policy for twenty one days in the first phase, which starts from 25 March 2020, and it also extended upto 17 May 2020. The COVID-19 is highly infectious worldwide spreading life-threatening disease. But there is no particular vaccine, medicine or anti-viral therapy to protect or recover from this infection. So the present important issue throughout the World is protecting the human society from this infection. In this context, some preventive measures such as maintaining social distance, wearing masks, frequently washing the hands with soap and water, etc., can be employed to protect human from this infection. Among the COVID-19 preventives, maintaining social distance plays an crucial role to protect from the infection. Social distancing means keeping a safe space among the peoples who are not their house hold [19]. The measurement of this social distance should be at least six feet [19]. 

In real field, a portion of the population always maintain social distance to avoid infection in the endemic period due to their awareness. In order to study the impact of this important factor (social distance) on the disease dynamics, we have incorporated it in our model. Our main goal of this work is to study the disease dynamics of COVID-2019 by studying a deterministic compartmental model for Indian scenario and access the preventive measures to control COVID-19 outbreak in India. Using the daily reported cases of India, we have estimated the model parameters, estimated the effective reproduction number and make some prediction about the prevalence of the disease. Organization of the paper is as follows: In Sect. 2, we have formulated the model. Basic properties and the basic reproduction number are given in Sects. 3 and 4, respectively. In Sect. 5, we study the steady state analysis of the disease-free equilibrium point. Model fitting, parameter estimation, model validation and prediction are done in Sect. 6. Study of sensitivity analysis is done in Sect. 7. Computation of basic reproduction number from initial growth rate and effective reproduction number are described in Sect. 8. Some preventive measures are presented in Sect. 9. Finally, the concluding remarks are given in Sect.