Mathematical Modelling on Spread and Optimal Control Techniques of Viral Infection

Abstract

scientists and mathematicians to solve the problems of mankind. In this thesis a newlineparticular interest has been given to analyse the dynamics of viral infection and predict newlinethe disease status in future. It also describes the performance of vaccination with newlinedeterministic compartmental models. Present viral infection COVID-19 have a newlinesignificant effect on human health and economy in Indian subcontinent. India is at high newlinerisk because of its dense population, patchy healthcare system and high rate of newlineregards to tackle the COVID-19 crisis. The present study consists of five chapters newlineaccording to various models and on its analyses bases. A brief description of chapters newlineis as follows. newlineChapter1: This chapter deals with the introduction part which includes mathematical newlinemodelling, viral infection, definitions of various important terms, methodology, history, newlinetreatment and consequences of coronavirus and literature review. newlineChapter 2: In this chapter mathematical modelling of dynamics of COVID-19 in India newlineand its states has been analysed and comparison with other countries has been done. newlineThis section deals with the analysis and simulation of dynamics of corona infection and newlinepredict the future trends of disease with SIRD model. The important epidemiological newlinemetric basic reproduction number has been evaluated using next generation matrix newlinemethod to study the size of outbreak. The jacobian matrix has been used to calculate newlineeigen values at disease free equilibrium point to study the stability of system. The newlineimportant parameters recovery rate and death rate have been evaluated with the help of newlineregression equation by using available data. MATLAB 2021a software has been used newlinefor model simulations that helps to compare predicted and actual values. In this part newlinedynamics of COVID-19 infection of India and Colombia has been compared and newlineanalysed with the help of SIRD model. Also parameters affecting COVID-19 and future newlinetrend of infection of both countries has been compared with the help of MATLAB newlinesimulation. The SIR model has been used to understand the dynamics of corona newlineinfection in states Odisha and West Bengal that are among the worst affected states in newlineIndia. The COVID-19 fatalities of India and other large population regions have been newlinecompared and analysed from the point of their land area and population density. SEIR newlinemodelling for states Punjab, Haryana and Himachal Pradesh has been done to newlineunderstand the future trend of viral infection COVID-19 by simulating the data for 250 newlinedays. newlineChapter 3: This section describes the application of SEIR model to study the newlinetransmission of COVID-19 in two states Andhra Pradesh and Tamil Nadu of India. This newlinemodel accounts for the spread of infection during latent period. Transmission rate, newlinerecovery rate, exposed rate and death rate for pre-lockdown period, lockdown period newlineand unlock period for these two states have been calculated. The important newlineepidemiological metric basic reproduction number and effective reproduction number newlinehas been formulated for Andhra Pradesh and Tamil Nadu which show that the exposure newlinerate play significant role in disease spread. Numerical simulation has been performed newlineiii newlineiv newlinewith the help of MATLAB software to analyse dynamics and future prediction of newlineCOVID-19. In next part of this section the COVID-19 status of worst affected states newlineincluding Andhra Pradesh and Tamil Nadu and other parts of country with parameters newlineactive case ratio and case fatality ratio has been compared and analysed by collecting newlinedata as on October 27, 2020. newlineChapter 4: In this section SIR and SEIR modelling of a campus outbreak has been newlinedone and impact of control measure social distancing has been analysed. The newlineoptimization of social distancing control parameter has been done to understand its newlineimpact on infection peak. newlineChapter 5: This section includes the model that divides the total population into four newlinecompartments susceptible, infected, recovered and vaccinated individuals. newlineCompartmental model has been used to study the impact of vaccination of COVID-19 newlinein India. Basic reproduction has been evaluated with the help of next generation matrix newlinemethod for vaccination at disease free equilibrium point that helps to understand prime newlineimportance of optimal control strategies. Local stability at steady state has been newlineanalysed by calculating eigen values of jacobian matrix of the model system. Lyapunov newlinefunction has been used to check the global stability for the model. The SIRV modelling newlinehas been done for AON vaccine and leaky vaccine to understand the effectiveness of newlineboth vaccines at different rates of vaccination. newlineKeywords: newlineSIRD model, Regression line method, Transmission rate, Exposed rate, Death rate, newlineRecovery rate, SEIR model, Eigen values, Basic reproduction number, Jacobian matrix, newlineNext generation matrix method, SIRV model, Local stability, Global stability, newlineLyapunov function. newline