This paper deals with the investigation on spread of COVID-19 and its stability analysis (both local and global stability) in Odisha, India. Being the second most populous country in the world, It is urgent need to investigate the spread and control of disease in India .However, due to diversity of vast population, uncertainty of infection, varying rate of recovery, state wise different COVID-19 induced death rate and non uniform quarantine policy of the states, it is strenuous to predict the spread and control of disease accurately in the country. So, it is crucial to study the aspects of disease in each state for the better prediction. We have considered the state Odisha (India) having population nearly equal to the population of Spain because the entry of huge migrants to the state suddenly enhanced the number of COVID-19 patients from below two hundred to more than eight hundred within one week even after forty days of lockdown period. We have developed SIAQR epidemic model fabricated with influx of out-migrants diagnosed at compartment (A) , then sent to the compartment (I) for treatment those have confirmed the disease and the remaining healthy individuals are sent to quarantine compartment (Q) for a period of twenty one days under surveillance and observation. The set of ordinary (nonlinear) differential equations are formulated and they are solved using Runge -Kutta fourth order method. The simulation of numerical data is performed using computer software MATLAB. As there is no specific treatment, vaccine or medicine available for the disease till the date, so the only intervention procedure called quarantine process is devised in this model to check the stability behaviour of the disease. The numerical and analytical results of the study show that the disease free equilibrium is locally stable when basic reproduction number is less than unity and unstable when it is more than unity. Further the study shows that it persists to endemic equilibrium for global stability when basic reproduction number greater than unity. As per the current trends ,this study shows that the prevalence of COVID -19 would remain nearly 250 to 300 days in Odisha as for as the infected migrants would have been entering to the state. This mathematical modelling embedded with important risk factor like migration could be adopted for each state that will be helpful for better prediction of the entire country and world.

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In this paper, a mathematical model is proposed on the spread and control of corona virus disease2019 (COVID19) to ascertain the impact of pre quarantine for suspected individuals having travel history ,immigrants and new born cases in the susceptible class following the lockdown or shutdown rules and adopted the post quarantine process for infected class. Set of nonlinear ordinary differential equations (ODEs) are generated and parameters like natural mortality rate, rate of COVID-19 induced death, rate of immigrants, rate of transmission and recovery rate are integrated in the scheme. A detailed analysis of this model is conducted analytically and numerically. The local and global stability of the disease is discussed mathematically with the help of Basic Reproduction Number. The ODEs are solved numerically with the help of Runge-Kutta 4th order method and graphs are drawn using MATLAB software to validate the analytical result with numerical simulation. It is found that both results are in good agreement with the results available in the existing literatures. The stability analysis is performed for both disease free equilibrium and endemic equilibrium points. The theorems based on Routh-Hurwitz criteria and Lyapunov function are proved .It is found that the system is locally asymptotically stable at disease free and endemic equilibrium points for basic reproduction number less than one and globally asymptotically stable for basic reproduction number greater than one. Finding of this study suggest that COVID-19 would remain pandemic with the progress of time but would be stable in the long-term if the pre and post quarantine policy for asymptomatic and symptomatic individuals are implemented effectively followed by social distancing, lockdown and containment.