Rapidly spreading disease, COVID-19 is classified as the human-to-human transmission-able disease and currently is a pandemic in the Globe. In this paper, we propose conceptual mathematical models for COVID-19 outbreak and it's control measurement; quarantine, hospitalization and the effect of panic and anxiety. In this situation, mathematical models are a important tool to employ an effective strategy in order to fight against this pandemic. We establish the positivity and boundedness of solutions, local and global stability analysis of equilibria to examine its epidemiological relevance. To validate the model and estimating the important model parameters and prediction about the disease, we consider the real cases of Italy from $15^{th}$ Feb to $13^{th} $ April 2020. In a series of graphical map, we have presented the comparative study to estimate the current scenarios and to predict the control measurement time boundary of the outbreak.

Background: Outbreak of the Covid-19 is now an ongoing global health emergency. At the end of December 2019, the first infection was reported in Wuhan and the world did not pay attention to this extremely contaminated disease and plucked to react rapidly. The World is in an vulnerable state in disease spreading, facing a great loss of lives and socio-economic aspects also. That is why we have proposed a potential mathematical model with data analysis to predict and control the outcome of this pandemic. Methods: The model presented the epidemic dynamics of multiple compartments. We collected available online data of Spain. In primary step, we estimated the parameters using either the data analysis or reference papers. Then we did the data fitting analysis in comparison with the outcome of our mathematical results. The results of the system depended not only the parameters also on social consciousness. Results: It is found that disease progression in this model is determined by the basic reproductive ratio, $R_0$, the actual epidemic of $R_0$ and effective $R(t)$ of each day. If $R_0>1$, the number of latently infected individuals grows exponentially; endemic solution is stable while infection rate decays if $R_0<1$. The optimal control theory stated that vaccination and treatment strategies are highly effective for reducing both susceptible and infected population and to increase the recover rate high. In Spain, after state of alarm (quarantine) on 14 March 2020, reported cases increasing for 13 days only and from the 14th day, daily reported cases started to decline albeit with small fluctuation. Our proposed model approximates that the disease in Spain could be fully under control by after July 2020. Conclusion: Outbreak will be in control of health care system, reduce the death rate and will ensure social-economic stability.

The present novel corona virus (2019-nCoV) infection has created a global emergency situation by spreading all over the world in a large scale within very short time period. The infection induced death rate is also very high. There is no vaccine or 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 a high population density country, where the corona virus disease (COVID-19) has started to spread from $1^{st}$ week of March, 2020 in a significant number of COVID-19 positive cases. Due to this high population density human to human social contact rate is very high in India. So control of the pandemic COVID-19 in early stage is very urgent and challenging problem. Mathematical models are employed in this paper to study the COVID-19 dynamics, to identify the influential parameters and to find the proper prevention strategies to reduce the 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 key 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 impacts on the disease dynamics have also been studied. Future trends of the disease transmission has been Predicted from our model with some control measures. Finally, the positive measures to control the disease have been summarized.