COVID-19 takes a gigantic form worldwide in a short time from December, 2019. For this reason, World Health Organization (WHO) declared COVID-19 as a pandemic outbreak. In the early days when this outbreak began, the coronavirus spread rapidly in the community due to a lack of knowledge about the virus and the unavailability of medical facilities. Therefore it becomes a significant challenge to control the influence of the disease outbreak. In this situation, mathematical models are an important tool to employ an effective strategy in order to fight against this pandemic. To study the disease dynamics and their influence among the people, we propose a deterministic mathematical model for the COVID-19 outbreak and validate the model with real data of Italy from 15th Feb 2020 to 14th July 2020. We establish the positivity and boundedness of solutions, local stability of equilibria to examine its epidemiological relevance. Sensitivity analysis has been performed to identify the highly influential parameters which have the most impact on basic reproduction number $\left(R_0\right)$. We estimate the basic reproduction number ($R_0$) from available data in Italy and also study effective reproduction numbers based on reported data per day from 15th Feb 2020 to 14th July 2020 in Italy. Finally, the disease control policy has been summarized in the conclusion section.

Besides maintaining health precautions, vaccination has been the only prevention from SARS-CoV-2, though no clinically proved 100% effective vaccine has been developed till date. At this stage, to withhold the debris of this pandemic-experts need to know the impact of the vaccine efficacy rates, the threshold level of vaccine effectiveness and how long this pandemic may extent with vaccines that have different efficacy rates. In this article, a mathematical model study has been done on the importance of vaccination and vaccine efficiency rate during an ongoing pandemic. We simulated a five compartment mathematical model to analyze the pandemic scenario in both California, and whole U.S. We considered four vaccines, Pfizer (95%), Moderna (94%), AstraZeneca (79%), and Johnson & Johnson (72%), which are being used rigorously to control the SARS-CoV-2 pandemic, in addition with two special cases: a vaccine with 100% efficacy rate and no vaccine under use. SARS-CoV-2 related data of California, and U.S. were used in this study. Both the infection and death rates are very high in California. Our model suggests that the pandemic situation in California will be under control in the last quartile of the year 2023 if vaccination program is continued with the Pfizer vaccine. During this time, six waves may happen from the beginning of the immunization where the case fatality and recovery rates will be 1.697% and 98.30%, respectively. However, according to the considered model, this period might be extended to the mid of 2024 when vaccines with lower efficacy rates are used. On the other hand, the daily cases and deaths in the U.S. will be under control at the end of 2026 with multiple waves. Although the number of susceptible people will fall down to none in the beginning of 2027, there is less chance to stop the vaccination program if vaccinated with a vaccine other than a 100% effective vaccine or Pfizer, and at that case vaccination program must run till the mid of 2028. According to this study, the unconfirmed-infectious and infected cases will be under control at the end of 2027 and at the mid of 2028, respectively. The more effective a vaccine is, the less people suffer from this malign infection. Vaccines which are less than 90% effective do not have notable contribution to control the pandemic besides hard immunity. Furthermore, specific groups of people are getting prioritized initially, mass vaccination and quick responses are required to control the spread of this disease.

Background: The world is now in an emergency of preventing the drastic spread of COVID-19. After the infection was first reported in December 2019, almost every country did not pay attention to this highly contaminated disease and failed to react swiftly. Now the whole planet is in an vulnerable state, resulting to increase the mortality rate and facing difficulties in all socio-economic aspects. That is why we have the urge to develop an efficient mathematical model (quarantine) based on social consciousness to control the epidemic. Methods: This is a quarantine mathematical model. The outcome of the system is dependent on social consciousness. We have calculated the awareness level by considering various socio-economic factor of each country. In our model, the parameters are Education Index, Gross Domestic Product (GDP) per capita, population density, high literacy and stable economy. To maximize the efficiency of the model, it has to be implemented in initial stage. However, strict application of the method in vigorous stage of epidemic will also bring a satisfactory outcome. Results: In Spain, quarantine was effected on March 14, 2020. Spain experienced an increase in reported cases for 13 days of quarantine enforcement and from the 14th day, daily reported cases started to decrease with small fluctuation. Government ensured the social isolation through quarantine. After imposition of a quarantine on March 9, 2020 in Italy, within 13 days of lock-down, the maximum number of infection started to decrease. Similar results observed in France. Higher social consciousness would decrease the number of infected population dramatically while minimal or lower awareness will do a outburst. Conclusion: Outbreak will be in control of health care system which yields to reduce the death rate and will ensure social and economic stability.

Lately, a multilevel inverter takes an emergent role for medium voltage and high-power applications. It has many advantages like low switching loss, high dv/dt stress, low THD. But it has also disadvantage like for higher level required more switching devices, huge size, high cost. This paper mainly proposed a newly invented topology with a reduced switch belongs to only eight switches, four voltage sources, and two power diodes to produce twenty-one level voltage waveforms. The major advantage of this proposed inverter ends in diminished size, lesser loss, along with low installation cost, and it is suitable for high power applications. The simulation results for different switching angle calculation methods especially Equal Phase (EP) method and Half Height (HH) method are analyzed and presented. The proposed inverter has been simulated through MATLAB software.