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• Mathematical model for coronavirus disease that fits well the spread in China. • New θ -SEIHRD model taking into account undetected infections. • Validation of the model with the reported data on China. • Estimation of errors when identifying parameters at early stages of the pandemic. • Different scenarios to show the impact of undetected cases on the pandemic. In this paper we develop a mathematical model for the spread of the coronavirus disease 2019 (COVID-19). It is a new θ -SEIHRD model (not a SIR, SEIR or other general purpose model), which takes into account the known special characteristics of this disease, as the existence of infectious undetected cases and the different sanitary and infectiousness conditions of hospitalized people. In particular, it includes a novel approach that considers the fraction θ of detected cases over the real total infected cases, which allows to study the importance of this ratio on the impact of COVID-19. The model is also able to estimate the needs of beds in hospitals. It is complex enough to capture the most important effects, but also simple enough to allow an affordable identification of its parameters, using the data that authorities report on this pandemic. We study the particular case of China (including Chinese Mainland, Macao, Hong-Kong and Taiwan, as done by the World Health Organization in its reports on COVID-19), the country spreading the disease, and use its reported data to identify the model parameters, which can be of interest for estimating the spread of COVID-19 in other countries. We show a good agreement between the reported data and the estimations given by our model. We also study the behavior of the outputs returned by our model when considering incomplete reported data (by truncating them at some dates before and after the peak of daily reported cases). By comparing those results, we can estimate the error produced by the model when identifying the parameters at early stages of the pandemic. Finally, taking into account the advantages of the novelties introduced by our model, we study different scenarios to show how different values of the percentage of detected cases would have changed the global magnitude of COVID-19 in China, which can be of interest for policy makers. [ABSTRACT FROM AUTHOR]

• Simulation of the Impact of SARS-CoV-2 variants and vaccines on the spread of COVID. • New θ -SIR type model ready (and freely available) to be used with real data. • Application to the case of Italy. A new variant may cause a new disease wave. • Current rates of vaccination might not be enough to avoid a new disease wave. • Effective reproduction number < 1 is not enough for having COVID-19 under control. The continuous mutation of SARS-CoV-2 opens the possibility of the appearance of new variants of the virus with important differences in its spreading characteristics, mortality rates, etc. On 14 December 2020, the United Kingdom reported a potentially more contagious coronavirus variant, present in that country, which is referred to as VOC 202012/01. On 18 December 2020, the South African government also announced the emergence of a new variant in a scenario similar to that of the UK, which is referred to as variant 501.V2. Another important milestone regarding this pandemic was the beginning, in December 2020, of vaccination campaigns in several countries. There are several vaccines, with different characteristics, developed by various laboratories and research centers. A natural question arises: what could be the impact of these variants and vaccines on the spread of COVID-19? Many models have been proposed to simulate the spread of COVID-19 but, to the best of our knowledge, none of them incorporates the effects of potential SARS-CoV-2 variants together with the vaccines in the spread of COVID-19. We develop here a θ − i j -SVEIHQRD mathematical model able to simulate the possible impact of this type of variants and of the vaccines, together with the main mechanisms influencing the disease spread. The model may be of interest for policy makers, as a tool to evaluate different possible future scenarios. We apply the model to the particular case of Italy (as an example of study case), showing different outcomes. We observe that the vaccines may reduce the infections, but they might not be enough for avoiding a new wave, with the current expected vaccination rates in that country, if the control measures are relaxed. Furthermore, a more contagious variant could increase significantly the cases, becoming the most common way of infection. We show how, even with the pandemic cases slowing down (with an effective reproduction number less than 1) and the disease seeming to be under control, the effective reproduction number of just the new variant may be greater than 1 and, eventually, the number of infections would increase towards a new disease wave. Therefore, a rigorous follow-up of the evolution of the number of infections with any potentially more dangerous new variant is of paramount importance at any stage of the pandemic. [ABSTRACT FROM AUTHOR]