Coronavirus disease (COVID-19) outbreak has affected billions of people, where millions of them have been infected and thousands of them have lost their lives. In addition, to constraint the spread of the virus, economies have been shut down, curfews and restrictions have interrupted the social lives. Currently, the key question in minds is the future impacts of the virus on the people. It is a fact that the parametric modelling and analyses of the pandemic viruses are able to provide crucial information about the character and also future behaviour of the viruses. This paper initially reviews and analyses the Susceptible-Infected-Recovered (SIR) model, which is extensively considered for the estimation of the COVID-19 casualties. Then, this paper introduces a novel comprehensive higher-order, multi-dimensional, strongly coupled, and parametric Suspicious-Infected-Death (SpID) model. The mathematical analysis results performed by using the casualties in Turkey show that the COVID-19 dynamics are inside the slightly oscillatory, stable (bounded) region, although some of the dynamics are close to the instability region (unbounded). However, analysis with the data just after lifting the restrictions reveals that the dynamics of the COVID-19 are moderately unstable, which would blow up if no actions are taken. The developed model estimates that the number of the infected and death individuals will converge zero around 300 days whereas the number of the suspicious individuals will require about a thousand days to be minimized under the current conditions. Even though the developed model is used to estimate the casualties in Turkey, it can be easily trained with the data from the other countries and used for the estimation of the corresponding COVID-19 casualties. INDEX TERMS COVID-19 casualties, parametric model, prediction, SpID model, SIR model.

Coronavirus Disease 2019 (COVID-19) has spread the world resulting in detrimental effects on human health, lives, societies, and economies. The state authorities mostly take non-pharmacological actions against the outbreak since there are no confirmed vaccines or treatments yet. In this paper, we developed Suspicious-Infected-Death with Non-Pharmacological policies (SpID-N) model to analyze the properties of the COVID-19 casualties and also estimate the future behavior of the outbreak. We can state the key contributions of the paper with three folds. Firstly, we propose the SpID-N model covering the higher-order internal dynamics which cause the peaks in the casualties. Secondly, we parametrize the non-pharmacological policies such as the curfews on people with chronic disease, people age over 65, people age under 20, restrictions on the weekends and holidays, and closure of the schools and universities. Thirdly, we explicitly incorporate the internal and coupled dynamics of the model with these multi-dimensional non-pharmacological policies. The corresponding higher-order and strongly coupled model has utterly unknown parameters and we construct a batch type Least Square (LS) based optimization algorithm to learn these unknown parameters from the available data. The parametric model and the predicted future casualties are analyzed extensively.

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Insights about the dominant dynamics, coupled structures and the unknown uncertainties of the pandemic diseases play an important role in determining the future characteristics of the pandemic diseases. To enhance the prediction capabilities of the models, properties of the unknown uncertainties in the pandemic disease, which can be utterly random, or function of the system dynamics, or it can be correlated with an unknown function, should be determined. The known structures and amount of the uncertainties can also help the state authorities to improve the policies based on the recognized source of the uncertainties. For instance, the uncertainties correlated with an unknown function imply existence of an undetected factor in the casualties. In this paper, we extend the SpID-N (Suspicious-Infected-Death with non-pharmacological policies) model as in the form of MIMO (Multi-Input-Multi-Output) structure by adding the multi-dimensional unknown uncertainties. The results confirm that the infected and death sub-models mostly have random uncertainties (due undetected casualties) whereas the suspicious sub-model has uncertainties correlated with the internal dynamics (governmental policy of increasing the number of the daily tests) for Turkey. However, since the developed MIMO model parameters are learned from the data (daily reported casualties), it can be easily adapted for other countries. Obtained model with the corresponding uncertainties predicts a distinctive second peak where the number of deaths, infected and suspicious casualties disappear in 240, 290, and more than 300 days, respectively, for Turkey.

In Algebraic Coding Theory, all linear codes are described by generator matrices. Any linear code has many generator matrices which are equivalent. It is important to find the number of the generator matrices for constructing of these codes. In this paper, we study Z_2 Z_4 Z_8-additive codes, which are the extension of recently introduced Z_2 Z_4-additive codes. We count the number of arbitrary Z_2 Z_4 Z_8-additive codes. Then we investigate connections to Z_2 Z_4 and Z_2 Z_8-additive codes with Z_2 Z_4 Z_8, and give some illustrative examples.