Heterogeneous COVID-19 Incidences in Diamond Princess cruise ship: Some Volatile Informatics

: This article centers around heterogeneity. It is popularly and briefly mentioned in literature but is not well defined, explained enough on how to compute or interpret. We here offer a formal definition and properties of heterogeneity. We propose an identifier of its existence. Other related ideas are discussed in this article. The concepts and new expressions of this article are illustrated using the COVID-19 data in a Diamond Princess cruise ship, during February 2020.

sample variance 2 s is large. In insurance applications, for example, the premium is assessed more if the insurer is heterogeneous compared to others and has hazard rate (Spreeuw, 1999). Should a large (small) sample variance be indicative of a homogeneity? See Ecochard (2006) for details. In healthcare disciplines, heterogeneity is referred in patient's outcome level sometimes or in the latent genetic level in other times. Should the heterogeneity be connected to a nonobservable hidden trait? Does the heterogeneity refer dissimilar attributes across the subgroups of the population itself even before sampling? Is heterogeneity really pointing out the non-identical nature in a random sample or population? Should heterogeneity imply a shifting population? In genetic studies, several authors refer that genetic heterogeneity is rather too difficult to ascertain. What does it really mean? If alleles in more than one locus exhibit susceptibility to a disease, there is a need to track the loci to infer their heterogeneity. So, in a sense, the application of heterogeneity is really a discussion of similarity or differences across loci (see Elston et al., 2003, pages 3404-344 for details). To realize how the heterogeneity plays a role to judge the crime victimization, see Hope and Norris (2013).
In the statistics literature, using a random sample 12 , ,..., n y y y from a population whose main parameter is , when the null hypothesis 12 : .... is not pertinent in statistical disciplines.
It is known that a binomial population possesses an under dispersion (that is, variance of the binomial distribution is smaller than its mean). In a Poisson population, the drawn random sample ought to reflect equality between the mean and variance. When the main (incidence rate) parameter of a Poisson chance mechanism is stochastically transient, the unconditional observation of the random variable convolutes to an inverse binomial model (Ross, 2002). The inverse binomial distribution is known to attest that the variance is larger than its mean.
See Stuart and Ord (2015) for details. Consequently, a comparison between the mean and variance characterizes only which type binomial, Poisson, or inverse binomial is the underlying chance mechanism we are sampling from but not anything about the heterogeneity.
See Shanmugam (2020) for details about the probabilistic patterns among coronavirus confirmed, cured and deaths in the thirty-two India's states/territories.
To be concise about the confusion with respect to the heterogeneity, let us consider the data (of  in Table 1 Also, the denominator . Realizing that their absolute difference is really 1 , we obtain after simplifications that Figure 3 for the Page 11 of 30 configuration of the distance, 1 ( , ) dy between the observable and non-observable in Poisson mechanism.
We now turn to discuss stochastic properties of the Poisson distribution. The survival function of the random number, 1 Y of COVID-19 cases is The hazard rate is a force of mortality. The hazard rate, 1 () hy for the COVID-19 occurrence is . Does the Poisson chance mechanism keep any a finite memory? For example, the geometric distribution is known to have no memory. The memory is really a conditional probability. That is, , confirming that there is a finite memory in the Poisson mechanism of COVID-19 incidences.
To be specific, with 0, 1 rs == in the above result, the memory between COVID-19 free situation and just one COVID-19 occurrence is revealed in the chance oriented Poisson mechanism. Such a memory is Likewise, the memory between at least one COVID-19 case situation and at least two COVID-19 cases situation is revealed with a substitution of 1 Table 1). However, the odds for COVID-19 free healthy situation to prevail is (see their values in Table 1). For details on how the chance for an incidence of a disease to occur from a disease-free scenario changes, see Shanmugam and Radhakrishnan (2011).

Binomial heterogeneity
In this section, we explore the heterogeneity for two sub binomial processes emanating from a Poisson process.
Hence, their correlation is 23 23 , 23 portrays the drift between the symptomatic observable, 2 Y and the asymptomatic observable, 3 Y and it is simplified to Table 3 for their values), due to applying The binomial distribution has a finite memory confirming that the usual binomial distribution does possess a finite memory. The conditional odds, for a fixed 1 y , for safe asymptomatic symptom are    . See Figure 5 for the configuration of the isomorphic factor, ( Lastly, we develop the Tango index and its significance level over the period. Tango (1984) proposed an index to detect disease cluster in grouped data. This  Table 1 or Table 2 or Table 3 Table 4 for other entities.

2020.
In this section we illustrate all the concepts and expressions of Section 2. Let us consider the COVID-19 data in Table 1

Conclusion with comments.
The risk of contracting COVID-19 in cruise is more than in a community as the social distancing is weakly implemented and the breathing air is tightly internalized. More nations are afraid to let the voyagers come inside at the seaports.
Not even the ships are permitted to dock at the port to avoid mitigation by the communities. The scenario seems to be ant humanistic. The crews are known to have committed suicide in the ship itself. The medical doctors and/or pharmaceutical service are strained for the infected and COVID-19 free voyagers.
Lack of clear symptoms among those that are infected adds to difficulties in managing the COVID-19 crisis in any ship. Most importantly, how do we dispose the COVID-19 death bodies, away from further infectivity.
In the midst of uncertainties about the root cause and/or the appearance of any symptoms, the best modelers can do (as it is done in this article) is to devise a methodology to address the observable as well as non-observable heterogeneity, estimate the proportion of COVID-19 cases to be asymptomatic, estimate the odds of becoming symptomatic, and also the odds ratio for asymptomatic in comparison to those symptomatic among the COVID-19 cases. Some of these are non-trivial to the professional experts dealing with the reduced spread of COVID-19 if not its total elimination. Much of COVID-19 is mystic.