HEALTH BROKEN WOVEN POISSON SPHERES TO MANAGE DEADLY EBOLA INCIDENCES

The deadly infectious Ebola incidences scare not on ly to the residents of western Africa but also to t he travelers and medical professionals who treat the p atients, as they became victims. Since 27 July unti l 13 August 2014 alone, about 2,127 Ebola cases occurred in just four Western African countries: Guinea, Liberia, Nigeria and Sierra Leone and more than 50% of them died . The mortality is extremely higher. No known medication exists. Though the virus is not ai rborne spreading, a contact with the patient’s flui ds, tissues, or bodies is known to transmit Ebola virus . There had been three categories: Suspected, probable, or confirmed in the collection of Ebola incidences and deaths. T heir data are quite informative if they are properly processed and it is exactly the aim of thi s article. For this purpose, the stochastic nature of the data is probed rationally. The Ebola incidences and deat hs in each category exhibit a separate Poisson chan ce environment and yet, they are connected. Therefore, suitable Poisson models are developed for each category and are then woven together to analyse the ntire pertinent data on Ebola incidences and deat hs in those four countries. Pictures are worth the thousa nd words to comprehend non-trivial findings. Hence,  innovatively the data analytic concepts for three-d imensional sphere for each country is developed and applied. By superimposing the four spheres (one for each cou ntry), this article points out the relative perform ance of the four countries with respect to the Ebola incide nces and deaths together in each category. One coun try does better than others in one category but poorly in ot her wo categories. A better performance by a count ry is a reflection of effective prevention and successful m edical treatment of Ebola cases.


MOTIVATION
Ebola is a deadly infectious disease and it is a nightmare even to the medical professionals, as some of them treating the patients became its victims. Lashley and Durham (2007;Magill et al., 2012) for a full list of infectious diseases. This epidemic was named Ebola because its first case, a school headmaster, was discovered near the African river called Ebola on 26 August 1976 in a village Yambuku, in Mongala district in northern democratic republic of Congo and he died on 8 th September 1976. BWHO (1978) report provides details. Pattyn (1978;Pourrut et al., 2005) illustrates the chronology of Ebola virus. Researchers believe that the Ebola virus originated in monkeys, pigs, or fruit bats but is never air borne. Leroy et al. (2005;BBC, 2005;Weingartl et al., 2012;Olival et al., 2013;BBC, 2014;Nossiter, 2014;Pollack, 2014;GCS, 2014) for details on the sources of Ebola virus.
Medical workers without wearing appropriate protective gloves and masks contract the virus from the Ebola patients. Other preventive action includes a quick disposal of the fluids, tissues and even bodies of the Ebola patients. Morvan et al. (1999) have catalogued both DNA and RNA of Ebola virus. In some countries, quarantine of the Ebola patients is allowed to prevent the disease from spreading. No effective vaccine is now available for humans. Johnson et al. (1995;Hoenen et al., 2012) for details on much needed vaccine for Ebola. No effective treatment now exists. Weingartl et al. (2013) contains an excellent review of Ebola virus.The disease has a high mortality. Given this nature of the disease, many fear that this Ebola virus could be abused as a bioterror weapon. On 31th July 2014, an experimental drug named ZMapp has been successfully tested on humans. The long-term complications in surviving patients include inflammation of the testicles, joint pains, skin peeling, blindness, hair loss among others. See CDC (2014;Cham, 2014;Roy-Macaulay, 2014) for further details about the recent Ebola incidences. A hope is created by the announcements CNN (2014a;2014b) The recent outbreak of occurred in Guinea on 6 th December 2013 and it was a 3-year-old boy in the village of Mellandou, Gueckedou Perfecture, Guinea. However, it was detected only March 2014. Just for the seriousness of the Ebola's infectivity, note that his mother, 3-year-old sister and grandmother also died soon with symptoms. The Ebola's symptoms are fever, sore throat, headache, muscle pain, vomiting, nausea, diarrhoea, decreased functioning of kidney, bleeding within the body and outwardly through nose and other organs. These symptoms could start as early as two days. The Ebola virus could be contracted in contact with blood or bodily fluids.

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However, the epidemic has already spread to the neighbouring countries: Liberia, Nigeria and Sierra Leone. Simpson (1977) for ways to prevent Ebola virus. The total cases exceeded 2,127 cases in these four countries alone and it is known to spread to other parts of the world. It is so sad to notice that more than half of them (about 1145 patients to be specific) died. It has become a great concern to the governing agencies of the countries and the World Health Organization (WHO). Both the infection goes through three identification stages: Suspect, probable and confirmed, according to the medical experts.
At any stage, death can happen to the patient. See Table 1 where, φ is interpreted as the Ebola mortality rate.
Unconditionally speaking (that is, without knowing how many Ebola cases occurred in the region in a period of time), what is the Ebola's mortality rate? To answer this question, we need the unconditional (that is, marginal) probability distribution of the random numbers, z. To find it, we first find the joint bivariate probability distribution of the number of Ebola cases, y and the number of Ebola deaths, z. That is Equation 3: [ (1 )] [ / (1 )] ( , , ) ; !( )! 0,1,2,..,; 0,1,2,.., ; From the joint probability distribution (3), the marginal probability distribution of the number of Ebola deaths in a region during a period is obtained and it is parameters compounded Poisson distribution:

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August 2014, which is quite alarming. Let us compare this trend with those of periods: 5 June through 23 July (Fig. 2), 2 April through 23 May (Fig. 3) and 25 March through 27 March (Fig. 4) of 2014. The mortality rate was larger than 60% in earlier periods. Interestingly, the correlation the Ebola incidences and deaths is -0.61, 0.56, 0.36 and -0.71 in the chronological order of time.

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Notice the correlation was negative to begin with, became positive and eventually negative. The negative correlation means that when the Ebola incidence rate is upward, its mortality rate is downward. The positive correlation means that the Ebola incidence and mortality rates were either upward or downward together. The correlations of the data analysis reveal that the healthcare management of Ebola is lately same as before but it must have been out of control in the interim periods. This clue motivates to conduct facts-finding operations by probing through the Ebola data for the four countries in Table 1 and it is the research aim of this article.

MAIN RESULTS: WOVEN POISSONS FOR EBOLA FACTS FINDING
A resident in the region with Ebola epidemic incidence rate θ might encounter the virus and becomes one of three and: The epidemic of Ebola has a strangeness unlike other epidemics such as cholera, smallpox etc. See Smith (2005) for reasons to worry about the Ebola incidences as a serious epidemic. The symptoms of Ebola are deceptive for a while for the family members and other community co-residents. After a lag period, a suspect with partial symptoms goes to a medical facility for counseling. The medical facilities may not be fully Science Publications AJID equipped to treat Ebola victims. Several announcements appear in online that the foundations and charities are forthcoming with donations to be used for Ebola patients. Such facilitations do probably help the medical professionals to confirm the Ebola cases much better now than before. There is no known medication to cure Ebola patients. With these limitations and blind spots in the data collection process, the number of Ebola cases is under-reported. A probabilistic estimate of the number of Ebola cases is there a necessity, practically viewing. For this purpose, we need to first identify an underlying probability pattern of the number of Ebola cases based on the number of Ebola deaths. This amounts to finding, from (3) and (4), the conditional probability distribution of Y given Z and it happens to be a shifted Poisson distribution Equation 11: (1 ) [ (1 )] ( , , ) ; ( )! , 1, 2,..,; 0,1, 2,.., ; The result (11) indicates that its conditional expected value is Equation 12: ( , , ) ( , , ) (1 ) ( ) The result (12) implies that the conditional projection of the number of Ebola incidences based on the number of Ebola deaths starts at z with an increment (1-φ), which is the survival chance from Ebola. It is well known that lesser the variance refers more efficiency of projection. The result (13) confirms that the conditional variance ( , , ) Var y z θ φ is less than the unconditional variance ( , ) Var y θ φ meaning that the conditional projection is more efficient. The conditional projection efficiency happens to be survival chance (1-φ).
Just for a contrast, if we have to project the number z of Ebola deaths without a knowledge of how many Ebola cases occurred, it will be ( , ) E z θ φ θφ = with a variance ( , ) Var z θ φ θφ = from (4 has been interpreted already. The second factor

refers the variance reduction relative to
Science Publications AJID the unconditional variance but it is due to the knowledge on the number of Ebola cases. The variance reduction (equivalently, it is the increase in projection efficiency) happens to be the survival chance (1-φ). Several Poisson distributions are oven to describe the chance mechanism in which the Ebola incidences and deaths occur. Yet, there is a symmetry in the Poisson chance environment. The conditional projection of the number of Ebola cases based on knowing the number of Ebola deaths or vice versa has an efficiency level (1-φ), which happens to be the survival chance from Ebola epidemic.
Consequently, the number of Ebola cases, y and deaths, z must be correlated. The value of the correlation explains the latent healthcare management of Ebola epidemic. From (3), the correlation between y and z is obtained and it is ( , , ) Analogous results and interpretations exist in suspected, probable and confirmed category. The conditional probability distribution of s Y given s Z is a shifted Poisson distribution

ILLUSTRATIONS OF EBOLA CASES AND DEATHS IN GUINEA, LIBERIA, NIGERIA AND SIERRA LEONE
The recent outbreak of the Ebola cases and deaths cover a period from 6th December 2013 until 13th August 2014, as occurred in Guinea, Liberia, Nigeria and Sierra Leone in African continent. The summarized data are displayed in Table 1. A synapsis of daily occurrence is given in Table 2. The derived expression of Section 2 are applied and the results are summarized in Table 3.
It is mysterious that the MLE of the daily Ebola incidence rates (with minimum of 21 and a maximum of 531) and the chance of dying due to Ebola (with minimum of 0.52 and maximum of 0.69) are negatively correlated (that is, -0.75) meaning the chance for death is lesser when the Ebola incidence rate is higher. A simple regression fit is performed (Fig. 5). The projected daily chance of dying due to Ebola is 0.65-0.0002θ , where θ is the MLE of the Ebola incidence. The regression fir is statistically significant with a multiple correlation coefficient 2 0.57 R = . The residual plot (Fig. 6) confirms the significance of the fit. The estimated chance of dying due to Ebola decreases by two for every estimated increase of 10,000 Ebola cases. This change is nominal. The data do not lie. The decrease is not intuitive. Is it a clue that medical facilitations of Ebola cases are quickly taken to a high alert level?
The odds of surviving from Ebola epidemic across the four countries are dramatically different (Fig. 7). Only in Nigeria, the odds are significantly higher. The odds in Guinea, Liberia and Sierra Leone are not good.
Why not probe further in each category among suspected, probable and confirmed Ebola cases? Figure  8 for the comparative estimates of the Ebola incidences and Fig. 9 for a comparison of the estimated chance of dying due to Ebola across countries: Guinea, Liberia, Nigeria and Sierra Leone. The suspected and probable Ebola cases are estimated to be more in Liberia than in other two countries. The confirmed Ebola cases are estimated to be more in Guinea than in other two countries. The suspected and probable Ebola cases most likely to die in Liberia than in other two countries, according to their MLEs. The confirmed Ebola cases are most likely to die in Nigeria than in other two countries, according to their MLEs. These numerical estimates are summarized in Table 3.

Ebola Geometric Proximities
Unfortunately, the collected data do not exhibit explicitly information how effective were the administrative practices to prevent or about how successful the medical treatment of Ebola cases. Yet, an insight into them needs to be developed for future operations. The data of the four countries with respect to the suspected, probable and confirmed Ebola incidences and deaths should be molded with an appropriate innovative three-dimensional graphical concept because pictures are worth thousand words.  For this purpose, let a vector ( , , ) i i θ π φ denote respectively the expected aggregate number of Ebola cases, the probability for an Ebola case is in the i th category (i = 1 for suspected), (i = 2 for probable) and (i = 3 for confirmed), the probability for an Ebola case in the i th category to die. The MLE of the vector is ˆ( , , )

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Sphere with a larger volume signifies more volatility with respect to the Ebola incidences and deaths. In each category, there are four concentric spheres at different locations and sizes depending their radius and the estimates ˆ( , , ) i i θ π φ . See Figure 10 through 11 for concentric spheres for the confirmed, probable and suspected categories. In the confirmed category (Fig. 10), the countries (the spheres) rank from smallest to largest in an order: Liberia,

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Guinea, Nigeria and Sierra Leone. The country Sierra Leone is the most volatile in the confirmed category.
In the probable category (Fig 11), the countries (the spheres) rank from smallest to largest in an order: Nigeria, Sierra Leone, Guinea and Liberia. The country Liberia is the most volatile in the probable category. In the suspected category (Fig. 12), the countries (the spheres) rank from smallest to largest in an order: Nigeria, Guinea, Sierra Leone, and Liberia. The country Liberia is again the most volatile in the suspected category.

CONCLUSION
Having learned that one country is more volatile than others in one category are but is better in other category. This transition is due to administrative, medical and other several infrastructures within the country. International travelers stop visiting such a volatile country. The ships or other commercial Science Publications AJID transportations cease to arrive or leave from such volatile country causing the economy of the volatile country suffers. To find out, how destructive the Ebola incidences and deaths of these four countries on their economies requires an expansion of pertinent database but it is worthwhile.

ACKNOWLEDGEMENT
The researcher appreciates and thanks Vithya, Ajay and Kathir for bringing the data in Table 1 and 2 to my attention.