Compound Intervened Poisson Distribution

The nature and characteristics of Intervened Poisson Distribution (IPD) has been well discussed by Shanmugam (1985). In this paper, Compound Intervened Poisson Distribution (CIPD) is introduced and its properties are studied.     

Impact of Ten-Valent Pneumococcal Conjugate Vaccination on Invasive Pneumococcal Disease in Finnish Children – A Population-Based Study

Background

The ten-valent pneumococcal conjugate vaccine (PCV10) was introduced into the Finnish National Vaccination Program (NVP) in September 2010 with a 2+1 schedule (3, 5, 12 months) without catch-up vaccinations. We evaluated the direct and indirect effects of PCV10 on invasive pneumococcal disease (IPD) among children ≤5 years of age during the first three years after NVP introduction.

Methods

We conducted a population-based, observational follow-up study. The cohort of vaccine-eligible children (all children born June 1, 2010 or later) was followed from 3 months of age until the end of 2013. For the indirect effect, another cohort of older children ineligible for PCV10 vaccination was followed from 2011 through 2013. Both cohorts were compared with season- and age-matched reference cohorts before NVP introduction. National, population-based laboratory surveillance data were used to compare culture-confirmed serotype-specific IPD rates in the vaccine target and reference cohorts by using Poisson regression models.

Results

The overall IPD rate among vaccine-eligible children was reduced by 80% (95%CI 72 to 85); the reduction in vaccine-type IPD was 92% (95%CI 86 to 95). However, a non-significant increase in non-vaccine type IPD was observed. During 2012–2013, we also observed a 48% (95%CI 18 to 69) reduction in IPD among unvaccinated children 2 to 5 years of age, which was mostly attributable to the ten vaccine serotypes.

Conclusions

This is the first population-based study investigating the impact of PCV10 introduction without prior PCV7 use. A substantial decrease in IPD rates among vaccine-eligible children was observed. A smaller and temporally delayed reduction among older, unvaccinated children suggests that PCV10 also provides indirect protection against vaccine-type IPD. Changes in serotype distribution warrant continuous monitoring of potential increases in non-vaccine serotypes.     

Effect of Pneumococcal Conjugate Vaccination in Uruguay,a Middle-Income Country

Background

In 2008, a 7-valent pneumococcal conjugate vaccine (PCV7) was introduced into the routine childhood immunization program in Uruguay, with a 2+1 schedule. In 2010, PCV13 replaced PCV7, and the same 2+1 schedule was used. The effect of these pneumococcal vaccines on the incidence of invasive pneumococcal infections (IPD) and on serotype distribution was analyzed retrospectively, based on passive national laboratory surveillance.

Methods

Data from 1,887 IPD isolates from 5 years before and 5 years after PCV7 introduction (7 before and 3 after PCV13 introduction) was examined to assess the incidence rate per 100,000 age-specific population of all IPD, PCV7-serotypes, and PCV13-serotypes associated IPD among children <2 years and 2 to 4 years old, and patients ≥5 years old. Trends of frequency for each serotype were also analyzed.

Results

Comparison of pre-vaccination (2003–2007) and post-vaccination (2008–2012) periods showed a significant decrease in IPD incidence among children <2 years old (IR 68.7 to IR 29.6, p<0.001) and children 2 to 4 years (p<0.04). IPD caused by serotypes in PCV7 was reduced by 95.6% and IPD caused by 6 serotypes added in PCV13 was reduced by 83.9% in children <5 years old. Indirect effects of both conjugate vaccines were observed among patients ≥5 years old one year after the introduction of each vaccine, in 2010 for PCV7 and in 2012 for PCV13. Nevertheless, for reasons that still need to be explained, perhaps due to ascertainment bias, total IPD in this group increased after 2007. In 2012, the relative frequency of vaccine serotypes among vaccinated and unvaccinated population declined, except for serotype 3. Non vaccine serotypes with increasing frequency were identified, in rank order: 12F, 8, 24F, 22F, 24A, 15C, 9N, 10A and 33.

Conclusion

Consecutive immunization with PCV7 and PCV13 has significantly reduced IPD in children <5 years of age in Uruguay.     

Mixtures of Distributions by the Poisson Distribution of Order K

The Poisson-binomial, the Poisson-negative binomial (or Pascal) and Neyman's Type A distribution, which are Poisson mixtures of the binomial, the negative binomial and the Poisson distribution, respectively, have received a lot of attention in statistical literature [see e.g. Katti and Gurland (1961, 1962), Anscombe (1950), and Neyman (1939)]. In the present paper, their respective generalizations are introduced and briefly studied, when the Poisson distribution of order k [see Philippou (1983), Philippou, Georghiou and Philippou (1983), and Charalambides (1986)] replaces the Poisson distribution in its mixing role.     

Blood levels of immunoglobulins G,A,M, circulating immune complexes, complement and proteins in patients on intermittent peritoneal dialysis

The study involved 17 patients on IPD. Blood serum levels of IgG, IgA, IgM, circulating immune complexes, complement and proteins were determinated at the beginning of therapy, after 3, 6, 12 months on IPD and after 1 year on hemodialysis. The frequency of peritonitis was noted during this time. Peritonitis was the most frequent during first 3 months on IPD. No differences in blood serum levels of IgG, IgA, IgM, in the specific periods of IPD were noted. A significant increase in blood serum circulating immune complexes in patients on IPD and hemodialysis compared to the control group was found. A significant decrease in blood serum of C3 complement in patients on IPD and hemodialysis in comparison with the controls were found. A significant decrease in blood serum proteins at the beginning of IPD and after 3 months IPD in comparison with proteins concentration in patients on hemodialysis was observed.     

The spatial distribution of neurons and glia in human cortex based on the poisson distribution

A method was devised that employs deviations from the Poisson distribution to analyze the spatial arrangement of neurons and glia in human cerebral cortex. A field of randomly distributed points equal in number of a sample field of neuronal or glial cells is generated by computer, and the proportion of cells in the sample field that are closer to the nearest neighboring cells than to the nearest randomly distributed point is determined. We call this proportion the "Poisson ratio." When the cells are randomly distributed, the Poisson ratio is equal to 0.5. If the Poisson ratio is less than 0.5, the cells are farther away from one another than a random distribution would predict (exclusionary pattern); if the Poisson ratio is greater than 0.5, the cells are closer to one another than a random distribution would predict (clustering). A simple nonparametric statistical test is used to determine the significance of differences in the ratios. This method was applied to samples of human cerebral cortex in order to test the hypothesis that patients with schizophrenic psychosis may have an altered pattern of neuronal clustering. The analysis revealed that there is no difference in the nearest-neighbor distribution of either neurons or glia between psychotic patients and controls. It was found, however, that there is a highly significant difference in the spatial distribution of neurons versus glia in human cerebral cortex. Neurons of layers II to VI in the human cortex show greater-than-expected distances among them and are distributed according to an exclusionary pattern, while neurons in layer I show a clustering pattern.(ABSTRACT TRUNCATED AT 250 WORDS)     

Zero-Inflated Poisson Models and C.A.MAN: A Tutorial Collection of Evidence

Analysis of count data is required in many areas of biometric interest. Often the simple Poisson distribution is not appropriate, since an extra-number of zero counts occur in the count data. Some current approaches for the problem at hand are reviewed. It will be argued that these situations can often be easily modeled using the zero-inflated Poisson distribution. A variety of applications are considered in which this occurs. Possibilities are outlined on how the validity of the zero-inflated Poisson can be validated including a comparison with the nonparametric Poisson mixture maximum likelihood estimator.     

7-Valent Pneumococcal Conjugate Vaccination in England and Wales: Is It Still Beneficial Despite High Levels of Serotype Replacement?

Background

The UK introduced the 7-valent pneumococcal conjugate vaccine (PCV7) into the national vaccination program in September 2006. Previous modelling assumed that the likely impact of PCV7 on invasive pneumococcal disease (IPD) would be similar to the US experience with PCV7. However, recent surveillance data show a more rapid replacement of PCV7 IPD cases by non-PCV7 IPD cases than was seen in the US.

Methods and Findings

A previous model of pneumococcal vaccination was re-parameterised using data on vaccine coverage and IPD from England and Wales between 2006 and 2009. Disease incidence was adjusted for the increasing trend in reported IPD cases prior to vaccination. Using this data we estimated that individuals carrying PCV7 serotypes have much higher protection (96%;95% CI 72%-100%) against acquisition of NVT carriage than the 15% previously estimated from US data, which leads to greater replacement. However, even with this level of replacement, the annual number of IPD cases may be 560 (95% CI, -100 to 1230) lower ten years after vaccine introduction compared to what it may have been without vaccination. A particularly marked fall of 39% in children under 15 years by 2015/6 is predicted.

Conclusion

Our model suggests that PCV7 vaccination could result in a decrease in overall invasive pneumococcal disease, particularly in children, even in an environment of rapid replacement with non-PCV7 serotypes within 5 years of vaccine introduction at high coverage.     

Persistent high burden of invasive pneumococcal disease in South African HIV-infected adults in the era of an antiretroviral treatment program

Background

Highly active antiretroviral treatment (HAART) programs have been associated with declines in the burden of invasive pneumococcal disease (IPD) in industrialized countries. The aim of this study was to evaluate trends in IPD hospitalizations in HIV-infected adults in Soweto, South Africa, associated with up-scaling of the HAART program from 2003 to 2008.

Methods

Laboratory-confirmed IPD cases were identified from 2003 through 2008 through an existing surveillance program. The period 2003-04 was designated as the early-HAART era, 2005–06 as the intermediate-HAART era and 2007–08 as the established-HAART era. The incidence of IPD was compared between the early-HAART and established-HAART eras in HIV-infected and–uninfected individuals.

Results

A total of 2,567 IPD cases among individuals older than 18 years were reported from 2003 through 2008. Overall incidence of IPD (per 100,000) did not change during the study period in HIV-infected adults (207.4 cases in the early-HAART and 214.0 cases in the established-HAART era; p = 0.55). IPD incidence, actually increased 1.16-fold (95% CI: 1.01; 1.62) in HIV-infected females between the early-and established-HAART eras (212.1 cases and 246.2 cases, respectively; p = 0.03). The incidence of IPD remained unchanged in HIV-uninfected adults across the three time periods.

Conclusion

Despite a stable prevalence of HIV and the increased roll-out of HAART for treatment of AIDS patients in our setting, the burden of IPD has not decreased among HIV-infected adults. The study indicates a need for ongoing monitoring of disease and HAART program effectiveness to reduce opportunistic infections in African adults with HIV/AIDS, as well as the need to consider alternate strategies including pneumococcal conjugate vaccine immunization for the prevention of IPD in HIV-infected adults.     

The Serotype Distribution among Healthy Carriers before Vaccination Is Essential for Predicting the Impact of Pneumococcal Conjugate Vaccine on Invasive Disease

Pneumococcal conjugate vaccines (PCVs) have substantially reduced morbidity and mortality of pneumococcal disease. The impact of the 7-valent PCV on all-serotype invasive pneumococcal disease (IPD) among children was reported to vary between high-income countries. We investigate the ability to predict this heterogeneity from pre-vaccination data. We propose a parsimonious model that predicts the impact of PCVs from the odds of vaccine serotype (VT) among carriers and IPD cases in the pre-PCV period, assuming that VT are eliminated in a mature PCV programme, that full serotype replacement occurs in carriage and that invasiveness of the NVT group is unchanged. We test model performance against the reported impact of PCV7 on childhood IPD in high-income countries from a recent meta-analysis. The odds of pre-PCV7 VT IPD, PCV schedule, PCV coverage and whether a catch up campaign was used for introduction was gathered from the same analysis. We conducted a literature review and meta-analysis to obtain the odds of pre-PCV7 VT carriage in the respective settings. The model predicted the reported impact on childhood IPD of mature PCV programmes; the ratio of predicted and observed incidence risk ratios was close to 1 in all settings. In the high income settings studied differences in schedule, coverage, and catch up campaigns were not associated with the observed heterogeneity in impact of PCV7 on childhood all-serotype IPD. The pre-PCV7 proportion of VT IPD alone also had limited predictive value. The pre-PCV7 proportion of VT carriage and IPD are the main determinants for the impact of PCV7 on childhood IPD and can be combined in a simple model to provide predictions of the vaccine preventable burden of IPD.     

Recurrent Invasive Pneumococcal Disease in Children: Underlying Clinical Conditions,and Immunological and Microbiological Characteristics

Purpose

Clinical, immunological and microbiological characteristics of recurrent invasive pneumococcal disease (IPD) in children were evaluated, differentiating relapse from reinfection, in order to identify specific risk factors for both conditions.

Methods

All patients <18 years-old with recurrent IPD admitted to a tertiary-care pediatric center from January 2004 to December 2011 were evaluated. An episode of IPD was defined as the presence of clinical findings of infection together with isolation and/or pneumococcal DNA detection by Real-Time PCR in any sterile body fluid. Recurrent IPD was defined as 2 or more episodes in the same individual at least 1 month apart. Among recurrent IPD, we differentiated relapse (same pneumococcal isolate) from reinfection.

Results

593 patients were diagnosed with IPD and 10 patients died. Among survivors, 23 episodes of recurrent IPD were identified in 10 patients (1.7%). Meningitis was the most frequent form of recurrent IPD (10 episodes/4 children) followed by recurrent empyema (8 episodes/4 children). Three patients with recurrent empyema caused by the same pneumococcal clone ST306 were considered relapses and showed high bacterial load in their first episode. In contrast, all other episodes of recurrent IPD were considered reinfections. Overall, the rate of relapse of IPD was 0.5% and the rate of reinfection 1.2%. Five out of 7 patients with reinfection had an underlying risk factor: cerebrospinal fluid leak (n = 3), chemotherapy treatment (n = 1) and a homozygous mutation in MyD88 gene (n = 1). No predisposing risk factors were found in the remainder.

Conclusions

recurrent IPD in children is a rare condition associated with an identifiable risk factor in case of reinfection in almost 80% of cases. In contrast, recurrent IPD with pleuropneumonia is usually a relapse of infection.     

Sharing Individual Participant Data from Clinical Trials: An Opinion Survey Regarding the Establishment of a Central Repository

Background

Calls have been made for increased access to individual participant data (IPD) from clinical trials, to ensure that complete evidence is available. However, despite the obvious benefits, progress towards this is frustratingly slow. In the meantime, many systematic reviews have already collected IPD from clinical trials. We propose that a central repository for these IPD should be established to ensure that these datasets are safeguarded and made available for use by others, building on the strengths and advantages of the collaborative groups that have been brought together in developing the datasets.

Objective

Evaluate the level of support, and identify major issues, for establishing a central repository of IPD.

Design

On-line survey with email reminders.

Participants

71 reviewers affiliated with the Cochrane Collaboration''s IPD Meta-analysis Methods Group were invited to participate.

Results

30 (42%) invitees responded: 28 (93%) had been involved in an IPD review and 24 (80%) had been involved in a randomised trial. 25 (83%) agreed that a central repository was a good idea and 25 (83%) agreed that they would provide their IPD for central storage. Several benefits of a central repository were noted: safeguarding and standardisation of data, increased efficiency of IPD meta-analyses, knowledge advancement, and facilitating future clinical, and methodological research. The main concerns were gaining permission from trial data owners, uncertainty about the purpose of the repository, potential resource implications, and increased workload for IPD reviewers. Restricted access requiring approval, data security, anonymisation of data, and oversight committees were highlighted as issues under governance of the repository.

Conclusion

There is support in this community of IPD reviewers, many of whom are also involved in clinical trials, for storing IPD in a central repository. Results from this survey are informing further work on developing a repository of IPD which is currently underway by our group.     

Pediatric Invasive Pneumococcal Disease Caused by Vaccine Serotypes following the Introduction of Conjugate Vaccination in Denmark

A seven-valent pneumococcal conjugate vaccine (PCV7) was introduced in the Danish childhood immunization program (2+1 schedule) in October 2007, followed by PCV13 starting from April 2010. The nationwide incidence of IPD among children younger than 5 years nearly halved after the introduction of PCV7 in the program, mainly due to a decline in IPD caused by PCV7-serotypes. We report the results from a nationwide population-based cohort study of laboratory confirmed IPD cases in children younger than 5 years during October 1, 2007 to December 31, 2010 and describe the characteristics of children suspected to present with a vaccine failure. The period between April 19 and December 31, 2010 was considered a PCV7/PCV13 transitional period, where both vaccines were offered. We identified 45 episodes of IPD caused by a PCV7 serotype (23% of the total number) and 105 (55%) caused by one of the 6 additional serotypes in PCV13. Ten children had received at least one PCV7 dose before the onset of IPD caused by a PCV7 serotype. Seven children were considered to be incompletely vaccinated before IPD, but only three cases fulfilled the criteria of vaccine failure (caused by serotypes 14, 19F and 23F). One case of vaccine failure was observed in a severely immunosuppressed child following three PCV7 doses, and two cases were observed in immunocompetent children following two infant doses before they were eligible for their booster. None of the IPD cases caused by the additional PCV13 serotypes had been vaccinated by PCV13 and there were therefore no PCV13-vaccine failures in the first 8-months after PCV13 introduction in Denmark.     

Melanopsin and rhodopsin mediate UVA-induced immediate pigment darkening: Unravelling the photosensitive system of the skin

The mammalian skin has a photosensitive system comprised by several opsins, including rhodopsin (OPN2) and melanopsin (OPN4). Recently, our group showed that UVA (4.4?kJ/m²) leads to immediate pigment darkening (IPD) in murine normal and malignant melanocytes. We show the role of OPN2 and OPN4 as UVA sensors: UVA-induced IPD was fully abolished when OPN4 was pharmacologically inhibited by AA9253 or when OPN2 and OPN4 were knocked down by siRNA in both cell lines. Our data, however, demonstrate that phospholipase C/protein kinase C pathway, a classical OPN4 pathway, is not involved in UVA-induced IPD in either cell line. Nonetheless, in both cell types we have shown that: a) intracellular calcium signal is necessary for UVA-induced IPD; b) the involvement of CaMK II, whose inhibition, abolished the UVA-induced IPD; c) the role of CAMK II/NOS/sGC/cGMP pathway in the process since inhibition of either NOS or sGC abolished the UVA-induced IPD. Taken altogether, we show that OPN2 and OPN4 participate in IPD induced by UVA in murine normal and malignant melanocytes through a conserved common pathway. Interestingly, upon knockdown of OPN2 or OPN4, the UVA-driven IPD is completely lost, which suggests that both opsins are required and cooperatively signal in murine both cell lines. The participation of OPN2 and OPN4 system in UVA radiation-induced response, if proven to take place in human skin, may represent an interesting pharmacological target for the treatment of depigmentary disorders and skin-related cancer.     

Levels of pneumococcal conjugate vaccine coverage and indirect protection against invasive pneumococcal disease and pneumonia hospitalisations in Australia: An observational study

BackgroundThere is limited empiric evidence on the coverage of pneumococcal conjugate vaccines (PCVs) required to generate substantial indirect protection. We investigate the association between population PCV coverage and indirect protection against invasive pneumococcal disease (IPD) and pneumonia hospitalisations among undervaccinated Australian children.Methods and findingsBirth and vaccination records, IPD notifications, and hospitalisations were individually linked for children aged <5 years, born between 2001 and 2012 in 2 Australian states (New South Wales and Western Australia; 1.37 million children). Using Poisson regression models, we examined the association between PCV coverage, in small geographical units, and the incidence of (1) 7-valent PCV (PCV7)-type IPD; (2) all-cause pneumonia; and (3) pneumococcal and lobar pneumonia hospitalisation in undervaccinated children. Undervaccinated children received <2 doses of PCV at <12 months of age and no doses at ≥12 months of age. Potential confounding variables were selected for adjustment a priori with the assistance of a directed acyclic graph.There were strong inverse associations between PCV coverage and the incidence of PCV7-type IPD (adjusted incidence rate ratio [aIRR] 0.967, 95% confidence interval [CI] 0.958 to 0.975, p-value < 0.001), and pneumonia hospitalisations (all-cause pneumonia: aIRR 0.991 95% CI 0.990 to 0.994, p-value < 0.001) among undervaccinated children. Subgroup analyses for children <4 months old, urban, rural, and Indigenous populations showed similar trends, although effects were smaller for rural and Indigenous populations. Approximately 50% coverage of PCV7 among children <5 years of age was estimated to prevent up to 72.5% (95% CI 51.6 to 84.4) of PCV7-type IPD among undervaccinated children, while 90% coverage was estimated to prevent 95.2% (95% CI 89.4 to 97.8). The main limitations of this study include the potential for differential loss to follow-up, geographical misclassification of children (based on residential address at birth only), and unmeasured confounders.ConclusionsIn this study, we observed substantial indirect protection at lower levels of PCV coverage than previously described—challenging assumptions that high levels of PCV coverage (i.e., greater than 90%) are required. Understanding the association between PCV coverage and indirect protection is a priority since the control of vaccine-type pneumococcal disease is a prerequisite for reducing the number of PCV doses (from 3 to 2). Reduced dose schedules have the potential to substantially reduce program costs while maintaining vaccine impact.

In an observational study, Jocelyn Chan and colleagues investigate associations between pneumococcal conjugate vaccine coverage and incidence of invasive pneumococcal disease and pneumonia among children under 5 years in Australia.     

Relative efficiency of using summary versus individual data in random-effects meta-analysis

Meta-analysis is a statistical methodology for combining information from diverse sources so that a more reliable and efficient conclusion can be reached. It can be conducted by either synthesizing study-level summary statistics or drawing inference from an overarching model for individual participant data (IPD) if available. The latter is often viewed as the “gold standard.” For random-effects models, however, it remains not fully understood whether the use of IPD indeed gains efficiency over summary statistics. In this paper, we examine the relative efficiency of the two methods under a general likelihood inference setting. We show theoretically and numerically that summary-statistics-based analysis is at most as efficient as IPD analysis, provided that the random effects follow the Gaussian distribution, and maximum likelihood estimation is used to obtain summary statistics. More specifically, (i) the two methods are equivalent in an asymptotic sense; and (ii) summary-statistics-based inference can incur an appreciable loss of efficiency if the sample sizes are not sufficiently large. Our results are established under the assumption that the between-study heterogeneity parameter remains constant regardless of the sample sizes, which is different from a previous study. Our findings are confirmed by the analyses of simulated data sets and a real-world study of alcohol interventions.     

Changes in the interpupillary distance (IPD) with ages and its effect on the near convergence/distance (NC/D) ratio

This study aims at determining the mean value of the ophthalmic anthropometrics parametar-IPD (Interpupillary distance) and the mean value of NC/D (Near convergence/distance) in 300 subjects aged 5 to 60 years. The influence of IPD on the NC/D ratio in the same subjects will also be investigated. The investigation showed that the mean value of IPD is 60.5 +/- 2.4 cm, and the mean value of NC/D ratio is 4.95 +/- 2.28 prD. Mean IPD of 5.1 cm in 5-year old children increases to 6.3 cm in adult over 20. In adulthood, IPD remains the same, i.e. 6.3 cm. Our study showed that the NC/D ratio is stable during lifetime. Since NC/D is stabile and IPD changes during lifetime, it is evident that factors other than IPD, such as convergence and accommodation, influence the stability of the NC/D ratio.     

Simulation of deformable models with the Poisson equation

In this paper, we present a new methodology for the deformation of soft objects by drawing an analogy between the Poisson equation and elastic deformation from the viewpoint of energy propagation. The potential energy stored due to a deformation caused by an external force is calculated and treated as the source injected into the Poisson system, as described by the law of conservation of energy. An improved Poisson model is developed for propagating the energy generated by the external force in a natural manner. An autonomous cellular neural network (CNN) model is established by using the analogy between the Poisson equation and CNN to solve the Poisson model for the real-time requirement of soft object deformation. A method is presented to derive the internal forces from the potential energy distribution. The proposed methodology models non-linear materials with the non-linear Poisson equation and thus non-linear CNN, rather than geometric non-linearity. It not only deals with large-range deformations, but also accommodates isotropic, anisotropic and inhomogeneous materials by simply modifying constitutive coefficients. A haptic virtual reality system has been developed for deformation simulation with force feedback. Examples are presented to demonstrate the efficiency of the proposed methodology.     

Simulation of deformable models with the Poisson equation

In this paper, we present a new methodology for the deformation of soft objects by drawing an analogy between the Poisson equation and elastic deformation from the viewpoint of energy propagation. The potential energy stored due to a deformation caused by an external force is calculated and treated as the source injected into the Poisson system, as described by the law of conservation of energy. An improved Poisson model is developed for propagating the energy generated by the external force in a natural manner. An autonomous cellular neural network (CNN) model is established by using the analogy between the Poisson equation and CNN to solve the Poisson model for the real-time requirement of soft object deformation. A method is presented to derive the internal forces from the potential energy distribution. The proposed methodology models non-linear materials with the non-linear Poisson equation and thus non-linear CNN, rather than geometric non-linearity. It not only deals with large-range deformations, but also accommodates isotropic, anisotropic and inhomogeneous materials by simply modifying constitutive coefficients. A haptic virtual reality system has been developed for deformation simulation with force feedback. Examples are presented to demonstrate the efficiency of the proposed methodology.     

Generalized Poisson distribution: the property of mixture of Poisson and comparison with negative binomial distribution

We prove that the generalized Poisson distribution GP(theta, eta) (eta > or = 0) is a mixture of Poisson distributions; this is a new property for a distribution which is the topic of the book by Consul (1989). Because we find that the fits to count data of the generalized Poisson and negative binomial distributions are often similar, to understand their differences, we compare the probability mass functions and skewnesses of the generalized Poisson and negative binomial distributions with the first two moments fixed. They have slight differences in many situations, but their zero-inflated distributions, with masses at zero, means and variances fixed, can differ more. These probabilistic comparisons are helpful in selecting a better fitting distribution for modelling count data with long right tails. Through a real example of count data with large zero fraction, we illustrate how the generalized Poisson and negative binomial distributions as well as their zero-inflated distributions can be discriminated.