In this paper, we derive and prove, by means of Binomial theorem and Faulhaber's formula, the following identity between $m$-order polynomials in $T$ $$\sum_{k=1}^{\ell}\sum_{j=0}^m A_{m,j}k^j(T-k)^j=\sum_{k=0}^{m}(-1)^{m-k}U_m(\ell,k)\cdot T^k=T^{2m+1}, \ \ell=T\in\mathbb{N}.$$

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k; 0 ≤ k ≤ n−1 or 1 ≤ k ≤ n, by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m = 5; 7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n^2m+1; m ≥ 0 as sum of row terms of corresponding triangle. In other words, in this manuscript are found and discussed the polynomials D_m(n,k) and U_m(n,k), such that, when being summed up over k in some range with respect to m and n returns the monomial n^2m+1.

There is no method of predicting the price of an option other than hedging strategies such as the binomial hedging strategy, the Black-Scholes hedging strategy and others. The price of an option is defined to be equal to the amount the seller needs to implement a hedging strategy. We will study these two basic hedging strategies in terms of their feasibility and we will see that the Black-Scholes hedging strategy is not feasible because this strategy demands instantaneously rebuilding the replicating portfolio. Consequently, the real world prices of the options are not relevant at all with the Black-Scholes hedging strategy! We will suitably redefine the binomial hedging strategy so that it will be practically useful and present other feasible and generally more effective hedging strategies with some of them are practically useful for options with no tradable underlying assets. Finally, we will mention some open questions related to the above.

The purpose of the present study is to explore the linkage between Hazardous Road Locations-based crash counts and a variety of geometric characteristics, roadway characteristics, traffic flow characteristics and spatial features in the region of Sousse, Tunisia. For this purpose, collision data was collected from at 52 hazardous road sections including 1397 crash records for a 11-year monitoring period from January 1, 2004 to December 31, 2014 obtained from National Observatory for Information, Training, Documentation and Studies on Road Safety in Tunisia (NOITDRS). The matrix of Pearson correlation was used in order to avoid inclusion of both variables, which were highly correlated. Both the Random Effects Negative Binomial model and the Negative Binomial model were estimated. To evaluate the models, the Random Effect Negative Binomial model improves the goodness-of-fit compared to the Negative Binomial model. Average Daily Traffic volume, Curved alignment, Presence of public lighting, Visibility, Number of lane, Presence of vertical/horizontal sign, Presence of rural segment, Presence of drainage system, Roadway surface condition, Presence of paved shoulder and presence of major road were found as significant variables influencing accident occurrences. Overall, the current research contributes to the literature from empirical, modeling methodological standpoints since it was the first study conducted in Tunisia to use crash prediction models for hazardous road locations, and that portrays Tunisian reality. The research findings present advantageous insights on hazardous road locations in the region of Sousse, Tunisia and present useful planning tools for public authorities in Tunisia.

This study aims to spatially analyze the distribution characteristics of vacant homes in Asan-si, Chungcheongnam-do, and understand the differences in factors causing vacant homes by type. We use a variety of statistical methods to examine these factors and identify key influencing factors such as the number of vacant stores, changes in overall population, land use complexity, and physical characteristics of land and buildings. The findings reveal that the factors contributing to the occurrence of vacant homes vary significantly by region and property type, highlighting the need for region-specific policies. For instance, in old town areas, small lot ratios and land use complexity are significant, while in rural village, the average number of floors and land use complexity play a major role. This study underscores the diversity of factors influencing vacant homes, suggesting that policies should be tailored to the unique characteristics of different regions to effectively address this issue.

ABSTRACT Background: To date, the COVID-19 pandemic has been the foremost health concern for humankind in the new millennia. This paper aims to describe the related hospitalizations in Italy during the first two years of the health emergency. Design and methods: This is a retrospective, population-based study of Italian hospitalizations for patients diagnosed with COVID-19 during the 2020-2021 period, extracted from the National Hospital Discharge Registry. The outcome variables considered include hospital admissions, costs, and length of stay, among other hospitalization-level variables. Costs were estimated through the charges associated with the Diagnosis-Related Group and Major Diagnostic Categories coding system, plus an additional cost attributed to COVID-19 admissions. To these charges was added an extra cost defined by the ministry of health. Results: In Italy, there were 357,354 hospitalizations for COVID-19 attributed to 298,856 patients in 2020 and 399,043 hospitalizations for COVID-19 attributed to 333,447 patients in 2021. COVID-19 patients faced a transfer rate thrice that of other patients. Hospitalizations were concentrated predominantly in the northern regions, particularly in the first year. Hospitalization rates varied by age in a sine wave pattern, peaking in the youngest and oldest age groups, with mortality rates escalating with age— tending to remain below 3% for those under 44 but surging to 40% in individuals over 75. The financial impact of COVID-19 hospitalizations was substantial, with total costs reaching €3.97 billion in 2020 and €4.99 billion in 2021. Costs per admission and per day also rose, from €11,112 and €807 in 2020 to €12,503 and €844 in 2021, respectively. Excluding the added financial burden of COVID-19, costs would have been notably lower. Hospitalizations involving continuous invasive mechanical ventilation were particularly costly (about 24.000 euros x admission), reflecting the significant resources required for these treatments. Conclusion: Implementing a protective pad around the entire health system that leverages networks of family doctors and nurses, connected in real time to the entire health system, can be crucial. Such a network, tasked with monitoring the local epidemiological situation, protecting vulnerable individuals also through telehealth visits, and establishing a triage system for infections in the initial phase, may effectively help in managing virus spread and protecting hospitals from unpredictable waves of admission demand. Strengthening primary health care could be a decisive factor for the future and enhance the system's resilience in facing new challenges.

In this note we study the binomial model applied to European, American and Bermudan type of derivatives. Our aim is to give the necessary and sufficient conditions under which we can define a fair value via replicating portfolios for any derivative using simple mathematical arguments and without using no arbitrage techniques. Giving suitable definitions we are able to define rigorously the fair value of any derivative without using concepts from probability theory or stochastic analysis therefore is suitable for students or young researchers. It will be clear in our analysis that if $e^{r \delta} \notin [d,u]$ then we can not define a fair value by any means for any derivative while if $d \leq e^{r \delta} \leq u$ we can. Therefore the definition of the fair value of a derivative is not so closely related with the absence of arbitrage. In the usual probabilistic point of view we assume that $d < e^{r \delta} < u$ in order to define the fair value but it is not clear what we can (or we can not) do in the cases where $e^{r \delta} \leq d$ or $e^{r \delta} \geq u$.

This study has developed a Matlab application for simulating statistical models project (SMp) probabilistic distributions that are similar to binomial and Poisson, which were created by mathematical procedures. The simulated distributions are graphically compared with these popular distributions. The application allows to obtain many probabilistic distributions, and shows the trend (τ ) for n trials with success probability p, i.e. the maximum probability as τ=np. While the Poisson distribution PD(x;µ) is a unique probabilistic distribution, where PD=0 in x=+∞, the application simulates many SMp(x;µ,Xmax) distributions, where µ is the Poisson parameter and value of x with generally the maximum probability, and Xmax is the upper limit of x with SMp(x;µ,Xmax) ≥ 0 and limit of the stochastic region of a random discrete variable. It is shown that by simulation via, one can get many and better probabilistic distributions than by mathematical one.

This study has developed a Matlab application for simulating statistical models project (SMp) probabilistic distributions that are similar to binomial and Poisson, which were created by mathematical procedures. The simulated distributions are graphically compared with these legendary distributions. The application allows to obtain many probabilistic distributions, and shows the trend (τ ) for n trials with success probability p, i.e. the maximum probability as τ=np. While the Poisson distribution PD(x;µ) is a unique probabilistic distribution, where PD=0 in x=+∞, the application simulates many SMp(x;µ,Xmax) distributions, where µ is the Poisson parameter and value of x with generally the maximum probability, and Xmax is upper limit of x with SMp(x;µ,Xmax) ≥ 0 and limit of the stochastic region of the random discrete variable X. It is shown that by simulation via, one can get many and better probabilistic distributions than by mathematical one.

The negative binomial regression model (NBRM) is a generalized linear model which relaxes the restrictive assumption by the Poisson regression model when the variance is equal to the mean. The estimation of the parameters of the NBRM is obtained using the maximum likelihood (ML) method. Maximum likelihood estimator becomes unstable when the explanatory variables are linearly dependent, a situation known as multicollinearity. Based on this, we developed a new estimator called modified jackknifed Negative Binomial Kibria-Lukman (MJNBKL) estimator for the radiation of multicollinearity in NBRM using four different biasing (shrinkage) parameters. We establish superiority condition for MJNBKL estimator over the ones. The performance MJNBKL estimator was ascertained by comparing it with the existing ones through a Monte Carlo simulation study and two real life application datasets. The results of the simulation and real life application show that MJNBKL estimator outperformed the other estimators compared with by having the smallest MSE across all sample sizes and for different levels of correlation for the four biasing parameters used and the third biasing parameter is the optimal shrinkage parameter with the lowest MSE.

We evaluate various binomial sums involving the powers of Fibonacci and Lucas numbers.

Shared micromobility has gained significant attention in the field of transportation engineering in recent years as an environmentally friendly, convenient, and easily accessible transportation mode. Like other medium-sized cities, Birmingham, Alabama implemented a shared micromobility pilot program in 2021 that captured the attention of local travelers. This study examined shared e-scooter usage and associated travel patterns in Birmingham using 2021-2022 field data. From these data, ArcGIS maps were used to showcase trip origins and destinations. To gain a further understanding of e-scooter travel patterns in the study area, zip code and block group densities were calculated. Additionally, a negative binomial regression model was constructed to identify determinants of shared e-scooter trips. The analysis results showed that the usage of shared e-scooters was the highest during the nighttime, on weekends, and in the fall season. Furthermore, the research findings indicated that shared e-scooters experienced their highest utilization rates in areas with a higher proportion of educated and higher-income individuals. These findings suggest that travelers’ mode choice related to the use of micromobility modes is influenced by environmental and demographic factors. Overall, this case study offers valuable contributions to the understanding of the role of shared e-scooters in Birmingham's transportation landscape and can guide transportation authorities in other medium-sized cities in their efforts to plan for micromobility options.

Wellness is a multidimensional concept that touches upon the various physical, mental, emotional, spiritual, social and environmental facets of health. Interest towards and importance of wellness have been growing constantly for the past two decades and thus makes it crucial to understand which factors affect public engagement with wellness information for multiple stakeholders. The Instagram account of New York Times (NYT) specifically for sharing wellness content with the handle nyt_well was selected as the object of study. 773 posts from this account between March of 2019 and December of 2022 were collected and analyzed to answer the research question of which factors are most influential to public engagement with wellness content. Two negative binomial regressions were run on features including the type of post, length, word count, sentiment score and topic with number of likes and comments as the dependent variables for each of those regression models. Results indicated that the type of post and its sentiment score were the two most influential determinants of public engagement with p-values smaller than 0.05. While the effects of some of these factors aligned with findings from previous studies conducted on social media content not related to wellness (e.g., marketing), some others affected the two separate public engagement metrics in opposite directions, warranting future studies to investigate further on the cause of this phenomenon.

The decapod fauna in intertidal rocky shore in continental Chile has high species diversity mainly in northern and central Chile due high productivity of these coast due influence of cold Humboldt stream. One of the abundant species in these ecosystems is decapod Cyclograpsus cinereus Dana, 1851. The aim of the present study was determine the spatial distribution pattern of C. cinereus along rocky shore in Antofagasta bay, north of Chile between spring and summer 2018-2020. Data was obtained counting individuals from random quadrants in intertidal zones, to the obtained data was applied the variance mean ratio for determine if the specimens have random, aggregate or uniform distribution, that are associated to Poisson, negative binomial or positive binomial distributions respectively. Data obtained revealed the existence of associated distribution with respective negative binomial distribution pattern in 16 observations, and uniform distribution for 3 observations. The sites corresponds to rocky shores in urban zones, and in a protected zone, and the density does not have significant differences for both kind of sites. The observed data about aggregated pattern agree with observations for decapods for rocky shore in central and southern Chile, specifically in interpretative probabilistic models.

This study investigated the trends of registered Death and Birth in Nigeria using Generalized Linear Models. Annual data on Death and Birth was collected from National Population Commission for the period of 2004 to 2017. The Natural increase calculated revealed a positive trend in the natural increase in Nigeria from 2004 to 2017. Evidence from summary statistics revealed some level of over dispersion (variance > mean). This study explored Poisson Regression Models and Negative Binomial Regression Models using two links (identity and log). The results revealed a positive increase in registration of birth and death rates in Nigeria and among the competing the models, Negative Binomial regression model with identity link emerged as the best model for modeling birth and death rates registration in Nigeria. Data on numbers of deaths and causes of death are essential if countries are to determine priorities, formulate and monitor policies for public health care as well as other government policies that may be based on such data

The purpose of this paper is to evaluate the timing of innovative investment in technology product life cycles using a compound binomial option with management flexibility. Considering the business cycles changes in the macroeconomic will affect consumer purchasing power. The focus is how to evaluate the optimal investment strategy and the project value. It was applied to different product stages (three stages including production innovation, manufacture innovation, and operation innovation) and factored to different risks to build a technology innovation strategy model. An aim of this study is the options premium of the best strategy timing for each innovation stage. Its application of the compound binomial options for the manufacture innovation will only be considered after the execution of the production innovation. The same condition is applied to the operation innovation, which will only be considered after the execution of the manufacture innovation. Then, this paper constructs the dynamic investment sequential decision model, assesses the feasibility of an investment strategy, and makes a decision on the appropriate project value and options premium for each stage under the possible change of Gross Domestic Product (GDP). This paper investigates the product life cycle innovation investment topic by using the compound binomial options method and will provide a more flexible strategy decision compared with other trend forecast criteria.

Leishmaniasis is the third most common vector-borne disease and a very important protozoan infection. Cutaneous leishmaniasis is one of the most common types of leishmaniasis infectious diseases with up to 2 million occurrences of new cases each year worldwide. A dynamic transmission multivariate time series model was applied to the data to account for overdispersion and evaluate the effects of three environmental layers as well as seasonality in the data. Furthermore, ecological niche modeling was used to investigate the geographical suitable conditions for cutaneous leishmaniasis using temperature, precipitation and altitude as environmental layers, together with the leishmaniasis presence data. A retrospective analysis of the cutaneous leishmaniasis spatial data in Afghanistan between 2003 and 2009 indicates a steady increase from 2003 to 2007, a small decrease in 2008, then another increase in 2009. An upward trend and regularly repeating patterns of highs and lows was observed related to the months of the year which suggests seasonality effect in the data. Two peaks were observed in the disease occurrence-- January to March and September to December -- which coincide with the cold period. Ecological niche modelling indicates that precipitation has the greatest contribution to the potential distribution of leishmaniasis.

Hypertension stands as the foremost modifiable risk factor for cardiovascular disorders in South Asia. This research demonstrates the superiority of statistical models over traditional approaches in identifying clinical risks related to hypertension. Leveraging individual-level data from the Bangladesh Demographic and Health Survey (2017-18), a unified dataset was created. Adhering to JNC7 and ACC/AHA guidelines for hypertension classification, two prominent regression models were employed to estimate the prevalence of undiagnosed hypertension. A substantial 16% of hypertension cases in Bangladesh remain undetected. The study identifies undiagnosed hypertension as a heightened risk, particularly among the youth and residents of Sylhet and Rangpur divisions. Statistical models exhibit efficacy in discerning undiagnosed hypertension and associated factors in South Asia. Subsequent investigations should incorporate biological markers to enhance statistical models and ascertain their practicality.

We evaluate some binomial cubic Fibonacci sums.

Let $a, b$ and $n > 1$ be three positive integers such that $a$ and $\sum_{j=0}^{n-1} b^j$ are relatively prime. In this paper, we prove that the toric ideal $I$ associated to the submonoid of $\mathbb{N}$ generated by $\{\sum_{j=0}^{n-1} b^j\} \cup \{\sum_{j=0}^{n-1} b^j + a\, \sum_{j=0}^{i-2} b^j \mid i = 2, \ldots, n\}$ is determinantal. Moreover, we prove that for $n > 3$, the ideal $I$ has a unique minimal system of generators if and only if $a < b-1$.

In this paper, we mainly show that generalized hyperharmonic number sums with reciprocal binomial coefficients can be expressed in terms of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.

This study introduces a new discrete distribution which is a weighted version of Poisson-Lindley distribution. The weighted distribution is obtained using the negative binomial weight function and can be fitted to count data with over-dispersion. The p.m.f., p.g.f. and simulation procedure of the new weighted distribution, namely weighted negative binomial Poisson-Lindley (WNBPL), are provided. The maximum likelihood method for parameter estimation is also presented. The WNBPL distribution is fitted to several insurance datasets, and is compared to the Poisson and negative binomial distributions in terms of several statistical tests.

In the paper, using two inversion theorems for the Stirling numbers and binomial coecients, employing properties of the Bell polynomials of the second kind, and utilizing a higher order derivative formula for the ratio of two dierentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, and recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.

Count data analysis presents unique challenges due to its discrete nature, often exhibiting excessive zeros and overdispersion. To address these complexities, count models, such as Poisson regression and Negative Binomial regression, have been developed, enabling modeling and prediction of count-based phenomena. Additionally, it’s important to notice that zero inflation, a phenomenon commonly observed in count data, requires specialized techniques for robust analysis. This article provides an overview of count data and count models, explores zero inflation, introduces Likelihood Ratio Tests, and explains how the Vuong Test can be used as a model selection criteria. Furthermore, we created a Vuong Test implementation from scratch using the Python programming language. This implementation enhances the accessibility and applicability of the Vuong Test in real-world scenarios, providing a valuable contribution to the academic community, since Python didn’t have an implementation of this statistical test.

It is quite easy to stochastically distort an original count variable to obtain a new count variable with relatively more variability than in the original variable. Many popular overdispersion models (variance greater than mean) can indeed be obtained by mixtures, compounding or randomlystopped sums. There is no analogous stochastic mechanism for the construction of underdispersed count variables (variance less than mean), starting from an original count distribution of interest. This work proposes a generic method to stochastically distort an original count variable to obtain a new count variable with relatively less variability than in the original variable. The proposed mechanism, termed condensation, attracts probability masses from the quantiles in the tails of the original distribution and redirect them toward quantiles around the expected value. If the original distribution can be simulated, then the simulation of variates from a condensed distribution is straightforward. Moreover, condensed distributions have a simple mean-parametrization, a characteristic useful in a count regression context. An application to the negative binomial distribution resulted in a distribution allowing under, equi and overdispersion. In addition to graphical insights, fields of applications of special cases of condensed Poisson and condensed negative binomial distributions were pointed out as an indication of the potential of condensation for a flexible analysis of count data

Transportation authorities aim to boost road safety by identifying risky locations and applying suitable safety measures. The Highway Safety Manual (HSM) is a vital resource for US transportation professionals, aiding in the creation of Safety Performance Functions (SPFs), which are predictive models for crashes. These models rely on Negative Binomial distribution-based regression and misinterpreting them due to unmet statistical assumptions can lead to erroneous conclusions, including inaccurately assessing crash rates or missing high-risk sites. The Florida Department of Transportation (FDOT) has introduced context classifications to HSM SPFs, complicating assumption violation identification. This study, part of an FDOT-sponsored project, investigates established statistical diagnostic tests to identify model violations and proposes a novel approach to determine optimal spatial regions for Empirical Bayes adjustment. This adjustment aligns HSM-SPFs with regression assumptions. The study employs a case study involving Florida roads. Results indicate that a 20-mile radius offers an optimal spatial sample size for modeling crashes of all injury levels, ensuring accurate assumptions. For severe injury crashes, which are less frequent and harder to predict, a 60-mile radius is suggested to fulfill statistical modeling assumptions. This methodology guides FDOT practitioners in assessing the conformity of HSM-SPFs with intended assumptions and determining appropriate region sizes.

In this article, the usual factorials and binomial coefficients have been generalized and extended to the negative integers. Basing on this generalization and extension, a new kind of polynomials has been proposed, which led directly to the non-classical hypergeometric orthogonal polynomials and the non-classical second-order hypergeometric linear DEs. The resulting polynomials can be used in non-relativistic and relativistic QM, particularly, in the case of the Schrödinger equation, and Dirac equations for an electron in a Coulomb potential field.

In this paper, we introduce two discrete degenerate random variables, namely the degenerate binomial and degenerate Poisson random variables. We deduce the expectations of the degenerate binomial random variables. We compute the generating function of the moments of the degenerate Poisson random variables, which leads us to define the new type degenerate Bell polynomials, and hence obtain explicit expressions for the moments of those random variables in terms of such polynomials. We also get the variances of the degenerate Poisson random variables. Finally, we illustrate two examples of the degenerate Poisson random variables.

In many practical applications, it is more convenient to characterize the quality of production processes or service operations throughout the count of nonconformities. In the context of SPC, nonconformities are usually assumed to appear according to the binomial probability model. The conventional way for monitoring nonconformities involves Shewhart-type control procedures based on both constant and time-varying sample sizes. In this article, an EWMA scheme is proposed for monitoring the fraction of nonconforming items with time-varying sample sizes. The proposed control chart is referred to as the EWMAG-B and can be easily adapted to work with a constant sample size by fixing it at a needed value. By means of simulation, it was found out that the EWMAG-B chart outperforms the conventional p control chart in Phase II while detecting changes in the process level is wanted.

Google Trends data can be informative for infectious disease incidences, including Lyme disease. However, the use of Google Trends for predictive purposes is underutilized. In this study, we tested the ability of Google Trends search data to predict monthly state-level Lyme disease case counts in the United States. We requested Lyme disease data for the years 2010-2021. We downloaded Google Trends search data on terms for Lyme disease, symptoms of Lyme disease, and diseases with similar symptoms as Lyme disease. We built mixed negative binomial models based on a training dataset (2010-2016) and tested the models on a test dataset (2017-2021). A model was built for each search term and monthly lags of search terms were included as predictors. The highest performing models had high predictive ability, indicated by low Root Mean Squared Errors (RMSEs) and close association between observed and predicted case counts. The highest performing model was for the search term “Summer Flu”, which indicates low specificity of some of the terms. We outline challenges of using Google Trends data, including data availability and a mismatch between geographic units. We discuss opportunities for Google Trends data, including prediction of additional zoonotic diseases and incorporating environmental and companion animal data.

The move to a low carbon economy is very important for enhancing international competitiveness. The eco-innovation is the critical factor of the green paradigm. This study is designed to investigate deeply the determinants of eco-innovation of manufacturing firms in Korea by suggesting anticipated regulations, self-regulations, and industry-specific characteristics as external factors and open information sources as internal factors. The data used in the analysis is 1946 sample firms from Korean Innovation Survey 2010 based on the Oslo Manual. Using the multi-variate probit analysis and the zero-inflated negative binomial (ZINB) regression analysis, we have found out that the anticipated regulations and self-regulations have significant influences both on eco-process innovation and eco-product innovation, while industrial characteristics have no effects. The empirical results also show that the breadth of information sources has a positive effect on businesses in implementing eco-innovations. Our findings show that the Korean government should provide a good platform where firms can better understand the future trends of environmental policies, particularly policies on anticipated and self-regulations. At the same time, Korean firms should establish a voluntary system to control environmental activities so that they can improve eco-innovations through integrating external information.

Crash prediction of the sharp horizontal curve segment (SHCS) of a freeway is an important tool in analyzing safety of SHCSs and in building a crash prediction model (CPM). The design and crash report data of 88 SHCSs from different institutions were surveyed and three negative binomial (NB) regression models and three generalized negative binomial (GNB) regression models were built to prove that the interactive influence of explanatory variables plays an important role in fitting goodness. The study demonstrates the effective use of the GNB model in analyzing the interactive influence of explanatory variables and in predicting freeway basic segments. Traffic volume, highway horizontal radius, and curve length have been formulated as explanatory variables. Subsequently, we performed statistical analysis to determine the model parameters and conducted sensitivity analysis. Among the six models, the result of model 6, which considered interactive influence, is much better than those of the other models by fitting rules. We also compared the actual results from crashes of 88 SHCSs with those predicted by models 1, 3, and 6. Results demonstrate that model 6 is much more reasonable than models 1 and 3.

Tuberculosis cause of death worldwide and the leading cause from a single infectious agent, ranking above Human immunodeficiency virus (HIV) and Acquired Immune Deficiency Syndrome (AIDS). The aim of this study is to ascertain the trend of tuberculosis prevalence and the effect of HIV prevalence onl Tuberculosis case in some West African countries from 2000 to 2016 using count panel data regression models. The data used annual HIV and Tuberculosis cases spanning from 2000 to 2016 extracted from online publication of World health Organization (WHO). Panel Poisson regression model and Negative binomial regression model for fixed and random effects were used to analyzed the count data, the result revealed a positive trend in TB cases while increased in HIV cases leads to increase in TB cases in West African countries. Among the competing models used in this study, Panel Negative Binomial Regression Model with fixed effect emerged the best model with log likelihood value of -1336.554. This study recommended that Government and NGOs need more strategies to fight against HIV menace in West Africa as this will in turn reduced TB cases in West Africa.

In Sub-Saharan African Countries such as Nigeria with high prevalence rate, Child HIV/AIDs acquired through Mother-to-Child transmission (MTCT) can be largely prevented by using a well-established prevention programme and scheme. This study examined factors that can enhanced Prevention of Mother-to-Child transmission (PMTCT) in Nasarawa State. To achieve this, structured questionnaire were used to collect data from one hundred and sixteen (116) women attending two (2) primary facilities and two (2) secondary facilities in the State. This study utilized methods of Poisson Regression, Negative Binomial Regression and Logistic Regression Analyses. Results revealed that women with at least a secondary school education, women with husband in military and women with perceived confidentiality of their HIV status significantly enhanced PMTCT of HIV in Nasarawa State while significant proportion of the women attest to the fact that drugs are available in the facilities (p-value=0.0000<0.05) . Other factors include mother income level, willingness to continue with PMTCT programme and women in support group can also enhanced PMTCT though they are not significant. This study recommends that the factors identified should be explored by NGOs, Ministry of Health and, Support groups and other relevant agencies since they have the capacity to enhanced PMTCT of HIV in Nasarawa State, Nigeria.

Let be a power function $f_{r,M}(s)$ defined for every $s$ within the finite set $M$ as follows $$f_{r,M}(s)= \begin{cases} s^r, \ &s\in M,\\ 0, \ &\mathrm{otherwise}. \end{cases} $$ Let a discrete convolution of $f_{r,M}(s)$ be denoted as follows $\mathrm{Conv}_{r,M}[n]=(f_{r,M}*f_{r,M})[n]$. Let a real coefficients $A_{m,j}$ be given by the following recurrence $$ A_{m,j} = \begin{cases} 0, & \mathrm{if } \ j<0 \ \mathrm{or } \ j>m, \\ (2j+1)\binom{2j}{j} \sum_{d=2j+1}^{m} A_{m,d} \binom{d}{2j+1} \frac{(-1)^{d-1}}{d-j} B_{2d-2j}, & \mathrm{if } \ 0 \leq j < m, \\ (2j+1)\binom{2j}{j}, & \mathrm{if } \ j=m. \end{cases} $$ In this paper we show that for every $n>0$ the following odd-power identities involving coefficients $A_{m,j}$ and convolution transform $\mathrm{Conv}_{r,M}[n]$ hold $$ \begin{split} n^{2m+1}+1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{N}}[n],\\ n^{2m+1}-1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{Z}_{>0}}[n],\\ n^{2m+1}&=\sum_{r=0}^{m}A_{m,r}\sum_{k=1}^{n} k^r(n-k)^r\\ &=\sum_{r=0}^{m}A_{m,r}\sum_{k=0}^{n-1} k^r(n-k)^r. \end{split} $$

We tested the possibility of using “adverse events count” (AEC) as a drug-risk indicator and side-effect severity indicator. Data from 3938 adverse event (AE) reports for COVID-19 vaccines (Comirnaty, Moderna, Vaxzevria, and Janssen) and 6869 AE reports for other medicines were collected from the Bulgarian Drug Agency database (01.01.2018–31.03.2022). AEC was modeled with zero-adjusted negative binomial (ZANBI) and zero-adjusted Poisson inverse Gaussian (ZAPIG) regression models, which account for zero absence and overdispersion. The models’ fit was checked with residual diagnostic plots and parametric correspondence to normality. Explanatory variables were: age, sex, sequence number (order of submission), severity of AE, and vaccine type. Average AEC was higher in severe vs. non-severe AE, and in females vs. males; it decreased with age and was lower in Comirnaty than other vaccines. Variability of AEC for COVID-19 data decreased with sequence number in severe AE and increased in non-severe AE. Full ZANBI models had greater sensitivity than parametric ZANBI or ZAPIG models. Results showed correlation between AEC and AE severity, suggesting AEC as a simple and reliable measure of drug-risk and side-effect severity for clinicians and regulators, especially for COVID-19 vaccines.