In this paper we propose the explicit formulas of Average Run Length (ARL) of Exponentially Weighted Moving Average (EWMA) control chart for Autoregressive Integrated Moving Average: ARIMA (p,d,q) (P, D, Q)_L process with exponential white noise. To check the accuracy, the ARL results were compared with numerical integral equations based on the Gauss-Legendre rule. There was an excellent agreement between the explicit formulas and the numerical solutions. Additionally, we compared the computational time between our explicit formulas for the ARL with the one obtained via Gauss-Legendre numerical scheme. The computational time for the explicit formulas was approximately one second that is much less than the numerical approximations. The explicit analytical formulas for evaluating ARL0 and ARL₁ can produce a set of optimal parameters which depend on the smoothing parameter (λ) and the width of control limit (H), for designing an EWMA chart with a minimum ARL₁.

The research work examined the trend of HIV/AIDS, Tuberculosis, and Hepatitis diseases in Plateau state. Annual data from 2003 to 2018 was collected from the department of biostatistics at Plateau State Specialist Hospital (PSSH), Jos. The methods of analysis used are the Poisson Autoregressive Model (PAR(1)) and the Poisson Exponentially Weighted Moving Average Model (PEWMA). The results revealed a significant annual decrease of 23.9% and 4% in Tuberculosis and HIV/AIDS respectively. Furthermore, the results showed a significant annual increase of 46% in Hepatitis. The PEWMA model used revealed that TB increased by 0.02% when there is an increase in HIV but not significant, while Hepatitis significantly aggravates TB by at least 0.24%. Also, there is a significant rise in HIV by 0.85% when TB increases but Hepatitis has no such effect on HIV. Lastly, PEWMA model indicated a rise of 0.5% in Hepatitis cases when there is an increase in TB, but a surge in HIV has no such effect on Hepatitis cases in Jos. The study recommended that fight against TB should be intensified since TB cases significantly affect both HIV and Hepatitis in Jos, Nigeria.

Classical numerical methods for solving ordinary differential equations often produce less accurate results when applied to problems with oscillatory or periodic behaviour. To adapt them for such problems, they are usually modified using the exponential fitting technique. This adaptation allows for the construction of new methods from their classical counterparts. The new methods are usually more accurate, efficient and suitable for handling the oscillatory or periodic behaviour of the problem. In this work, we construct a two-parameter exponentially-fitted Taylor method suitable for solving oscillatory or periodic problems that possess two frequencies. The construction algorithm is based on a proposed six-step flowchart discussed by authors in related literature. Two standard test problems were used to illustrate the accuracy and performance of the proposed method.

The quest for accurate and more efficient methods for solving periodic/oscillatory problems is gaining more attention in recent time. This paper presents the construction and implementation of a family of exponentially–fitted Obrechkoff methods. A single–step Obrechkoff method involving terms up to the fourth derivatives was used as the base method. We also present the stability and convergence properties of the constructed family of methods. Two numerical examples were use to illustrate the performance of the constructed methods.

In this paper, we consider the development of algorithms for the solution of first order initial value problems whose solution exhibits exponential behaviour. Method of interpolation and collocation of basis function to give system of nonlinear equations which is solved for the unknown parameters to give a continuous scheme. The discrete methods recovered from the continuous scheme are implemented in block form. The stability properties of the method is verified and numerical experiments show that our method is efficient in handling these problems

Generalized distributions have been studied a lot recently because of their flexibility and relia-bility in modeling lifetime data. The two-parameter exponentially-modified logistic distribution is a new flexible modified distribution that was introduced recently. It is regarded as a strong competitor for widely used classical symmetrical and non-symmetrical distributions such as normal, logistic, lognormal, and log-logistic. In this study, the unknown parameters of the dis-tribution are estimated using the maximum likelihood method. The Grey Wolf Optimization al-gorithm, which is a new meta-heuristic algorithm, is applied in order to solve the nonlinear like-lihood equations of the study model. The performance of the Grey Wolf Optimization method is compared to that of the other meta-heuristic algorithms used in this study, which include the Whale Optimization Algorithm, the Sine Cosine Algorithm, and the Particle Swarm Optimiza-tion Algorithm. The efficiencies of maximum likelihood estimates for all algorithms are com-pared via an extensive Monte-Carlo simulation study. The likelihood estimates for the location α and scale β parameters of the exponentially-modified logistic distribution developed with the Grey Wolf Optimization algorithm are the most efficient among others, according to simulation findings. Four real datasets are analyzed to show the flexibility of this distribution.