An integral equation method for solving neumann problems on simply and multiply connected regions with smooth boundaries

This research presents several new boundary integral equations for the solution of Laplace’s equation with the Neumann boundary condition on both bounded and unbounded multiply connected regions. The integral equations are uniquely solvable Fredholm integral equations of the second kind with the generalized Neumann kernel. The complete discussion of the solvability of the integral equations is also presented. Numerical results obtained show the efficiency of the proposed method when the boundaries of the regions are sufficiently smooth

Epidemiology of measles in the metropolitan setting, Addis Ababa, Ethiopia, 2005–2014: a retrospective descriptive surveillance data analysis

Abstract Background Measles is a highly infectious and serious respiratory viral disease which caused by a virus. It is a significant cause of illness and death worldwide. This data analysis was conducted to describe the trend and determine the reporting rate of measles cases in Addis Ababa to make recommendation for the government of the city to strengthening measles control interventions. Methods We obtained and extracted ten years (2005–2014) Addis Ababa city’s measles surveillance data from national database. We carried out retrospective descriptive data analysis by time, place and person variables. We calculated cumulative and specific reporting rates by dividing measles cases (lab confirmed, epidemiologically linked and compatible cases) to respective population and multiplying by 100,000. We divided average of ten years measles cases to midyear population and multiplied by 100,000 to calculate annualized reporting rate. We analyzed non-measles febrile rash rate by dividing laboratory negative cases to total population and multiplying by 100,000. Results A total of 4203 suspected measles cases were identified. Among them 1154 (27.5%) were laboratory confirmed, 512 (12.2%) were clinically compatible, 52 (1.2%) were epidemiologically linked cases and the rest 2485 (59.1%) were IgM negative for measles which makes total measles cases 1718 (40.9%). Median age was 5 years with 2–18 years interquartile-range. The annualized measles reporting rate was 5.9, which was 40.2 among > 1 year, 11.5 among 1–4 years, 6.0 among 5–14 years, 4.1 among 15–44 years and 0.01 among ≥ 45 years per 100,000 population. Among the total measles cases; 380 (22%) were received at least one dose of measles containing vaccine (MCV) while 415 (24%) cases were not vaccinated and the vaccination status of 923 (54%) cases were not known. Conclusion Our analysis revealed that the reporting rate was higher among young children than older age group. Among all the patients 22% were received at least one dose of measles vaccine whereas 13% were not vaccinated against measles antigen. Routine immunization should be strengthened to reach all children through well monitored vaccine cold chain management