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Cholera, characterized by severe diarrhea and rapid dehydration, is a water-borne infectious disease caused by the bacterium Vibrio cholerae. Haiti offers the most recent example of the tragedy that can befall a country and its people when cholera strikes. While cholera has been a recognized disease for two centuries, there is no strategy for its effective control. We formulate and analyze a mathematical model that includes two essential and affordable control measures: water chlorination and education. We calculate the basic reproduction number and determine the global stability of the disease-free equilibrium for the model without chlorination. We use Latin Hypercube Sampling to demonstrate that the model is most sensitive to education. We also derive the minimal effective chlorination period required to control the disease for both fixed and variable chlorination. Numerical simulations suggest that education is more effective than chlorination in decreasing bacteria and the number of cholera cases.

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We study a mixed boundary value problem for the quasilinear elliptic equation div A(x, ∇u(x)) = 0 in an open infinite circular half-cylinder with prescribed continuous Dirichlet data on a part of the boundary and zero conormal derivative on the rest. We prove the existence and uniqueness of bounded weak solutions to the mixed problem and characterize the regularity of the point at infinity in terms of p-capacities. For solutions with only Neumann data near the point at infinity we show that they behave in exactly one of three possible ways, similar to the alternatives in the Phragmén-Lindelöf principle.

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Ebola Virus disease (EVD) is an emerging and re-emerging zoonotic disease which mostly occur in Africa. Both prediction of the next EVD and controlling an ongoing outbreak remain challenging to disease prone countries. Depending on previous experiences to curb an outbreak is subjective and often inadequate as temporal socioeconomic advances are dynamic and complex at each disease. We hypothesize that a scientific model would predict EVD disease outbreak control. In this work, a mathematical model with a convex incidence rate for an optimal control model of Ebola Virus Disease is formulated and analyzed. An optimal control strategy which aims at reducing the number of infected individuals in the population and increasing the number of recovered through treatment is evaluated. Three control measures: tracing of contacts, lock-down and treatment have been considered. A qualitative analysis and numerical experiments are performed on the model and the findings reveal that the most expensive strategy involved imposing lock-down and contact tracing of the infected while the cheapest alternative was lock-down and treatment of the infected. Hence, policy makers should concentrate on treatment and lock down to combat the disease.

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Regularity of a boundary point x 0 ∈ ∂Ω for the Laplace equation ∆u = 0 was characterised by the celebrated Wiener criterion which was established in 1924 by Wiener [23]. This was done through exhaustions by regular open sets. With Wiener's criterion, one measures the thickness of the complement of Ω near x 0 in terms of capacities, see (2.7). If the complement is too thin, then the boundary point x 0 ∈ ∂Ω is irregular.