There are major advances which have been made to understand the epidemiology of infectious diseases. However, more than 2 million children in the developing countries still die from pneumonia each year. The efforts to promptly detect, effectively treat and control the spread of pneumonia is possible if its dynamics is understood. In this paper, we develop a mathematical model for pneumonia among children under five years of age. The model is analyzed using the theory of ordinary differential equations and dynamical systems. We derive the basic reproduction number, R 0 , analyze the stability of equilibrium points and bifurcation analysis. The results of the analysis shows that there exist a locally stable disease free equilibrium point, E f when R 0 < 1 and a unique endemic equilibrium, E e when R 0 > 1.The analysis also shows that there is a possibility of a forward bifurcation.
In this paper, we give a characterization of scalar operators. In particular we show that a densely defined closed linear operator H acting on a reflexive Banach space X is scalar if it is of (0, 1) type R and f (H) ≤ f ∞ for f in the algebra of smooth functions U.
In this paper, we characterize the scalar operators by using the semigroup theory and the corresponding generators of (α, α + 1) type R operators. In particular, we show that if a densely defined operator H generates a contraction semigroup, then both H and H * are scalar operators and if H admits a U(Algebra of smooth functions) functional calculus of scalar type, then H * also admits a U functional calculus of scalar type.
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