A transmission model for dengue fever is discussed here. Restricting the dynamics for the constant host and vector populations, the model is reduced to a two-dimensional planar equation. In this model the endemic state is stable if the basic reproductive number of the disease is greater than one. A trapping region containing the heteroclinic orbit connecting the origin (as a saddle point) and the endemic fixed point occurs. By the use of the heteroclinic orbit, we estimate the time needed for an initial condition to reach a certain number of infectives. This estimate is shown to agree with the numerical results computed directly from the dynamics of the populations.

/pdf/a-two-dimensional-model-for-the-transmission-of-dengue-fever-gckj9umkyc.pdf

A Two-dimensional Model for the Transmission of Dengue Fever Disease *

Forest-pest interaction dynamics: the simplest mathematical models.

Amethodfordeterminationandtwomethodsforapproximationofthedomainofattraction Da(0) of the asymptotically stable zero steady state of an autonomous, R-analytical, discrete dynamical system are presented. The method of determination is based on the construction of a Lyapunov function V, whose domain of analyticity is Da(0). The first method of approximation uses a sequence of Lyapunov functions Vp, which converge to the Lyapunov function V on Da(0). Each Vp defines an estimate Np of Da(0). For any x ∈ Da(0), there exists an estimate Npx which contains x. The second method of approximation uses a ball B(R) ⊂ Da(0) which generates the sequence of estimates Mp = f −p (B(R)). For any x ∈ Da(0),thereexistsanestimate Mpx whichcontains x.Thecases � ∂0 f � <1and

/pdf/methods-for-determination-and-approximation-of-the-domain-of-korvrb05oq.pdf

Methods for determination and approximation of the domain of attraction in the case of autonomous discrete dynamical systems

The long history and little progress of efforts to systematically evaluate and certify numerical software is recounted. Although this continues to be a serious problem, traditional numerical software libraries themselves are becoming increasingly irrelevant in today's computing environment. The reasons for this are described and some prescriptions for overcoming this crisis are presented.

Quality of Numerical Software

This analysis investigates the mass dynamics of competitive electoral politics with respect to the presidential vote during the 1928–36 realigning period for the United States. A formal system of three interdependent differential equations is employed to characterize the dynamic processes of the aggregate voter shifts between the Republican, Democratic, and nonvoter populations. The modeling strategy is used to locate substantial variations in the mass dynamics between large subgroups in the electorate as well as to identify both national and socially interactive components to the patterns of voter movements. The results show that the overall realignment period was quite complex. Vote switching from the Republican party to the Democratic party was the dominant characteristic of the 1932 election, whereas additional Democratic gains in 1936 came mostly from new voters.

Mass Dynamics of U.S. Presidential Competitions, 1928–1936

Differential and difference equations through computer experiments

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Differential and difference equations through computer experiments