Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number $R_{0}$, the contact number $\sigma$, and the replacement number $R$ are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of $R_{0}$ and $\sigma$ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.

https://epubs.siam.org/doi/pdf/10.1137/S0036144500371907

The Mathematics of Infectious Diseases

This paper analyses the stability and fairness of two classes of rate control algorithm for communication networks. The algorithms provide natural generalisations to large-scale networks of simple additive increase/multiplicative decrease schemes, and are shown to be stable about a system optimum characterised by a proportional fairness criterion. Stability is established by showing that, with an appropriate formulation of the overall optimisation problem, the network's implicit objective function provides a Lyapunov function for the dynamical system defined by the rate control algorithm. The network's optimisation problem may be cast in primal or dual form: this leads naturally to two classes of algorithm, which may be interpreted in terms of either congestion indication feedback signals or explicit rates based on shadow prices. Both classes of algorithm may be generalised to include routing control, and provide natural implementations of proportionally fair pricing.

https://link.springer.com/content/pdf/10.1057%2Fpalgrave.jors.2600523.pdf

Rate control for communication networks: shadow prices, proportional fairness and stability

Symmetric diffeomorphic image registration with cross-correlation: evaluating automated labeling of elderly and neurodegenerative brain.

Finite-time stability is defined for equilibria of continuous but non-Lipschitzian autonomous systems. Continuity, Lipschitz continuity, and Holder continuity of the settling-time function are studied and illustrated with several examples. Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability. It is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov results can only assure the existence of continuous Lyapunov functions. Finally, the sensitivity of finite-time-stable systems to perturbations is investigated.

/pdf/finite-time-stability-of-continuous-autonomous-systems-1e0bmid7w7.pdf

Finite-Time Stability of Continuous Autonomous Systems

A reproducible evaluation of ANTs similarity metric performance in brain image registration.

Ordinary differential equations

together with the initial condition y(t0) = y0 A numerical solution to this problem generates a sequence of values for the independent variable, t0, t1, . . . , and a corresponding sequence of values for the dependent variable, y0, y1, . . . , so that each yn approximates the solution at tn yn ≈ y(tn), n = 0, 1, . . . Modern numerical methods automatically determine the step sizes hn = tn+1 − tn so that the estimated error in the numerical solution is controlled by a specified tolerance. The Fundamental Theorem of Calculus gives us an important connection between differential equations and integrals.

Wilfred Kaplan

Papers

Ordinary differential equations

Ordinary Differential Equations.

Operational Methods for Linear Systems

Advanced mathematics for engineers

Regular curve-families filling the plane, II