We formulate an SIS epidemic model on two patches. In each patch, media coverage about the cases present in the local population leads individuals to limit the number of contacts they have with others, inducing a reduction in the rate of transmission of the infection. A global qualitative analysis is carried out, showing that the typical threshold behavior holds, with solutions either tending to an equilibrium without disease, or the system being persistent and solutions converging to an endemic equilibrium. Numerical analysis is employed to gain insight in both the analytically tractable and intractable cases; these simulations indicate that media coverage can reduce the burden of the epidemic and shorten the duration of the disease outbreak.

An SIR model with vaccination and varying population is formulated. The global dynamics of this model and its corresponding proportionate system are investigated. The correlations between the two systems in terms of disease eradication and persistence are presented.Three critical vaccination rates φ 1c , φ 2c and φ 3c are obtained. It is found that when φ > φ 1c the disease can be eradicated by increasing the vaccination rate until it exceeds φ 3c . When φ < φ 1c , the disease can be controlled to an endemic level by taking the appropriate vaccination rate φ 2c .

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In this paper, we first show that the second-order cone linear complementarity problem (SOCLCP) can be solved by finding a positive zero s * ∈ R of a particular rational function h(s), and we then propose a Krylov subspace method to reduce h(s) to h (s) as in the model reduction. The zero s * of h(s) can be accurately approximated by that of h (s) = 0, which itself can be cast as a small eigenvalue problem. The new method is made possible by a complete characterization of the curve of h(s), and it has several advantages over the bisection-Newton (BN) iteration recently proposed by [L.-H. Zhang and W. H. Yang, Math. Comp., 83 (2013), pp. 1701-1720] and shown to be very efficient for small-to medium-size problems. The method is tested and compared against the BN iteration and two other state-of-the-art packages: SDPT3 and SeDuMi. Our numerical results show that the method is very efficient for both small to medium dense problems and large-scale ones.

In this paper, we mainly study various notions of regularity for a finite collection {C 1 , • • • , C m } of closed convex subsets of a Banach space X and their relations with other fundamental concepts. We show that a proper lower semicontinuous function f on X has a Lipschitz error bound (resp., Υ-error bound) if and only if the pair {epi(f), X ×{0}} of sets in the product space X × R is linearly regular (resp., regular). Similar results for multifunctions are also established. Next, we prove that {C 1 , • • • , C m } is linearly regular if and only if it has the strong CHIP and the collection {N C 1 (z), • • • , N Cm (z)} of normal cones at z has property (G) for each z ∈ C := ∩ m i=1 C i. Provided that C 1 is a closed convex cone and that C 2 = Y is a closed vector subspace of X, we show that {C 1 , Y } is linearly regular if and only if there exists α > 0 such that each positive (relative to the order induced by C 1) linear functional on Y of norm one can be extended to a positive linear functional on X with norm bounded by α. Similar characterization is given in terms of normal cones.

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