Stimuli-responsive hybrid nanoparticles used for controllable catalysis have been attracting increasing attention. This study aims to prepare hybrid microgels with excellent temperature-sensitive colorimetric and catalytic properties through combining the surface plasmon resonance properties of gold nanoparticles (AuNPs) with the temperature-sensitive properties of poly(N-isopropylacrylamide) (PNIPAM)-based microgels. Microgels with hydroxy groups (MG-OH) were prepared by soap-free emulsion polymerization, using N-isopropylacrylamide as the main monomer, hydroxyethyl methylacrylate as the functional monomer, N,N′-methylene bisacrylamide as the crosslinker, and 2,2′-azobis(2-methylpropionamidine) dihydrochloride as an initiator to ensure the microgels are positively charged. Furthermore, chemical modification on the surface of MG-OH was carried out by 3-mercaptopropyltriethoxysilane to obtain thiolated microgels (MG-SH). Two kinds of hybrid nanoparticles, AuNPs@MG-OH and AuNPs@MG-SH, were self-assembled, through electrostatic interaction between positive MG-OH and negative citrate-stabilized AuNPs as well as through synergistic bonding of electrostatic interaction and Au–S bonding between positive MG-SH and negative AuNPs. The morphology, stability, temperature-sensitive colorimetric properties, and catalytic properties of hybrid microgels were systematically investigated. Results showed that although both AuNPs@MG-OH and AuNPs@MG-SH exhibit good temperature-sensitive colorimetric properties and controllable catalytic properties for the reduction reaction of p-nitrophenol, AuNPs@MG-SH with synergistic bonding has better stability and higher catalytic performance than AuNPs@MG-OH. This work has good competitiveness against known PNIPAM-based materials and may provide an effective method for preparing smart catalysts by self-assembly with stimuli-responsive polymers, which has a great potential application for catalyzing a variety of reactions.

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The basic reproduction number (R 0 ) often cannot be explicitly computed when dealing with continuously age-structured epidemic models. In this paper, we numerically compute R 0 of a PDE model for a human immunodeficiency virus (HIV) infection, defined as the spectral radius of a next-generation operator. Since R 0 cannot be analytically obtained, on the one hand, we discretize the linearized PDE model into a system of linear ODEs (Euler method), and on the other hand, we discretize the eigenvalue problem (pseudo-spectral method). In both cases, we approximate R 0 by the largest eigenvalue R 0, n of a next-generation matrix, and we show the convergence of R 0, n to R 0 as the discretization index n increases to infinity. Finally, we present several tests to check/compare the accuracy of both numerical methods.

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In general, the basic reproduction number (R0) cannot be explicitly calculated for HIV(Human Immunodeficiency Virus) infection model with age-structured in a infinite dimensional spaces. To find R0, we need to transform the HIV model into a finite-dimensional space. In this paper, we are absorbed in numerical approximation of R0, which is the non-negative dominant eigenvalues of the positive irreducible matrices whose spectrum radius is defined as the next generation matrix. The linear operators generated by infected population are discretized into ordinary differential equations in a finite n-dimensional space. Thus, the abstract problem is transformed to find the positive dominant eigenvalues of the next generation matrix, we obtain a threshold R 0,n. Based on the spectral approximation theory, we show that R 0,n -R0 as n -+[?]. Finally, by virtue of a numerical simulation, we demonstrate the results of the theorem.