We develop and analyze an ordinary differential equation model to assess the potential effectiveness of infecting mosquitoes with the Wolbachia bacteria to control the ongoing mosquito-borne epidemics, such as dengue fever, chikungunya, and Zika. Wolbachia is a natural parasitic microbe that stops the proliferation of the harmful viruses inside the mosquito and reduces disease transmission. It is difficult to sustain an infection of the maternal transmitted Wolbachia in a wild mosquito population because of the reduced fitness of the Wolbachia-infected mosquitoes and cytoplasmic incompatibility limiting maternal transmission. The infection will only persist if the fraction of the infected mosquitoes exceeds a minimum threshold. Our two-sex mosquito model captures the complex transmission-cycle by accounting for heterosexual transmission, multiple pregnant states for female mosquitoes, and the aquatic-life stage. We identify important dimensionless numbers and analyze the critical threshold condition for obtaining a sustained Wolbachia infection in the natural population. This threshold effect is characterized by a backward bifurcation with three coexisting equilibria of the system of differential equations: a stable disease-free equilibrium, an unstable intermediate-infection endemic equilibrium and a stable high-infection endemic equilibrium. We perform sensitivity analysis on epidemiological and environmental parameters to determine their relative importance to Wolbachia transmission and prevalence. We also compare the effectiveness of different integrated mitigation strategies and observe that the most efficient approach to establish the Wolbachia infection is to first reduce the natural mosquitoes and then release both infected males and pregnant females. The initial reduction of natural population could be accomplished by either residual spraying or ovitraps.

Numerical simulations of phase-field models require long time computations and therefore it is necessary to develop efficient and highly accurate numerical methods. In this paper, we propose fast and stable explicit operator splitting methods for both one-and two-dimensional nonlinear diffusion equations for thin film epitaxy with slope selection and the Cahn-Hilliard equation. The equations are split into nonlinear and linear parts. The nonlinear part is solved using a method of lines together with an efficient large stability domain explicit ODE solver. The linear part is solved by a pseudo-spectral method, which is based on the exact solution and thus has no stability restriction on the time-step size. We demonstrate the performance of the proposed methods on a number of one-and two-dimensional numerical examples, where different stages of coarsening such as the initial preparation, alternating rapid structural transition and slow motion can be clearly observed.

We introduce adaptive moving mesh central-upwind schemes for one-and two-dimensional hyperbolic systems of conservation and balance laws. The proposed methods consist of three steps. First, the solution is evolved by solving the studied system by the second-order semi-discrete central-upwind scheme on either the one-dimensional nonuniform grid or the two-dimensional structured quadrilateral mesh. When the evolution step is complete, the grid points are redistributed according to the moving mesh differential equation. Finally, the evolved solution is projected onto the new mesh in a conservative manner. The resulting adaptive moving mesh methods are applied to the one-and two-dimensional Euler equations of gas dynamics and granular hydrodynamics systems. Our numerical results demonstrate that in both cases, the adaptive moving mesh central-upwind schemes outperform their uniform mesh counterparts. Keywords Adaptive moving mesh methods • Finite-volume methods • Central-upwind schemes • Moving mesh differential equations • Euler equations of gas dynamics • Granular hydrodynamics • Singular solutions

Two-layer shallow water equations describe flows that consist of two layers of inviscid fluid of different, constant densities flowing over bottom topography. Unlike the singe-layer shallow water system, the two-layer one is only conditionally hyperbolic: It loses its hyperbolicity because of the momentum exchange terms between the layers and as the results its solutions may develop instabilities. We study a three-layer approximation of the two-layer shallow water equations by introducing an intermediate layer of a small depth. We examine the hyperbolicity range of the three-layer model and demonstrate that while it still may lose hyperbolicity, in some cases, the three-layer approximation may improve stability properties of the two-layer shallow water system.

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