Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates

This paper concerns with detailed analysis of a reaction-diffusion host-pathogen model with space-dependent parameters in a bounded domain. By considering the fact the mobility of host individuals playing a crucial role in disease transmission, we formulate the model by a system of degenerate reaction-diffusion equations, where host individuals disperse at distinct rates and the mobility of pathogen is ignored in the environment.We first establish the well-posedness of the model, including the global existence of solution and the existence of the global compact attractor. The basic reproduction number is identified, and also characterized by some equivalent principal spectral conditions, which establishes the threshold dynamical result for pathogen extinction and persistence. When the positive steady state is confirmed, we investigate the asymptotic profiles of positive steady state as host individuals disperse at small and large rates. Our result suggests that small and large diffusion rate of hosts have a great impacts in formulating the spatial distribution of the pathogen.

An Age-Structured Model of HIV Latent Infection with Two Transmission Routes: Analysis and Optimal Control

In addition to direct virus infection of target cells, HIV can also be transferred from infected to uninfected cells (cell-to-cell transmission). These two routes might facilitate viral production and the establishment of the latent virus pool, which is considered as a major obstacle to HIV cure. We studied an HIV infection model including the two infection routes and the time since latent infection. The basic reproductive ratio R0 was derived. The existence, positivity, and boundedness of the solution are proved. We investigated the existence of steady states and their stability, which were shown to depend on R0. We established the global asymptotic dynamical behavior by proving the existence of the global compact attractor and uniform persistence of the system and by applying the method of Lyapunov functionals. In the end, we formulated and solved the optimal control problem for the age-structured model. The necessary condition for minimization of the viral level and the cost of drug treatment was obtained, and numerical simulations of various optimal control strategies were performed.

Attractors for 2D quasi-geostrophic equations with and without colored noise in W2α−,p(ℝ2)

The asymptotic behavior of stochastic modified quasi-geostrophic equations with damping driven by colored noise is analyzed. In fact, the existence of random attractors is established in [Formula: see text] In particular, we prove also the existence of a global compact attractor for autonomous quasi-geostrophic equations with damping in [Formula: see text] Here, we do not add any modifying factor on the nonlinear term.

Semidiscretization for a Doubly Nonlinear Parabolic Equation Related to the p(x)-Laplacian

This paper studies a time discretization for a doubly nonlinear parabolic equation related to the p(x)-Laplacian by using Euler-forward scheme. We investigate existence, uniqueness, and stability questions and prove existence of the global compact attractor.

Asymptotic Behavior of the Coupled Nonlinear Schrödinger Lattice System

This paper studies asymptotic behavior of solutions for the coupled nonlinear Schrödinger lattice system. We obtain the existence and stability of compact attractor by means of tail estimates method and finite-dimensional approximations.

Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

We prove the asymptotic regularity of global solutions for a class of semilinear evolution equations in H01(Ω)×H01(Ω). Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor 𝒜 which is bounded in H2(Ω)×H2(Ω), where the nonlinear term f satisfies a critical exponential growth condition.

PEANO'S THEOREM AND ATTRACTORS FOR LATTICE DYNAMICAL SYSTEMS

In this paper, we prove the existence of solutions for first order lattice dynamical systems with continuous nonlinear term obtained via discretization of a reaction–diffusion system. Since the uniqueness of the Cauchy problem is not guaranteed, we define a multivalued semiflow and prove the existence of a global compact attractor.