Threshold Dynamics of a Diffusive Herpes Model Incorporating Fixed Relapse Period in a Spatial Heterogeneous Environment

In this paper, we aim to establish the threshold-type dynamics of a diffusive herpes model that assumes a fixed relapse period and nonlinear recovery rate. It turns out that when considering diseases with a fixed relapse period, the diffusion of recovered individuals will lead to nonlocal recovery term. We characterize the basic reproduction number, ℜ 0 , for the model through the next generation operator approach. Moreover, in a homogeneous case, we calculate the ℜ 0 explicitly. By utilizing the principal eigenvalue of the associated eigenvalue problem or equivalently by ℜ 0 , we establish the threshold-type dynamics of the model in the sense that the herpes is either extinct or close to the epidemic value. Numerical simulations are performed to verify the theoretical results and the effects of the spatial heterogeneity on disease transmission.

On the lower bound of the principal eigenvalue of a nonlinear operator

We prove sharp lower bound estimates for the first nonzero eigenvalue of the non-linear elliptic diffusion operator $$L_p$$ L p on a smooth metric measure space, without boundary or with a convex boundary and Neumann boundary condition, satisfying $$BE(\kappa ,N)$$ B E ( κ , N ) for $$\kappa \ne 0$$ κ ≠ 0 . Our results extends the work of Koerber Valtorta (Calc Vari Partial Differ Equ. 57(2), 49 2018) for case $$\kappa =0$$ κ = 0 and Naber–Valtorta (Math Z 277(3–4):867–891, 2014) for the p-Laplacian.

The effect of self-limiting on the prevention and control of the diffuse COVID-19 epidemic with delayed and temporal-spatial heterogeneous

Background The global spread of the novel coronavirus pneumonia is still continuing, and a new round of more serious outbreaks has even begun in some countries. In this context, this paper studies the dynamics of a type of delayed reaction-diffusion novel coronavirus pneumonia model with relapse and self-limiting treatment in a temporal-spatial heterogeneous environment. Methods First, focus on the self-limiting characteristics of COVID-19, incorporate the relapse and self-limiting treatment factors into the diffusion model, and study the influence of self-limiting treatment on the diffusion of the epidemic. Second, because the traditional Lyapunov stability method is difficult to determine the spread of the epidemic with relapse and self-limiting treatment, we introduce a completely different method, relying on the existence conditions of the exponential attractor of our newly established in the infinite-dimensional dynamic system to determine the diffusion of novel coronavirus pneumonia. Third, relapse and self-limiting treatment have led to a change in the structure of the delayed diffusion COVID-19 model, and the traditional basic reproduction number $$R_0$$ R 0 no longer has threshold characteristics. With the help of the Krein-Rutman theorem and the eigenvalue method, we studied the threshold characteristics of the principal eigenvalue and found that it can be used as a new threshold to describe the diffusion of the epidemic. Results Our results prove that the principal eigenvalue $$\uplambda ^{*}$$ λ ∗ of the delayed reaction-diffusion COVID-19 system with relapse and self-limiting treatment can replace the basic reproduction number $$R_0$$ R 0 to describe the threshold effect of disease transmission. Combine with the latest official data and the prevention and control strategies, some numerical simulations on the stability and global exponential attractiveness of the diffusion of the COVID-19 epidemic in China and the USA are given. Conclusions Through the comparison of numerical simulations, we find that self-limiting treatment can significantly promote the prevention and control of the epidemic. And if the free activities of asymptomatic infected persons are not restricted, it will seriously hinder the progress of epidemic prevention and control.

The principal eigenvalue of some nth order linear boundary value problems

The purpose of this paper is to present a procedure for the estimation of the smallest eigenvalues and their associated eigenfunctions of nth order linear boundary value problems with homogeneous boundary conditions defined in terms of quasi-derivatives. The procedure is based on the iterative application of the equivalent integral operator to functions of a cone and the calculation of the Collatz–Wielandt numbers of such functions. Some results on the sign of the Green functions of the boundary value problems are also provided.

Generalized Two-Dimensional Principal Component Analysis and Two Artificial Neural Networks to Detect Traveling Ionospheric Disturbances Using the Tsunami-Atmosphere-Ionosphere Coupling Mechanism

A weak tsunami was induced by the 2016 Mw = 7.8 Sumatra earthquake, which occurred at 12:49 on March 2, 2016 (UTC). The epicenter was at 5.060°S, 94.170°E at a depth of 10 km. At 15.02 on March 2 (UTC), the weak tsunami (amplitude: 0.11 m) arrived at the station located at 10.40°S, 105.67°E. The largest first principal eigenvalue derived using the bilateral projection-based two-dimensional principal component analysis (B2DPCA) indicated a spatial traveling ionospheric disturbance (TID), which was caused by internal gravity waves (IGWs), at 13:20 on March 2 (UTC). The largest second principal eigenvalue represented another TID expanding to the southwest. The two largest principal eigenvalues were associated with the TIDs, which were also determined using two back-propagation neural network (BPNN) models and two convolutional neural network (CNN) models, called the BPNN-B2DPCA and CNN-B2DPCA methods, respectively. These two methods yielded the same results as the B2DPCA. Therefore, the robustness and reliability of the B2DPCA were validated.

Effect of the protection zone in a diffusive ratio-dependent predator–prey model with fear and Allee effect

In this paper, we study the influence of a protection zone for the prey on a diffusive predator–prey model with fear factor and Allee effect. The prior estimate, global existence, nonexistence of nonconstant positive solutions and bifurcation from semitrivial solutions are well discussed. We show the existence of a critical patch value $\lambda ^{D}_{1}(\Omega _{0})$ λ 1 D ( Ω 0 ) of the protection zone, described by the principal eigenvalue of the Laplacian operator over $\Omega _{0}$ Ω 0 with Neumann boundary conditions. When the mortality rate of the predator $\mu \geq d_{2}\lambda ^{D}_{1}(\Omega _{0})$ μ ≥ d 2 λ 1 D ( Ω 0 ) , we show that the semitrivial solutions $(1,0)$ ( 1 , 0 ) and $(\theta,0)$ ( θ , 0 ) are unstable and there is no bifurcation occurring along respective semitrivial branches.

Global Directed Dynamic Behaviors of a Lotka-Volterra Competition-Diffusion-Advection System

This paper investigates the problem of the global directed dynamic behaviors of a Lotka-Volterra competition-diffusion-advection system between two organisms in heterogeneous environments. The two organisms not only compete for different basic resources, but also the advection and diffusion strategies follow the dispersal towards a positive distribution. By virtue of the principal eigenvalue theory, the linear stability of the co-existing steady state is established. Furthermore, the classification of dynamical behaviors is shown by utilizing the monotone dynamical system theory. This work can be seen as a further development of a competition-diffusion system.