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Discrete & Continuous Dynamical Systems - B

September 2021 , Volume 26 , Issue 9

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Modeling error of $ \alpha $-models of turbulence on a two-dimensional torus
Luigi C. Berselli, Argus Adrian Dunca, Roger Lewandowski and Dinh Duong Nguyen
2021, 26(9): 4613-4643 doi: 10.3934/dcdsb.2020305 +[Abstract](1172) +[HTML](284) +[PDF](440.58KB)
Abstract:

This paper is devoted to study the rate of convergence of the weak solutions \begin{document}$ {\bf u}_\alpha $\end{document} of \begin{document}$ \alpha $\end{document}-regularization models to the weak solution \begin{document}$ {\bf u} $\end{document} of the Navier-Stokes equations in the two-dimensional periodic case, as the regularization parameter \begin{document}$ \alpha $\end{document} goes to zero. More specifically, we will consider the Leray-\begin{document}$ \alpha $\end{document}, the simplified Bardina, and the modified Leray-\begin{document}$ \alpha $\end{document} models. Our aim is to improve known results in terms of convergence rates and also to show estimates valid over long-time intervals. The results also hold in the case of bounded domain with homogeneous Dirichlet boundary conditions.

Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems
Everaldo de Mello Bonotto, Matheus Cheque Bortolan, Rodolfo Collegari and José Manuel Uzal
2021, 26(9): 4645-4661 doi: 10.3934/dcdsb.2020306 +[Abstract](762) +[HTML](275) +[PDF](342.63KB)
Abstract:

In this paper we investigate the long time behavior of a nonautonomous dynamical system (cocycle) when its driving semigroup is subjected to impulses. We provide conditions to ensure the existence of global attractors for the associated impulsive skew-product semigroups, uniform attractors for the coupled impulsive cocycle and pullback attractors for the associated evolution processes. Finally, we illustrate the theory with an application to cascade systems.

Analytic solution to an interfacial flow with kinetic undercooling in a time-dependent gap Hele-Shaw cell
Xuming Xie
2021, 26(9): 4663-4680 doi: 10.3934/dcdsb.2020307 +[Abstract](786) +[HTML](271) +[PDF](381.76KB)
Abstract:

Hele-Shaw cells where the top plate is lifted uniformly at a prescribed speed and the bottom plate is fixed have been used to study interface related problems. This paper focuses on an interfacial flow with kinetic undercooling regularization in a radial Hele-Shaw cell with a time dependent gap. We obtain the local existence of analytic solution of the moving boundary problem when the initial data is analytic. The methodology is to use complex analysis and reduce the free boundary problem to a Riemann-Hilbert problem and an abstract Cauchy-Kovalevskaya evolution problem.

Non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions
Pengyu Chen and Xuping Zhang
2021, 26(9): 4681-4695 doi: 10.3934/dcdsb.2020308 +[Abstract](830) +[HTML](286) +[PDF](349.07KB)
Abstract:

In this paper, we study the non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions in Hilbert spaces, where the operators in linear part (possibly unbounded) depend on time \begin{document}$ t $\end{document} and generate an evolution family. New existence result of mild solutions is established under more weaker conditions by introducing a new Green's function. The discussions are based on Schauder's fixed-point theorem as well as the theory of evolution family. At last, an example is also given to illustrate the feasibility of our theoretical results. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay
Tongtong Liang and Yejuan Wang
2021, 26(9): 4697-4726 doi: 10.3934/dcdsb.2020309 +[Abstract](907) +[HTML](262) +[PDF](434.92KB)
Abstract:

In this paper, we investigate a stochastic fractionally dissipative quasi-geostrophic equation driven by a multiplicative white noise, whose external forces contain hereditary characteristics. The existence and uniqueness of both local martingale and local pathwise solutions are established in \begin{document}$ H^s $\end{document} with \begin{document}$ s\geq2-2\alpha $\end{document}, where \begin{document}$ \alpha\in(\frac{1}{2}, 1) $\end{document}. For the critical case \begin{document}$ \alpha = \frac12 $\end{document}, we obtain the similar results in \begin{document}$ H^s $\end{document} with \begin{document}$ s>1 $\end{document}.

Existence-uniqueness and stability of the mild periodic solutions to a class of delayed stochastic partial differential equations and its applications
Qi Yao, Linshan Wang and Yangfan Wang
2021, 26(9): 4727-4743 doi: 10.3934/dcdsb.2020310 +[Abstract](927) +[HTML](301) +[PDF](732.11KB)
Abstract:

In this paper, we focus on the mild periodic solutions to a class of delayed stochastic reaction-diffusion differential equations. First, the key issues of Markov property in Banach space \begin{document}$ C $\end{document}, \begin{document}$ p $\end{document}-uniformly boundedness, and \begin{document}$ p $\end{document}-point dissipativity of mild solutions \begin{document}$ \boldsymbol{u}_t $\end{document} to the equations are discussed. Then, the theorems of existence-uniqueness and exponential stability in the mean-square sense of the mild periodic solutions are established by using the dissipative theory and the operator semigroup technique, and the relevant results about the existence of mild periodic solutions in the quoted literature are generalized. Next, the given theoretical results are successfully applied to the delayed stochastic reaction-diffusion Hopfield neural networks, and some easy-to-test criteria of exponential stability for the mild periodic solution to the networks are obtained. Finally, some examples are presented to demonstrate the feasibility of our results.

Local structure-preserving algorithms for the molecular beam epitaxy model with slope selection
Lin Lu, Qi Wang, Yongzhong Song and Yushun Wang
2021, 26(9): 4745-4765 doi: 10.3934/dcdsb.2020311 +[Abstract](1115) +[HTML](346) +[PDF](8422.29KB)
Abstract:

Based on the local energy dissipation property of the molecular beam epitaxy (MBE) model with slope selection, we develop three, second order fully discrete, local energy dissipation rate preserving (LEDP) algorithms for the model using finite difference methods. For periodic boundary conditions, we show that these algorithms are global energy dissipation rate preserving (GEDP). For adiabatic, physical boundary conditions, we construct two GEDP algorithms from the three LEDP ones with consistently discretized physical boundary conditions. In addition, we show that all the algorithms preserve the total mass at the discrete level as well. Mesh refinement tests are conducted to confirm the convergence rates of the algorithms and two benchmark examples are presented to show the accuracy and performance of the methods.

Invasion dynamics of a diffusive pioneer-climax model: Monotone and non-monotone cases
Yuxiang Zhang and Shiwang Ma
2021, 26(9): 4767-4788 doi: 10.3934/dcdsb.2020312 +[Abstract](865) +[HTML](247) +[PDF](447.24KB)
Abstract:

In this paper, we study the invasion dynamics of a diffusive pioneer-climax model in monotone and non-monotone cases. For parameter ranges in which the system admits monotone properties, we establish the existence of spreading speeds and their coincidence with the minimum wave speeds by monotone dynamical system theories. The linear determinacy of the minimum wave speeds is also studied by constructing suitable upper solutions. For parameter ranges in which the system is non-monotone, we further determine the existence of spreading speeds and traveling waves by the sandwich technique and upper-lower solution method. Our results generalize the existing results established under monotone assumptions to more general cases.

Propagation phenomena for a criss-cross infection model with non-diffusive susceptible population in periodic media
Liangliang Deng and Zhi-Cheng Wang
2021, 26(9): 4789-4814 doi: 10.3934/dcdsb.2020313 +[Abstract](770) +[HTML](262) +[PDF](434.99KB)
Abstract:

This paper is concerned with propagation phenomena for an epidemic model describing the circulation of a disease within two populations or two subgroups in periodic media, where the susceptible individuals are assumed to be motionless. The spatial dynamics for the cooperative system obtained by a classical transformation are investigated, including spatially periodic steady state, spreading speeds and pulsating travelling fronts. It is proved that the minimal wave speed is linearly determined and given by a variational formula involving linear eigenvalue problem. Further, we prove that the existence and non-existence of travelling wave solutions of the model are entirely determined by the basic reproduction ratio \begin{document}$ \mathcal{R}_{0} $\end{document}. As an application, we prove that if the localized amount of infectious individuals are introduced at the beginning, then the solution of such a system has an asymptotic spreading speed in large time and that is exactly coincident with the minimal wave speed.

Novel entire solutions in a nonlocal 2-D discrete periodic media for bistable dynamics
Zhixian Yu, Rong Yuan and Shaohua Gan
2021, 26(9): 4815-4838 doi: 10.3934/dcdsb.2020314 +[Abstract](711) +[HTML](266) +[PDF](454.1KB)
Abstract:

This paper is concerned with novel entire solutions originating from three pulsating traveling fronts for nonlocal discrete periodic system (NDPS) on 2-D Lattices

More precisely, let \begin{document}$ \varphi_{i,j;k}(i cos\theta +j sin\theta+v_{k}t)\,\,(k = 1,2,3) $\end{document} be the pulsating traveling front of NDPS with the wave speed \begin{document}$ v_k $\end{document} and connecting two different constant states, then NDPS admits an entire solution \begin{document}$ u_{i,j}(t) $\end{document}, which satisfies

where \begin{document}$ \xi = :i \cos\theta +j \sin\theta $\end{document}, \begin{document}$ v_1<v_2<v_3 $\end{document} and \begin{document}$ \theta_{k}\,(k = 1,2) $\end{document} is some constant, \begin{document}$ p_0 = -\infty $\end{document}, \begin{document}$ p_k(t): = -(v_k+v_{k+1})t/2\,\,(k = 1,2) $\end{document} and \begin{document}$ p_3 = +\infty $\end{document}.

On a matrix-valued PDE characterizing a contraction metric for a periodic orbit
Peter Giesl
2021, 26(9): 4839-4865 doi: 10.3934/dcdsb.2020315 +[Abstract](738) +[HTML](258) +[PDF](662.6KB)
Abstract:

The stability and the basin of attraction of a periodic orbit can be determined using a contraction metric, i.e., a Riemannian metric with respect to which adjacent solutions contract. A contraction metric does not require knowledge of the position of the periodic orbit and is robust to perturbations.

In this paper we characterize such a Riemannian contraction metric as matrix-valued solution of a linear first-order Partial Differential Equation. This enables the explicit construction of a contraction metric by numerically solving this equation in [7]. In this paper we prove existence and uniqueness of the solution of the PDE and show that it defines a contraction metric.

Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model
Weiwei Liu, Jinliang Wang and Yuming Chen
2021, 26(9): 4867-4885 doi: 10.3934/dcdsb.2020316 +[Abstract](1042) +[HTML](296) +[PDF](376.66KB)
Abstract:

Taking account of spatial heterogeneity, latency in infected individuals, and time for shed bacteria to the aquatic environment, we build a delayed nonlocal reaction-diffusion cholera model. A feature of this model is that the incidences are of general nonlinear forms. By using the theories of monotone dynamical systems and uniform persistence, we obtain a threshold dynamics determined by the basic reproduction number \begin{document}$ \mathcal {R}_0 $\end{document}. Roughly speaking, the cholera will die out if \begin{document}$ \mathcal{R}_0<1 $\end{document} while it persists if \begin{document}$ \mathcal{R}_0>1 $\end{document}. Moreover, we derive the explicit formulae of \begin{document}$ \mathcal{R}_0 $\end{document} for two concrete situations.

A stochastic differential equation SIS epidemic model with regime switching
Siyang Cai, Yongmei Cai and Xuerong Mao
2021, 26(9): 4887-4905 doi: 10.3934/dcdsb.2020317 +[Abstract](1206) +[HTML](287) +[PDF](582.77KB)
Abstract:

In this paper, we combined the previous model in [2] with Gray et al.'s work in 2012 [8] to add telegraph noise by using Markovian switching to generate a stochastic SIS epidemic model with regime switching. Similarly, threshold value for extinction and persistence are then given and proved, followed by explanation on the stationary distribution, where the \begin{document}$ M $\end{document}-matrix theory elaborated in [20] is fully applied. Computer simulations are clearly illustrated with different sets of parameters, which support our theoretical results. Compared to our previous work in 2019 [2, 3], our threshold value are given based on the overall behaviour of the solution but not separately specified in every state of the Markov chain.

A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation
Leilei Wei and Yinnian He
2021, 26(9): 4907-4926 doi: 10.3934/dcdsb.2020319 +[Abstract](983) +[HTML](244) +[PDF](803.73KB)
Abstract:

The tempered fractional diffusion equation could be recognized as the generalization of the classic fractional diffusion equation that the truncation effects are included in the bounded domains. This paper focuses on designing the high order fully discrete local discontinuous Galerkin (LDG) method based on the generalized alternating numerical fluxes for the tempered fractional diffusion equation. From a practical point of view, the generalized alternating numerical flux which is different from the purely alternating numerical flux has a broader range of applications. We first design an efficient finite difference scheme to approximate the tempered fractional derivatives and then a fully discrete LDG method for the tempered fractional diffusion equation. We prove that the scheme is unconditionally stable and convergent with the order \begin{document}$ O(h^{k+1}+\tau^{2-\alpha}) $\end{document}, where \begin{document}$ h, \tau $\end{document} and \begin{document}$ k $\end{document} are the step size in space, time and the degree of piecewise polynomials, respectively. Finally numerical experimets are performed to show the effectiveness and testify the accuracy of the method.

On existence and uniqueness properties for solutions of stochastic fixed point equations
Christian Beck, Lukas Gonon, Martin Hutzenthaler and Arnulf Jentzen
2021, 26(9): 4927-4962 doi: 10.3934/dcdsb.2020320 +[Abstract](989) +[HTML](290) +[PDF](481.08KB)
Abstract:

The Feynman–Kac formula implies that every suitable classical solution of a semilinear Kolmogorov partial differential equation (PDE) is also a solution of a certain stochastic fixed point equation (SFPE). In this article we study such and related SFPEs. In particular, the main result of this work proves existence of unique solutions of certain SFPEs in a general setting. As an application of this main result we establish the existence of unique solutions of SFPEs associated with semilinear Kolmogorov PDEs with Lipschitz continuous nonlinearities even in the case where the associated semilinear Kolmogorov PDE does not possess a classical solution.

A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition
Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine and Omar Oubbih
2021, 26(9): 4963-4998 doi: 10.3934/dcdsb.2020321 +[Abstract](1356) +[HTML](381) +[PDF](2292.79KB)
Abstract:

This paper is devoted to the mathematical and numerical study of a new proposed model based on a fractional diffusion equation coupled with a nonlinear regularization of the Total Variation operator. This model is primarily intended to introduce a weak norm in the fidelity term, where this norm is considered more appropriate for capturing very oscillatory characteristics interpreted as a texture. Furthermore, our proposed model profits from the benefits of a variable exponent used to distinguish the features of the image. By using Faedo-Galerkin method, we prove the well-posedness (existence and uniqueness) of the weak solution for the proposed model. Based on the alternating direction implicit method of Peaceman-Rachford and the approximations of the Gr\begin{document}$ \ddot{u} $\end{document}nwald-Letnikov operators, we develop the numerical discretization of our fractional diffusion equation. Experimental results claim that our model provides high-quality results in cartoon-texture-edges decomposition and image denoising. In particular, our model can successfully reduce the staircase phenomenon during the image denoising. Furthermore, small details, texture and fine structures still maintained in the restored image. Finally, we compare our numerical results with the existing models in the literature.

A comparative study of atomistic-based stress evaluation
Shuyang Dai, Fengru Wang, Jerry Zhijian Yang and Cheng Yuan
2021, 26(9): 4999-5021 doi: 10.3934/dcdsb.2020322 +[Abstract](763) +[HTML](330) +[PDF](950.84KB)
Abstract:

This paper presents a comparative study on several issues of the microscopic stress definitions. Firstly, we derived an Irving-Kirkwood formulation for Cauchy stress evaluation in Eulerian coordinates. We showed that quantities, such as density and momentum, should to be defined properly on microscopic level in order to guarantee the conservation relations on macroscopic level. Secondly, the relation between Cauchy and first Piola-Kirchhoff stress was investigated both theoretically and numerically. At zero temperature, classical pointwise relation between these two stress is satisfied both in Virial and Hardy formulation. While at finite temperature, temporal averaging is required to guarantee this relation for Virial formulation. For Hardy formulation, an additional term need to be included in the classical relation between the Cauchy stress and the first Piola-Kirchhoff stress. Meanwhle, the linear relation between the Cauchy stress and the first Piola-Kirchhoff stress with respect to the temperature are obtained in both Virial and Hardy formulations. The thermal expansion coefficients are also studied by using quasi-harmonic approximation. Thirdly, different from that in the Lagrangian coordinates case, where the time averaging procedure can be performed in a post-processing manner when the kernel function is separable, the stress evaluation in Eulerian system must be evaluated spatially and temporally at the same time, even in separable kernel case. This can be seen from the comparison of the two procedures. Numerical examples were provided to illustrate our investigations.

Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations
Zhenzhen Wang and Tianshou Zhou
2021, 26(9): 5023-5045 doi: 10.3934/dcdsb.2020323 +[Abstract](866) +[HTML](252) +[PDF](369.82KB)
Abstract:

Fisher-KPP equations are an important class of mathematical models with practical background. Previous studies analyzed the asymptotic behaviors of the front and back of the wavefront and proved the existence of stochastic traveling waves, by imposing decrease constraints on the growth function. For the Fisher-KPP equation with a stochastically fluctuated growth rate, we find that if the decrease restrictions are removed, the same results still hold. Moreover, we show that with increasing the noise intensity, the original equation with Fisher-KPP nonlinearity evolves into first the one with degenerated Fisher-KPP nonlinearity and then the one with Nagumo nonlinearity. For the Fisher-KPP equation subjected to the environmental noise, the established asymptotic behavior of the front of the wavefront still holds even if the decrease constraint on the growth function is ruled out. If this constraint is removed, however, the established asymptotic behavior of the back of the wavefront will no longer hold, implying that the decrease constraint on the growth function is a sufficient and necessary condition to ensure the asymptotic behavior of the back of the wavefront. In both cases of noise, the systems can allow stochastic traveling waves.

Stochastic modelling and analysis of harvesting model: Application to "summer fishing moratorium" by intermittent control
Xiaoling Zou and Yuting Zheng
2021, 26(9): 5047-5066 doi: 10.3934/dcdsb.2020332 +[Abstract](844) +[HTML](262) +[PDF](622.73KB)
Abstract:

As we all know, "summer fishing moratorium" is an internationally recognized management measure of fishery, which can protect stock of fish and promote the balance of marine ecology. In this paper, "intermittent control" is used to simulate this management strategy, which is the first attempt in theoretical analysis and the intermittence fits perfectly the moratorium. As an application, a stochastic two-prey one-predator Lotka-Volterra model with intermittent capture is considered. Modeling ideas and analytical skills in this paper can also be used to other stochastic models. In order to deal with intermittent capture in stochastic model, a new time-averaged objective function is proposed. Besides, the corresponding optimal harvesting strategies are obtained by using the equivalent method (equivalency between time-average and expectation). Theoretical results show that intermittent capture can affect the optimal harvesting effort, but it cannot change the corresponding optimal time-averaged yield, which are accord with observations. Finally, the results are illustrated by practical examples of marine fisheries and numerical simulations.

Traveling wave solutions for a cancer stem cell invasion model
Caleb Mayer and Eric Stachura
2021, 26(9): 5067-5093 doi: 10.3934/dcdsb.2020333 +[Abstract](867) +[HTML](271) +[PDF](482.8KB)
Abstract:

In this paper we analyze the dynamics of a cancer invasion model that incorporates the cancer stem cell hypothesis. In particular, we develop a model that includes a cancer stem cell subpopulation of tumor cells. Traveling wave analysis and Geometric Singular Perturbation Theory are used in order to determine existence and persistence of solutions for the model.

Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity
Xiangdong Zhao
2021, 26(9): 5095-5100 doi: 10.3934/dcdsb.2020334 +[Abstract](823) +[HTML](248) +[PDF](294.04KB)
Abstract:

We consider a chemotaxis system with singular sensitivity and logistic-type source: \begin{document}$ u_t = \Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+ru-\mu u^k $\end{document}, \begin{document}$ v_t = \epsilon\Delta v-v+u $\end{document} in a smooth bounded domain \begin{document}$ \Omega\subset\mathbb{R}^n $\end{document} with \begin{document}$ \chi,r,\mu,\epsilon>0 $\end{document}, \begin{document}$ k>1 $\end{document} and \begin{document}$ n\ge 2 $\end{document}. It is proved that the system possesses a globally bounded classical solution when \begin{document}$ \epsilon+\chi<1 $\end{document}. This shows that the diffusive coefficient \begin{document}$ \epsilon $\end{document} of the chemical substance \begin{document}$ v $\end{document} properly small benefits the global boundedness of solutions, without the restriction on the dampening exponent \begin{document}$ k>1 $\end{document} in logistic source.

Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps
Shangzhi Li and Shangjiang Guo
2021, 26(9): 5101-5134 doi: 10.3934/dcdsb.2020335 +[Abstract](1012) +[HTML](250) +[PDF](1446.2KB)
Abstract:

This paper is devoted to a stochastic regime-switching susceptible-infected-susceptible epidemic model with nonlinear incidence rate and Lévy jumps. A threshold \begin{document}$ \lambda $\end{document} in terms of the invariant measure, different from the usual basic reproduction number, is obtained to completely determine the extinction and prevalence of the disease: if \begin{document}$ \lambda>0 $\end{document}, the disease is persistent and there is a stationary distribution; if \begin{document}$ \lambda<0 $\end{document}, the disease goes to extinction and the susceptible population converges weakly to a boundary distribution. Moreover, some numerical simulations are performed to illustrate our theoretical results. It is very interesting to notice that random fluctuations (including the white noise and Lévy noise) acting the infected individuals can prevent the outbreak of disease, that the disease of a regime-switching model may have the opportunity to persist eventually even if it is extinct in one regime, and that the prevalence of the disease can also be controlled by reducing the value of transmission rate of disease.

Long time localization of modified surface quasi-geostrophic equations
Guido Cavallaro, Roberto Garra and Carlo Marchioro
2021, 26(9): 5135-5148 doi: 10.3934/dcdsb.2020336 +[Abstract](787) +[HTML](255) +[PDF](320.36KB)
Abstract:

We discuss the time evolution of a two-dimensional active scalar flow, which extends some properties valid for a two-dimensional incompressible nonviscous fluid. In particular we study some characteristics of the dynamics when the field is initially concentrated in \begin{document}$ N $\end{document} small disjoint regions, and we discuss the conservation in time of this localization property. We discuss also how long this localization persists, showing that in some cases this happens for quite long times.

Recurrent solutions of the Schrödinger-KdV system with boundary forces
Mo Chen
2021, 26(9): 5149-5170 doi: 10.3934/dcdsb.2020337 +[Abstract](622) +[HTML](247) +[PDF](413.39KB)
Abstract:

In this paper, we consider the Schrödinger-KdV system with time-dependent boundary external forces. We give conditions on the external forces sufficient for the unique existence of small solutions bounded for all time. Then, we investigate the existence of bounded solution, periodic solution, quasi-periodic solution and almost periodic solution for the Schrödinger-KdV system. The main difficulty is the nonlinear terms in the equations, in order to overcome this difficulty, we establish some properties for the semigroup associated with linear operator which is a crucial tool.

Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases
Hiroshi Ito
2021, 26(9): 5171-5196 doi: 10.3934/dcdsb.2020338 +[Abstract](815) +[HTML](266) +[PDF](620.22KB)
Abstract:

This paper demonstrates input-to-state stability (ISS) of the SIR model of infectious diseases with respect to the disease-free equilibrium and the endemic equilibrium. Lyapunov functions are constructed to verify that both equilibria are individually robust with respect to perturbation of newborn/immigration rate which determines the eventual state of populations in epidemics. The construction and analysis are geometric and global in the space of the populations. In addition to the establishment of ISS, this paper shows how explicitly the constructed level sets reflect the flow of trajectories. Essential obstacles and keys for the construction of Lyapunov functions are elucidated. The proposed Lyapunov functions which have strictly negative derivative allow us to not only establish ISS, but also get rid of the use of LaSalle's invariance principle and popular simplifying assumptions.

2020 Impact Factor: 1.327
5 Year Impact Factor: 1.492
2020 CiteScore: 2.2

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