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Discrete & Continuous Dynamical Systems - B

July 2021 , Volume 26 , Issue 7

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Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency
Yanling Shi and Junxiang Xu
2021, 26(7): 3479-3490 doi: 10.3934/dcdsb.2020241 +[Abstract](628) +[HTML](280) +[PDF](363.61KB)

In this paper, one dimensional nonlinear wave equation

with Dirichlet boundary condition is considered, where \begin{document}$ \varepsilon $\end{document} is small positive parameter, \begin{document}$ \omega = \xi \bar{\omega}, $\end{document} \begin{document}$ \bar{\omega} $\end{document} is weak Liouvillean frequency. It is proved that there are many quasi-periodic solutions with Liouvillean frequency for the above equation. The proof is based on an infinite dimensional KAM Theorem.

Moran process and Wright-Fisher process favor low variability
Jan Rychtář and Dewey T. Taylor
2021, 26(7): 3491-3504 doi: 10.3934/dcdsb.2020242 +[Abstract](666) +[HTML](312) +[PDF](463.91KB)

We study evolutionary dynamics in finite populations. We assume the individuals are one of two competing genotypes, \begin{document}$ A $\end{document} or \begin{document}$ B $\end{document}. The genotypes have the same average fitness but different variances and/or third central moments. We focus on two frequency-independent stochastic processes: (1) Wright-Fisher process and (2) Moran process. Both processes have two absorbing states corresponding to homogeneous populations of all \begin{document}$ A $\end{document} or all \begin{document}$ B $\end{document}. Despite the fact that types \begin{document}$ A $\end{document} and \begin{document}$ B $\end{document} have the same average fitness, both stochastic dynamics differ from a random drift. In both processes, the selection favors \begin{document}$ A $\end{document} replacing \begin{document}$ B $\end{document} and opposes \begin{document}$ B $\end{document} replacing \begin{document}$ A $\end{document} if the fitness variance for \begin{document}$ A $\end{document} is smaller than the fitness variance for \begin{document}$ B $\end{document}. In the case the variances are equal, the selection favors \begin{document}$ A $\end{document} replacing \begin{document}$ B $\end{document} and opposes \begin{document}$ B $\end{document} replacing \begin{document}$ A $\end{document} if the third central moment of \begin{document}$ A $\end{document} is larger than the third central moment of \begin{document}$ B $\end{document}. We show that these results extend to structured populations and other dynamics where the selection acts at birth. We also demonstrate that the selection favors a larger variance in fitness if the selection acts at death.

Extremum estimates of the $ L^1 $-norm of weights for eigenvalue problems of vibrating string equations based on critical equations
Jiangang Qi and Bing Xie
2021, 26(7): 3505-3516 doi: 10.3934/dcdsb.2020243 +[Abstract](516) +[HTML](284) +[PDF](300.79KB)

The present paper is concerned with the extremal problem of the \begin{document}$ L^1 $\end{document}-norm of the weights for non-left-definite eigenvalue problems of vibrating string equations with separated boundary conditions. Applying the critical equations of the weights, the infimum is obtained in terms of the given eigenvalue and the parameter in boundary conditions.

Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains
Fuzhi Li and Dongmei Xu
2021, 26(7): 3517-3542 doi: 10.3934/dcdsb.2020244 +[Abstract](498) +[HTML](268) +[PDF](437.77KB)

This paper is devoted to bi-spatial random attractors of the stochastic Fitzhugh-Nagumo equations with additive noise on thin domains when the terminate space is the Sobolev space. We first established the existence of random attractor on regular space and then show that the upper semi-continuity of these attractors under the Sobolev norm when a family of \begin{document}$ (n+1) $\end{document}-dimensional thin domains degenerates onto an \begin{document}$ n $\end{document}-dimensional domain.

Modeling within-host viral dynamics: The role of CTL immune responses in the evolution of drug resistance
Qi Deng, Zhipeng Qiu, Ting Guo and Libin Rong
2021, 26(7): 3543-3562 doi: 10.3934/dcdsb.2020245 +[Abstract](762) +[HTML](309) +[PDF](1337.35KB)

To study the emergence and evolution of drug resistance during treatment of HIV infection, we study a mathematical model with two strains, one drug-sensitive and the other drug-resistant, by incorporating cytotoxic T lymphocyte (CTL) immune response. The reproductive numbers for each strain with and without the CTL immune response are obtained and shown to determine the stability of the steady states. By sensitivity analysis, we evaluate how the changes of parameters influence the reproductive numbers. The model shows that CTL immune response can suppress the development of drug resistance. There is a dynamic relationship between antiretroviral drug administration, the prevalence of drug resistance, the total level of viral production, and the strength of immune responses. We further investigate the scenario under which the drug-resistant strain can outcompete the wild-type strain. If drug efficacy is at an intermediate level, the drug-resistant virus is likely to arise. The slower the immune response wanes, the slower the drug-resistant strain grows. The results suggest that immunotherapy that aims to enhance immune responses, combined with antiretroviral drug treatment, may result in a functional control of HIV infection.

Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations
Xin Zhong
2021, 26(7): 3563-3578 doi: 10.3934/dcdsb.2020246 +[Abstract](597) +[HTML](294) +[PDF](323.85KB)

The present paper concerns an initial boundary value problem of two-dimensional (2D) nonhomogeneous magnetohydrodynamic (MHD) equations with non-negative density. We establish the global existence and exponential decay of strong solutions. In particular, the initial data can be arbitrarily large. The key idea is to use a lemma due to Desjardins (Arch. Rational Mech. Anal. 137:135–158, 1997).

Analytical study of resonance regions for second kind commensurate fractional systems
Assia Boubidi, Sihem Kechida and Hicham Tebbikh
2021, 26(7): 3579-3594 doi: 10.3934/dcdsb.2020247 +[Abstract](566) +[HTML](281) +[PDF](2269.04KB)

The aim of this paper is to determine analytically the resonance limits for second kind commensurate fractional systems in terms of the pseudo damping factor \begin{document}$ \xi $\end{document} and the commensurate order \begin{document}$ v $\end{document} and in addition specify the different resonance regions. In the literature, these limits and regions have never been discussed mathematically, they are determined numerically. Second kind commensurate fractional systems are resonant if the equation : \begin{document}$ \Omega^{3v}+3\xi cos(v \pi/2)\Omega^{2v}+(2\xi^{2}+cos(v\pi))\Omega{^v}+\xi cos(v\pi/2) = 0 $\end{document}, obtained by setting the first derivative of the amplitude-frequency response equal to zero, has at last one strictly positive root. As in the conventional case, resonance limits correspond to zero discriminant of the last equation. This discriminant is a cubic equation in \begin{document}$ \xi{^2} $\end{document} whose coefficients change depending on \begin{document}$ v $\end{document}. To resolve this equation, the tangent trigonometric solving method is used and the relationship between \begin{document}$ \xi $\end{document} and \begin{document}$ v $\end{document} is established, which represents the resonance limits expression. To search resonance regions, a mathematical study is conducted on the first equation to find the positive roots number for each (\begin{document}$ v $\end{document}, \begin{document}$ \xi $\end{document}) combination. Compared to works already achieved, a new region appeared in the region of single resonant frequency with an anti-resonant one. The results are tested through numerical examples and applied to a fractional filter.

Finite-time cluster synchronization of coupled dynamical systems with impulsive effects
Tianhu Yu, Jinde Cao and Chuangxia Huang
2021, 26(7): 3595-3620 doi: 10.3934/dcdsb.2020248 +[Abstract](593) +[HTML](309) +[PDF](2302.24KB)

In our paper, the finite-time cluster synchronization problem is investigated for the coupled dynamical systems in networks. Based on impulsive differential equation theory and differential inequality method, two novel Lyapunov-based finite-time stability results are proposed and be used to obtain the finite-time cluster synchronization criteria for the coupled dynamical systems with synchronization and desynchronization impulsive effects, respectively. The settling time with respect to the average impulsive interval is estimated according to the sufficient synchronization conditions. It is illustrated that the introduced settling time is not only dependent on the initial conditions, but also dependent on the impulsive effects. Compared with the results without stabilizing impulses, the attractive domain of the finite-time stability can be enlarged by adding impulsive control input. Conversely, the smaller attractive domain can be obtained when the original system is subject to the destabilizing impulses. By using our criteria, the continuous feedback control can always be designed to finite-time stabilize the unstable impulsive system. Several existed results are extended and improved in the literature. Finally, typical numerical examples involving the large-scale complex network are outlined to exemplify the availability of the impulsive control and continuous feedback control, respectively.

Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems
Xiongxiong Bao and Wan-Tong Li
2021, 26(7): 3621-3641 doi: 10.3934/dcdsb.2020249 +[Abstract](627) +[HTML](289) +[PDF](468.63KB)

The current paper is devoted to the study of the existence and stability of generalized transition waves of the following time-dependent reaction-diffusion cooperative system

Here \begin{document}$ \mathbf{F}(t,\mathbf{u}(t,x)) $\end{document} depends on \begin{document}$ t\in\Bbb{R} $\end{document} in a general way. Recently, the spreading speeds and linear determinacy of the above time-dependent system have been studied by Bao et al. [J. Differential Equations 265 (2018) 3048-3091]. In this paper, using the principal Lyapunov exponent and principal Floquent bundle theory of linear cooperative systems, we prove the existence of generalized transition waves in any given direction with speed greater than the spreading speed by constructing appropriate subsolutions and supersolutions. When the initial value is uniformly bounded with respect to a weighted maximum norm, we further show that all solutions converge to the generalized transition wave solutions exponentially in time.

Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory
Yangrong Li, Fengling Wang and Shuang Yang
2021, 26(7): 3643-3665 doi: 10.3934/dcdsb.2020250 +[Abstract](482) +[HTML](278) +[PDF](415.07KB)

We establish a new robustness theorem of delayed random attractors at zero-memory and the criteria are given by part convergence of cocycles along with regularity, recurrence and eventual compactness of attractors, where we relax the convergence condition of cocycles in all known robustness theorem of attractors, especially by Wang et al.(Siam-jads, 2015). As an application, we consider the stochastic non-autonomous 2D-Ginzburg-Landau delay equation, whose solutions seem not to be convergent for all initial data as the memory time goes to zero, but we can show the convergence of solutions toward zero-memory for part initial data in the lower-regular space. As a further result, we show that, for each memory time, the delay equation has a pullback random attractor such that it is upper semi-continuous at zero-memory.

Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes
Matheus C. Bortolan and José Manuel Uzal
2021, 26(7): 3667-3692 doi: 10.3934/dcdsb.2020252 +[Abstract](545) +[HTML](271) +[PDF](431.94KB)

In this paper, following the work done in [11], we deal with the upper and weak-lower semicontinuity of pullback attractors for impulsive evolution processes. We first deal with the upper semicontinuity, presenting the abstract theory and applying it to uniform perturbations of a nonautonomous integrate-and-fire neuron model. We also present the abstract theory of weak-lower semicontinuity, and finish with an improvement of [11,Subsection 4.2], proving an invariance property for impulsive pullback \begin{document}$ \omega $\end{document}-limits with weaker assumptions.

Flocking and line-shaped spatial configuration to delayed Cucker-Smale models
Zhisu Liu, Yicheng Liu and Xiang Li
2021, 26(7): 3693-3716 doi: 10.3934/dcdsb.2020253 +[Abstract](556) +[HTML](267) +[PDF](639.9KB)

As we known, it is popular for a designed system to achieve a prescribed performance, which have remarkable capability to regulate the flow of information from distinct and independent components. Also, it is not well understand, in both theories and applications, how self propelled agents use only limited environmental information and simple rules to organize into an ordered motion. In this paper, we focus on analysis the flocking behaviour and the line-shaped pattern for collective motion involving time delay effects. Firstly, we work on a delayed Cucker-Smale-type system involving a general communication weight and a constant delay \begin{document}$ \tau>0 $\end{document} for modelling collective motion. In a result, by constructing a new Lyapunov functional approach, combining with two delayed differential inequalities established by \begin{document}$ L^2 $\end{document}-analysis, we show that the flocking occurs for the general communication weight when \begin{document}$ \tau $\end{document} is sufficiently small. Furthermore, to achieve the prescribed performance, we introduce the line-shaped inner force term into the delayed collective system, and analytically show that there is a flocking pattern with an asymptotic flocking velocity and line-shaped pattern. All results are novel and can be illustrated by numerical simulations using some concrete influence functions. Also, our results significantly extend some known theorems in the literature.

Analysis of non-Markovian effects in generalized birth-death models
Zhenquan Zhang, Meiling Chen, Jiajun Zhang and Tianshou Zhou
2021, 26(7): 3717-3735 doi: 10.3934/dcdsb.2020254 +[Abstract](545) +[HTML](298) +[PDF](459.06KB)

Birth-death processes are a fundamental reaction module for which we can find its prototypes in many scientific fields. For such a kind of module, if all the reaction events are Markovian, the reaction kinetics is simple. However, experimentally observable quantities are in general consequences of a series of reactions, implying that the synthesis of a macromolecule in general involve multiple middle reaction steps with some reactions that would not be specified by experiments. This multistep process can create molecular memory between reaction events, leading to non-Markovian behavior. Based on the theoretical framework established in a recent paper published in [39], we find that the effect of non-Markovianity is equivalent to the introduction of a feedback, non-Markovianity always amplifies the mean level of the product if the death reaction is non-Markovian but always reduces the mean level if the birth reaction is non-Markovian, and in contrast to Markovianity, non-Markovianity can reduce or amplify the product noise, depending on the details of waiting-time distributions characterizing reaction events. Examples analysis indicates that non-Markovianity, whose effects were neglected in previous studies, can significantly impact gene expression.

The coupled 1:2 resonance in a symmetric case and parametric amplification model
Reza Mazrooei-Sebdani and Zahra Yousefi
2021, 26(7): 3737-3765 doi: 10.3934/dcdsb.2020255 +[Abstract](540) +[HTML](269) +[PDF](2598.21KB)

This paper deals with the coupled Hamiltonian \begin{document}$ 1 $\end{document}:\begin{document}$ 2 $\end{document} resonance, i.e. the Hamiltonian \begin{document}$ 1 $\end{document}:\begin{document}$ 2 $\end{document}:\begin{document}$ 1 $\end{document}:\begin{document}$ 2 $\end{document} resonance. This resonance is of the first order. We isolate several integrable cases. Our main focus is on two models. In the first part of the paper, we present a discrete symmetric normal form truncated to order three and we compute the relative equilibria for its corresponding system. In the second part, the paper is devoted to the study of the Hamiltonian describing the four-wave mixing (FWM) model. In addition to the Hamiltonian, the corresponding system possesses three more independent integrals. We use these integrals to obtain estimates for the phase space and total energy. Further, we compute the relative equilibria of the FWM system for the \begin{document}$ 1 $\end{document}:\begin{document}$ 2 $\end{document}:\begin{document}$ 1 $\end{document}:\begin{document}$ 2 $\end{document} resonance. Finally, we carry out some numerical experiments for the detuned system.

Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment
Weiyi Zhang, Zuhan Liu and Ling Zhou
2021, 26(7): 3767-3784 doi: 10.3934/dcdsb.2020256 +[Abstract](696) +[HTML](294) +[PDF](447.43KB)

In this paper we study a nonlocal diffusion model with double free boundaries in time periodic environment, which is the natural extension of the free boundary model in [17], where local diffusion is used to describe the population dispersal. We give the existence and uniqueness of global solution and consider the properties of principle eigenvalue of time-periodic parabolic-type eigenvalue problem. With the help of attractivity of time periodic solutions, we establish a spreading-vanishing dichotomy. The sharp criteria for spreading and vanishing are also obtained.

Evolutionary de Rham-Hodge method
Jiahui Chen, Rundong Zhao, Yiying Tong and Guo-Wei Wei
2021, 26(7): 3785-3821 doi: 10.3934/dcdsb.2020257 +[Abstract](784) +[HTML](300) +[PDF](5618.5KB)

The de Rham-Hodge theory is a landmark of the 20\begin{document}$ ^\text{th} $\end{document} Century's mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of unique evolutionary Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exactly the persistent Betti numbers of dimensions 0, 1 and 2. Additionally, three sets of non-zero eigenvalues further reveal both topological persistence and geometric progression during the manifold evolution. Extensive numerical experiments are carried out via the discrete exterior calculus to demonstrate the potential of the proposed paradigm for data representation and shape analysis of both point cloud data and density maps. To demonstrate the utility of the proposed method, the application is considered to the protein B-factor predictions of a few challenging cases for which existing biophysical models break down.

Convergence of quasilinear parabolic equations to semilinear equations
Flank D. M. Bezerra, Jacson Simsen and Mariza Stefanello Simsen
2021, 26(7): 3823-3834 doi: 10.3934/dcdsb.2020258 +[Abstract](488) +[HTML](257) +[PDF](319.84KB)

In this work we consider a family of reaction-diffusion equations with variable exponents reaching as a limit problem a semilinear equation. We provide uniform estimates for the solutions and we prove that the solutions of the family of quasilinear equations with variable exponents converge to the solution of a limit semilinear equation when the exponents go to 2. Moreover, the robustness of the global attractors is also studied.

Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission
Zhikun She and Xin Jiang
2021, 26(7): 3835-3861 doi: 10.3934/dcdsb.2020259 +[Abstract](567) +[HTML](258) +[PDF](609.07KB)

In this paper, a general viral infection model with humoral immunity is investigated. The model describes the interaction of uninfected target cells, infected cells, free viruses and humoral immune response, incorporating two virus transmission modes and intracellular delay. Some reasonable hypothesises are made for the general incidence rates. Through stability analysis of equilibria under these hypothesises, the model exhibits threshold dynamics with respect to the immune-inactivated reproduction rate \begin{document}$ \mathfrak{R}_0 $\end{document} and the immune-activated reproduction rate \begin{document}$ \mathfrak{R}_1 $\end{document}. The theoretical results and corresponding numerical simulations show that the intracellular latency, both of virus-to-cell infection and cell-to-cell infection have direct effects on the global dynamics of the general viral infection model. Our results improve and generalize some known results on within-host virus dynamics.

Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations
Anhui Gu
2021, 26(7): 3863-3878 doi: 10.3934/dcdsb.2020266 +[Abstract](514) +[HTML](262) +[PDF](361.3KB)

In this paper, we obtain the existence and uniqueness of weak pullback mean random attractors for non-autonomous deterministic \begin{document}$ p $\end{document}-Laplacian equations with random initial data and non-autonomous stochastic \begin{document}$ p $\end{document}-Laplacian equations with general diffusion terms in Bochner spaces, respectively.

A class of stochastic Fredholm-algebraic equations and applications in finance
Zheng Liu and Tianxiao Wang
2021, 26(7): 3879-3903 doi: 10.3934/dcdsb.2020267 +[Abstract](457) +[HTML](281) +[PDF](434.0KB)

A class of stochastic Fredholm-algebraic equations (SFAEs) is introduced and investigated. Like backward stochastic differential equations (BSDEs), its solution includes two parts. The interesting thing is that the first part is deterministic and constrained, even though the whole system is stochastic. Our study is mainly motivated by risk indifference pricing problem. Actually, the existing risk indifference price always keeps unchangeable with respect to initial wealth, which is economically unsatisfying. Nevertheless, here a new wealth dependent risk indifference price is proposed by particular SFAEs.

On the reducibility of a class of almost periodic Hamiltonian systems
Jia Li and Junxiang Xu
2021, 26(7): 3905-3919 doi: 10.3934/dcdsb.2020268 +[Abstract](460) +[HTML](257) +[PDF](349.12KB)

In this paper we consider the following linear almost periodic hamiltonian system

where \begin{document}$ A $\end{document} is a constant matrix with different eigenvalues, and \begin{document}$ Q(t, \varepsilon) $\end{document} is analytic almost periodic with respect to \begin{document}$ t $\end{document} and analytic with respect to \begin{document}$ \varepsilon $\end{document}. Without any non-degeneracy condition, we prove that the linear hamiltonian system is reducible for most of sufficiently small parameter \begin{document}$ \varepsilon $\end{document} by an almost periodic symplectic mapping.

Numerical analysis of two new finite difference methods for time-fractional telegraph equation
Xiaozhong Yang and Xinlong Liu
2021, 26(7): 3921-3942 doi: 10.3934/dcdsb.2020269 +[Abstract](577) +[HTML](263) +[PDF](4398.04KB)

Fractional telegraph equations are an important class of evolution equations and have widely applications in signal analysis such as transmission and propagation of electrical signals. Aiming at the one-dimensional time-fractional telegraph equation, a class of explicit-implicit (E-I) difference methods and implicit-explicit (I-E) difference methods are proposed. The two methods are based on a combination of the classical implicit difference method and the classical explicit difference method. Under the premise of smooth solution, theoretical analysis and numerical experiments show that the E-I and I-E difference schemes are unconditionally stable, with 2nd order spatial accuracy, \begin{document}$ 2-\alpha $\end{document} order time accuracy, and have significant time-saving, their calculation efficiency is higher than the classical implicit scheme. The research shows that the E-I and I-E difference methods constructed in this paper are effective for solving the time-fractional telegraph equation.

On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies
Manil T. Mohan and Arbaz Khan
2021, 26(7): 3943-3988 doi: 10.3934/dcdsb.2020270 +[Abstract](782) +[HTML](320) +[PDF](738.2KB)

In this work, we consider the forced generalized Burgers-Huxley equation and establish the existence and uniqueness of a global weak solution using a Faedo-Galerkin approximation method. Under smoothness assumptions on the initial data and external forcing, we also obtain further regularity results of weak solutions. Taking external forcing to be zero, a positivity result as well as a bound on the classical solution are also established. Furthermore, we examine the long-term behavior of solutions of the generalized Burgers-Huxley equations. We first establish the existence of absorbing balls in appropriate spaces for the semigroup associated with the solutions and then show the existence of a global attractor for the system. The inviscid limits of the Burgers-Huxley equations to the Burgers as well as Huxley equations are also discussed. Next, we consider the stationary Burgers-Huxley equation and establish the existence and uniqueness of weak solution by using a Faedo-Galerkin approximation technique and compactness arguments. Then, we discuss about the exponential stability of stationary solutions. Concerning numerical studies, we first derive error estimates for the semidiscrete Galerkin approximation. Finally, we present two computational examples to show the convergence numerically.

Sufficient conditions for global dynamics of a viral infection model with nonlinear diffusion
Wei Wang, Wanbiao Ma and Xiulan Lai
2021, 26(7): 3989-4011 doi: 10.3934/dcdsb.2020271 +[Abstract](544) +[HTML](246) +[PDF](929.23KB)

In this paper, we study the global dynamics of a viral infection model with spatial heterogeneity and nonlinear diffusion. For the spatially heterogeneous case, we first derive some properties of the basic reproduction number \begin{document}$ R_0 $\end{document}. Then for the auxiliary system with quasilinear diffusion, we establish the comparison principle under some appropriate conditions. Some sufficient conditions are derived to ensure the global stability of the virus-free steady state. We also show the existence of the positive non-constant steady state and the persistence of virus. For the spatially homogeneous case, we show that \begin{document}$ R_0 $\end{document} is the only determinant of the global dynamics when the derivative of the function \begin{document}$ g $\end{document} with respect to \begin{document}$ V $\end{document} (the rate of change of infected cells for the repulsion effect) is small enough. Our simulation results reveal that pyroptosis and Beddington-DeAngelis functional response function play a crucial role in the controlling of the spreading speed of virus, which are some new phenomena not presented in the existing literature.

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