Wave propagation in a diffusive SAIV epidemic model with time delays

2021 ◽  
pp. 1-27
Author(s):  
JIANGBO ZHOU ◽  
JINGHUAN LI ◽  
JINGDONG WEI ◽  
LIXIN TIAN
Keyword(s):  
Epidemic Model ◽  
Time Delays ◽  
Wave Speed ◽  
Reproduction Number ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Epidemic Disease ◽  
Incubation Periods ◽  
And Control ◽  
Theoretical Results

Based on the fact that the incubation periods of epidemic disease in asymptomatically infected and infected individuals are inevitable and different, we propose a diffusive susceptible, asymptomatically infected, symptomatically infected and vaccinated (SAIV) epidemic model with delays in this paper. To see whether epidemic disease can propagate spatially with a constant speed, we focus on the travelling wave solution for this model. When the basic reproduction number of the corresponding spatial-homogenous delayed differential system is greater than one and the wave speed is greater than or equal to the critical speed, we prove that this model admits nontrivial positive travelling wave solutions. Our theoretical results are of benefit to the prevention and control of epidemic.

scholarly journals Pattern Dynamics of Nonlocal Delay SI Epidemic Model with the Growth of the Susceptible following Logistic Mode

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Zun-Guang Guo ◽  
Jing Li ◽  
Can Li ◽  
Juan Liang ◽  
Yiwei Yan
Keyword(s):  
Time Delay ◽  
Epidemic Model ◽  
Turing Instability ◽  
Average Density ◽  
Linear Stability Theory ◽  
Susceptible Population ◽  
Nonlocal Delay ◽  
Pattern Dynamics ◽  
And Control ◽  
Theoretical Results

In this paper, we investigate pattern dynamics of a nonlocal delay SI epidemic model with the growth of susceptible population following logistic mode. Applying the linear stability theory, the condition that the model generates Turing instability at the endemic steady state is analyzed; then, the exact Turing domain is found in the parameter space. Additionally, numerical results show that the time delay has key effect on the spatial distribution of the infected, that is, time delay induces the system to generate stripe patterns with different spatial structures and affects the average density of the infected. The numerical simulation is consistent with the theoretical results, which provides a reference for disease prevention and control.

scholarly journals Travelling Waves of a Diffusive Epidemic Model with Latency and Relapse

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Zhiping Wang ◽  
Rui Xu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Epidemic Model ◽  
Wave Solution ◽  
Local Stability ◽  
Travelling Waves ◽  
Upper And Lower Solutions ◽  
Travelling Wave ◽  
Spatial Diffusion ◽  
Travelling Wave Solution

An SEIR epidemic model with relapse and spatial diffusion is studied. By analyzing the corresponding characteristic equations, the local stability of each of the feasible steady states to this model is discussed. The existence of a travelling wave solution is established by using the technique of upper and lower solutions and Schauder's fixed point theorem. Numerical simulations are carried out to illustrate the main results.

Hopf bifurcation of a delay SIRS epidemic model with novel nonlinear incidence: Application to scarlet fever

2018 ◽  
Vol 11 (07) ◽  
pp. 1850091 ◽  
Author(s):  
Yong Li ◽  
Xianning Liu ◽  
Lianwen Wang ◽  
Xingan Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Epidemic Model ◽  
Scarlet Fever ◽  
Reproduction Number ◽  
Control Strategies ◽  
Nonlinear Incidence ◽  
Sirs Epidemic Model ◽  
Intervention Measures ◽  
Theoretical Results

An [Formula: see text] epidemic model incorporating incubation time delay and novel nonlinear incidence is proposed and analyzed to seek for the control strategies of scarlet fever, where the contact rate which can reflect the regular behavior and habit changes of children is non-monotonic with respect to the number of susceptible. The model without delay may exhibit backward bifurcation and bistable states even though the basic reproduction number is less than unit. Furthermore, we derive the conditions for occurrence of Hopf bifurcation when the time delay is considered as a bifurcation parameter. The data of scarlet fever of China are simulated to verify our theoretical results. In the end, several effective preventive and intervention measures of scarlet fever are found out.

scholarly journals Analysis of COVID-19 based on SEIR epidemic models in a multi-patch environment

2021 ◽  
Author(s):  
Lan Meng ◽  
Wei Zhu
Keyword(s):  
Numerical Simulations ◽  
Dynamic Behavior ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Migration Rate ◽  
Reproduction Number ◽  
Population Migration ◽  
Theoretical Results ◽  
Disease Free ◽  
The Basic Reproduction Number

Abstract In this paper, an n-patch SEIR epidemic model for the coronavirus disease 2019 (COVID-19) is presented. It is shown that there is unique disease-free equilibrium for this model. Then, the dynamic behavior is studied by the basic reproduction number. Some numerical simulations with three patches are given to validate the effectiveness of the theoretical results. The influence of quarantined rate and population migration rate on the basic reproduction number is also discussed by simulation.

THE DYNAMICS AND THERAPEUTIC STRATEGIES OF A SEIS EPIDEMIC MODEL

2013 ◽  
Vol 06 (05) ◽  
pp. 1350029 ◽  
Author(s):  
XINZHU MENG ◽  
ZHITAO WU ◽  
TONGQIAN ZHANG
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Therapeutic Strategy ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Epidemic Disease ◽  
Globally Asymptotically Stable ◽  
Effective Therapeutic Strategy ◽  
The Impact ◽  
The Basic Reproduction Number

Based on an epidemic model which Manvendra and Vinay [Mathematical model to simulate infections disease, VSRD-TNTJ3(2) (2012) 60–68] have proposed, we consider the dynamics and therapeutic strategy of a SEIS epidemic model with latent patients and active patients. First, the basic reproduction number is established by applying the method of the next generation matrix. By means of appropriate Lyapunov functions, it is proven that while the basic reproduction number 0 < R0 < 1, the disease-free equilibrium is globally asymptotically stable and the disease eliminates; and if the basic reproduction number R0 > 1, the endemic equilibrium is globally asymptotically stable and therefore the disease becomes endemic. Numerical investigations of their basin of attraction indicate that the locally stable equilibria are global attractors. Second, we consider the impact of treatment on epidemic disease and analytically determine the most effective therapeutic strategy. We conclude that the most effective therapeutic strategy consists of treating both the exposed and the infectious, while treating only the exposed is the least effective therapeutic strategy. Finally, numerical simulations are given to illustrate the effectiveness of the proposed results.

scholarly journals Travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion

2021 ◽  
Vol 477 (2256) ◽  
Author(s):  
Chloé Colson ◽  
Faustino Sánchez-Garduño ◽  
Helen M. Byrne ◽  
Philip K. Maini ◽  
Tommaso Lorenzi
Keyword(s):  
Wave Solution ◽  
Degradation Rate ◽  
Wave Speed ◽  
Propagation Speed ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Tumour Invasion ◽  
Wave Analysis ◽  
Shooting Argument ◽  
Minimal Wave Speed

In this paper, we carry out a travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion. We consider two types of invasive fronts of tumour tissue into extracellular matrix (ECM), which represents healthy tissue. These types differ according to whether the density of ECM far ahead of the wave front is maximal or not. In the former case, we use a shooting argument to prove that there exists a unique travelling-wave solution for any positive propagation speed. In the latter case, we further develop this argument to prove that there exists a unique travelling-wave solution for any propagation speed greater than or equal to a strictly positive minimal wave speed. Using a combination of analytical and numerical results, we conjecture that the minimal wave speed depends monotonically on the degradation rate of ECM by tumour cells and the ECM density far ahead of the front.

scholarly journals Global Dynamics for a Novel Differential Infectivity Epidemic Model with Stage Structure

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Yunguo Jin
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Reproduction Number ◽  
Stage Structure ◽  
Threshold Dynamics ◽  
Uniform Persistence ◽  
Global Dynamics ◽  
Differential Infectivity ◽  
The Stability ◽  
Theoretical Results

A novel differential infectivity epidemic model with stage structure is formulated and studied. Under biological motivation, the stability of equilibria is investigated by the global Lyapunov functions. Some novel techniques are applied to the global dynamics analysis for the differential infectivity epidemic model. Uniform persistence and the sharp threshold dynamics are established; that is, the reproduction number determines the global dynamics of the system. Finally, numerical simulations are given to illustrate the main theoretical results.

Travelling wave solutions of a parabolic-hyperbolic system for contact inhibition of cell-growth

2015 ◽  
Vol 26 (3) ◽  
pp. 297-323 ◽  
Author(s):  
M. BERTSCH ◽  
D. HILHORST ◽  
H. IZUHARA ◽  
M. MIMURA ◽  
T. WAKASA
Keyword(s):  
Partial Differential Equations ◽  
Cell Growth ◽  
Large Class ◽  
Wave Solution ◽  
Wave Speed ◽  
Large Time ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Travelling Wave Solutions ◽  
Wave Solutions

We consider a cell growth model involving a nonlinear system of partial differential equations which describes the growth of two types of cell populations with contact inhibition. Numerical experiments show that there is a parameter regime where, for a large class of initial data, the large time behaviour of the solutions is described by a segregated travelling wave solution with positive wave speed c. Here, the word segregated expresses the fact that the different types of cells are spatially segregated, and that the single densities are discontinuous at the moving interface which separates the two populations. In this paper, we show that, for each wave speed c > c, there exists an overlapping travelling wave solution, whose profile is continuous and no longer segregated. We also show that, for a large class of initial functions, the overlapping travelling wave solutions cannot represent the large time profile of the solutions of the system of partial differential equations. The structure of the travelling wave solutions strongly resembles that of the scalar Fisher-KPP equation, for which the special role played by the travelling wave solution with minimal speed has been extensively studied.

scholarly journals On an SE(Is)(Ih)AR epidemic model with combined vaccination and antiviral controls for COVID-19 pandemic

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
M. De la Sen ◽  
A. Ibeas
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Antiviral Treatment ◽  
Stability Properties ◽  
Local Asymptotic Stability ◽  
Proposed Model ◽  
And Control ◽  
The Basic Reproduction Number

AbstractIn this paper, we study the nonnegativity and stability properties of the solutions of a newly proposed extended SEIR epidemic model, the so-called SE(Is)(Ih)AR epidemic model which might be of potential interest in the characterization and control of the COVID-19 pandemic evolution. The proposed model incorporates both asymptomatic infectious and hospitalized infectious subpopulations to the standard infectious subpopulation of the classical SEIR model. In parallel, it also incorporates feedback vaccination and antiviral treatment controls. The exposed subpopulation has three different transitions to the three kinds of infectious subpopulations under eventually different proportionality parameters. The existence of a unique disease-free equilibrium point and a unique endemic one is proved together with the calculation of their explicit components. Their local asymptotic stability properties and the attainability of the endemic equilibrium point are investigated based on the next generation matrix properties, the value of the basic reproduction number, and nonnegativity properties of the solution and its equilibrium states. The reproduction numbers in the presence of one or both controls is linked to the control-free reproduction number to emphasize that such a number decreases with the control gains. We also prove that, depending on the value of the basic reproduction number, only one of them is a global asymptotic attractor and that the solution has no limit cycles.

Asymptotic propagations of a nonlocal dispersal population model with shifting habitats

2021 ◽  
pp. 1-28
Author(s):  
SHAO-XIA QIAO ◽  
WAN-TONG LI ◽  
JIA-BING WANG
Keyword(s):  
Growth Rate ◽  
Population Model ◽  
Wave Speed ◽  
Travelling Wave ◽  
Spreading Speed ◽  
Nonlocal Dispersal ◽  
Spatial Direction ◽  
Habitat Edge ◽  
Species Declines ◽  
Theoretical Results

This paper is concerned with the asymptotic propagations for a nonlocal dispersal population model with shifting habitats. In particular, we verify that the invading speed of the species is determined by the speed c of the shifting habitat edge and the behaviours near infinity of the species’ growth rate which is nondecreasing along the positive spatial direction. In the case where the species declines near the negative infinity, we conclude that extinction occurs if c > c*(∞), while c < c*(∞), spreading happens with a leftward speed min{−c, c*(∞)} and a rightward speed c*(∞), where c*(∞) is the minimum KPP travelling wave speed associated with the species’ growth rate at the positive infinity. The same scenario will play out for the case where the species’ growth rate is zero at negative infinity. In the case where the species still grows near negative infinity, we show that the species always survives ‘by moving’ with the rightward spreading speed being either c*(∞) or c*(−∞) and the leftward spreading speed being one of c*(∞), c*(−∞) and −c, where c*(−∞) is the minimum KPP travelling wave speed corresponding to the growth rate at the negative infinity. Finally, we give some numeric simulations and discussions to present and explain the theoretical results. Our results indicate that there may exists a solution like a two-layer wave with the propagation speeds analytically determined for such type of nonlocal dispersal equations.

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