Global Dynamics of Infectious Disease with Arbitrary Distributed Infectious Period on Complex Networks

Most of the current epidemic models assume that the infectious period follows an exponential distribution. However, due to individual heterogeneity and epidemic diversity, these models fail to describe the distribution of infectious periods precisely. We establish a SIS epidemic model with multistaged progression of infectious periods on complex networks, which can be used to characterize arbitrary distributions of infectious periods of the individuals. By using mathematical analysis, the basic reproduction numberR0for the model is derived. We verify that theR0depends on the average distributions of infection periods for different types of infective individuals, which extend the general theory obtained from the single infectious period epidemic models. It is proved that ifR0<1, then the disease-free equilibrium is globally asymptotically stable; otherwise the unique endemic equilibrium exists such that it is globally asymptotically attractive. Finally numerical simulations hold for the validity of our theoretical results is given.

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Dynamical Analysis of an SEIRS Model With Incomplete Recovery and Relapse on Networks and Its Optimal Vaccination Control

10.21203/rs.3.rs-510785/v1 ◽

2021 ◽

Author(s):

Lei Zhang ◽

Maoxing Liu ◽

Qiang Hou ◽

Boli Xie

Keyword(s):

Control Strategy ◽

Reproduction Number ◽

Dynamical Analysis ◽

Global Dynamics ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Seirs Epidemic Model ◽

Theoretical Results ◽

Disease Free ◽

The Basic Reproduction Number

Abstract For some infectious diseases, such as herpes and tuberculosis, there is incomplete recovery and relapse. These phenomena make them difficult to control. In consequence of this status, an SEIRS epidemic model with incomplete recovery and relapse on networks is established and the global dynamics is studied. The results show that when the basic reproduction number R 0 <=1 the disease-free equilibrium is globally asymptotically stable; when R 0 > 1, the endemic equilibrium is globally asymptotically stable. In addition, in consideration of vaccination control strategy, an SVEIRS model is introduced and the optimal control is solved. At last, the theoretical results are illustrated with numerical simulations.

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Threshold Dynamics in a Model for Zika Virus Disease with Seasonality

Bulletin of Mathematical Biology ◽

10.1007/s11538-020-00844-6 ◽

2021 ◽

Vol 83 (4) ◽

Cited By ~ 1

Author(s):

Mahmoud A. Ibrahim ◽

Attila Dénes

Keyword(s):

Zika Virus ◽

Reproduction Number ◽

Virus Disease ◽

Threshold Dynamics ◽

Global Dynamics ◽

Threshold Parameter ◽

Globally Asymptotically Stable ◽

Linear Integral ◽

The Impact ◽

Disease Free

AbstractWe present a compartmental population model for the spread of Zika virus disease including sexual and vectorial transmission as well as asymptomatic carriers. We apply a non-autonomous model with time-dependent mosquito birth, death and biting rates to integrate the impact of the periodicity of weather on the spread of Zika. We define the basic reproduction number $${\mathscr {R}}_{0}$$ R 0 as the spectral radius of a linear integral operator and show that the global dynamics is determined by this threshold parameter: If $${\mathscr {R}}_0 < 1,$$ R 0 < 1 , then the disease-free periodic solution is globally asymptotically stable, while if $${\mathscr {R}}_0 > 1,$$ R 0 > 1 , then the disease persists. We show numerical examples to study what kind of parameter changes might lead to a periodic recurrence of Zika.

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Threshold Dynamics of a Huanglongbing Model with Logistic Growth in Periodic Environments

Abstract and Applied Analysis ◽

10.1155/2014/841367 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

Jianping Wang ◽

Shujing Gao ◽

Yueli Luo ◽

Dehui Xie

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Logistic Growth ◽

Threshold Dynamics ◽

Global Dynamics ◽

Globally Asymptotically Stable ◽

Linear Integral ◽

The Impact ◽

Disease Free ◽

The Basic Reproduction Number

We analyze the impact of seasonal activity of psyllid on the dynamics of Huanglongbing (HLB) infection. A new model about HLB transmission with Logistic growth in psyllid insect vectors and periodic coefficients has been investigated. It is shown that the global dynamics are determined by the basic reproduction numberR0which is defined through the spectral radius of a linear integral operator. IfR0< 1, then the disease-free periodic solution is globally asymptotically stable and ifR0> 1, then the disease persists. Numerical values of parameters of the model are evaluated taken from the literatures. Furthermore, numerical simulations support our analytical conclusions and the sensitive analysis on the basic reproduction number to the changes of average and amplitude values of the recruitment function of citrus are shown. Finally, some useful comments on controlling the transmission of HLB are given.

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Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

Mathematical Problems in Engineering ◽

10.1155/2015/745732 ◽

2015 ◽

Vol 2015 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Yanju Xiao ◽

Weipeng Zhang ◽

Guifeng Deng ◽

Zhehua Liu

Keyword(s):

Sufficient Conditions ◽

Global Dynamics ◽

Sis Model ◽

Delayed Treatment ◽

Sis Epidemic Model ◽

Treatment Function ◽

Lyapunov Coefficient ◽

Saturated Treatment ◽

Theoretical Results ◽

Disease Free

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

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A Delay Differential Equation Model of HIV Infection, with Therapy and CTL Response

Bulletin of Mathematical Sciences and Applications ◽

10.18052/www.scipress.com/bmsa.9.53 ◽

2014 ◽

Vol 9 ◽

pp. 53-68 ◽

Cited By ~ 1

Author(s):

B. El Boukari ◽

N. Yousﬁ

Keyword(s):

Reproduction Number ◽

Equation Model ◽

Asymptotically Stable ◽

Differential Equation Model ◽

Globally Asymptotically Stable ◽

Intracellular Delay ◽

Immunodeficiency Virus ◽

Delay Differential ◽

Theoretical Results ◽

Disease Free

In this work we investigate a new mathematical model that describes the interactions betweenCD4+ T cells, human immunodeﬁciency virus (HIV), immune response and therapy with two drugs.Also an intracellular delay is incorporated into the model to express the lag between the time thevirus contacts a target cell and the time the cell becomes actively infected. The model dynamicsis completely deﬁned by the basic reproduction number R0. If R0 ≤ 1 the disease-free equilibriumis globally asymptotically stable, and if R0 > 1, two endemic steady states exist, and their localstability depends on value of R0. We show that the intracellular delay aﬀects on value of R0 becausea larger intracellular delay can reduce the value of R0 to below one. Finally, numerical simulationsare presented to illustrate our theoretical results.

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Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Mathematics ◽

10.3390/math6120328 ◽

2018 ◽

Vol 6 (12) ◽

pp. 328 ◽

Cited By ~ 3

Author(s):

Yanli Ma ◽

Jia-Bao Liu ◽

Haixia Li

Keyword(s):

Lyapunov Function ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Global Dynamics ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Disease Free ◽

The Basic Reproduction Number

In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

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Global Dynamics of an SIS Model on Metapopulation Networks with Demographics

Complexity ◽

10.1155/2021/8884236 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Maoxing Liu ◽

Xinjie Fu ◽

Jie Zhang ◽

Donghua Zhao

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Sufficient Conditions ◽

Active Role ◽

Endemic Equilibrium ◽

Global Dynamics ◽

Sis Model ◽

Globally Asymptotically Stable ◽

Sis Epidemic Model ◽

The Basic Reproduction Number

In this paper, we propose a susceptible-infected-susceptible (SIS) epidemic model with demographics on heterogeneous metapopulation networks. We analytically derive the basic reproduction number, which determines not only the existence of endemic equilibrium but also the global dynamics of the model. The model always has the disease-free equilibrium, which is globally asymptotically stable when the basic reproduction number is less than unity and otherwise unstable. We also provide sufficient conditions on the global stability of the unique endemic equilibrium. Numerical simulations are performed to illustrate the theoretical results and the effects of the connectivity and diffusion. Furthermore, we find that diffusion rates play an active role in controlling the spread of infectious diseases.

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Global Dynamics of a Discrete-Time MERS-Cov Model

Mathematics ◽

10.3390/math9050563 ◽

2021 ◽

Vol 9 (5) ◽

pp. 563

Author(s):

Mahmoud H. DarAssi ◽

Mohammad A. Safi ◽

Morad Ahmad

Keyword(s):

Discrete Time ◽

Global Attractivity ◽

Endemic Equilibrium ◽

Global Dynamics ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Proposed Model ◽

Threshold Conditions ◽

Theoretical Results ◽

Disease Free

In this paper, we have investigated the global dynamics of a discrete-time middle east respiratory syndrome (MERS-Cov) model. The proposed discrete model was analyzed and the threshold conditions for the global attractivity of the disease-free equilibrium (DFE) and the endemic equilibrium are established. We proved that the DFE is globally asymptotically stable when R0≤1. Whenever R˜0>1, the proposed model has a unique endemic equilibrium that is globally asymptotically stable. The theoretical results are illustrated by a numerical simulation.

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Global Stability of an Epidemic Model with Incomplete Treatment and Vaccination

Discrete Dynamics in Nature and Society ◽

10.1155/2012/530267 ◽

2012 ◽

Vol 2012 ◽

pp. 1-14 ◽

Cited By ~ 2

Author(s):

Hai-Feng Huo ◽

Li-Xiang Feng

Keyword(s):

Global Stability ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Reproduction Number ◽

Global Dynamics ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Incomplete Treatment ◽

Disease Free ◽

The Basic Reproduction Number

An epidemic model with incomplete treatment and vaccination for the newborns and susceptibles is constructed. We establish that the global dynamics are completely determined by the basic reproduction numberR0. IfR0≤1, then the disease-free equilibrium is globally asymptotically stable. IfR0>1, the endemic equilibrium is globally asymptotically stable. Some numerical simulations are also given to explain our conclusions.

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Global Stability of a SLIT TB Model with Staged Progression

Journal of Applied Mathematics ◽

10.1155/2012/571469 ◽

2012 ◽

Vol 2012 ◽

pp. 1-15

Author(s):

Yakui Xue ◽

Xiaohong Wang

Keyword(s):

Mathematical Model ◽

Latent Period ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Infectious Period ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

N Stage ◽

Disease Free ◽

The Basic Reproduction Number

Because the latent period and the infectious period of tuberculosis (TB) are very long, it is not reasonable to consider the time as constant. So this paper formulates a mathematical model that divides the latent period and the infectious period into n-stages. For a general n-stage stage progression (SP) model with bilinear incidence, we analyze its dynamic behavior. First, we give the basic reproduction numberR0. Moreover, ifR0≤1, the disease-free equilibriumP0is globally asymptotically stable and the disease always dies out. IfR0>1, the unique endemic equilibriumP∗is globally asymptotically stable and the disease persists at the endemic equilibrium.

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