Global Stability of a Host-Vector Model for Pine Wilt Disease with Nonlinear Incidence Rate

Based on classical epidemic models, this paper considers a deterministic epidemic model for the spread of the pine wilt disease which has vector mediated transmission. The analysis of the model shows that its dynamics are completely determined by the basic reproduction numberR0. Using a Lyapunov function and a LaSalle's invariant set theorem, we proved the global asymptotical stability of the disease-free equilibrium. We find that ifR0≤1, the disease free equilibrium is globally asymptotically stable, and the disease will be eliminated. IfR0>1, a unique endemic equilibrium exists and is shown to be globally asymptotically stable, under certain restrictions on the parameter values, using the geometric approach method for global stability, due to Li and Muldowney and the disease persists at the endemic equilibrium state if it initially exists.

Download Full-text

Global Stability of a HIV-1 Model with General Nonlinear Incidence and Delays

Journal of Applied Mathematics ◽

10.1155/2013/324546 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6

Author(s):

Yaping Wang ◽

Fuqin Sun

Keyword(s):

Global Stability ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Threshold Behavior ◽

Globally Asymptotically Stable ◽

Incidence Functions ◽

The Given ◽

Disease Free ◽

Hiv 1 ◽

Intracellular Delays

A HIV-1 model with two distributed intracellular delays and general incidence function is studied. Conditions are given under which the system exhibits the threshold behavior: the disease-free equilibriumE0is globally asymptotically stable ifR0≤1; ifR0>1, then the unique endemic equilibriumE1is globally asymptotically stable. Finally, it is shown that the given conditions are satisfied by several common forms of the incidence functions.

Download Full-text

Analysis of the Mathematical Model for the Spread of Pine Wilt Disease

Journal of Applied Mathematics ◽

10.1155/2013/184054 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 7

Author(s):

Xiangyun Shi ◽

Guohua Song

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Pine Wilt Disease ◽

Wilt Disease ◽

Endemic Equilibrium ◽

Feasible Region ◽

Global Dynamics ◽

Pine Wilt ◽

Theoretical Results ◽

The Basic Reproduction Number

This paper formulates and analyzes a pine wilt disease model. Mathematical analyses of the model with regard to invariance of nonnegativity, boundedness of the solutions, existence of nonnegative equilibria, permanence, and global stability are presented. It is proved that the global dynamics are determined by the basic reproduction numberℛ0and the other valueℛcwhich is larger thanℛ0. Ifℛ0andℛcare both less than one, the disease-free equilibrium is asymptotically stable and the pine wilt disease always dies out. If one is between the two values, though the pine wilt disease could occur, the outbreak will stop. If the basic reproduction number is greater than one, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at the endemic equilibrium state if it initially exists. Numerical simulations are carried out to illustrate the theoretical results, and some disease control measures are especially presented by these theoretical results.

Download Full-text

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.

Download Full-text

A mathematical analysis of a tuberculosis epidemic model with two treatments and exogenous re-infection

International Journal of Biomathematics ◽

10.1142/s1793524520500825 ◽

2020 ◽

pp. 2050082

Author(s):

Mehdi Lotfi ◽

Azizeh Jabbari ◽

Hossein Kheiri

Keyword(s):

Mathematical Model ◽

Asymptotic Stability ◽

Stable Disease ◽

Reproduction Number ◽

Geometric Approach ◽

Backward Bifurcation ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Bifurcation Phenomenon ◽

Disease Free

In this paper, we propose a mathematical model of tuberculosis with two treatments and exogenous re-infection, in which the treatment is effective for a number of infectious individuals and it fails for some other infectious individuals who are being treated. We show that the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibria when the related basic reproduction number is less than unity. Also, it is shown that under certain conditions the model cannot exhibit backward bifurcation. Furthermore, it is shown in the absence of re-infection, the backward bifurcation phenomenon does not exist, in which the disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity. The global asymptotic stability of the endemic equilibrium, when the associated reproduction number is greater than unity, is established using the geometric approach. Numerical simulations are presented to illustrate our main results.

Download Full-text

Modeling Transmission of Tuberculosis with MDR and Undetected Cases

Discrete Dynamics in Nature and Society ◽

10.1155/2011/296905 ◽

2011 ◽

Vol 2011 ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Yongqi Liu ◽

Zhendong Sun ◽

Guiquan Sun ◽

Qiu Zhong ◽

Li Jiang ◽

...

Keyword(s):

Mathematical Model ◽

Theoretical Analysis ◽

Control Strategies ◽

Multidrug Resistant ◽

Vital Role ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Tb Control ◽

Disease Free

This paper presents a novel mathematical model with multidrug-resistant (MDR) and undetected TB cases. The theoretical analysis indicates that the disease-free equilibrium is globally asymptotically stable ifR0<1; otherwise, the system may exist a locally asymptotically stable endemic equilibrium. The model is also used to simulate and predict TB epidemic in Guangdong. The results imply that our model is in agreement with actual data and the undetected rate plays vital role in the TB trend. Our model also implies that TB cannot be eradicated from population if it continues to implement current TB control strategies.

Download Full-text

MODELLING THE TRANSMISSION DYNAMICS OF CONTAGIOUS BOVINE PLEUROPNEUMONIA IN THE PRESENCE OF ANTIBIOTIC TREATMENT WITH LIMITED MEDICAL SUPPLY

Mathematical Modelling and Analysis ◽

10.3846/mma.2021.11795 ◽

2021 ◽

Vol 26 (1) ◽

pp. 1-20

Author(s):

Achamyelesh A. Aligaz ◽

Justin M. W. Munganga

Keyword(s):

Antibiotic Treatment ◽

Backward Bifurcation ◽

Endemic Equilibrium ◽

Transmission Dynamics ◽

Asymptotically Stable ◽

Contagious Bovine Pleuropneumonia ◽

Globally Asymptotically Stable ◽

Medical Supply ◽

Delayed Treatment ◽

Disease Free

We present and analyze a mathematical model of the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP) in the presence of antibiotic treatment with limited medical supply. We use a saturated treatment function to model the effect of delayed treatment. We prove that there exist one disease free equilibrium and at most two endemic equilibrium solutions. A backward bifurcation occurs for small values of delay constant such that two endemic equilibriums exist if Rt (R*t,1); where, Rt is the treatment reproduction number and R*t is a threshold such that the disease dies out if and persists in the population if Rt > R*t. However, when a backward bifurcation occurs, a disease free system may easily be shifted to an epidemic. The bifurcation turns forward when the delay constant increases; thus, the disease free equilibrium becomes globally asymptotically stable if Rt < 1, and there exist unique and globally asymptotically stable endemic equilibrium if Rt > 1. However, the amount of maximal medical resource required to control the disease increases as the value of the delay constant increases. Thus, antibiotic treatment with limited medical supply setting would not successfully control CBPP unless we avoid any delayed treatment, improve the efficacy and availability of medical resources or it is given along with vaccination.

Download Full-text

Global Stability of Vector-Host Disease with Variable Population Size

BioMed Research International ◽

10.1155/2013/710917 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 4

Author(s):

Muhammad Altaf Khan ◽

Saeed Islam ◽

Sher Afzal Khan ◽

Gul Zaman

Keyword(s):

Global Stability ◽

Population Size ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Host Disease ◽

Variable Population ◽

Globally Stable ◽

Disease Free ◽

Disease Persistence ◽

Variable Population Size

The paper presents the vector-host disease with a variability in population. We assume, the disease is fatal and for some cases the infected individuals become susceptible. We first show the local and global stability of the disease-free equilibrium, for the case when, the disease free-equilibrium of the model is both locally as well as globally stable. For , the disease persistence occurs. The endemic equilibrium is locally as well as globally asymptotically stable for . Numerical results are presented for the justifications of theoratical results.

Download Full-text

Dynamical Behavior of a New Epidemiological Model

Journal of Applied Mathematics ◽

10.1155/2014/854528 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9

Author(s):

Zizi Wang ◽

Zhiming Guo

Keyword(s):

Time Delay ◽

Dynamical Behavior ◽

Endemic Equilibrium ◽

Epidemiological Model ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Sufficient Condition ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Disease Free

A new epidemiological model is introduced with nonlinear incidence, in which the infected disease may lose infectiousness and then evolves to a chronic noninfectious disease when the infected disease has not been cured for a certain timeτ. The existence, uniqueness, and stability of the disease-free equilibrium and endemic equilibrium are discussed. The basic reproductive numberR0is given. The model is studied in two cases: with and without time delay. For the model without time delay, the disease-free equilibrium is globally asymptotically stable provided thatR0≤1; ifR0>1, then there exists a unique endemic equilibrium, and it is globally asymptotically stable. For the model with time delay, a sufficient condition is given to ensure that the disease-free equilibrium is locally asymptotically stable. Hopf bifurcation in endemic equilibrium with respect to the timeτis also addressed.

Download Full-text

ON THE GLOBAL STABILITY OF CHOLERA MODEL WITH PREVENTION AND CONTROL

MALAYSIAN JOURNAL OF COMPUTING ◽

10.24191/mjoc.v3i1.4812 ◽

2018 ◽

Vol 3 (1) ◽

pp. 28

Author(s):

M O Ibrahim ◽

A A Ayoade ◽

O J Peter ◽

F A Oguntolu

Keyword(s):

Global Stability ◽

Prevention And Control ◽

Control Measures ◽

Extended Model ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

First Order ◽

Prevention And Control Measures ◽

And Control ◽

Disease Free

In this study, a system of first order ordinary differential equations is used to analyse the dynamics of cholera disease via a mathematical model extended from Fung (2014) cholera model. The global stability analysis is conducted for the extended model by suitable Lyapunov function and LaSalle’s invariance principle. It is shown that the disease free equilibrium (DFE) for the extended model is globally asymptotically stable if 𝑅0 𝑞 < 1 and the disease eventually disappears in the population with time while there exists a unique endemic equilibrium that is globally asymptotically stable whenever 𝑅0 𝑞 > 1 for the extended model or 𝑅0 > 1 for the original model and the disease persists at a positive level though with mild waves (i.e few cases of cholera) in the case of𝑅0 𝑞 > 1. Numerical simulations for strong, weak, and no prevention and control measures are carried out to verify the analytical results and Maple 18 is used to carry out the computations.

Download Full-text

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.

Download Full-text