scholarly journals Multi-dimensional discrete Halanay inequalities and the global stability of the disease free equilibrium of a discrete delayed malaria model

2016 ◽  
Vol 2016 (1) ◽  
Author(s):  
Chunqing Wu ◽  
Patricia JY Wong
Keyword(s):  
Global Stability ◽  
Malaria Model ◽  
Disease Free

scholarly journals Volterra–Lyapunov Stability Analysis of the Solutions of Babesiosis Disease Model

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1272
Author(s):  
Fengsheng Chien ◽  
Stanford Shateyi
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Global Stability ◽  
Numerical Simulations ◽  
Lyapunov Stability ◽  
Disease Model ◽  
Transmission Dynamics ◽  
Global Stability Analysis ◽  
Lyapunov Stability Analysis ◽  
Disease Free

This paper studies the global stability analysis of a mathematical model on Babesiosis transmission dynamics on bovines and ticks populations as proposed by Dang et al. First, the global stability analysis of disease-free equilibrium (DFE) is presented. Furthermore, using the properties of Volterra–Lyapunov matrices, we show that it is possible to prove the global stability of the endemic equilibrium. The property of symmetry in the structure of Volterra–Lyapunov matrices plays an important role in achieving this goal. Furthermore, numerical simulations are used to verify the result presented.

scholarly journals Analysis of a malaria model with mosquito host choice and bed-net control

2015 ◽  
Vol 08 (06) ◽  
pp. 1550077 ◽  
Author(s):  
Bruno Buonomo
Keyword(s):  
Reproduction Number ◽  
Host Choice ◽  
Bed Nets ◽  
Bed Net ◽  
Host Manipulation ◽  
Host Preferences ◽  
Insecticide Treated Bed Nets ◽  
The Impact ◽  
Malaria Model ◽  
Disease Free

A malaria model is formulated which includes the enhanced attractiveness of infectious humans to mosquitoes, as result of host manipulation by malaria parasite, and the human behavior, represented by insecticide-treated bed-nets usage. The occurrence of a backward bifurcation at R0 = 1 is shown to be possible, which implies that multiple endemic equilibria co-exist with a stable disease-free equilibrium when the basic reproduction number is less than unity. This phenomenon is found to be caused by disease-induced human mortality. The global asymptotic stability of the endemic equilibrium for R0 > 1 is proved, by using the geometric method for global stability. Therefore, the disease becomes endemic for R0 > 1 regardless of the number of initial cases in both the human and vector populations. Finally, the impact on system dynamics of vector's host preferences and bed-net usage behavior is investigated.

scholarly journals Dynamical Models for Infectious Diseases with Varying Population Size and Vaccinations

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Peilin Shi ◽  
Lingzhen Dong
Keyword(s):  
Infectious Diseases ◽  
Global Stability ◽  
Periodic Solutions ◽  
Dynamical Models ◽  
Bifurcation Theorem ◽  
Time Period ◽  
Population Sizes ◽  
Variable Population ◽  
Disease Free ◽  
Varying Population

We formulate and discuss models for the spread of infectious diseases with variable population sizes and vaccinations on the susceptible individuals. First, we assume that the susceptible individuals are vaccinated continuously. We establish the threshold-like results for the existence and global stability of the disease-free and the endemic equilibriums for these systems. Especially, we prove the global stability of the endemic equilibriums by converting the systems into integrodifferential equations. Second, we suppose that vaccinations occur once per time period. We obtain the existence and global stability of the disease-free periodic solutions for such systems with impulsive effects. By a useful bifurcation theorem, we acquire the existence of the endemic periodic solutions when the disease-related deaths do not occur. At last, we compare the results with vaccinations and without vaccinations and illustrate our results by numerical simulations.

scholarly journals Global Stability of Pneumococcal Pneumonia with Awareness and Saturated Treatment

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Fulgensia Kamugisha Mbabazi ◽  
J. Y. T. Mugisha ◽  
Mark Kimathi
Keyword(s):  
Sensitivity Analysis ◽  
Antibiotic Resistance ◽  
Steady State ◽  
Global Stability ◽  
Influenza A ◽  
Pneumococcal Pneumonia ◽  
Effective Rate ◽  
Model Parameters ◽  
Saturated Treatment ◽  
Disease Free

Pneumocccal pneumonia, a secondary bacterial infection that follows influenza A infection, is responsible for morbidity and mortality in children, elderly, and immunocomprised groups. A mathematical model to study the global stability of pneumococcal pneumonia with awareness and saturated treatment is presented. The basic reproduction number, R0, is computed using the next generation matrix method. The results show that if R0<1, the disease-free steady state is locally asymptotically stable; thus, pneumococcal pneumonia would be eradicated in the population. On the other hand, if R0>1 the endemic steady state is globally attractive; thus, the disease would persist in the population. The quadratic-linear and Goh–Voltera Lyapunov functionals approach are used to prove the global stabilities of the disease-free and endemic steady states, respectively. The sensitivity analysis of R0 on model parameters shows that, it is positively sensitive to the maximal effective rate before antibiotic resistance awareness, rate of relapse encountered in administering treatment, and loss of information by aware susceptible individuals. Contrarily, the sensitivity analysis of R0 on model parameters is negatively sensitive to recovery rate due to treatment and the rate at which unaware susceptible individuals become aware. The numerical analysis of the model shows that awareness about antibiotic resistance and treatment plays a significant role in the control of pneumococcal pneumonia.

A spatial echinococcosis transmission model with time delays: Stability and traveling waves

2017 ◽  
Vol 10 (06) ◽  
pp. 1750081 ◽  
Author(s):  
Zhiting Xu ◽  
Cuihua Ai
Keyword(s):  
Global Stability ◽  
Traveling Wave ◽  
Time Delays ◽  
Traveling Wave Solutions ◽  
Spatial Domain ◽  
Transmission Model ◽  
Initial Value ◽  
Wave Solutions ◽  
Number Formula ◽  
Disease Free

In this paper, we derive a time-delayed and diffusive echinococcosis transmission model. We first address the well-posedness to the initial-value problem for the model and give the basic reproduction number [Formula: see text]. In the case of a bounded spatial domain, we establish the local stability as well as the global stability of the disease-free and disease equilibria of the model. The methods to prove the local and the global stability are to analyze the corresponding characteristic equations and construct Lyapunov functionals, respectively. In the case of an unbounded spatial domain, by applying Schauder’s fixed point theorem and the limiting arguments, we show that when [Formula: see text], there exists a constant [Formula: see text] such that the model admits positive traveling wave solutions connecting the disease-free and endemic equilibrium for [Formula: see text], and when [Formula: see text] and [Formula: see text], the model has no positive traveling wave solutions connecting them.

scholarly journals On Global Stability of Disease-Free Equilibrium in Epidemiological Models

2021 ◽  
Vol 2 (3) ◽  
pp. 37-42
Author(s):  
Tunde Tajudeen Yusuf
Keyword(s):  
Lyapunov Function ◽  
Global Stability ◽  
Direct Approach ◽  
Epidemiological Models ◽  
Homogeneous Population ◽  
Disease Free

This paper considers the problem of constructing appropriate Lyapunov function for establishing the global stability of a disease-free equilibrium in epidemiological models. A generalised algorithm is proposed and it is tested for some selected epidemiological models. Experience from the application of the algorithm on test examples shows that the algorithm is easy to use, less cumbersome, and yielded the desired result, particularly in models with homogeneous population. Thus, the proposed algorithm provides a direct approach for establishing global stability of disease-free equilibrium.

scholarly journals Analysis of a Malaria Transmission Model in Children

2020 ◽  
Vol 24 (5) ◽  
pp. 789-798
Author(s):  
F.Y. Eguda ◽  
A.C. Ocheme ◽  
M.M. Sule ◽  
J. Andrawus ◽  
I.B. Babura
Keyword(s):  
Malaria Transmission ◽  
Compartmental Model ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Transmission Model ◽  
Acquired Immunity ◽  
Asymptotically Stable ◽  
Threshold Parameter ◽  
Malaria Model ◽  
Disease Free

In this paper, a nine compartmental model for malaria transmission in children was developed and a threshold parameter called control reproduction number which is known to be a vital threshold quantity in controlling the spread of malaria was derived. The model has a disease free equilibrium which is locally asymptotically stable if the control reproduction number is less than one and an endemic equilibrium point which is also locally asymptotically stable if the control reproduction number is greater than one. The model undergoes a backward bifurcation which is caused by loss of acquired immunity of recovered children and the rate at which exposed children progress to the mild stage of infection. Keywords: Malaria, Model, Backward Bifurcation, Local Stability.

scholarly journals Global Stability of Vector-Host Disease with Variable Population Size

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
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.

MODELING THE SCHISTOSOMIASIS ON THE ISLETS IN NANJING

2012 ◽  
Vol 05 (04) ◽  
pp. 1250037 ◽  
Author(s):  
LONGXING QI ◽  
JING-AN CUI ◽  
YUAN GAO ◽  
HUAIPING ZHU
Keyword(s):  
Differential Equations ◽  
Global Stability ◽  
Field Study ◽  
Rattus Norvegicus ◽  
Compartmental Model ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Parasitic Diseases ◽  
Disease Free

A compartmental model is established for schistosomiasis infection in Qianzhou and Zimuzhou, two islets in the center of Yangtzi River near Nanjing, P. R. China. The model consists of five differential equations about the susceptible and infected subpopulations of mammalian Rattus norvegicus and Oncomelania snails. We calculate the basic reproductive number R0 and discuss the global stability of the disease free equilibrium and the unique endemic equilibrium when it exists. The dynamics of the model can be characterized in terms of the basic reproductive number. The parameters in the model are estimated based on the data from the field study of the Nanjing Institute of Parasitic Diseases. Our analysis shows that in a natural isolated area where schistosomiasis is endemic, killing snails is more effective than killing Rattus norvegicus for the control of schistosomiasis.

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