scholarly journals Modeling Transmission of Tuberculosis with MDR and Undetected Cases

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
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.

scholarly journals Mathematical Modeling and Analysis of Transmission Dynamics and Control of Schistosomiasis

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Ebrima Kanyi ◽  
Ayodeji Sunday Afolabi ◽  
Nelson Owuor Onyango
Keyword(s):  
Equilibrium Point ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Modeling And Analysis ◽  
Disease Free

This paper presents a mathematical model that describes the transmission dynamics of schistosomiasis for humans, snails, and the free living miracidia and cercariae. The model incorporates the treated compartment and a preventive factor due to water sanitation and hygiene (WASH) for the human subpopulation. A qualitative analysis was performed to examine the invariant regions, positivity of solutions, and disease equilibrium points together with their stabilities. The basic reproduction number, R 0 , is computed and used as a threshold value to determine the existence and stability of the equilibrium points. It is established that, under a specific condition, the disease-free equilibrium exists and there is a unique endemic equilibrium when R 0 > 1 . It is shown that the disease-free equilibrium point is both locally and globally asymptotically stable provided R 0 < 1 , and the unique endemic equilibrium point is locally asymptotically stable whenever R 0 > 1 using the concept of the Center Manifold Theory. A numerical simulation carried out showed that at R 0 = 1 , the model exhibits a forward bifurcation which, thus, validates the analytic results. Numerical analyses of the control strategies were performed and discussed. Further, a sensitivity analysis of R 0 was carried out to determine the contribution of the main parameters towards the die out of the disease. Finally, the effects that these parameters have on the infected humans were numerically examined, and the results indicated that combined application of treatment and WASH will be effective in eradicating schistosomiasis.

scholarly journals Global Stability of a SLIT TB Model with Staged Progression

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.

Dynamical system of a SEIQV epidemic model with nonlinear generalized incidence rate arising in biology

2017 ◽  
Vol 10 (07) ◽  
pp. 1750096 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Yasir Khan ◽  
Taj Wali Khan ◽  
Saeed Islam
Keyword(s):  
Dynamical System ◽  
Mathematical Model ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, a dynamical system of a SEIQV mathematical model with nonlinear generalized incidence arising in biology is investigated. The stability of the disease-free and endemic equilibrium is discussed. The basic reproduction number of the model is obtained. We found that the disease-free and endemic equilibrium is stable locally as well as globally asymptotically stable. For [Formula: see text], the disease-free equilibrium is stable both locally and globally and for [Formula: see text], the endemic equilibrium is stable globally asymptotically. Finally, some numerical results are presented.

scholarly journals Analysis of a mathematical model of the dynamics of contagious bovine pleuropneumonia

2017 ◽  
Vol 1 ◽  
pp. 64 ◽  
Author(s):  
Achamyelesh Amare Aligaz ◽  
Justin Manango W. Munganga
Keyword(s):  
Mathematical Model ◽  
Compartmental Model ◽  
Water Buffalo ◽  
Control Strategies ◽  
Asymptotically Stable ◽  
Contagious Bovine Pleuropneumonia ◽  
Globally Asymptotically Stable ◽  
Effective Contact ◽  
Mycoplasma Mycoides ◽  
Disease Free

Contagious bovine pleuropneumonia (CBPP) is a disease of cattle and water buffalo caused by Mycoplasma mycoides subspecies mycoides (Mmm). It attacks the lungs and the membranes that line the thoracic cavity. The disease is transmitted by inhaling droplets disseminated through coughing by infected cattle. In this paper a deterministic mathematical model for the transmission of Contagious Bovine plueropnemonia is presented. The model is a five compartmental model consisting of susceptible, Exposed, Infectious, Persistently infected and Recovered compartments. We derived a formula for the basic reproduction number R0. For R0 ≤ 1, the disease free equilibrium is globally asymptotically stable, thus CBPP dies out; whereas for R0 > 1, the unique endemic equilibrium is globally asymptotically stable and hence the disease persists. Elasticity indices for R0 with respect to different parameters are calculated; indicating parameters that are important for control strategies to bring R0 below 1, the effective contact rate β has the largest elasticity index. As the disease control options are associated to these parameters, for some values of these parameters, R0 < 1, thus the disease can be controlled.

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

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 328 ◽  
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.

scholarly journals A Mathematical Model of a Tuberculosis Transmission Dynamics Incorporating First and Second Line Treatment

2020 ◽  
Vol 24 (5) ◽  
pp. 917-922
Author(s):  
J. Andrawus ◽  
F.Y. Eguda ◽  
I.G. Usman ◽  
S.I. Maiwa ◽  
I.M. Dibal ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Equilibrium Point ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Line Treatment ◽  
Second Line ◽  
Tuberculosis Transmission ◽  
Disease Free

This paper presents a new mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment. We calculated a control reproduction number which plays a vital role in biomathematics. The model consists of two equilibrium points namely disease free equilibrium and endemic equilibrium point, it has been shown that the disease free equilibrium point was locally asymptotically stable if thecontrol reproduction number is less than one and also the endemic equilibrium point was locally asymptotically stable if the control reproduction number is greater than one. Numerical simulation was carried out which supported the analytical results. Keywords: Mathematical Model, Biomathematics, Reproduction Number, Disease Free Equilibrium, Endemic Equilibrium Point

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

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.

scholarly journals A MODEL FOR CONTROL OF HIV/AIDS WITH PARENTAL CARE

2013 ◽  
Vol 06 (02) ◽  
pp. 1350006 ◽  
Author(s):  
GBENGA JACOB ABIODUN ◽  
NIZAR MARCUS ◽  
KAZEEM OARE OKOSUN ◽  
PETER JOSEPH WITBOOI
Keyword(s):  
Mathematical Model ◽  
Parental Care ◽  
Optimal Strategies ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Qualitative And Quantitative ◽  
Aids Epidemic ◽  
Quantitative Analyses ◽  
Disease Free ◽  
Hiv Aids

In this study we investigate the HIV/AIDS epidemic in a population which experiences a significant flow of immigrants. We derive and analyze a mathematical model that describes the dynamics of HIV infection among the immigrant youths and how parental care can minimize or prevent the spread of the disease in the population. We analyze the model with both screening control and parental care, then investigate its stability and sensitivity behavior. We also conduct both qualitative and quantitative analyses. It is observed that in the absence of infected youths, disease-free equilibrium is achievable and is globally asymptotically stable. We establish optimal strategies for the control of the disease with screening and parental care, and provide numerical simulations to illustrate the analytic results.

scholarly journals Stability Analysis of a Vector-Borne Disease with Variable Human Population

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Muhammad Ozair ◽  
Abid Ali Lashari ◽  
Il Hyo Jung ◽  
Young Il Seo ◽  
Byul Nim Kim
Keyword(s):  
Mathematical Model ◽  
Human Population ◽  
Feasible Region ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Vector Borne Disease ◽  
Vector Borne ◽  
Theoretical Results ◽  
Disease Free ◽  
Varying Population

A mathematical model of a vector-borne disease involving variable human population is analyzed. The varying population size includes a term for disease-related deaths. Equilibria and stability are determined for the system of ordinary differential equations. IfR0≤1, the disease-“free” equilibrium is globally asymptotically stable and the disease always dies out. IfR0>1, a unique “endemic” equilibrium is globally asymptotically stable in the interior of feasible region and the disease persists at the “endemic” level. Our theoretical results are sustained by numerical simulations.

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

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.

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