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 Mathematical Model of COVID-19 Pandemic: A Case Study of Bangkok, Thailand

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Pakwan Riyapan ◽  
Sherif Eneye Shuaib ◽  
Arthit Intarasit
Keyword(s):  
Mathematical Model ◽  
Reproduction Number ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Proposed Model ◽  
Next Generation Matrix ◽  
Disease Free ◽  
The Basic Reproduction Number

In this study, we propose a new mathematical model and analyze it to understand the transmission dynamics of the COVID-19 pandemic in Bangkok, Thailand. It is divided into seven compartmental classes, namely, susceptible S , exposed E , symptomatically infected I s , asymptomatically infected I a , quarantined Q , recovered R , and death D , respectively. The next-generation matrix approach was used to compute the basic reproduction number denoted as R cvd 19 of the proposed model. The results show that the disease-free equilibrium is globally asymptotically stable if R cvd 19 < 1 . On the other hand, the global asymptotic stability of the endemic equilibrium occurs if R cvd 19 > 1 . The mathematical analysis of the model is supported using numerical simulations. Moreover, the model’s analysis and numerical results prove that the consistent use of face masks would go on a long way in reducing the COVID-19 pandemic.

scholarly journals Stability of a Mathematical Model of Malaria Transmission with Relapse

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Hai-Feng Huo ◽  
Guang-Ming Qiu
Keyword(s):  
Mathematical Model ◽  
Basic Reproduction Number ◽  
Recovery Rate ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Next Generation Matrix ◽  
The Government ◽  
Disease Free ◽  
The Basic Reproduction Number

A more realistic mathematical model of malaria is introduced, in which we not only consider the recovered humans return to the susceptible class, but also consider the recovered humans return to the infectious class. The basic reproduction numberR0is calculated by next generation matrix method. It is shown that the disease-free equilibrium is globally asymptotically stable ifR0≤1, and the system is uniformly persistence ifR0>1. Some numerical simulations are also given to explain our analytical results. Our results show that to control and eradicate the malaria, it is very necessary for the government to decrease the relapse rate and increase the recovery rate.

scholarly journals Epidemic Model and Mathematical Study of Impact of Vaccination for the Control of Malware in Computer Network

2021 ◽  
pp. 72-96
Author(s):  
Titus Ifeanyi Chinebu ◽  
Ikechukwu Valentine Udegbe ◽  
Adanma Cecilia Eberendu
Keyword(s):  
Compartmental Model ◽  
Computer Network ◽  
Reproduction Number ◽  
Nonlinear Ordinary Differential Equation ◽  
Backward Bifurcation ◽  
Asymptotically Stable ◽  
Mathematical Study ◽  
Globally Asymptotically Stable ◽  
Bifurcation Phenomenon ◽  
Asymptotical Stable

Malware remains a significant threat to computer network.  In this paper, we consideredthe problem which computer malware cause to personal computers with its control by proposing a compartmental model SVEIRS (Susceptible Vaccinated-Exposed-infected-Recovered-Susceptible) for malware transmission in computer network using nonlinear ordinary differential equation. Through the analysis of the model, the basic reproduction number  were obtained, and the malware free equilibrium was proved to be locally asymptotical stable if  is less than unity and globally asymptotically stable if Ro is less than some threshold using a Lyapunov function. Also, the unique endemic equilibrium exists under certain conditions and the model underwent backward bifurcation phenomenon. To illustrate our theoretical analysis, some numerical simulation of the system was performed with RungeKutta fourth order (KR4) method in Mathlab. This was used in analyzing the behavior of different compartments of the model and the results showed that vaccination and treatment is very essential for malware control.

scholarly journals Impact of Treatment on HIV-Malaria CoInfection Based on Mathematical Modeling

2019 ◽  
Vol 39 ◽  
pp. 45-62
Author(s):  
Amit Kumar Saha ◽  
Ashrafi Meher Niger ◽  
Chandra Nath Podder
Keyword(s):  
Treatment Strategy ◽  
Reproduction Number ◽  
Treatment Strategies ◽  
Backward Bifurcation ◽  
Asymptotically Stable ◽  
Single Treatment ◽  
Globally Asymptotically Stable ◽  
Mixed Treatment ◽  
The Impact ◽  
Disease Free

The distribution of HIV and malaria overlap globally. So there is always a chance of co-infection. In this paper the impact of medication on HIV-Malaria co-infection has been analyzed and we have developed a mathematical model using the idea of the models of Mukandavire, et al. [13] and Barley, et al. [3] where treatment classes are included. The disease-free equilibrium (DFE) of the HIV-only model is globally-asymptotically stable (GAS) when the reproduction number is less than one. But it is shown that in the malaria-only model, there is a coexistence of stable disease-free equilibrium and stable endemic equilibrium, for a certain interval of the reproduction number less than unity. This indicates the existence of backward bifurcation. Numerical simulations of the full model are performed to determine the impact of treatment strategies. It is shown that malaria-only treatment strategy reduces more new cases of the mixed infection than the HIV-only treatment strategy. Moreover, mixed treatment strategy reduces the least number of new cases compared to single treatment strategies. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 45-62

The backward bifurcation of a model for malaria infection

2018 ◽  
Vol 11 (02) ◽  
pp. 1850018 ◽  
Author(s):  
Juan Wang ◽  
Xue-Zhi Li ◽  
Souvik Bhattacharya
Keyword(s):  
Malaria Infection ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Global Asymptotical Stability ◽  
Globally Asymptotically Stable ◽  
Vector Borne ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an epidemic model of a vector-borne disease, namely, malaria, is considered. The explicit expression of the basic reproduction number is obtained, the local and global asymptotical stability of the disease-free equilibrium is proved under certain conditions. It is shown that the model exhibits the phenomenon of backward bifurcation where the stable disease-free equilibrium coexists with a stable endemic equilibrium. Further, it is proved that the unique endemic equilibrium is globally asymptotically stable under certain conditions.

Global stability of an SEIQV epidemic model with general incidence rate

2015 ◽  
Vol 08 (02) ◽  
pp. 1550020 ◽  
Author(s):  
Yu Yang ◽  
Cuimei Zhang ◽  
Xunyan Jiang
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Geometric Approach ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
General Incidence Rate ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, a class of SEIQV epidemic model with general nonlinear incidence rate is investigated. By constructing Lyapunov function, it is shown that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number ℛ0≤ 1. If ℛ0> 1, we show that the endemic equilibrium is globally asymptotically stable by applying Li and Muldowney geometric approach.

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.

scholarly journals Dynamics of A Multi-stage Epidemic Model with and without Treatment

2020 ◽  
Vol 8 (5) ◽  
pp. 5293-5300
Keyword(s):  
Mathematical Model ◽  
Basic Reproduction Number ◽  
Numerical Solutions ◽  
Reproduction Number ◽  
Dynamical Behaviour ◽  
Model Parameters ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, a non-linear mathematical model is proposed with the thought of treatment to depict the spread of infectious illness and assessed with three contamination stages. We talk about the dynamical behaviour and analytical study of the framework for the mathematical model which shows that it has two non-negative equilibrium points i.e., disease-free equilibrium (DFE) and interior(endemic) equilibrium. The outcomes show that the dynamical behaviour of the model is totally determined by the basic reproduction number. For the basic reproduction number , the disease-free equilibrium is locally as well as globally asymptotically stable under a particular parameter set. In case , the model at the interior equilibrium is locally as well as globally asymptotically stable. Finally, numerical solutions of the model corroborate the analytical results and facilitate a sensitivity analysis of the model parameters.

scholarly journals Mathematical Model of Cholera Transmission with Education Campaign and Treatment Through Quarantine

2019 ◽  
pp. 1-12
Author(s):  
H. O. Nyaberi ◽  
D. M. Malonza
Keyword(s):  
Mathematical Model ◽  
Health Education ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Watery Diarrhea ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Education Campaign ◽  
Next Generation Matrix ◽  
Disease Free

Cholera, a water-borne disease characterized by intense watery diarrhea, affects people in the regions with poor hygiene and untreated drinking water. This disease remains a menace to public health globally and it indicates inequity and lack of community development. In this research, SIQR-B mathematical model based on a system of ordinary differential equations is formulated to study the dynamics of cholera transmission with health education campaign and treatmentthrough quarantine as controls against epidemic in Kenya. The effective basic reproduction number is computed using the next generation matrix method. The equilibrium points of the model are determined and their stability is analysed. Results of stability analysis show that the disease free equilibrium is both locally and globally asymptotically stable R0 < 1 while the endemic equilibrium is both locally and globally asymptotically stable R0 > 1. Numerical simulation carried out using MATLAB software shows that when health education campaign is efficient, the number of cholera infected individuals decreases faster, implying that health education campaign is vital in controlling the spread of cholera disease.

DYNAMICS ANALYSIS OF A VACCINATION MODEL FOR HPV TRANSMISSION

2014 ◽  
Vol 22 (04) ◽  
pp. 555-599 ◽  
Author(s):  
ALIYA A. ALSALEH ◽  
ABBA B. GUMEL
Keyword(s):  
Deterministic Model ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Dynamics Analysis ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Effective Community ◽  
Hpv Types ◽  
Special Case ◽  
Disease Free

A new deterministic model for the transmission dynamics of human papillomavirus (HPV) and related cancers, in the presence of the Gardasil vaccine (which targets four HPV types), is presented. In the absence of routine vaccination in the community, the model is shown to undergo the phenomenon of backward bifurcation. This phenomenon, which has important consequences on the feasibility of effective disease control in the community, arises due to the re-infection of recovered individuals. For the special case when backward bifurcation does not occur, the disease-free equilibrium (DFE) of the model is shown to be globally-asymptotically stable (GAS) if the associated reproduction number is less than unity. The model with vaccination is also rigorously analyzed. Numerical simulations of the model with vaccination show that, with the assumed 90% efficacy of the Gardasil vaccine, the effective community-wide control of the four Gardasil-preventable HPV types is feasible if the Gardasil coverage rate is high enough (in the range 78–88%).

Sign in / Sign up

Export Citation Format

Share Document