scholarly journals Classical Evaluation of Zero, Incomplete and Complete Treatment Effects on SEIRS Model Dynamics

2020 ◽  
Vol 24 (6) ◽  
pp. 975-978
Author(s):  
A.O. Sangotola ◽  
O. Oyewole
Keyword(s):  
Lyapunov Function ◽  
Global Stability ◽  
Equilibrium Point ◽  
Reproduction Number ◽  
Research Work ◽  
Asymptotically Stable ◽  
Model Dynamics ◽  
Complete Treatment ◽  
Next Generation Matrix ◽  
Disease Free

In this research work, we extend the classical SEIR model to accommodate the effect of zero, incomplete and complete treatment on the dynamics of the model. The basic reproduction number (Ro) of the model dynamics is obtained by using the next generation matrix approach. The disease free equilibrium point of the model is found to be locally asymptotically stable if Ro < 1. A suitable Lyapunov function is constructed to determine the global stability of the disease free equilibrium point. Numerical simulation is carried out to determine the effect on the compliance to the treatment  prescription. Keywords: Equilibrium, Lyapunov function, Global stability, Simulation  

scholarly journals Dynamical System of the Mathematical Model for Tuberculosis with Vaccination

2019 ◽  
Vol 10 (2) ◽  
Author(s):  
Dian Grace Ludji ◽  
Paian Sianturi ◽  
Endar Nugrahani
Keyword(s):  
Dynamical System ◽  
Equilibrium Point ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Prompt Treatment ◽  
Latently Infected ◽  
Next Generation Matrix ◽  
The Mathematical Model ◽  
Disease Free

This research focused on the modification of deterministic mathematical models for tuberculosis with vaccination. It also aimed to see the effect of giving the vaccine. It was done by adding vaccine compartments to people who were given the vaccine in the susceptible compartment. The population was divided into nine different groups. Those were susceptible individuals (S), vaccine (V), new latently infected (E1), diagnosed latently infected (E2), undiagnosed latently infected (E3), undiagnosed actively infected (l), diagnosed actively infected with prompt treatment (Dr), diagnosed actively infected with delay treatment (Dp), and treated (T). Basic reproduction number was constructed using next-generation matrix. Sensitivity analysis was also conducted. The results show that the model comprises two equilibriums: diseasefree equilibrium (T0) and endemic equilibrium (T*). It also shows that there is a relationship between R0 and two equilibriums. Moreover, the disease-free equilibrium point is asymptotically stable local when it is R0 < 1. Then, the disease-endemic equilibrium point is asymptotically stable local when it is R0 > 1. Furthermore, the parameters of β, ρ, and γ are the most important parameter.

THE EFFECTS OF VACCINATION AND TREATMENT ON THE SPREAD OF HIV/AIDS

2004 ◽  
Vol 12 (04) ◽  
pp. 399-417 ◽  
Author(s):  
M. KGOSIMORE ◽  
E. M. LUNGU
Keyword(s):  
Equilibrium Point ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Treatment Programs ◽  
Asymptotically Stable ◽  
Reproduction Numbers ◽  
Threshold Conditions ◽  
Existence And Stability ◽  
Disease Free ◽  
Hiv Aids

This study investigates the effects of vaccination and treatment on the spread of HIV/AIDS. The objectives are (i) to derive conditions for the success of vaccination and treatment programs and (ii) to derive threshold conditions for the existence and stability of equilibria in terms of the effective reproduction number R. It is found, firstly, that the success of a vaccination and treatment program is achieved when R0t<R0, R0t<R0v and γeRVT(σ)<RUT(α), where R0t and R0v are respectively the reproduction numbers for populations consisting entirely of treated and vaccinated individuals, R0 is the basic reproduction number in the absence of any intervention, RUT(α) and RVT(σ) are respectively the reproduction numbers in the presence of a treatment (α) and a combination of vaccination and treatment (σ) strategies. Secondly, that if R<1, there exists a unique disease free equilibrium point which is locally asymptotically stable, while if R>1 there exists a unique locally asymptotically stable endemic equilibrium point, and that the two equilibrium points coalesce at R=1. Lastly, it is concluded heuristically that the stable disease free equilibrium point exists when the conditions R0t<R0, R0t<R0v and γeRVT(σ)<RUT(α) are satisfied.

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

scholarly journals Analisis Model Matematika Penyebaran Demam Berdarah Dengue dengan Fungsi Lyapunov

2019 ◽  
Vol 1 (2) ◽  
pp. 125
Author(s):  
Syafruddin Side ◽  
Ahmad Zaki ◽  
Nurwahidah Sari
Keyword(s):  
Lyapunov Function ◽  
Equilibrium Point ◽  
Dengue Fever ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Secondary Data ◽  
Hemorrhagic Fever ◽  
Endemic Equilibrium ◽  
Sirs Model ◽  
Disease Free

Abstrak. Artike lini adalah penelitian teori dan terapan. Artikelini bertujuan untuk membahas mengenai model matematika SIRS untuk penyebaran Demam Berdarah Dengue. Data yang digunakanadalah data sekunder jumlah penderita penyakit Demam Berdarah Dengue dari Side pada tahun 2014. Pembahasan di mulai dari membangun model matematika SIRS penyakit Demam Berdarah Dengue, menentukan eksistensi model SIRS menggunakan fungsi Lyapunov, penentuan titik ekuilibrium, kemudian mencari analisis kestabilan titik ekuilibrium menggunakan fungsi Lyapunov, menentukan nilai bilangan reproduksi dasar , membuat simulasi model, dan menginterpretasikannya. Dalam artikel ini diperoleh model matematika SIRS untuk penyakit Demam Berdarah Dengue, eksistensi model SIRS, dua titik ekuilibrium bebas penyakit dan endemik dari model SIRS, kestabilan global keseimbangan bebas penyakit dan endemik dari model SIRS dengan nilai bilangan reproduksi dasar , ini menunjukkan bahwa penyakit Demam Berdarah Dengue berstatus epidemik.Kata Kunci: Model Matematika, Penyebaran Penyakit, Demam Berdarah Dengue, Model  SIRS, Fungsi LyapunovAbstract. This paper is theorethical and applied research. This paper aims to discus about SIRS mathematical models for the spread of dengue fever. The data used is a secondary data about the number of people with dengue fever disease from Side (2014). The discussion start from constructing SIRS models of dengue fever disease, determining the existence of SIRS models using Lyapunov function, determining equilibrium point, then looking for stability analysis of equilibrium point using Lyapunov function, determining reproduction number , making models simulation, and interpreting it. In this paper, we obtained mathemathical models of SIRS for dengue fever disease, existence of SIRS models, disease-free and endemic equilibrium points of SIRS models, global stability of disease-free and endemic equilibrium of SIRS models with basic reproduction number , it shows that dengue fever disease is epidemic status. , This shows that Dengue Hemorrhagic Fever is an epidemic.Keyword: Mathematical Model, Spread of Disease, Dengue Fever, SIRS Model, Lyapunov Function

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 Drug-sensitivity and passive immunity mathematical epidemiological model for tuberculosis

2021 ◽  
Vol 25 (9) ◽  
pp. 1661-1670
Author(s):  
A.A. Danhausa ◽  
E.E. Daniel ◽  
C.J. Shawulu ◽  
A.M. Nuhu ◽  
L. Philemon
Keyword(s):  
Control Strategy ◽  
Drug Sensitivity ◽  
Reproduction Number ◽  
Passive Immunity ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Latently Infected ◽  
Widespread Availability ◽  
Next Generation Matrix ◽  
Disease Free

Regardless of many decades of research, the widespread availability of a vaccine and more recently highly visible WHO efforts to promote a unified global control strategy, Tuberculosis remains a leading cause of infectious mortality. In this paper, a Mathematical Model for Tuberculosis Epidemic with Passive Immunity and Drug-Sensitivity is presented. We carried out analytical studies of the model where the population comprises of eight compartments: passively immune infants, susceptible, latently infected with DS-TB. The Disease Free Equilibrium (DFE) and the Endemic Equilibrium (EE) points were established. The next generation matrix method was used to obtain the reproduction number for drug sensitive (𝑅𝑜𝑠) Tuberculosis. We obtained the disease-free equilibrium for drug sensitive TB which is locally asymptotically stable when 𝑅𝑜𝑠 < 1 indicating that tuberculosis eradication is possible within the population. We also obtained the global stability of the disease-free equilibrium and results showed that the disease-free equilibrium point is globally asymptotically stable when 𝑅𝑜𝑠 ≤ 1 which indicates that tuberculosis naturally dies out.

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.

Global stability and vaccination of an SEIVR epidemic model with saturated incidence rate

2016 ◽  
Vol 09 (05) ◽  
pp. 1650068 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Yasir Khan ◽  
Sehra Khan ◽  
Saeed Islam
Keyword(s):  
Global Stability ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Disease Free ◽  
The Basic Reproduction Number

This study considers SEIVR epidemic model with generalized nonlinear saturated incidence rate in the host population horizontally to estimate local and global equilibriums. By using the Routh–Hurwitz criteria, it is shown that if the basic reproduction number [Formula: see text], the disease-free equilibrium is locally asymptotically stable. When the basic reproduction number exceeds the unity, then the endemic equilibrium exists and is stable locally asymptotically. The system is globally asymptotically stable about the disease-free equilibrium if [Formula: see text]. The geometric approach is used to present the global stability of the endemic equilibrium. For [Formula: see text], the endemic equilibrium is stable globally asymptotically. Finally, the numerical results are presented to justify the mathematical results.

scholarly journals Stability Analysis of SEIL Tuberculosis Epidemic Model with Logistic Growth in Susceptible Compartment

2021 ◽  
Vol 16 ◽  
pp. 1-9
Author(s):  
Joko Harianto
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Asymptotically Stable ◽  
Tuberculosis Disease ◽  
Disease Free ◽  
The Basic Reproduction Number

This article discusses modifications to the SEIL model that involve logistical growth. This model is used to describe the dynamics of the spread of tuberculosis disease in the population. The existence of the model's equilibrium points and its local stability depends on the basic reproduction number. If the basic reproduction number is less than unity, then there is one equilibrium point that is locally asymptotically stable. The equilibrium point is a disease-free equilibrium point. If the basic reproduction number ranges from one to three, then there are two equilibrium points. The two equilibrium points are disease-free equilibrium and endemic equilibrium points. Furthermore, for this case, the endemic equilibrium point is locally asymptotically stable.

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