scholarly journals Stability Analysis of HIV/AIDS Model with Educated Subpopulation

CAUCHY ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 188-199
Author(s):  
Ummu Habibah
Keyword(s):  
Disease Transmission ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Hiv Positive ◽  
Asymptotically Stable ◽  
Transmission Coefficients ◽  
Two Stages ◽  
Susceptible Subpopulations ◽  
Disease Free ◽  
Hiv Aids

We had constructed mathematical model of HIV/AIDS with seven compartments. There were two different stages of infection and susceptible subpopulations. Two stages in infection subpopulation were an HIV-positive with consuming ARV such that this subpopulation can survive longer and an HIV-positive not consuming ARV.  The susceptible subpopulation was divided into two, uneducated and educated susceptible subpopulations.  The transmission coefficients from educated and uneducated subpopulations to infection stages were  where  ((  and ) (  and )) In this paper, we consider the case of  and  were zero.  We investigated local stability of the model solutions according to the basic reproduction number as a threshold of disease transmission. The disease-free and endemic equilibrium points were locally asymptotically stable when  and  respectively. To support the analytical results, numerical simulation was conducted.

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 HIV/AIDS-Pneumonia Coinfection Model with Treatment at Each Infection Stage: Mathematical Analysis and Numerical Simulation

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Shewafera Wondimagegnhu Teklu ◽  
Temesgen Tibebu Mekonnen
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Transmission Rates ◽  
Reproduction Numbers ◽  
Disease Free ◽  
The Basic Reproduction Number ◽  
Hiv Aids

In the paper, we have considered a nonlinear compartmental mathematical model that assesses the effect of treatment on the dynamics of HIV/AIDS and pneumonia coinfection in a human population at different infection stages. Our model revealed that the disease-free equilibrium points of the HIV/AIDS and pneumonia submodels are both locally and globally asymptotically stable whenever the associated basic reproduction numbers ( R H and R P ) are less than unity. Both the submodel endemic equilibrium points are locally and globally asymptotically stable whenever the associated basic reproduction numbers ( R P and R H ) are greater than unity. The full HIV/AIDS-pneumonia coinfection model has both locally and globally asymptotically stable disease-free equilibrium points whenever the basic reproduction number of the coinfection model R H P is less than unity. Using standard values of parameters collected from different kinds of literature, we found that the numerical values of the basic reproduction numbers of the HIV/AIDS-only submodel and pneumonia-only submodel are 17 and 7, respectively, and the basic reproduction number of the HIV/AIDS-pneumonia coinfection model is max 7 , 17 = 17 . Applying sensitive analysis, we identified the most influential parameters to change the behavior of the solution of the considered coinfection dynamical system are the HIV/AIDS and pneumonia transmission rates β 1 and β 2 , respectively. The coinfection model was numerically simulated to investigate the stability of the coinfection endemic equilibrium point, the impacts of transmission rates, and treatment strategies for HIV/AIDS-only, pneumonia-only, and HIV/AIDS-pneumonia coinfected individuals. Finally, we observed that numerical simulations indicate that treatment against infection at every stage lowers the rate of infection or disease prevalence.

scholarly journals On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation

2020 ◽  
Vol 10 (22) ◽  
pp. 8296 ◽  
Author(s):  
Malen Etxeberria-Etxaniz ◽  
Santiago Alonso-Quesada ◽  
Manuel De la Sen
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Disease Transmission ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Impulsive Vaccination ◽  
Disease Free ◽  
The Basic Reproduction Number

This paper investigates a susceptible-exposed-infectious-recovered (SEIR) epidemic model with demography under two vaccination effort strategies. Firstly, the model is investigated under vaccination of newborns, which is fact in a direct action on the recruitment level of the model. Secondly, it is investigated under a periodic impulsive vaccination on the susceptible in the sense that the vaccination impulses are concentrated in practice in very short time intervals around a set of impulsive time instants subject to constant inter-vaccination periods. Both strategies can be adapted, if desired, to the time-varying levels of susceptible in the sense that the control efforts be increased as those susceptible levels increase. The model is discussed in terms of suitable properties like the positivity of the solutions, the existence and allocation of equilibrium points, and stability concerns related to the values of the basic reproduction number. It is proven that the basic reproduction number lies below unity, so that the disease-free equilibrium point is asymptotically stable for larger values of the disease transmission rates under vaccination controls compared to the case of absence of vaccination. It is also proven that the endemic equilibrium point is not reachable if the disease-free one is stable and that the disease-free equilibrium point is unstable if the reproduction number exceeds unity while the endemic equilibrium point is stable. Several numerical results are investigated for both vaccination rules with the option of adapting through ime the corresponding efforts to the levels of susceptibility. Such simulation examples are performed under parameterizations related to the current SARS-COVID 19 pandemic.

scholarly journals Stability of an HIV/AIDS Treatment Model with Different Stages

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Hai-Feng Huo ◽  
Rui Chen
Keyword(s):  
Hiv Infection ◽  
Hiv Positive ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Treatment Model ◽  
Globally Asymptotically Stable ◽  
Asymptomatic Stage ◽  
Mathematical Analyses ◽  
Two Stages ◽  
Hiv Aids

An HIV/AIDS treatment model with different stages is proposed in this paper. The stage of the HIV infection is divided into two stages, that is, HIV-positive in the asymptomatic stage of HIV infection and HIV-positive individuals in the pre-AIDS stage. The fact that some individuals with HIV-positive individuals after the treatment can be transformed into the compartment of HIV-positive individuals in the asymptomatic stage of HIV infection, the compartment of HIV-positive individuals in the pre-AIDS stage, or the compartment of individuals with full-blown AIDS is also considered. Mathematical analyses establish the idea that the global dynamics of the HIV/AIDS model are determined by the basic reproduction numberR0. The disease-free equilibrium is globally asymptotically stable ifR0<1. The endemic equilibrium is globally asymptotically stable ifR0>1for a special case. Numerical simulations are also conducted to support the analytic results.

scholarly journals Complex mathematical SIR model for spreading of COVID-19 virus with Mittag-Leffler kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
F. Talay Akyildiz ◽  
Fehaid Salem Alshammari
Keyword(s):  
Fractional Order ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Sir Model ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Exponential Kernel ◽  
Disease Free ◽  
The Basic Reproduction Number

AbstractThis paper investigates a new model on coronavirus-19 disease (COVID-19), that is complex fractional SIR epidemic model with a nonstandard nonlinear incidence rate and a recovery, where derivative operator with Mittag-Leffler kernel in the Caputo sense (ABC). The model has two equilibrium points when the basic reproduction number $R_{0} > 1$ R 0 > 1 ; a disease-free equilibrium $E_{0}$ E 0 and a disease endemic equilibrium $E_{1}$ E 1 . The disease-free equilibrium stage is locally and globally asymptotically stable when the basic reproduction number $R_{0} <1$ R 0 < 1 , we show that the endemic equilibrium state is locally asymptotically stable if $R_{0} > 1$ R 0 > 1 . We also prove the existence and uniqueness of the solution for the Atangana–Baleanu SIR model by using a fixed-point method. Since the Atangana–Baleanu fractional derivative gives better precise results to the derivative with exponential kernel because of having fractional order, hence, it is a generalized form of the derivative with exponential kernel. The numerical simulations are explored for various values of the fractional order. Finally, the effect of the ABC fractional-order derivative on suspected and infected individuals carefully is examined and compared with the real data.

scholarly journals Mathematical Model for MERS-COV Disease Transmission with Medical Mask Usage and Vaccination

2019 ◽  
Vol 1 (2) ◽  
Author(s):  
Muhammad Manaqib ◽  
Irma Fauziah ◽  
Mujiyanti Mujiyanti
Keyword(s):  
Equilibrium Point ◽  
Disease Transmission ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Local Equilibrium ◽  
Preventive Measure ◽  
Simulation Models ◽  
Seir Model ◽  
Model Stability ◽  
Disease Free

AbstractThis study developed a model of the spread of MERS-CoV disease using the SEIR model which was added by a health mask and vaccination factor as a preventive measure. The population is divided into six subpopulations namely susceptible subpopulations not using health masks and using health masks, exposed subpopulations, infected subpopulations not using health masks and using health masks, and recovering subpopulations. The results are obtained two equilibrium points, namely disease-free equilibrium points and endemic equilibrium points. Analysis of the stability of the disease-free equilibrium point using linearization around the equilibrium point. As a result, the asymptotic stable disease-free local equilibrium point if the base reproduction number is less than one. Numerical simulation models for MERS-CoV disease are carried out in line with the analysis of model behavior.Keywords: MERS-CoV, SEIR Model, Stability Equilibrium Point, Basic Reproduction Number. AbstrakPenelitian ini mengembangkan model penyebaran penyakit MERS-CoV menggunakan model SEIR yang ditambahkan faktor masker kesehatan dan vaksinasi sebagai upaya pencegahan. Populasi dibagi menjadi enam subpopulasi yaitu subpopulasi rentan tidak menggunakan masker kesehatan dan menggunakan masker kesehatan, subpopulasi laten, subpopulasi terinfeksi tidak menggunakan masker kesehatan dan menggunakan masker kesehatan, serta subpopulasi sembuh. Hasilnya diperoleh dua titik ekuilibrium yaitu titik ekulibrium bebas penyakit dan endemik. Analisis kestabilan titik ekuilibrium bebas penyakit menggunakan linearisasi disekitar titik ekuilibrium. Hasilnya, titik ekuilibrium bebas penyakit stabil asimtotik lokal jika bilangan reproduksi dasar kurang dari satu. Simulasi numerik model untuk penyakit MERS-CoV yang dilakukan sejalan dengan analisis perilaku model.Kata kunci: MERS-CoV, Model SEIR, Kestabilan Titik Ekuilibrium, Bilangan Reproduksi Dasar.

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.

scholarly journals Kestabilan Model Epidemi Seir dengan Matriks Hurwitz

2016 ◽  
Vol 11 (2) ◽  
pp. 74
Author(s):  
Roni Tri Putra ◽  
Sukatik - ◽  
Sri Nita
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Local Stability ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Stability Of Equilibrium ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, it will be studied local stability of equilibrium points of  a SEIR epidemic model with infectious force in latent, infected and immune period. From the model it will be found investigated the existence and its stability of points its equilibrium by Hurwitz matrices. The local stability of equilibrium points is depending on the value of the basic reproduction number  If   the disease free equilibrium is local asymptotically stable.

scholarly journals Dynamics of COVID-19 Epidemic Model with Asymptomatic Infection, Quarantine, Protection and Vaccination

2021 ◽  
Vol 4 (2) ◽  
pp. 106-124
Author(s):  
Raqqasyi Rahmatullah Musafir ◽  
Agus Suryanto ◽  
Isnani Darti
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Asymptomatic Infection ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Proposed Model ◽  
Disease Free ◽  
The Basic Reproduction Number

We discuss the dynamics of new COVID-19 epidemic model by considering asymptomatic infections and the policies such as quarantine, protection (adherence to health protocols), and vaccination. The proposed model contains nine subpopulations: susceptible (S), exposed (E), symptomatic infected (I), asymptomatic infected (A), recovered (R), death (D), protected (P), quarantined (Q), and vaccinated (V ). We first show the non-negativity and boundedness of solutions. The equilibrium points, basic reproduction number, and stability of equilibrium points, both locally and globally, are also investigated analytically. The proposed model has disease-free equilibrium point and endemic equilibrium point. The disease-free equilibrium point always exists and is globally asymptotically stable if basic reproduction number is less than one. The endemic equilibrium point exists uniquely and is globally asymptotically stable if the basic reproduction number is greater than one. These properties have been confirmed by numerical simulations using the fourth order Runge-Kutta method. Numerical simulations show that the disease transmission rate of asymptomatic infection, quarantine rates, protection rate, and vaccination rates affect the basic reproduction number and hence also influence the stability of equilibrium points.

scholarly journals Analisis dinamik model SVEIR pada penyebaran penyakit campak

2020 ◽  
Vol 1 (2) ◽  
pp. 57-64
Author(s):  
Sitty Oriza Sativa Putri Ahaya ◽  
Emli Rahmi ◽  
Nurwan Nurwan
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Total Population ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Transmission Model ◽  
Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

In this article, we analyze the dynamics of measles transmission model with vaccination via an SVEIR epidemic model. The total population is divided into five compartments, namely the Susceptible, Vaccinated, Exposed, Infected, and Recovered populations. Firstly, we determine the equilibrium points and their local asymptotically stability properties presented by the basic reproduction number R0. It is found that the disease free equilibrium point is locally asymptotically stable if satisfies R01 and the endemic equilibrium point is locally asymptotically stable when R01. We also show the existence of forward bifurcation driven by some parameters that influence the basic reproduction number R0 i.e., the infection rate α or proportion of vaccinated individuals θ. Lastly, some numerical simulations are performed to support our analytical results.

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