scholarly journals Modelling the effect of vertical transmission in the dynamics of HIV/AIDS in an age-structured population

2003 ◽  
Vol 21 (1) ◽  
pp. 82 ◽  
Author(s):  
J. Y. T. Mugisha ◽  
L. S. Luboobi
Keyword(s):  
Vertical Transmission ◽  
Equilibrium Points ◽  
Age Groups ◽  
Endemic Equilibrium ◽  
Mass Action ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
First Case ◽  
Age Structured ◽  
Hiv Aids

We use a continuous age-structured model of McKendrick-von-Foerster type to derive a two-age groups HIV/AIDS epidemic model. In the analysis of the model, keen interest is put on the role of vertical transmission in the dynamics of the spread of the epidemic. The model is analysed in two scenarios: the case when the force of infection is a constant and the case when we have it as a mass action. In the first case, the only possible equilibrium is the endemic equilibrium. In this situation, we show that if all babies born to infected mothers are HIV-free we have the basic reproductive number R0 = 0 and as such the epidemic will die out. In the second case, we show that both the disease-free and endemic equilibrium points exist. We also derive conditions for their stability.

scholarly journals Stability Analysis and Clinic Phenomenon Simulation of a Fractional-Order HBV Infection Model

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Yongmei Su ◽  
Sinuo Liu ◽  
Shurui Song ◽  
Xiaoke Li ◽  
Yongan Ye
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Mass Action ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Infection Model ◽  
Order Model ◽  
Infected Cells ◽  
Set Up ◽  
Hbv Model

In this paper, a fractional-order HBV model was set up based on standard mass action incidences and quasisteady assumption. The basic reproductive number R0 and the cytotoxic T lymphocytes’ immune-response reproductive number R1 were derived. There were three equilibrium points of the model, and stable analysis of each equilibrium point was given with corresponding hypothesis about R0 or R1. Some numerical simulations were also given based on HBeAg clinical data, and the simulation showed that there existed positive logarithmic correlation between the number of infected cells and HBeAg, which was consistent with the clinical facts. The simulation also showed that the clinical individual differences should be reflected by the fractional-order model.

scholarly journals An SEIRS Epidemic Model with Immigration and Vertical Transmission

2020 ◽  
pp. 48-53
Author(s):  
Ruksana Shaikh ◽  
Pradeep Porwal ◽  
V. K. Gupta
Keyword(s):  
Vertical Transmission ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Seirs Epidemic Model ◽  
Model Equations ◽  
The Stability ◽  
Disease Free

The study indicates that we should improve the model by introducing the immigration rate in the model to control the spread of disease. An SEIRS epidemic model with Immigration and Vertical Transmission and analyzed the steady state and stability of the equilibrium points. The model equations were solved analytically. The stability of the both equilibrium are proved by Routh-Hurwitz criteria. We see that if the basic reproductive number R0<1 then the disease free equilibrium is locally asymptotically stable and if R0<1 the endemic equilibrium will be locally asymptotically stable.

scholarly journals SEIPR-Mathematical Model of the Pneumonia Spreading in Toddlers with Immunization and Treatment Effects

2020 ◽  
Vol 17 (2) ◽  
pp. 202-218
Author(s):  
Rusniwati S. Imran ◽  
Resmawan Resmawan ◽  
Novianita Achmad ◽  
Agusyarif Rezka Nuha
Keyword(s):  
Mathematical Model ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Treatment Success ◽  
Stable State ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Disease Spread ◽  
Reproductive Number

This research discussed the SEIPR mathematical model on the spread of pneumonia among children under five years old. The development of the model was done by considering factors of immunization and treatment factors, in an effort to reduce the rate of spread of pneumonia. In this research, mathematical model construction, stability analysis, and numerical simulation were carried out to see the dynamics of pneumonia cases in the population. The model analysis produces two equilibrium points, which are the equilibrium point without the disease, the endemic equilibrium point, and the basic reproduction number ( ) as the threshold value for disease spread. The point of equilibrium without disease reaches a stable state at the moment , which indicates that pneumonia will disappear from the population, while the endemic equilibrium point reaches a stable state at that time , which indicates that the disease will spread in the population. Furthermore, numerical simulations show that increasing the rate parameters of infected individuals undergoing treatment ( ), the treatment success rate ( ), and the immunization proportion ( ), could suppress the basic reproductive number so that control of the disease spread rate can be accelerated.

scholarly journals Modelling and Simulating a Transmission of COVID-19 Disease: Niger Republic Case

2020 ◽  
Vol 13 (3) ◽  
pp. 549-566
Author(s):  
Abba Mahamane Oumarou ◽  
Saley Bisso
Keyword(s):  
Matrix Method ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Contact Rate ◽  
Asymptotically Stable ◽  
Next Generation Matrix ◽  
The Impact ◽  
Disease Free

This paper focuses on the dynamics of spreads of a coronavirus disease (Covid-19).Through this paper, we study the impact of a contact rate in the transmission of the disease. We determine the basic reproductive number R0, by using the next generation matrix method. We also determine the Disease Free Equilibrium and Endemic Equilibrium points of our model. We prove that the Disease Free Equilibrium is asymptotically stable if R0 < 1 and unstable if R0 > 1. The asymptotical stability of Endemic Equilibrium is also establish. Numerical simulations are made to show the impact of contact rate in the spread of disease.

scholarly journals Lockdown Measures and their Impact on Single- and Two-age-structured Epidemic Model for the COVID-19 Outbreak in Mexico

2020 ◽  
Author(s):  
Jesús Cuevas-Maraver ◽  
Panayotis Kevrekidis ◽  
Qian-Yong Chen ◽  
George Kevrekidis ◽  
Víctor Villalobos-Daniel ◽  
...  
Keyword(s):  
Time Series ◽  
Age Groups ◽  
Optimization Procedure ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Ode Model ◽  
Age Structured ◽  
And Behavior ◽  
The Impact

The role of lockdown measures in mitigating COVID-19 in Mexico is investigated using a comprehensive nonlinear ODE model. The model includes both asymptomatic and presymptomatic populations with the latter leading to sickness (with recovery, hospitalization and death possibilities). We consider the situation involving the imposed application of partial social distancing measures in the time series of interest and find optimal parametric fits to the time series of deaths (only), as well as to that of deaths and cumulative infections. We discuss the merits and disadvantages of each approach, we interpret the parameters of the model and assess the realistic nature of the parameters resulting from the optimization procedure. Importantly, we explore a model involving two sub-populations (younger and older than a specific age), to more accurately reflect the observed impact as concerns symptoms and behavior in different age groups. For definitiveness and to separate people that are (typically) in the active workforce, our partition of population is with respect to members younger vs. older than the age of 65. The basic reproductive number of the model is computed for both the single- and the two-population variant. Finally, we consider what would be the impact on the number of deaths and cumulative infections upon imposition of partial lockdown (involving only the older population) and full lockdown (involving the entire population).

Modeling and stability analysis of the spread of novel coronavirus disease COVID-19

2021 ◽  
pp. 2150035
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
D. Abraham Vianny ◽  
Mary Jacintha ◽  
Fatma Bozkurt Yousef
Keyword(s):  
Fractional Order ◽  
Disease Transmission ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Basic Reproductive Number ◽  
Phase Portraits ◽  
Transmission Model ◽  
Reproductive Number ◽  
Discrete Version ◽  
Novel Coronavirus

Towards the end of 2019, the world witnessed the outbreak of Severe Acute Respiratory Syndrome Coronavirus-2 (COVID-19), a new strain of coronavirus that was unidentified in humans previously. In this paper, a new fractional-order Susceptible–Exposed–Infected–Hospitalized–Recovered (SEIHR) model is formulated for COVID-19, where the population is infected due to human transmission. The fractional-order discrete version of the model is obtained by the process of discretization and the basic reproductive number is calculated with the next-generation matrix approach. All equilibrium points related to the disease transmission model are then computed. Further, sufficient conditions to investigate all possible equilibria of the model are established in terms of the basic reproduction number (local stability) and are supported with time series, phase portraits and bifurcation diagrams. Finally, numerical simulations are provided to demonstrate the theoretical findings.

Global properties for an age-structured within-host model with Crowley–Martin functional response

2017 ◽  
Vol 10 (02) ◽  
pp. 1750030 ◽  
Author(s):  
Shaoli Wang ◽  
Xinyu Song
Keyword(s):  
Functional Response ◽  
Viral Infections ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Multi Scale ◽  
Global Properties ◽  
Age Structured

Based on a multi-scale view, in this paper, we study an age-structured within-host model with Crowley–Martin functional response for the control of viral infections. By means of semigroup and Lyapunov function, the global asymptotical property of infected steady state of the model is obtained. The results show that when the basic reproductive number falls below unity, the infection dies out. However, when the basic reproductive number exceeds unity, there exists a unique positive equilibrium which is globally asymptotically stable. This model can be deduced to different viral models with or without time delay.

scholarly journals Stability and Hopf Bifurcation of a Vector-Borne Disease Model with Saturated Infection Rate and Reinfection

2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Zhixing Hu ◽  
Shanshan Yin ◽  
Hui Wang
Keyword(s):  
Hopf Bifurcation ◽  
Infection Rate ◽  
Disease Model ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Cure Rate ◽  
Vector Borne Disease ◽  
Vector Borne ◽  
The Stability

This paper established a delayed vector-borne disease model with saturated infection rate and cure rate. First of all, according to the basic reproductive number R0, we determined the disease-free equilibrium E0 and the endemic equilibrium E1. Through the analysis of the characteristic equation, we consider the stability of two equilibriums. Furthermore, the effect on the stability of the endemic equilibrium E1 by delay was studied, the existence of Hopf bifurcations of this system in E1 was analyzed, and the length of delay to preserve stability was estimated. The direction and stability of the Hopf bifurcation were also been determined. Finally, we performed some numerical simulation to illustrate our main results.

scholarly journals Optimal Control of HIV/AIDS in the Workplace in the Presence of Careless Individuals

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Baba Seidu ◽  
Oluwole D. Makinde
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Strategies ◽  
Nonlinear Dynamical System ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Nonlinear Dynamical ◽  
Basic Model ◽  
Hiv Aids

A nonlinear dynamical system is proposed and qualitatively analyzed to study the dynamics of HIV/AIDS in the workplace. The disease-free equilibrium point of the model is shown to be locally asymptotically stable if the basic reproductive number,R0, is less than unity and the model is shown to exhibit a unique endemic equilibrium when the basic reproductive number is greater than unity. It is shown that, in the absence of recruitment of infectives, the disease is eradicated whenR0<1, whiles the disease is shown to persist in the presence of recruitment of infected persons. The basic model is extended to include control efforts aimed at reducing infection, irresponsibility, and nonproductivity at the workplace. This leads to an optimal control problem which is qualitatively analyzed using Pontryagin’s Maximum Principle (PMP). Numerical simulation of the resulting optimal control problem is carried out to gain quantitative insights into the implications of the model. The simulation reveals that a multifaceted approach to the fight against the disease is more effective than single control strategies.

A MODEL FOR VECTOR TRANSMITTED DISEASES WITH SATURATION INCIDENCE

2001 ◽  
Vol 09 (04) ◽  
pp. 235-245 ◽  
Author(s):  
LOURDES ESTEVA ◽  
MARIANO MATIAS
Keyword(s):  
Endemic Equilibrium ◽  
Saturation Effect ◽  
Basic Reproductive Number ◽  
Infected Host ◽  
Reproductive Number ◽  
The Stability ◽  
Stability Results ◽  
Disease Free

A model for a disease that is transmitted by vectors is formulated. All newborns are assumed susceptible, and human and vector populations are assumed to be constant. The model assumes a saturation effect in the incidences due to the response of the vector to change in the susceptible and infected host densities. Stability of the disease free equilibrium and existence, uniqueness and stability of the endemic equilibrium is investigated. The stability results are given in terms of the basic reproductive number R0.

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