scholarly journals Analysis of an Age-structured SIL model with demographics process and vertical transmission

2014 ◽  
Vol Volume 17 - 2014 - Special... ◽  
Author(s):  
Ramses Djidjou Demasse ◽  
Jean Jules Tewa ◽  
Samuel Bowong
Keyword(s):  
Vertical Transmission ◽  
Steady States ◽  
Asymptotically Stable ◽  
Basic Reproduction Ratio ◽  
Reproduction Ratio ◽  
Structured Population ◽  
Semigroup Approach ◽  
International Audience ◽  
Age Structured ◽  
Disease Free

International audience We consider a mathematical SIL model for the spread of a directly transmitted infectious disease in an age-structured population; taking into account the demographic process and the vertical transmission of the disease. First we establish the mathematical well-posedness of the time evolution problem by using the semigroup approach. Next we prove that the basic reproduction ratio R0 is given as the spectral radius of a positive operator, and an endemic state exist if and only if the basic reproduction ratio R0 is greater than unity, while the disease-free equilibrium is locally asymptotically stable if R0<1. We also show that the endemic steady states are forwardly bifurcated from the disease-free steady state when R0 cross the unity. Finally we examine the conditions for the local stability of the endemic steady states.

scholarly journals A transmission model of Bilharzia : A mathematical analysis of an heterogeneous model

2011 ◽  
Vol Volume 14 - 2011 - Special... ◽  
Author(s):  
Riveau Gilles ◽  
Sallet Gauthier ◽  
Tendeng Lena
Keyword(s):  
Real Data ◽  
Transmission Model ◽  
Asymptotically Stable ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Heterogeneous Model ◽  
Nous Montrons ◽  
International Audience ◽  
Disease Free

International audience We consider an heterogeneous model of transmission of bilharzia. We compute the basic reproduction ratio R 0. We prove that if R 0 < 1, then the disease free equilibrium is globally asymptotically stable. If R 0 > 1 then there exists an unique endemic equilibrium, which is globally asymptotically stable. We will then consider possible applications to real data On considère un modèle de transmission de la bilharziose prenant en compte les hétérogénéités. Nous calculons le taux de reproduction de base Nous montrons que si R0 < 1, alors l’équilibre sans maladie est globalement asymptotiquement stable. Si R0 > 1, alors il existe un unique équilibre endémique et celui-ci est globalement asymptotiquement stable. Nous considérons ensuite les applications possibles à des données réelles.

scholarly journals Mathematical Analysis of a General Two-Patch Model of Tuberculosis Disease with Lost Sight Individuals

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Abdias Laohombé ◽  
Isabelle Ngningone Eya ◽  
Jean Jules Tewa ◽  
Alassane Bah ◽  
Samuel Bowong ◽  
...  
Keyword(s):  
Sub Saharan Africa ◽  
Asymptotically Stable ◽  
Basic Reproduction Ratio ◽  
Reproduction Ratio ◽  
Patch Model ◽  
Latently Infected ◽  
Sub Saharan ◽  
Tuberculosis Disease ◽  
Theoretical Results ◽  
Disease Free

A two-patch model,SEi1,…,EinIiLi,  i=1,2, is used to analyze the spread of tuberculosis, with an arbitrary numbernof latently infected compartments in each patch. A fraction of infectious individuals that begun their treatment will not return to the hospital for the examination of sputum. This fact usually occurs in sub-Saharan Africa, due to many reasons. The model incorporates migrations from one patch to another. The existence and uniqueness of the associated equilibria are discussed. A Lyapunov function is used to show that when the basic reproduction ratio is less than one, the disease-free equilibrium is globally and asymptotically stable. When it is greater than one, there exists at least one endemic equilibrium. The local stability of endemic equilibria can be illustrated using numerical simulations. Numerical simulation results are provided to illustrate the theoretical results and analyze the influence of lost sight individuals.

scholarly journals STABILITY ANALYSIS OF DISEASE FREE EQUILIBRIUM STATE (E0) FOR AN ENDEMIC DETERMINISTIC MODEL FOR HIV/AIDS

2020 ◽  
Author(s):  
A. O. Victor ◽  
H. K. Oduwole
Keyword(s):  
Equilibrium State ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Asymptotically Stable ◽  
Treatment Rate ◽  
Structured Population ◽  
Threshold Conditions ◽  
Age Structured ◽  
Disease Free

AbstractThis paper focused on determining the asymptotical stability of the new model which establishes that a disease-free equilibrium state exists and is locally asymptotically stable when the basic reproduction number 0 ≤ R0 < 1 and the following threshold conditions (0 ≤ R1 < 1, 0 ≤ R2 < 1, 0 ≤ R3 < 1, 0 ≤ R4 < 1 and 0 ≤ R5 < 1, 0 ≤ R6 < 1, 0 ≤ R7 < 1, 0 ≤ R8 < 1) are satisfied. Results from the model analysis shows that the proportion of infected juvenile and adult sub-population in the presence of High Active Anteritroviral Therapy (HAART) drastically reduce to a zero (R0 = 0) as compared to the infected age-structured population when treatment rate is low and the net transmission rate is near zero.

GLOBAL DYNAMICS OF A SEASONAL MATHEMATICAL MODEL OF SCHISTOSOMIASIS TRANSMISSION WITH GENERAL INCIDENCE FUNCTION

2019 ◽  
Vol 27 (01) ◽  
pp. 19-49 ◽  
Author(s):  
BAKARY TRAORÉ ◽  
OUSMANE KOUTOU ◽  
BOUREIMA SANGARÉ
Keyword(s):  
Global Dynamics ◽  
Asymptotically Stable ◽  
Basic Reproduction Ratio ◽  
Periodic Functions ◽  
Rigorous Analysis ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Incidence Function ◽  
Theoretical Results ◽  
Disease Free

In this paper, we investigate a nonautonomous and an autonomous model of schistosomiasis transmission with a general incidence function. Firstly, we formulate the nonautonomous model by taking into account the effect of climate change on the transmission. Through rigorous analysis via theories and methods of dynamical systems, we show that the nonautonomous model has a globally asymptotically stable disease-free periodic equilibrium when the associated basic reproduction ratio [Formula: see text] is less than unity. Otherwise, the system admits at least one positive periodic solutions if [Formula: see text] is greater than unity. Secondly, using the average of periodic functions, we further derive the autonomous model associated with the nonautonomous model. Therefore, we show that the disease-free equilibrium of the autonomous model is locally and globally asymptotically stable when the associated reproduction ratio [Formula: see text] is less than unity. When [Formula: see text] is greater than unity, the existence and global asymptotic stability of the endemic equilibrium is established under certain conditions. Finally, using linear and nonlinear specific incidence function, we perform some numerical simulations to illustrate our theoretical results.

scholarly journals The Global Behavior of a Periodic Epidemic Model with Travel between Patches

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Luosheng Wen ◽  
Bin Long ◽  
Xin Liang ◽  
Fengling Zeng
Keyword(s):  
Epidemic Model ◽  
Transmission Rate ◽  
Global Behavior ◽  
Uniform Persistence ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Periodic Transmission ◽  
Theoretical Results ◽  
Disease Free

We establish an SIS (susceptible-infected-susceptible) epidemic model, in which the travel between patches and the periodic transmission rate are considered. As an example, the global behavior of the model with two patches is investigated. We present the expression of basic reproduction ratioR0and two theorems on the global behavior: ifR0< 1 the disease-free periodic solution is globally asymptotically stable and ifR0> 1, then it is unstable; ifR0> 1, the disease is uniform persistence. Finally, two numerical examples are given to clarify the theoretical results.

GLOBAL DYNAMICS OF A DELAYED HIV-1 INFECTION MODEL WITH ABSORPTION AND SATURATION INFECTION

2012 ◽  
Vol 05 (03) ◽  
pp. 1260012 ◽  
Author(s):  
RUI XU
Keyword(s):  
Viral Entry ◽  
Chronic Infection ◽  
Infection Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Virus Particles ◽  
Hiv 1

In this paper, an HIV-1 infection model with absorption, saturation infection and an intracellular delay accounting for the time between viral entry into a target cell and the production of new virus particles is investigated. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and LaSalle's invariance principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable; and if the basic reproduction ratio is greater than unity, sufficient condition is derived for the global stability of the chronic-infection equilibrium.

scholarly journals Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays

2019 ◽  
Vol 2019 ◽  
pp. 1-21
Author(s):  
Fulgensia Kamugisha Mbabazi ◽  
Joseph Y. T. Mugisha ◽  
Mark Kimathi
Keyword(s):  
Hopf Bifurcation ◽  
Time Delays ◽  
Pneumococcal Pneumonia ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Reproduction Ratio ◽  
Transversality Conditions ◽  
The Stability ◽  
Delay Differential ◽  
Disease Free

In this paper, a mathematical model of pneumococcal pneumonia with time delays is proposed. The stability theory of delay differential equations is used to analyze the model. The results show that the disease-free equilibrium is asymptotically stable if the control reproduction ratioR0is less than unity and unstable otherwise. The stability of equilibria with delays shows that the endemic equilibrium is locally stable without delays and stable if the delays are under conditions. The existence of Hopf-bifurcation is investigated and transversality conditions are proved. The model results suggest that, as the respective delays exceed some critical value past the endemic equilibrium, the system loses stability through the process of local birth or death of oscillations. Further, a decrease or an increase in the delays leads to asymptotic stability or instability of the endemic equilibrium, respectively. The analytical results are supported by numerical simulations.

Dynamics of an SIRS model with age structure and two delays

2021 ◽  
pp. 2150056
Author(s):  
Hongquan Sun ◽  
Hong Li ◽  
Jin Li ◽  
Zhangsheng Zhu
Keyword(s):  
Age Structure ◽  
Semigroup Theory ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Sirs Model ◽  
Number Formula ◽  
Two Delays ◽  
Age Structured ◽  
The Stability ◽  
Disease Free

In this paper, we propose and investigate an SIRS model with age structure and two delays. Both the infected and the recovered individuals have age structure, the infection rate (from the infective to the susceptible) and the immune loss rate (from the recovered to the susceptible) are related to two independent time delays, respectively. We prove that the proposed age structured SIRS model is well-posed by using the [Formula: see text]-semigroup theory. The basic reproduction number [Formula: see text] is given, and the unique endemic equilibrium exists when [Formula: see text], while the disease-free equilibrium always exists. A rigorous mathematical analysis for the stability of two equilibria is provided. The disease-free equilibrium is local asymptotically stable if [Formula: see text], and the endemic equilibrium is local asymptotically stable if [Formula: see text] and [Formula: see text]. Finally, we give numerical simulations to verify our results.

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