scholarly journals Global stability in a competitive infection-age structured model

2020 ◽  
Vol 15 ◽  
pp. 54
Author(s):  
Quentin Richard
Keyword(s):  
Global Stability ◽  
Basin Of Attraction ◽  
Exclusion Principle ◽  
Globally Asymptotically Stable ◽  
Infection Age ◽  
Age Structured ◽  
The Stability ◽  
Globally Attractive ◽  
Age Structured Model ◽  
Disease Free

We study a competitive infection-age structured SI model between two diseases. The well-posedness of the system is handled by using integrated semigroups theory, while the existence and the stability of disease-free or endemic equilibria are ensured, depending on the basic reproduction number R0x and R0y of each strain. We then exhibit Lyapunov functionals to analyse the global stability and we prove that the disease-free equilibrium is globally asymptotically stable whenever max{R0x, R0y} ≤ 1. With respect to explicit basin of attraction, the competitive exclusion principle occurs in the case where R0x ≠ R0y and max{R0x, R0y} > 1, meaning that the strain with the largest R0 persists and eliminates the other strain. In the limit case R0x = Ry0 > 1, an infinite number of endemic equilibria exists and constitute a globally attractive set.

scholarly journals Optimal intervention strategies of staged progression HIV infections through an age-structured model with probabilities of ART drop out

2021 ◽  
Author(s):  
Ramsès Djidjou-Demasse
Keyword(s):  
Global Analysis ◽  
Hiv Infections ◽  
Host Population ◽  
Drop Out ◽  
Globally Asymptotically Stable ◽  
Optimal Intervention ◽  
Age Structured ◽  
Age Structured Model ◽  
The Impact ◽  
Disease Free

In this paper, we construct a model to describe the transmission of HIV in a homogeneous host population. By considering the specific mechanism of HIV, we derive a model structured in three successive stages: (i) primary infection, (ii) long phase of latency without symptoms and (iii) AIDS. Each HIV stage is stratified by the duration for which individuals have been in the stage, leading to a continuous age-structure model. In the first part of the paper, we provide a global analysis of the model depending upon the basic reproduction number R0. When R0<=1, then the disease-free equilibrium is globally asymptotically stable and the infection is cleared in the host population. On the contrary, if R0>1, we prove the epidemic's persistence with the asymptotic stability of the endemic equilibrium. By performing the sensitivity analysis, we then determine the impact of control-related parameters of the outbreak severity. For the second part, the initial model is extended with intervention methods. By taking into account ART interventions and the probability of treatment drop out, we discuss optimal interventions methods which minimize the number of AIDS cases.

STABILITY OF AN AGE-STRUCTURED SEIS EPIDEMIC MODEL WITH INFECTIVITY IN INCUBATIVE PERIOD

2010 ◽  
Vol 03 (03) ◽  
pp. 299-312 ◽  
Author(s):  
SHU-MIN GUO ◽  
XUE-ZHI LI ◽  
XIN-YU SONG
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Stability Conditions ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Structured ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an age-structured SEIS epidemic model with infectivity in incubative period is formulated and studied. The explicit expression of the basic reproduction number R0 is obtained. It is shown that the disease-free equilibrium is globally asymptotically stable if R0 < 1, at least one endemic equilibrium exists if R0 > 1. The stability conditions of endemic equilibrium are also given.

Study of an epidemic model with Z-type control

2018 ◽  
Vol 11 (07) ◽  
pp. 1850084 ◽  
Author(s):  
Sudip Samanta
Keyword(s):  
Epidemic Model ◽  
Control Mechanism ◽  
Basin Of Attraction ◽  
Endemic Equilibrium ◽  
Successful Implementation ◽  
Asymptotically Stable ◽  
Infectious Disease Outbreak ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Disease Free

In the present paper, an epidemic model with Z-type control mechanism has been proposed and analyzed to explore the disease control strategy on an infectious disease outbreak. The uncontrolled model can have a disease-free equilibrium and an endemic equilibrium. The expression of the basic reproduction number and the conditions for the stability of the equilibria are derived. It is also observed that the disease-free equilibrium is globally asymptotically stable if [Formula: see text], whereas the endemic equilibrium is globally asymptotically stable if [Formula: see text]. The model is further improved by considering Z-control mechanism and investigated. Disease can be controlled by using Z-control while the basic reproduction of the uncontrolled system is greater than unity. The positivity conditions of the solutions are derived and the basin of attraction for successful implementation of Z-control mechanism is also investigated. To verify the analytical findings, extensive numerical simulations on the model are carried out.

Global stability for an SEI model of infectious diseases with immigration and age structure in susceptibility

2019 ◽  
Vol 12 (04) ◽  
pp. 1950042
Author(s):  
Gang Huang ◽  
Chenguang Nie ◽  
Yueping Dong
Keyword(s):  
Infectious Diseases ◽  
Global Stability ◽  
Age Structure ◽  
Lyapunov Functional ◽  
Asymptotically Stable ◽  
Structured Model ◽  
Globally Asymptotically Stable ◽  
Age Structured ◽  
Age Structured Model ◽  
Sei Model

In this paper, we propose an SEI age-structured model for infectious diseases where the susceptibility depends on the age and with immigration of new individuals into the susceptible, exposed and infectious classes. The existence of a global attractor and the asymptotic smoothness of the solution semi-flow generated by the model are addressed. Using a Lyapunov functional, we show that the unique endemic equilibrium is globally asymptotically stable.

scholarly journals Mathematical Analysis and Stochastic Stability of Nonlinear Epidemic Model with Incidence Rate

2020 ◽  
pp. 8-19
Author(s):  
Laid Chahrazed
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Globally Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Stability ◽  
Disease Free ◽  
Temporary Immunity

In this work, we consider a nonlinear epidemic model with temporary immunity and saturated incidence rate. Size N(t) at time t, is divided into three sub classes, with N(t)=S(t)+I(t)+Q(t); where S(t), I(t) and Q(t) denote the sizes of the population susceptible to disease, infectious and quarantine members with the possibility of infection through temporary immunity, respectively. We have made the following contributions: The local stabilities of the infection-free equilibrium and endemic equilibrium are; analyzed, respectively. The stability of a disease-free equilibrium and the existence of other nontrivial equilibria can be determine by the ratio called the basic reproductive number, This paper study the reduce model with replace S with N, which does not have non-trivial periodic orbits with conditions. The endemic -disease point is globally asymptotically stable if R0 ˃1; and study some proprieties of equilibrium with theorems under some conditions. Finally the stochastic stabilities with the proof of some theorems. In this work, we have used the different references cited in different studies and especially the writing of the non-linear epidemic mathematical model with [1-7]. We have used the other references for the study the different stability and other sections with [8-26]; and sometimes the previous references.

Stability analysis and optimal control of an epidemic model with vaccination

2015 ◽  
Vol 08 (03) ◽  
pp. 1550030 ◽  
Author(s):  
Swarnali Sharma ◽  
G. P. Samanta
Keyword(s):  
Optimal Control ◽  
Stability Analysis ◽  
Epidemic Model ◽  
Asymptotically Stable ◽  
Control Pair ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
The Cost ◽  
Objective Functional ◽  
Disease Free

In this paper, we have developed a compartment of epidemic model with vaccination. We have divided the total population into five classes, namely susceptible, exposed, infective, infective in treatment and recovered class. We have discussed about basic properties of the system and found the basic reproduction number (R0) of the system. The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at disease-free equilibrium E0when R0< 1. When R0> 1 endemic equilibrium E1exists and the system becomes locally asymptotically stable at E1under some conditions. We have also discussed the epidemic model with two controls, vaccination control and treatment control. An objective functional is considered which is based on a combination of minimizing the number of exposed and infective individuals and the cost of the vaccines and drugs dose. Then an optimal control pair is obtained which minimizes the objective functional. Our numerical findings are illustrated through computer simulations using MATLAB. Epidemiological implications of our analytical findings are addressed critically.

THE DYNAMICS OF A DISCRETE SEIT MODEL WITH AGE AND INFECTION AGE STRUCTURES

2012 ◽  
Vol 05 (03) ◽  
pp. 1260004 ◽  
Author(s):  
HUI CAO ◽  
YANNI XIAO ◽  
YICANG ZHOU
Keyword(s):  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Disease Spread ◽  
Threshold Parameter ◽  
Infection Age ◽  
The Stability ◽  
Globally Stable ◽  
Disease Free ◽  
The Basic Reproduction Number ◽  
Age Structures

Age and infection age have significant influence on the transmission of infectious diseases, such as HIV/AIDS and TB. A discrete SEIT model with age and infection age structures is formulated to investigate the dynamics of the disease spread. The basic reproduction number R0 is defined and used as the threshold parameter to characterize the disease extinction or persistence. It is shown that the disease-free equilibrium is globally stable if R0 < 1, and it is unstable if R0 > 1. When R0 > 1, there exists an endemic equilibrium, and the disease is uniformly persistent. The stability of the endemic equilibrium is investigated numerically.

scholarly journals Mathematical analysis of an age structured epidemic model with a quarantine class

2021 ◽  
Author(s):  
Bedreddine AINSEBA ◽  
Tarik Touaoula ◽  
Zakia Sari
Keyword(s):  
Reproduction Number ◽  
Asymptotic Behavior Of Solutions ◽  
Model Parameters ◽  
Qualitative Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Structured ◽  
The Stability ◽  
The Impact

In this paper, an age structured epidemic Susceptible-Infected-Quarantined-Recovered-Infected (SIQRI) model is proposed, where we will focus on the role of individuals that leave their class of quarantine before being completely recovered and thus will participate again to the transmission of the disease. We investigate the asymptotic behavior of solutions by studying the stability of both trivial and positive equilibria. In order to see the impact of the different model parameters like the relapse rate on the qualitative behavior of our system, we firstly, give the explicit expression of the epidemic reproduction number $R_{0}.$ This number is a combination of the classical epidemic reproduction number for the SIQR model and a new epidemic reproduction number corresponding to the individuals infected by a relapsed person from the R-class. It is shown that, if $R_{0}\leq 1$, the disease free equilibrium is globally asymptotically stable and becomes unstable for $R_{0}>1$. Secondly, while $R_{0}>1$, a suitable Lyapunov functional is constructed to prove that the unique endemic equilibrium is globally asymptotically stable on some subset $\Omega_{0}.$

scholarly journals Human Mobility and Epidemic Evolution

2021 ◽  
Author(s):  
Fabio Vanni ◽  
David Lambert ◽  
Luigi Palatella ◽  
Paolo Grigolini
Keyword(s):  
Reproduction Number ◽  
Human Mobility ◽  
Theoretical Framework ◽  
Infection Age ◽  
Mobility Data ◽  
Risk Of Disease ◽  
Age Structured ◽  
Using Data ◽  
Age Structured Model ◽  
The Impact

Abstract The CoViD-19 pandemic ceased to be describable by a susceptible-infected-recovered (SIR) model when lockdowns were enforced. We introduce a theoretical framework to explain and predict changes in the reproduction number of SARS-CoV-2 (Sudden Acute Respiratory Syndrome Coronavirus 2) in terms of individual mobility and interpersonal proximity (alongside other epidemiological and environmental variables) during and after the lockdown period. We use an infection-age structured model described by a renewal equation. The model predicts the evolution of the reproduction number up to a week ahead of well-established estimates used in the literature. We show how lockdown policies, via reduction of proximity and mobility, reduce the impact of CoViD-19 and mitigate the risk of disease resurgence. We validate our theoretical framework using data from Google, Voxel51, Unacast, The CoViD-19 Mobility Data Network, and Analisi Distribuzione Aiuti.

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