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.

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 Threshold Dynamics in a Model for Zika Virus Disease with Seasonality

2021 ◽  
Vol 83 (4) ◽  
Author(s):  
Mahmoud A. Ibrahim ◽  
Attila Dénes
Keyword(s):  
Zika Virus ◽  
Reproduction Number ◽  
Virus Disease ◽  
Threshold Dynamics ◽  
Global Dynamics ◽  
Threshold Parameter ◽  
Globally Asymptotically Stable ◽  
Linear Integral ◽  
The Impact ◽  
Disease Free

AbstractWe present a compartmental population model for the spread of Zika virus disease including sexual and vectorial transmission as well as asymptomatic carriers. We apply a non-autonomous model with time-dependent mosquito birth, death and biting rates to integrate the impact of the periodicity of weather on the spread of Zika. We define the basic reproduction number $${\mathscr {R}}_{0}$$ R 0 as the spectral radius of a linear integral operator and show that the global dynamics is determined by this threshold parameter: If $${\mathscr {R}}_0 < 1,$$ R 0 < 1 , then the disease-free periodic solution is globally asymptotically stable, while if $${\mathscr {R}}_0 > 1,$$ R 0 > 1 , then the disease persists. We show numerical examples to study what kind of parameter changes might lead to a periodic recurrence of Zika.

scholarly journals Threshold Dynamics of a Huanglongbing Model with Logistic Growth in Periodic Environments

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jianping Wang ◽  
Shujing Gao ◽  
Yueli Luo ◽  
Dehui Xie
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Logistic Growth ◽  
Threshold Dynamics ◽  
Global Dynamics ◽  
Globally Asymptotically Stable ◽  
Linear Integral ◽  
The Impact ◽  
Disease Free ◽  
The Basic Reproduction Number

We analyze the impact of seasonal activity of psyllid on the dynamics of Huanglongbing (HLB) infection. A new model about HLB transmission with Logistic growth in psyllid insect vectors and periodic coefficients has been investigated. It is shown that the global dynamics are determined by the basic reproduction numberR0which is defined through the spectral radius of a linear integral operator. IfR0< 1, then the disease-free periodic solution is globally asymptotically stable and ifR0> 1, then the disease persists. Numerical values of parameters of the model are evaluated taken from the literatures. Furthermore, numerical simulations support our analytical conclusions and the sensitive analysis on the basic reproduction number to the changes of average and amplitude values of the recruitment function of citrus are shown. Finally, some useful comments on controlling the transmission of HLB are given.

scholarly journals Seasonality Impact on the Transmission Dynamics of Tuberculosis

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Yali Yang ◽  
Chenping Guo ◽  
Luju Liu ◽  
Tianhua Zhang ◽  
Weiping Liu
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Autonomous Systems ◽  
Positive Periodic Solution ◽  
Transmission Dynamics ◽  
Uniform Persistence ◽  
Globally Asymptotically Stable ◽  
The Impact ◽  
Disease Free ◽  
The Basic Reproduction Number

The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show the seasonality fluctuations in Shaanxi of China. A seasonality TB epidemic model with periodic varying contact rate, reactivation rate, and disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB. Simulations show that the basic reproduction number of time-averaged autonomous systems may underestimate or overestimate infection risks in some cases, which may be up to the value of period. The basic reproduction number of the seasonality model is appropriately given, which determines the extinction and uniform persistence of TB disease. If it is less than one, then the disease-free equilibrium is globally asymptotically stable; if it is greater than one, the system at least has a positive periodic solution and the disease will persist. Moreover, numerical simulations demonstrate these theorem results.

GLOBAL ANALYSIS OF A VIRAL INFECTION MODEL WITH APPLICATION TO HBV INFECTION

2010 ◽  
Vol 18 (02) ◽  
pp. 325-337 ◽  
Author(s):  
YU JI ◽  
LEQUAN MIN ◽  
YONGAN YE
Keyword(s):  
Viral Infection ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Global Analysis ◽  
Hiv Infections ◽  
Hbv Infection ◽  
Infection Model ◽  
Globally Asymptotically Stable ◽  
Reproduction Numbers ◽  
The Basic Reproduction Number

The basic models of within-host viral infection, proposed by Nowak and May2 and Perelson and Nelson,5 have been widely used in the studies of HBV and HIV infections. The basic reproduction numbers of the two models are proportional to the number of total cells of the host's organ prior to the infection. In this paper, we formulate an amended Perelson and Nelson's model with standard incidence. The basic reproduction number of the amended model is independent of total cells of the host's organ. If the basic reproduction number R0 < 1, then the infection-free equilibrium is globally asymptotically stable and the virus is cleared; if R0 > 1, then the virus persists in the host, and solutions approach either an endemic equilibrium or a periodic orbit. Numerical simulations of this model agree well with the clinical HBV infection data. This can provide a possible interpretation for the viral oscillation behaviors, which were observed in chronic HBV infection patients.

scholarly journals Mathematical Modeling of the Coinfection Dynamics of Malaria-Toxoplasmosis in the Tropics

2019 ◽  
Vol 56 (2) ◽  
pp. 139-163
Author(s):  
Oluwatayo M. Ogunmiloro
Keyword(s):  
Reproduction Number ◽  
Model System ◽  
Host Population ◽  
Globally Asymptotically Stable ◽  
Susceptible Host ◽  
Tropical Regions ◽  
First Order ◽  
Mathematical Techniques ◽  
The Tropics ◽  
The Impact

SummaryCoinfection by Plasmodium species and Toxoplasma gondii in humans is widespread, with its endemic impact mostly felt in the tropics. A mathematical model is formulated as a first-order nonlinear system of ordinary differential equations to describe the coinfection dynamics of malaria-toxoplasmosis in the mainly human and feline susceptible host population in tropical regions. Comprehensive mathematical techniques are applied to show that the model system is bounded, positive and realistic in an epidemiological sense. Also, the basic reproduction number (Romt) of the coinfection model is obtained. It is shown that if Romt < 1, the model system at its malaria-toxoplasmosis absent equilibrium is both locally and globally asymptotically stable. The impact of toxoplasmosis and its treatment on malaria, and vice versa, is studied and analyzed. Sensitivity analysis was performed to investigate the impact of the model system parameters on the reproduction number of the transmission of malaria-toxoplasmosis coinfection. Simulations and graphical illustrations were made to validate the results obtained from the theoretical model.

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.

scholarly journals On the use of aggregated human mobility data to estimate the reproduction number

2021 ◽  
Vol 11 (1) ◽  
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

AbstractThe reproduction number of an infectious disease, such as CoViD-19, can be described through a modified version of the susceptible-infected-recovered (SIR) model with time-dependent contact rate, where mobility data are used as proxy of average movement trends and interpersonal distances. We introduce a theoretical framework to explain and predict changes in the reproduction number of SARS-CoV-2 in terms of aggregated 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.

TRANSMISSION DYNAMICS AND OPTIMAL CONTROL OF AN INFLUENZA MODEL WITH QUARANTINE AND TREATMENT

2012 ◽  
Vol 05 (03) ◽  
pp. 1260011 ◽  
Author(s):  
WEI-WEI SHI ◽  
YUAN-SHUN TAN
Keyword(s):  
Optimal Control ◽  
Disease Transmission ◽  
Reproduction Number ◽  
Influenza Pandemic ◽  
Control Measures ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
Parameter Values ◽  
The Impact ◽  
Disease Free

We develop an influenza pandemic model with quarantine and treatment, and analyze the dynamics of the model. Analytical results of the model show that, if basic reproduction number [Formula: see text], the disease-free equilibrium (DFE) is globally asymptotically stable, if [Formula: see text], the disease is uniformly persistent. The model is then extended to assess the impact of three anti-influenza control measures, precaution, quarantine and treatment, by re-formulating the model as an optimal control problem. We focus primarily on controlling disease with a possible minimal the systemic cost. Pontryagin's maximum principle is used to characterize the optimal levels of the three controls. Numerical simulations of the optimality system, using a set of reasonable parameter values, indicate that the precaution measure is more effective in reducing disease transmission than the other two control measures. The precaution measure should be emphasized.

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