EXOGENOUS REINFECTION AND RESURGENCE OF TUBERCULOSIS: A THEORETICAL FRAMEWORK

2004 ◽  
Vol 12 (02) ◽  
pp. 231-247 ◽  
Author(s):  
S. M. MOGHADAS ◽  
M. E. ALEXANDER
Keyword(s):  
Multiple Equilibria ◽  
Incidence Rates ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
High Prevalence ◽  
Saturated Incidence ◽  
Stable Equilibria ◽  
Parameter Values ◽  
Multiple Stable Equilibria ◽  
Exogenous Reinfection

The importance of exogenous reinfection versus endogenous reactivation for the resurgence of tuberculosis (TB) has been a matter of ongoing debate. Previous mathematical models of TB give conflicting results on the possibility of multiple stable equilibria in the presence of reinfection, and hence the failure to control the disease even when the basic reproductive number is less than unity. The present study reconsiders the effect of exogenous reinfection, by extending previous studies to incorporate a generalized rate of reinfection as a function of the number of actively infected individuals. A mathematical model is developed to include all possible routes to the development of active TB (progressive primary infection, endogenous reactivation, and exogenous reinfection). The model is qualitatively analyzed to show the existence of multiple equilibria under realistic assumptions and plausible range of parameter values. Two examples, of unbounded and saturated incidence rates of reinfection, are given, and simulation results using estimated parameter values are presented. The results reflect exogenous reinfection as a major cause of TB emergence, especially in high prevalence areas, with important public health implications for controlling its spread.

scholarly journals Subthreshold domain of bistable equilibria for a model of HIV epidemiology

2003 ◽  
Vol 2003 (58) ◽  
pp. 3679-3698 ◽  
Author(s):  
B. D. Corbett ◽  
S. M. Moghadas ◽  
A. B. Gumel
Keyword(s):  
Hiv Transmission ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Global Stability Analysis ◽  
Hiv Epidemiology ◽  
Preventive Vaccine ◽  
Hiv Eradication ◽  
Stable Equilibria ◽  
Multiple Stable Equilibria ◽  
Anti Hiv

A homogeneous-mixing population model for HIV transmission, which incorporates an anti-HIV preventive vaccine, is studied qualitatively. The local and global stability analysis of the associated equilibria of the model reveals that the model can have multiple stable equilibria simultaneously. The epidemiological consequence of this (bistability) phenomenon is that the disease may still persist in the community even when the classical requirement of the basic reproductive number of infection (ℛ0) being less than unity is satisfied. It is shown that under specific conditions, the community-wide eradication of HIV is feasible ifℛ0<ℛ∗, whereℛ∗is some threshold quantity less than unity. Furthermore, for the bistability case (which occurs whenℛ∗<ℛ0<1), it is shown that HIV eradication is dependent on the initial sizes of the subpopulations of the model.

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.

scholarly journals Global Stability of an SEIS Epidemic Model with General Saturation Incidence

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Hui Zhang ◽  
Li Yingqi ◽  
Wenxiong Xu
Keyword(s):  
Latent Period ◽  
Epidemic Model ◽  
Convergence Theorem ◽  
Incidence Rates ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free

We present an SEIS epidemic model with infective force in both latent period and infected period, which has different general saturation incidence rates. It is shown that the global dynamics are completely determined by the basic reproductive number R0. If R0≤1, the disease-free equilibrium is globally asymptotically stable in T by LaSalle’s Invariance Principle, and the disease dies out. Moreover, using the method of autonomous convergence theorem, we obtain that the unique epidemic equilibrium is globally asymptotically stable in T0, and the disease spreads to be endemic.

scholarly journals Global Stability for a Viral Infection Model with Saturated Incidence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Huaqin Peng ◽  
Zhiming Guo
Keyword(s):  
Viral Infection ◽  
Global Stability ◽  
Incidence Rate ◽  
Sufficient Conditions ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Infection Model ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Viral Infection Model

A viral infection model with saturated incidence rate and viral infection with delay is derived and analyzed; the incidence rate is assumed to be a specific nonlinear formβxv/(1+αv). The existence and uniqueness of equilibrium are proved. The basic reproductive numberR0is given. The model is divided into two cases: with or without delay. In each case, by constructing Lyapunov functionals, necessary and sufficient conditions are given to ensure the global stability of the models.

scholarly journals A case study of 2019-nCOV cases in Argentina with the real data based on daily cases from March 03, 2020 to March 29, 2021 using classical and fractional derivatives

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Pushpendra Kumar ◽  
Vedat Suat Erturk ◽  
Marina Murillo-Arcila ◽  
Ramashis Banerjee ◽  
A. Manickam
Keyword(s):  
Fractional Order ◽  
Trust Region ◽  
Simulated Data ◽  
Real Data ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Order Model ◽  
The Real ◽  
The Stability ◽  
Parameter Values

AbstractIn this study, our aim is to explore the dynamics of COVID-19 or 2019-nCOV in Argentina considering the parameter values based on the real data of this virus from March 03, 2020 to March 29, 2021 which is a data range of more than one complete year. We propose a Atangana–Baleanu type fractional-order model and simulate it by using predictor–corrector (P-C) method. First we introduce the biological nature of this virus in theoretical way and then formulate a mathematical model to define its dynamics. We use a well-known effective optimization scheme based on the renowned trust-region-reflective (TRR) method to perform the model calibration. We have plotted the real cases of COVID-19 and compared our integer-order model with the simulated data along with the calculation of basic reproductive number. Concerning fractional-order simulations, first we prove the existence and uniqueness of solution and then write the solution along with the stability of the given P-C method. A number of graphs at various fractional-order values are simulated to predict the future dynamics of the virus in Argentina which is the main contribution of this paper.

scholarly journals Stochastic Stability and Analytical Solution with Homotopy Perturbation Method of Multicompartment Non-Linear Epidemic Model with Saturated Rate

2021 ◽  
pp. 149-457
Author(s):  
Laid Chahrazed
Keyword(s):  
Analytical Solution ◽  
Perturbation Method ◽  
Epidemic Model ◽  
Homotopy Perturbation Method ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Homotopy Perturbation ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Stability

In this work, we consider a nonlinear epidemic model with a saturated incidence rate. we consider a population of size N(t) at time t, this population is divided into six subclasses, with N(t)=S(t)+I(t)+I₁(t)+I₂(t)+I₃(t)+Q(t). Where S(t), I(t), I₁(t), I₂(t), I₃(t), and Q(t) denote the sizes of the population susceptible to disease, infectious members, and quarantine members, respectively. We have made the following contributions: 1. 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 determined by the ratio called the basic reproductive number. 2. We find the analytical solution of the nonlinear epidemic model by Homotopy perturbation method. 3. Finally the stochastic stabilities. The study of its sections are justified with theorems and demonstrations under certain conditions. In this work, we have used the different references cited in different studies in the three sections already mentioned.

A Note on Trial Delay and Social Welfare: The Impact of Multiple Equilibria

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
Tim Friehe ◽  
Thomas J. Miceli
Keyword(s):  
Social Welfare ◽  
Multiple Equilibria ◽  
Policy Implications ◽  
Important Policy ◽  
Deterrence Effect ◽  
Stable Equilibria ◽  
The Impact ◽  
Multiple Stable Equilibria

AbstractGreater trial delay is commonly associated with decreasing demand for trials, thereby bringing about an equilibrium for a given trial capacity. This note highlights that – in contrast to this premise – trial delay may in fact increase trial demand. Such an outcome is established for a scenario in which the number of cases is endogenous based on the deterrence effect of lawsuits. That trial demand may increase with longer delay makes multiple stable equilibria possible. This reality has important policy implications, which are discussed.

scholarly journals Behavior of a Discrete Fractional Order SIR Epidemic Model

2018 ◽  
Vol 7 (4.10) ◽  
pp. 675 ◽  
Author(s):  
A. George Maria Selvam ◽  
D. Abraham Vianny
Keyword(s):  
Equilibrium Point ◽  
Fractional Order ◽  
Epidemic Model ◽  
Dynamical Behavior ◽  
Basic Reproductive Number ◽  
Phase Portraits ◽  
Reproductive Number ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Parameter Values

In this paper we investigate the dynamical behavior of a SIR epidemic model of fractional order. Disease Free Equilibrium point, Endemic Equilibrium point and basic reproductive number are obtained. Time series plots, phase portraits and bifurcation diagrams are presented for suitable parameter values. Also some numerical examples are provided to illustrate the dynamics of the system.  

scholarly journals Using proper mean generation intervals in modelling of COVID-19

2021 ◽  
Author(s):  
Xiujuan Tang ◽  
Salihu S Musa ◽  
Shi Zhao ◽  
Daihai He
Keyword(s):  
Time Lag ◽  
Basic Reproductive Number ◽  
Infectious Period ◽  
Reproductive Number ◽  
Secondary Case ◽  
Primary Case ◽  
Control Efficacy ◽  
The Mean ◽  
And Control ◽  
Parameter Values

In susceptible-exposed-infectious-recovered (SEIR) epidemic models, with the exponentially distributed duration of exposed/infectious statuses, the mean generation interval (GI, time lag between infections of a primary case and its secondary case) equals the mean latent period (LP) plus the mean infectious period (IP). It was widely reported that the GI for COVID-19 is as short as 5 days. However, many works in top journals used longer LP or IP with the sum (i.e., GI), e.g., > 7 days. This discrepancy will lead to overestimated basic reproductive number, and exaggerated expectation of infectious attack rate and control efficacy, since all these quantities are functions of basic reproductive number. We argue that it is important to use suitable epidemiological parameter values.

A NOVEL APPLICATION OF A CLASSICAL METHOD FOR CALCULATING THE BASIC REPRODUCTIVE NUMBER, R0 FOR A GENDER AND RISK STRUCTURED TRANSMISSION DYNAMIC MODEL OF HUMAN PAPILLOMAVIRUS INFECTION

2013 ◽  
Vol 06 (06) ◽  
pp. 1350046 ◽  
Author(s):  
KATY TOBIN ◽  
CATHERINE COMISKEY
Keyword(s):  
Human Papillomavirus ◽  
Dynamic Model ◽  
Classical Method ◽  
Control Strategies ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Population Estimates ◽  
Transmission Dynamic ◽  
Papillomavirus Infection ◽  
Parameter Values

Mathematical models are increasingly being used in the evaluation of control strategies for infectious disease such as the vaccination program for the Human Papillomavirus (HPV). Here, an ordinary differential equation (ODE) transmission dynamic model for HPV is presented and analyzed. Parameter values for a gender and risk structured model are estimated by calibrating the model around the known prevalence of infection. The effect on gender and risk sub-group prevalence induced by varying the epidemiological parameters are investigated. Finally, the outcomes of this model are applied using a classical mathematical method for calculating R0 in a heterogeneous mixing population. Estimates for R0 under various gender and mixing scenarios are presented.

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