Threshold Quantity Research Articles

Mathematical analysis of hepatitis C model for intravenous drug misusers: Impact of antiviral therapy, abstinence and relapse

Despite advances in hepatitis C therapy and better knowledge of viral/host factors related to disease progression, hepatitis C virus (HCV) remains the leading cause of chronic liver disease, causing progression to end-stage liver disease (ESLD) as well as the development of hepatocellular carcinoma. In this paper a mathematical model for assessing the impact of antiviral therapy, abstinence and relapse on the transmission dynamics of HCV is formulated and analyzed. A threshold quantity known as the reproductive number has been computed, and the stability of the steady states has been investigated. The dynamical analysis reveals that the model has globally asymptotically stable steady states. The impacts of antiviral therapy, abstinence and relapse on the transmission dynamics of HCV are discussed through the basic reproductive number and numerical simulations.

Global dynamics of treatment models with time delay

The problem of the asymptomatic dynamics of a treatment model with time delay is considered, subject to two incidence functions, namely standard incidence and Holling type II (saturated) incidence function. Rigorous qualitative analysis of the model shows that for each of the two incidence functions, the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold quantity is less than unity. Further, it has a unique endemic equilibrium when the threshold quantity exceeds unity. For the case with Holling type II incidence function, it is shown that the unique endemic equilibrium of the model is globally-asymptotically stable for a special case. Finally, for each of the two incidence functions, the disease burden decreases with increasing time delay (incubation period). In summary the results in this article is similar to those established in Safi and Gumel (Nonlinear Anal Ser B Real World Appl 12:215–235, 2011) (i.e., treatment models considered here have the same dynamics of quarantine-isolation models in Safi and Gumel (Nonlinear Anal Ser B Real World Appl 12:215–235, 2011).

Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation

This paper characterizes the qualitative dynamics of a stochastic SIRS epidemic model. The study shows that the dynamics of the model are determined by a certain threshold quantity RS expressed in terms of the model parameters and the intensity of the noise. If RS<1, the disease will be eliminated from the community; whereas an epidemic occurs if RS>1. Our results recover the known results in the earlier literature as special cases. The presented results are illustrated by numerical simulations.

HEPATITIS C VIRUS AND INTRAVENOUS DRUG MISUSE: A MODELING APPROACH

Hepatitis C virus (HCV) is a blood-borne infection that can lead to progressive liver failure, cirrhosis, hepatocellular carcinoma and death. A deterministic mathematical model for assessing the impact of daily intravenous drug misuse on the transmission dynamics of HCV is presented and analyzed. A threshold quantity known as the reproductive number has been computed. Stability of the steady states has been investigated. The dynamical analysis reveals that the model has globally asymptotically stable steady states. The impact of daily intravenous drug misuse on the transmission dynamics of HCV has been discussed through the basic reproductive number and numerical simulations.

A Mathematical study on the vaccination impact on the disease dynamics of HBV

We develop a deterministic model to understand the underlying dynamics of HBV infection at population level. The model, which incorporates the vaccination and treatment of individuals, the re-infections of latent, carrier and recovery individuals, is rigorously analyzed to gain insight into its dynamical features. The mathematical analysis reveals that the model exhibits a backward bifurcation. This phenomenon resulted due to the exogenous re-infection of HBV disease . It is shown that, in the absence of such re-infection, the model has a disease-free equilibrium (DFE) which is globally asymptotically stable (GAS) using Lyapunov function and LaSalle Invariance Principle whenever the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium(EEP), for a special case, whenever the associate threshold quantity exceeds unity. This EEP is shown to be GAS, for a special case, using a non-linear Lyapunov function of Goh- Volterra type.

An analysis of transmission dynamics of drug-resistant disease on scale-free networks

In this paper we present an SIR drug-resistant model on scale-free networks. The model has a disease-free equilibrium, which is globally asymptotically stable when a threshold quantity is less than one. The model has a wild strain equilibrium and a resistant strain equilibrium, which are asymptotically stable when their respective reproduction number is great than one. When treatment leads to resistance development, the model can have a unique coexistence equilibrium, and we also prove the persistence of infection. The effects of two immunization schemes, including proportional immunization scheme and targeted immunization scheme, are studied and compared. We find that targeted strategy compare favorably to a proportional immunization has prominent effect to control the disease spread on scale-free networks.

Mathematical Analysis of West Nile Virus Model with Discrete Delays

The paper presents the basic model for the transmission dynamics of West Nile virus (WNV). The model, which consists of seven mutually-exclusive compartments representing the birds and vector dynamics, has a locally-asymptotically stable disease-free equilibrium whenever the associated reproduction number (ℝ0) is less than unity. As reveal in [3, 20], the analyses of the model show the existence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity). It is shown, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. Analysis of the reproduction number of the model shows that, the disease will persist, whenever ℝ0 > 1, and increase in the length of incubation period can help reduce WNV burden in the community if a certain threshold quantities, denoted by Δb and Δv are negative. On the other hand, increasing the length of the incubation period increases disease burden if Δb > 0 and Δv > 0. Furthermore, it is shown that adding time delay to the corresponding autonomous model with standard incidence (considered in [2]) does not alter the qualitative dynamics of the autonomous system (with respect to the elimination or persistence of the disease).

Assessing the impact of temperature on malaria transmission dynamics

A mathematical model to assess the impact of temperature on malaria transmission dynamics is explored and analysed. Threshold quantities of the model are determined and analysed. The model is shown to exhibit backward bifurcation. Analysis of the reproduction number suggests that increase in temperature to about $$32~{}^{\circ }\mathrm{C}$$ has the potential to increase the epidemic. The burden of the disease increases with increase in temperature with an optimal temperature window of 30– $$32~{}^{\circ }\mathrm{C}$$ for malaria transmission. However as temperatures approach $$40~{}^{\circ }\mathrm{C}$$ , infected human and mosquito populations decline to asymptotically low levels.

Selection never dominates drift (nor vice versa)

The probability that the fitter of two alleles will increase in frequency in a population goes up as the product of N (the effective population size) and s (the selection coefficient) increases. Discovering the distribution of values for this product across different alleles in different populations is a very important biological task. However, biologists often use the product Ns to define a different concept; they say that drift “dominates” selection or that drift is “stronger than” selection when Ns is much smaller than some threshold quantity (e.g., ½) and that the reverse is true when Ns is much larger than that threshold. We argue that the question of whether drift dominates selection for a single allele in a single population makes no sense. Selection and drift are causes of evolution, but there is no fact of the matter as to which cause is stronger in the evolution of any given allele.

Dynamics Analysis of a Multi-strain Cholera Model with an Imperfect Vaccine

A new two-strain model, for assessing the impact of basic control measures, treatment and dose-structured mass vaccination on cholera transmission dynamics in a population, is designed. The model has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. The model has a unique, and locally-asymptotically stable, endemic equilibrium when the threshold quantity exceeds unity and another condition holds. Numerical simulations of the model show that, with the expected 50% minimum efficacy of the first vaccine dose, vaccinating 55% of the susceptible population with the first vaccine dose will be sufficient to effectively control the spread of cholera in the community. Such effective control can also be achieved if 50% of the first vaccine dose recipients take the second dose. It is shown that a control strategy that emphasizes the use of antibiotic treatment is more effective than one that emphasizes the use of basic (non-pharmaceutical) anti-cholera control measures only. Numerical simulations show that, while the universal strategy (involving all three control measures) gives the best outcome in minimizing cholera burden in the community, the combined basic anti-cholera control measures and treatment strategy also has very effective community-wide impact.

Closed-Form Approximations to the Threshold Quantities of Distributed-Feedback Lasers With Varying Phase Shifts and Positions

We derive analytical approximations to the threshold gain, detuning, and power distribution ratios of distributed-feedback lasers with a variable phase shift at a variable longitudinal position. These closed-form approximations exhibit the direct influence of the grating parameters, and are in excellent agreement with the exact solution of the coupled mode equations for a wide range of phase shifts, positions, grating strengths and lengths. It is shown that the product of grating strength and phase shift offset distance from the grating center has a dominant influence on most threshold parameters. The threshold gain and output power splitting ratio grow exponentially with this product, whereas the intracavity peak power and the Q-factor of the cavity decay exponentially with it. It is also shown that a nonvanishing sine of the phase shift leads to a linear change of the detuning, a quadratic increase of the threshold gain, and a quadratic decrease of the output power splitting ratio, intracavity peak power, and Q-factor. Our results are independent of the source of the gain, such as semiconductor, rare-earth, and Raman fiber lasers, provided that the gain is approximately constant along the grating length and that reflections from facets can be neglected. As an example, we simulate a Raman DFB laser and demonstrate that our closed-form approximations are in perfect agreement with steady state results from involved time-domain simulations.

Competence of Frankliniella occidentalis and Frankliniella intonsa strains as vectors for Chrysanthemum stem necrosis virus

The vector competence of Frankliniella occidentalis for Chrysanthemum stem necrosis virus (CSNV) was evaluated. Three vector strains with distinct competences for Tomato spotted wilt virus (TSWV) transmission were investigated, including an artificially selected strain (TsH) that has a particularly high competence (>90 %). Newly hatched larvae of F. occidentalis were given an acquisition access period of 5 days on CSNV-infected D. stramonium leaves, and reared to maturity. Their transmission efficiencies were examined using a leaf disk assay using Petunia x hybrida leaves. Following the leaf disk assay, the virus accumulation in the vectors was examined via a double antibody sandwich enzyme-linked immunosorbent assay (DAS-ELISA) of their bodies. The results showed that the CSNV acquisition and transmission efficiency of the TsH strain did not differ from those of the others, indicating that the competence of F. occidentalis as a vector for CSNV is not related to that for TSWV. The CSNV transmission and acquisition efficiencies of two F. intonsa strains (Hiroshima and Fukuoka) were also evaluated. In Hiroshima strain, 35 % of adults were viruliferous, but only two transmitters (3 %) were observed. In Fukuoka strain, 6 % were viruliferous, and no transmitters were observed. These results indicate that F. intonsa cannot be a major vector for CSNV. The accumulation of CSNV in the adults of F. occidentalis and F. intonsa evaluated using DAS-ELISA showed a significant difference in ELISA values among transmitter, viruliferous non-transmitter, and non-viruliferous individuals. These results clearly demonstrated that only transmitters that accumulated a threshold quantity of virus can transmit CSNV to plants.

Qualitative analysis of an age- and sex-structured vaccination model for human papillomavirus

A new model for the transmission dynamics of human pappilomavirus (HPV) is designed and analysed. The model, which stratifies the total population in terms of age and gender, incorporates an imperfect anti-HPV vaccine with some therapeutic benefits. Rigorous qualitative analysis of the resulting age-structured model, which takes the form of a deterministic system of non-linear partial differential equations with separable transmission coefficients, shows that the disease-free equilibrium of the model is locally-asymptotically stable whenever the effective reproduction number (denoted by $\mathcal{R}_v$) is less than unity. It is shown to be globally-asymptotically stable if certain additional conditions hold. Furthermore, it is shown that the model has at least one endemic equilibrium when $\mathcal{R}_v$ exceeds unity. Hence, the effective control of HPV spread in a community, using a vaccine, is governed by the threshold quantity $\mathcal{R}_v$ (the use of the vaccine will lead to effective disease control or elimination only if it reduces the threshold quantity to a value less than unity; and the use of such vaccine will not lead to effective disease control if it fails to make the threshold quantity to be less than unity).

On a Certainty Equivalence Design of Continuous-Time Stochastic Systems

This paper presents a certainty equivalence design method with application for continuous-time stochastic systems with unknown parameters. The first aim is to approximate the observed process $x=(x_t)_{t\geq0},$ satisfying a stochastic differential equation with control, by choosing a control process $(u_t)_{t\geq0}$ to a stable Ornstein--Uhlenbeck process $(x^0_t)_{t\geq0},$ driven by the same Wiener process. Based on observation of $x$ for a given $\varepsilon>0$ an adaptive control process $u^\varepsilon=(u^\varepsilon_t)_{t\geq0}$ is constructed such that the corresponding observed process $x(u^\varepsilon)=(x^\varepsilon_t)_{t\geq0}$ satisfies the relation $ \sup\nolimits_{t\geq t_\varepsilon}\ E(x^\varepsilon_t-x^0_t)^2\leq \varepsilon$. The time $t_\varepsilon$ $\mbox{(}\varepsilon$ is a threshold quantity) is found. A similar problem is solved for a stochastic delay differential equation with an unknown parameter. It is shown that the time $t_\varepsilon$ has equal rates of increase $\varepsilon^{-1}\ln \varepsilon^{-1}$ in both problems. Moreover, a target inequality, which ensures the stability of the controlled processes in the sense of the $L_2$-norm, is established. Similar problems for more general controlled multidimensional systems can be solved using sequential estimators of unknown parameters.

An Epidemic Model for Tick-Borne Disease with Two Delays

We have considered an epidemic model of a tick-borne infection which has nonviraemic transmission in addition to the viremic transmission. The basic reproduction number , which is a threshold quantity for stability of equilibria, is calculated. If , then the infection-free equilibrium is globally asymptotically stable, and this is the only equilibrium. On the contrary, if , then an infection equilibrium appears which is globally asymptotically stable, when one time delay is absent. By applying a permanence theorem for infinite dimensional systems, we obtain that the disease is always present when .

Transmission model of hepatitis B virus with the migration effect.

Hepatitis B is a globally infectious disease. Mathematical modeling of HBV transmission is an interesting research area. In this paper, we present characteristics of HBV virus transmission in the form of a mathematical model. We analyzed the effect of immigrants in the model to study the effect of immigrants for the host population. We added the following flow parameters: “the transmission between migrated and exposed class” and “the transmission between migrated and acute class.” With these new features, we obtained a compartment model of six differential equations. First, we find the basic threshold quantity Ro and then find the local asymptotic stability of disease-free equilibrium and endemic equilibrium. Furthermore, we find the global stability of the disease-free and endemic equilibria. Previous similar publications have not added the kind of information about the numerical results of the model. In our case, from numerical simulation, a detailed discussion of the parameters and their numerical results is presented. We claim that with these assumptions and by adding the migrated class, the model informs policy for governments, to be aware of the immigrants and subject them to tests about the disease status. Immigrants for short visits and students should be subjected to tests to reduce the number of immigrants with disease.

北東北地域の飼料イネ栽培におけるタイヌビエの許容残存量

北東北地域の飼料イネ栽培におけるタイヌビエの許容残存量

Production Dynamic Method to Determine the Critical Production of Bottom Water Reservoir

Tahe oilfield of Triassic oil under low amplitude anticline group monolithic sandstone reservoirs with bottom water features, having a uniform oil-water interface, water energy sufficient. The current production wells generally see water, water cut rising too fast. Horizontal well bottom water coning phenomenon is outstanding, later period of high water cut stage manages difficulty to increase, water blocking measures have a low efficiency. Determine critical output, to improve production well life, efficient development reservoir has important significance .Based on the study of dynamic characteristics of single well and geological analysis of factors determining the critical output, and the application of numerical simulation method to verify the reasonable threshold quantity production. This method is simple and practical, can accurately make the critical output of single well.

Interdistrict Variation in the Implementation of the Crack Retroactivity Policy by U.S. District Courts

In 2007, after years of dissatisfaction with the 100-to-1 disparity between crack cocaine and powder cocaine, the U.S. Sentencing Commission amended the U.S. Sentencing Guidelines, reducing by two levels the base offense level assigned to threshold quantities of crack cocaine. The amendment was subsequently made retroactive, which required the federal courts to process approximately 20,000 prisoner motions for sentence reduction. Using data collected from a case study of one particular district as well as a survey of representatives from the 94 federal judicial districts, this study examines the process of “crack retroactivity” implementation. The authors found that court communities varied greatly with respect to their organizational arrangements, procedures, and motivations. Although U.S. Probation and Pretrial Services played an important role in every district, how and why districts organized to respond to crack retroactivity varied. The study findings also indicate that sentencing procedures may be another mechanism through which location affects sentencing outcomes.

A curve of thresholds governs plague epizootics in Central Asia

A core concept of infectious disease epidemiology is the abundance threshold, below which an infection is unable to invade or persist. There have been contrasting theoretical predictions regarding the nature of this threshold for vector-borne diseases, but for infections with an invertebrate vector, it is common to assume a threshold defined by the ratio of vector and host abundances. Here, we show in contrast, both from field data and model simulations, that for plague (Yersinia pestis) in Kazakhstan, the invasion threshold quantity is based on the product of its host (Rhombomys opimus) and vector (mainly Xenopsylla spp.) abundances, resulting in a combined threshold curve with hyperbolic shape. This shape implies compensation between host and vector abundances in permitting infection, which has important implications for disease control. Realistic joint thresholds, supported by data, should promote improved understanding, prediction and management of disease occurrence in this and other vector-borne disease systems.