Dynamics of an HIV-1 therapy model of fighting a virus with another virus

In this paper, we rigorously analyse an ordinary differential equation system that models fighting the HIV-1 virus with a genetically modified virus. We show that when the basic reproduction ratio ?(0)<1, then the infection-free equilibrium E (0) is globally asymptotically stable; when ?(0)>1, E (0) loses its stability and there is the single-infection equilibrium E (s). If ?(0)∈(1, 1+δ) where δ is a positive constant explicitly depending on system parameters, then the single-infection equilibrium E (s) that is globally asymptotically stable, while when ?(0)>1+δ, E (s) becomes unstable and the double-infection equilibrium E (d) comes into existence. When ?(0) is slightly larger than 1+δ, E (d) is stable and it loses its stability via Hopf bifurcation when ?(0) is further increased in some ways. Through a numerical example and by applying a normal form theory, we demonstrate how to determine the bifurcation direction and stability, as well as the estimates of the amplitudes and the periods of the bifurcated periodic solutions. We also perform numerical simulations which agree with the theoretical results. The approaches we use here are a combination of analysis of characteristic equations, fluctuation lemma, Lyapunov function and normal form theory.     

Global Dynamics of an In-host Viral Model with Intracellular Delay

The dynamics of a general in-host model with intracellular delay is studied. The model can describe in vivo infections of HIV-I, HCV, and HBV. It can also be considered as a model for HTLV-I infection. We derive the basic reproduction number R ₀ for the viral infection, and establish that the global dynamics are completely determined by the values of R ₀. If R ₀≤1, the infection-free equilibrium is globally asymptotically stable, and the virus are cleared. If R ₀>1, then the infection persists and the chronic-infection equilibrium is locally asymptotically stable. Furthermore, using the method of Lyapunov functional, we prove that the chronic-infection equilibrium is globally asymptotically stable when R ₀>1. Our results shows that for intercellular delays to generate sustained oscillations in in-host models it is necessary have a logistic mitosis term in target-cell compartments.     

Dynamical behavior of a hepatitis B virus transmission model with vaccination

Hepatitis B virus (HBV) infection is a globally health problem. In 2005, the WHO Western Pacific Regional Office set a goal of reducing chronic HBV infection rate to less than 2% among children five years of age by 2012, as an interim milestone towards the final goal of less than 1%. Many countries made some plans (such as free HBV vaccination program for all neonates in China now) to control the transmission HBV. We develop a model to explore the impact of vaccination and other controlling measures of HBV infection. The model has simple dynamical behavior which has a globally asymptotically stable disease-free equilibrium when the basic reproduction number R₀≤1, and a globally asymptotically stable endemic equilibrium when R₀>1. Numerical simulation results show that the vaccination is a very effective measure to control the infection and they also give some useful comments on controlling the transmission of HBV.     

An SIS patch model with variable transmission coefficients

In this paper, an SIS patch model with non-constant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number R₀ is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if R₀?1, and the disease is uniformly persistent and there exists at least one endemic equilibrium if R₀>1. In particular, when the disease is non-fatal and the travel rates of susceptible and infectious individuals in each patch are the same, the endemic equilibrium is unique and is globally asymptotically stable as R₀>1. Numerical calculations are performed to illustrate some results for the case with two patches.     

A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage

We formulate a deterministic epidemic model for the spread of Hepatitis C containing an acute, chronic and isolation class and analyse the effects of the isolation class on the transmission dynamics of the disease. We calculate the basic reproduction number R₀ and show that for R₀≤1, the disease-free equilibrium is globally asymptotically stable. In addition, it is shown that for a special case when R₀>1, the endemic equilibrium is locally asymptotically stable. Furthermore, an analogous stochastic epidemic model for Hepatitis C is formulated using a continuous time Markov chain. Numerical simulations are used to estimate the mean, variance and probability distributions of the discrete random variables and these are compared to the steady-state solutions of the deterministic model. Finally, the expected time to disease extinction is estimated for the stochastic model and the impact of isolation on the time to extinction is explored.     

A Tuberculosis Model with Seasonality

The statistical data of tuberculosis (TB) cases show seasonal fluctuations in many countries. A TB model incorporating seasonality is developed and the basic reproduction ratio R ₀ is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the disease eventually disappears if R ₀<1, and there exists at least one positive periodic solution and the disease is uniformly persistent if R ₀>1. Numerical simulations indicate that there may be a unique positive periodic solution which is globally asymptotically stable if R ₀>1. Parameter values of the model are estimated according to demographic and epidemiological data in China. The simulation results are in good accordance with the seasonal variation of the reported cases of active TB in China.     

Qualitative analysis of models with different treatment protocols to prevent antibiotic resistance

This paper is concerned with the qualitative analysis of two models [S. Bonhoeffer, M. Lipsitch, B.R. Levin, Evaluating treatment protocols to prevent antibiotic resistance, Proc. Natl. Acad. Sci. USA 94 (1997) 12106] for different treatment protocols to prevent antibiotic resistance. Detailed qualitative analysis about the local or global stability of the equilibria of both models is carried out in term of the basic reproduction number R₀. For the model with a single antibiotic therapy, we show that if R₀ < 1, then the disease-free equilibrium is globally asymptotically stable; if R₀ > 1, then the disease-endemic equilibrium is globally asymptotically stable. For the model with multiple antibiotic therapies, stabilities of various equilibria are analyzed and combining treatment is shown better than cycling treatment. Numerical simulations are performed to show that the dynamical properties depend intimately upon the parameters.     

Global Properties of <Emphasis Type="Italic">SIR</Emphasis> and <Emphasis Type="Italic">SEIR</Emphasis> Epidemic Models with Multiple Parallel Infectious Stages

We consider global properties of compartment SIR and SEIR models of infectious diseases, where there are several parallel infective stages. For instance, such a situation may arise if a fraction of the infected are detected and treated, while the rest of the infected remains undetected and untreated. We assume that the horizontal transmission is governed by the standard bilinear incidence rate. The direct Lyapunov method enables us to prove that the considered models are globally stable: There is always a globally asymptotically stable equilibrium state. Depending on the value of the basic reproduction number R ₀, this state can be either endemic (R ₀>1), or infection-free (R ₀≤1).     

Oscillations in a patchy environment disease model

For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R(0) is identified. It is shown that the disease-free equilibrium is globally asymptotically stable if R(0)<1. For R(0)>1, local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided. For a two patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that travel can reduce oscillations in both patches; travel may enhance oscillations in both patches; or travel can switch oscillations from one patch to another.     

Stability analysis of pathogen-immune interaction dynamics

The paper considers models of dynamics of infectious disease in vivo from the standpoint of the mathematical analysis of stability. The models describe the interaction of the target cells, the pathogens, and the humoral immune response. The paper mainly focuses on the interior equilibrium, whose components are all positive. If the model ignores the absorption of the pathogens due to infection, the interior equilibrium is always asymptotically stable. On the other hand, if the model does consider it, the interior equilibrium can be unstable and a simple Hopf bifurcation can occur. A sufficient condition that the interior equilibrium is asymptotically stable is obtained. The condition explains that the interior equilibrium is asymptotically stable when experimental parameter values are used for the model. Moreover, the paper considers the models in which uninfected cells are involved in the immune response to pathogens, and are removed by the immune complexes. The effect of the involvement strongly affects the stability of the interior equilibria. The results are shown with the aid of symbolic calculation software.     

具有一般形式饱和接触率SEIS模型渐近分析

研究具有一般形式饱和接触率SEIS模型渐近性态,得到决定疾病绝灭和持续的阈值-基本再生数R0。当R0 ≤ 1时,仅存在无病平衡点P^0;当R0＞1时,除存在无病平衡点P^0外,还存在惟一的地方病平衡点P^＊。当R0＜1时,无病平衡点P^0全局渐近稳定;当R0＞1时,地方病平衡点P^＊局部渐近稳定。特别地,无因病死亡时,极限方程地方病平衡点P^-＊全局渐近稳定。     

Three stage semelparous Leslie models

Nonlinear Leslie matrix models have a long history of use for modeling the dynamics of semelparous species. Semelparous models, as do nonlinear matrix models in general, undergo a transcritical equilibrium bifurcation at inherent net reproductive number R ₀ = 1 where the extinction equilibrium loses stability. Semelparous models however do not fall under the purview of the general theory because this bifurcation is of higher co-dimension. This mathematical fact has biological implications that relate to a dichotomy of dynamic possibilities, namely, an equilibration with over lapping age classes as opposed to an oscillation in which age classes are periodically missing. The latter possibility makes these models of particular interest, for example, in application to the well known outbreaks of periodical insects. While the nature of the bifurcation at R ₀ = 1 is known for two-dimensional semelparous Leslie models, only limited results are available for higher dimensional models. In this paper I give a thorough accounting of the bifurcation at R ₀ = 1 in the three-dimensional case, under some monotonicity assumptions on the nonlinearities. In addition to the bifurcation of positive equilibria, there occurs a bifurcation of invariant loops that lie on the boundary of the positive cone. I describe the geometry of these loops, classify them into three distinct types, and show that they consist of either one or two three-cycles and heteroclinic orbits connecting (the phases of) these cycles. Furthermore, I determine stability and instability properties of these loops, in terms of model parameters, as well as those of the positive equilibria. The analysis also provides the global dynamics on the boundary of the cone. The stability and instability conditions are expressed in terms of certain measures of the strength and the symmetry/asymmetry of the inter-age class competitive interactions. Roughly speaking, strong inter-age class competitive interactions promote oscillations (not necessarily periodic) with separated life-cycle stages, while weak interactions promote stable equilibration with overlapping life-cycle stages. Methods used include the theory of planar monotone maps, average Lyapunov functions, and bifurcation theory techniques.      

Dynamics of an epidemic model with non-local infections for diseases with latency over a patchy environment

In this paper, with the assumptions that an infectious disease in a population has a fixed latent period and the latent individuals of the population may disperse, we formulate an SIR model with a simple demographic structure for the population living in an n-patch environment (cities, towns, or countries, etc.). The model is given by a system of delay differential equations with a fixed delay accounting for the latency and a non-local term caused by the mobility of the individuals during the latent period. Assuming irreducibility of the travel matrices of the infection related classes, an expression for the basic reproduction number R₀{\mathcal{R}_0} is derived, and it is shown that the disease free equilibrium is globally asymptotically stable if R₀ < 1{\mathcal{R}_0 < 1} , and becomes unstable if ${\mathcal{R}_0 > 1}${\mathcal{R}_0 > 1} . In the latter case, there is at least one endemic equilibrium and the disease will be uniformly persistent. When n = 2, two special cases allowing reducible travel matrices are considered to illustrate joint impact of the disease latency and population mobility on the disease dynamics. In addition to the existence of the disease free equilibrium and interior endemic equilibrium, the existence of a boundary equilibrium and its stability are discussed for these two special cases.     

Thresholds for macroparasite infections

We analyse here the equilibria of an infinite system of partial differential equations modelling the dynamics of a population infected by macroparasites. We find that it is possible to define a reproduction number R₀ that satisfies the intuitive definition, and yields a sharp threshold in the behaviour of the system: if R₀ < 1, the parasite-free equilibrium (PFE) is asymptotically stable and there are no endemic equilibria; if R₀ > 1, the PFE is unstable and there exists a unique endemic equilibrium. The results mainly confirm what had been obtained in simplified models, except for the fact that no backward bifurcation occurs in this model. The stability of equilibria is established by extending an abstract linearization principle and by analysing the spectra of appropriate operators.Revised version: 14 November 2003Supported in part by CNR under Grant n. 00.0142.ST74 Metodi e modelli matematici nello studio dei fenomeni biologici     

Seasonal dynamics in the stable carbon isotope composition δ¹³C from non-leafy branch, trunk and coarse root CO₂ efflux of adult deciduous (Fagus sylvatica) and evergreen (Picea abies) trees

Respiration is a substantial driver of carbon (C) flux in forest ecosystems and stable C isotopes provide an excellent tool for its investigation. We studied seasonal dynamics in δ¹³C of CO₂ efflux (δ¹³C_E) from non‐leafy branches, upper and lower trunks and coarse roots of adult trees, comparing deciduous Fagus sylvatica (European beech) with evergreen Picea abies (Norway spruce). In both species, we observed strong and similar seasonal dynamics in the δ¹³C_E of above‐ground plant components, whereas δ¹³C_E of coarse roots was rather stable. During summer, δ¹³C_E of trunks was about ?28.2‰ (Beech) and ?26.8‰ (Spruce). During winter dormancy, δ¹³C_E increased by 5.6–9.1‰. The observed dynamics are likely related to a switch from growth to starch accumulation during fall and remobilization of starch, low TCA cycle activity and accumulation of malate by PEPc during winter. The seasonal δ¹³C_E pattern of branches of Beech and upper trunks of Spruce was less variable, probably because these organs were additionally supplied by winter photosynthesis. In view of our results and pervious studies, we conclude that the pronounced increases in δ¹³C_E of trunks during the winter results from interrupted access to recent photosynthates.     

Numerical solutions of the Lamm equation. II. Equilibrium sedimentation

The Lamm equation has been solved numerically for conditions corresponding to equilibrium runs for a nonlinear concentration dependence of the form s/s₀ = (1 + kc)^?1. It is shown that the approach to equilibrium is very close to being exponential (in time) as in the case k = 0. We also compare results for the nonlinear case given above with results obtained for linear c-dependence of the form s/s₀ = 1 – kc. For relatively high speeds the time required to attain equilibrium may be greatly underestimated by use of the latter approximation. Finally, we present analytic approximations for the concentration distribution at equilibrium and as a function of time.     

Mathematical model for Zika virus dynamics with sexual transmission route

Zika virus is a flavivirus transmitted to humans primarily through the bite of infected Aedes mosquitoes. In addition to vector-borne spread, however, the virus can also be transmitted through sexual contact. In this paper, we formulate and analyze a new system of ordinary differential equations which incorporates both vector and sexual transmission routes. Theoretical analysis of this model when there is no disease induced mortality shows that the disease-free equilibrium is locally and globally asymptotically stable whenever the associated reproduction number is less than unity and unstable otherwise. However, when we extend this same model to include Zika induced mortality, which have been documented in Latin America, we find that the model exhibits a backward bifurcation. Specifically, a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction number is less than unity. To further explore model predictions, we use numerical simulations to assess the importance of sexual transmission to disease dynamics. This analysis shows that risky behavior involving multiple sexual partners, particularly among male populations, substantially increases the number of infected individuals in the population, contributing significantly to the disease burden in the community.     

Equilibria in structured populations

The existence of a stable positive equilibrium state for the density of a population which is internally structured by means of a single scalar such as age, size, etc. is studied as a bifurcation problem. Using an inherent birth modulus n as a bifurcation parameter it is shown for very general nonlinear model equations, in which vital birth and growth processes depend on population density, that a global unbounded continuum of nontrivial equilibrium pairs (n, ) bifurcates from the unique (normalized) critical point (1, 0). The pairs are locally positive and conditions are given under which the continuum is globally positive. Local stability is shown to depend on the direction of bifurcation. For the important case when density dependence is a nonlinear expression involving a linear functional of density (such as total population size) it is shown how a detailed global bifurcation diagram is easily constructed in applications from the graph of a certain real valued function obtained from an invariant on the continuum. Uniqueness and nonuniqueness of positive equilibrium states are studied. The results are illustrated by several applications to models appearing in the literature.This research was done while the author was on leave at the Lehrstuhl für Biomathematik, Universität Tübingen, Auf der Morgenstelle 10, 7400 Tübingen 1, Federal Republic of Germany     

Asymptotical Tests on the Equivalence,Substantial Difference and Non‐inferiority Problems with Two Proportions

Let d = p₂ ? p₁ be the difference between two binomial proportions obtained from two independent trials. For parameter d, three pairs of hypothesis may be of interest: H₁: d ≤ δ vs. K₁: d > δ; H₂: d ? (δ₁, δ₂) vs. K₂: d ∈ (δ₁, δ₂); and H₃: d ∈ [δ₁, δ₂] vs. K₃: d ? [δ₁, δ₂], where H_i is the null hypothesis and K_i is the alternative hypothesis. These tests are useful in clinical trials, pharmacological and vaccine studies and in statistics generally. The three problems may be investigated by exact unconditional tests when the sample sizes are moderate. Otherwise, one should use approximate (or asymptotical) tests generally based on a Z‐statistics like those suggested in the paper. The article defines a new procedure for testing H₂ or H₃, demonstrates that this is more powerful than tests based on confidence intervals (the classic TOST – two one sided tests – test), defines two corrections for continuity which reduce the liberality of the three tests, and selects the one that behaves better. The programs for executing the unconditional exact and asymptotic tests described in the paper can be loaded at http://www.ugr.es/~bioest/software.htm. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evolution under the multilocus Levene model without epistasis

Evolution under the multilocus Levene model is investigated. The linkage map is arbitrary, but epistasis is absent. The geometric-mean fitness, , depends only on the vector of gene frequencies, ρ; it is nondecreasing, and the single-generation change is zero only on the set, Λ, of gametic frequencies at gene-frequency equilibrium. The internal gene-frequency equilibria are the stationary points of . If the equilibrium points of ρ(t) (where t denotes time in generations) are isolated, as is generic, then ρ(t) converges as t→∞ to some . Generically, ρ(t) converges to a local maximum of . Write the vector of gametic frequencies, p, as , where d represents the vector of linkage disequilibria. If is a local maximum of , then the equilibrium point is asymptotically stable. If either there are only two loci or there is no dominance, then d(t)→0 globally as t→∞. In the second case, has a unique maximum and is globally asymptotically stable. If underdominance and overdominance are excluded, and if at each locus, the degree of dominance is deme independent for every pair of alleles, then the following results also hold. There exists exactly one stable gene-frequency equilibrium (point or manifold), and it is globally attracting. If an internal gene-frequency equilibrium exists, it is globally asymptotically stable. On Λ, (i) the number of demes, Γ, is a generic upper bound on the number of alleles present per locus; and (ii) if every locus is diallelic, generically at most Γ−1 loci can segregate. Finally, if migration and selection are completely arbitrary except that the latter is uniform (i.e., deme independent), then every uniform selection equilibrium is a migration-selection equilibrium and generically has the same stability as under pure selection.