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 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.     

Impact of group mixing on disease dynamics

A general mathematical model is proposed to study the impact of group mixing in a heterogeneous host population on the spread of a disease that confers temporary immunity upon recovery. The model contains general distribution functions that account for the probabilities that individuals remain in the recovered class after recovery. For this model, the basic reproduction number R₀ is identified. It is shown that if R₀<1, then the disease dies out in the sense that the disease free equilibrium is globally asymptotically stable; whereas if R₀>1, this equilibrium becomes unstable. In this latter case, depending on the distribution functions and the group mixing strengths, the disease either persists at a constant endemic level or exhibits sustained oscillatory behavior.     

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.     

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.     

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).     

Modelling Hospitalization,Home-Based Care,and Individual Withdrawal for People Living with HIV/AIDS in High Prevalence Settings

In sub-Saharan Africa, the model of care for people who are living with HIV/AIDS has changed from hospital care to home-based care. In this paper, a mathematical model describing the dynamics of HIV transmission, hospitalization, and home-based care is constructed and analysed. The model reproduction number R _e is determined and discussed. The equilibria are determined and analysed in terms of R _e . It is shown that if R _e <1, the disease free equilibrium is both locally and globally asymptotically stable. The model has a unique endemic equilibrium and is locally asymptotically stable whenever R _e >1. Five cases arise in the discussion of R _e pertaining to intervention strategies. Numerical simulations are done to compare the impact of each strategy on the dynamics of HIV/AIDS. The model is fitted to the prevalence data estimates from UNAIDS on Zimbabwe. The implications of some key epidemiological parameters are investigated numerically. Projections are made to determine the possible long term trends of the prevalence of HIV in Zimbabwe.     

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     

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。     

Epidemic dynamics on semi-directed complex networks

In this paper an SIS model for epidemic spreading on semi-directed networks is established, which can be used to examine and compare the impact of undirected and directed contacts on disease spread. The model is analyzed for the case of uncorrelated semi-directed networks, and the basic reproduction number _R0$R_0$ is obtained analytically. We verify that the _R0$R_0$ contains the outbreak threshold on undirected networks and directed networks as special cases. It is proved that if _R0<1$R_0<1$ then the disease-free equilibrium is globally asymptotically stable, otherwise the disease-free equilibrium is unstable and the unique endemic equilibrium exists, which is globally asymptotically stable. Finally the numerical simulations holds for these analytical results are given.     

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.     

The effects of human movement on the persistence of vector-borne diseases

With the recent resurgence of vector-borne diseases due to urbanization and development there is an urgent need to understand the dynamics of vector-borne diseases in rapidly changing urban environments. For example, many empirical studies have produced the disturbing finding that diseases continue to persist in modern city centers with zero or low rates of transmission. We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian models that mimic human commuting behavior and (ii) Eulerian models that mimic human migration. We determine the basic reproduction number R₀ for both modeling approaches. We show that for both approaches that if the disease-free equilibrium is stable (R₀<1) then it is globally stable and if the disease-free equilibrium is unstable (R₀>1) then there exists a unique positive (endemic) equilibrium that is globally stable among positive solutions. Finally, we prove in general that Lagrangian and Eulerian modeling approaches are not equivalent. The modeling approaches presented provide a framework to explore spatial vector-borne disease dynamics and control in heterogeneous environments. As an example, we consider two patches in which the disease dies out in both patches when there is no movement between them. Numerical simulations demonstrate that the disease becomes endemic in both patches when humans move between the two patches.     

Stochastic models for virus and immune system dynamics

New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number, R₀, is calculated and it is shown that if R₀<1, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case R₀>1, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.     

Polymorphism in multiallelic migration–selection models with dominance

The evolution of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection with dominance is investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Underdominance and overdominance are excluded. If the degree of dominance is deme independent for every pair of alleles, then under the Levene model, the qualitative evolution of the gene frequencies (i.e., the existence and stability of the equilibria) is the same as without dominance. In particular: (i) the number of demes is a generic upper bound on the number of alleles present at equilibrium; (ii) there exists exactly one stable equilibrium, and it is globally attracting; and (iii) if there exists an internal equilibrium, it is globally asymptotically stable. Analytic examples demonstrate that if either the Levene model does not apply or the degree of dominance is deme dependent, then the above results can fail. A complete global analysis of weak migration and weak selection on a recessive allele in two demes is presented.     

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.     

From heroin epidemics to methamphetamine epidemics: Modelling substance abuse in a South African province

The global rise in the use of methamphetamine has been documented to have reached epidemic proportions. Researchers have focussed on the social implications of the epidemic. A typical drug use cycle consists of concealed drugs use after initiation, addiction, treatment-recovery-relapse cycle, whose dynamics are not well understood. The model by White and Comiskey [41], on heroin epidemics, treatment and ODE modelling, is modified to model the dynamics of methamphetamine use in a South African province. The analysis of the model is presented in terms of the methamphetamine epidemic threshold R₀. It is shown that the model has multiple equilibria and using the center manifold theory, the model exhibits the phenomenon of backward bifurcation where a stable drug free equilibrium co-exists with a stable drug persistent equilibrium for a certain defined range of R₀. The stabilities of the model equilibria are ascertained and persistence conditions established. Furthermore, numerical simulations are performed; these include fitting the model to the available data on the number of patients with methamphetamine problems. The implications of the results to drug policy, treatment and prevention are discussed.     

A mathematical model for assessing control strategies against West Nile virus

Since its incursion into North America in 1999, West Nile virus (WNV) has spread rapidly across the continent resulting in numerous human infections and deaths. Owing to the absence of an effective diagnostic test and therapeutic treatment against WNV, public health officials have focussed on the use of preventive measures in an attempt to halt the spread of WNV in humans. The aim of this paper is to use mathematical modelling and analysis to assess two main anti-WNV preventive strategies, namely: mosquito reduction strategies and personal protection. We propose a single-season ordinary differential equation model for the transmission dynamics of WNV in a mosquito-bird-human community, with birds as reservoir hosts and culicine mosquitoes as vectors. The model exhibits two equilibria; namely the disease-free equilibrium and a unique endemic equilibrium. Stability analysis of the model shows that the disease-free equilibrium is globally asymptotically stable if a certain threshold quantity , which depends solely on parameters associated with the mosquito-bird cycle, is less than unity. The public health implication of this is that WNV can be eradicated from the mosquito-bird cycle (and, consequently, from the human population) if the adopted mosquito reduction strategy (or strategies) can make . On the other hand, it is shown, using a novel and robust technique that is based on the theory of monotone dynamical systems coupled with a regular perturbation argument and a Liapunov function, that if , then the unique endemic equilibrium is globally stable for small WNV-induced avian mortality. Thus, in this case, WNV persists in the mosquito-bird population.     

Clines with partial panmixia in an unbounded unidimensional habitat

In geographically structured populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into diallelic single-locus clines maintained by migration and selection in an unbounded unidimensional habitat is investigated. Migration and selection are both weak. The former is homogenous and isotropic; the latter has no dominance. The population density is uniform. A simple, explicit formula is derived for the maximum value β₀ of the scaled panmictic rate β for which a cline exists. The former depends only on the asymptotic values of the scaled selection coefficient. If the two alleles have the same average selection coefficient, there exists a unique, globally asymptotically stable cline for every β≥0. Otherwise, if β≥β₀, the allele with the greater average selection coefficient is ultimately fixed. If β<β₀, there exists a unique, globally asymptotically stable cline, and some polymorphism is retained even infinitely far from its center. The gene frequencies at infinity are determined by a continuous-time, two-deme migration-selection model. An explicit expression is deduced for the monotone cline in a step-environment. These results differ fundamentally from those for the classical cline without panmixia.     

Global behaviour of an age-infection-structured HIV model with impulsive drug-treatment strategy

In this paper, we develop an HIV model that considers the dependence of HIV/AIDS progress on infection age (duration since infection), chronological age and impulsive antiretroviral treatment. The analytical results show that there exists a globally asymptotically stable infection-free state when the impulsive period T and drug-treatment proportion p satisfy R₀(p,T)<1. The numerical simulation results indicate that there exist T and p such that R₀(p,T)<1. Due to pulses, it would be difficult to obtain the optimal pulse interval in the age distribution of population. However, our results demonstrate the effect of the impulsive drug-treatment strategy on the dynamics of HIV/AIDS.     

Interconversion between cyclodimer and cyclotrimer: Synthesis and characterization of cyclo-[Pd(II)Cl2(N-N)] complexes

The reaction of (COD)PdCl₂ (COD = 1,5-cyclooctadiene) with (3-Py)₂SiR₁R₂ (3-Py = 3-pyridyl; R₁ = Ph, R₂ = Ph (m-pdps); R₁ = Ph, R₂ = Me (m-pmps)) in acetone affords single crystals consisting of cyclodimers, [PdCl₂((3-Py)₂SiR₁R₂)]₂, whereas the same reaction in a mixture of dichloromethane and ethanol yields amorphous spheres consisting of cyclotrimers, [PdCl₂((3-Py)₂SiR₁R₂)]₃. In a boiling chloroform solution, the cyclodimers are completely converted to cyclotrimers. These cyclotrimers, in the 10−60 °C range, are partly returned to cyclodimers. By contrast, the reaction of (COD)PdCl₂ with (3-Py)₂SiR₁R₂ (R₁ = Bu, R₂ = Me (m-pbms); R₁ = dodecyl, R₂ = Me (m-pddms)) yields amorphous spheres consisting of cyclotrimers irrespective of solvents. Both [PdCl₂(m-pbms)]₃ and [PdCl₂(m-pddms)]₃ are initially cyclotrimers in chloroform, but they exist as a mixture of cyclodimers and cyclotrimers in solution in the 10−60 °C range. The metallacycles tend to form cyclodimers in the order m-pdps > m-pmps > m-pbms > m-pddms. The equilibrium between cyclodimers and the cyclotrimers is sensitive to solvent, temperature, and concentration as well as molecular structure.