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.     

带有年龄结构的捕食者-食饵模型研究

研究带有年龄结构的捕食者-食饵模型的渐近行为.本文所研究的模型假定捕食者从幼年阶段到成年阶段的转变率依赖于幼年种群的密度,还假定幼年捕食者捕食食饵.本文最终给出了有年龄结构的捕食者-食饵模型的捕食者持久和灭绝的若干条件.     

A note on epidemics in heterogeneous populations.

Recent analysis has shown the importance of heterogeneity for understanding the course of epidemics. However, the results generally rely on computer models or the assumption that the population consists of internally homogeneous subgroups. This note presents some analytic results for the more general case, in which any distribution can characterize population heterogeneity in susceptibility under proportionate mixing. At any moment, epidemics in such a situation resemble classic epidemics, with rate of spread governed by the average susceptibility of those not yet infected. But, over time, this average susceptibility falls at a rate proportional to the dispersion of susceptibility among those not yet infected. The author concludes by noting some implications of heterogeneity for understanding epidemics.     

Recurrent epidemics in small world networks

The effect of spatial correlations on the spread of infectious diseases was investigated using a stochastic susceptible-infective-recovered (SIR) model on complex networks. It was found that in addition to the reduction of the effective transmission rate, through the screening of infectives, spatial correlations have another major effect through the enhancement of stochastic fluctuations, which may become considerably larger than in the homogeneously mixed stochastic model. As a consequence, in finite spatially structured populations significant differences from the solutions of deterministic models are to be expected, since sizes even larger than those found for homogeneously mixed stochastic models are required for the effects of fluctuations to be negligible. Furthermore, time series of the (unforced) model provide patterns of recurrent epidemics with slightly irregular periods and realistic amplitudes, suggesting that stochastic models together with complex networks of contacts may be sufficient to describe the long-term dynamics of some diseases. The spatial effects were analysed quantitatively by modelling measles and pertussis, using a susceptible-exposed-infective-recovered (SEIR) model. Both the period and the spatial coherence of the epidemic peaks of pertussis are well described by the unforced model for realistic values of the parameters.     

一类具有饱和发生率的SEIS模型的全局稳定性

建立并分析了一类具有饱和发生率、在潜伏期具有传染性的SEIS模型.得到了模型的基本再生数R_0和无病平衡点与地方病平衡点全局渐近稳定的充分条件.     

A general model for stochastic SIR epidemics with two levels of mixing

This paper is concerned with a general stochastic model for susceptible→infective→removed epidemics, among a closed finite population, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution ‘centred’ on that infective, and each global contact is with an individual chosen independently and uniformly from the whole population. The asymptotic situation in which the local contact distribution remains fixed as the population becomes large is considered. The concepts of local infectious clump and local susceptibility set are used to develop a unified approach to the threshold behaviour of this class of epidemic models. In particular, a threshold parameter R_* governing whether or not global epidemics can occur, the probability that a global epidemic occurs and the mean proportion of initial susceptibles ultimately infected by a global epidemic are all determined. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective’s household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; and (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour.     

Meteorological method based on wet foliage to determine areas at high risk of fungal epidemics

 The occurrence of fungal diseases is closely linked to specific atmospheric conditions affecting the life cycle of the fungus. Popular beliefs as well as scientific knowledge are in agreement on the effect of high temperature and moisture on fungal epidemics. The presence of a film of water on a plant, during an uninterrupted number of threshold hours is in many cases a necessary, and sometimes sufficient condition for fungal infection to develop. In this paper, we present a method to estimate the sequence of hours of wet and dry vegetal surface. Conventional hourly meteorological data are used. The accuracy of this method has been studied over an ideal sunflower field located in Balcarce (Buenos Aires, Argentina) with promising results. Received: 1 July 1996 / Revised: 2 December 1996 / Accepted: 17 December 1996     

海洛因成瘾模型建立及中脑腹侧被盖区Bax的表达

目的探索海洛因对中脑腹侧被盖区细胞Bax表达的影响。方法肌肉注射海洛因,建立成瘾大鼠模型,用免疫组化方法检测中脑腹侧被盖区细胞Bax的表达。结果连续给大鼠注射海洛因7d后,大鼠出现明显的戒断症状;中脑腹侧被盖区细胞Bax表达阳性细胞比对照组明显增多,与对照组相比差异有显著性（P〈0.01）。结论海洛因具有诱导Bax基因表达、损伤脑组织细胞的作用。     

Backward bifurcations in dengue transmission dynamics

A deterministic model for the transmission dynamics of a strain of dengue disease, which allows transmission by exposed humans and mosquitoes, is developed and rigorously analysed. The model, consisting of seven mutually-exclusive compartments representing the human and vector dynamics, has a locally-asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number(R(0)) is less than unity. Further, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making R(0) less than unity is no longer sufficient, although necessary, for effectively controlling the spread of dengue in a community. The model is extended to incorporate an imperfect vaccine against the strain of dengue. Using the theory of centre manifold, the extended model is also shown to undergo backward bifurcation. In both the original and the extended models, it is shown, using Lyapunov function theory and LaSalle Invariance Principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. In other words, in addition to establishing the presence of backward bifurcation in models of dengue transmission, this study shows that the use of standard incidence in modelling dengue disease causes the backward bifurcation phenomenon of dengue disease.     

A note on the occasional instability of detrending in correspondence analysis

When the two first eigenvalues of correspondence analysis are close to each other, their order can be reversed due to random variation in the data. The first axis can actually be in any direction in the plane defined by the two axes. However, the configuration of the points in the plane can remain unchanged but their projections onto any line in the plane can be very variable. The ordering in the first axis is preserved in detrending. The second axis is detrended with respect to the first one and therefore very variable configurations result when the orientation of the first axis in the plane is changed. This can lead to a situation where the detrended solutions are very unstable under random variation and therefore they can be only casually interpretable.     

Modelling the effect of a booster vaccination on disease epidemiology

Despite the effectiveness of vaccines in dramatically decreasing the number of new infectious cases and severity of illnesses, imperfect vaccines may not completely prevent infection. This is because the immunity afforded by these vaccines is not complete and may wane with time, leading to resurgence and epidemic outbreaks notwithstanding high levels of primary vaccination. To prevent an endemic spread of disease, and achieve eradication, several countries have introduced booster vaccination programs. The question of whether this strategy could eventually provide the conditions for global eradication is addressed here by developing a seasonally-forced mathematical model. The analysis of the model provides the threshold condition for disease control in terms of four major parameters: coverage of the primary vaccine; efficacy of the vaccine; waning rate; and the rate of booster administration. The results show that if the vaccine provides only temporary immunity, then the infection typically cannot be eradicated by a single vaccination episode. Furthermore, having a booster program does not necessarily guarantee the control of a disease, though the level of epidemicity may be reduced. In addition, these findings strongly suggest that the high coverage of primary vaccination remains crucial to the success of a booster strategy. Simulations using estimated parameters for measles illustrate model predictions. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). One of the authors (P.R.) acknowledges the support of the Ellison Medical Foundation.     

Models for the effect of toxicant in single-species and predator-prey systems

Models of single-species and predator-prey systems in a polluted closed environment are developed and partially analyzed. Three cases are considered: a single influx of toxicant, a constant influx of toxicant, and a periodic pollution of the environment. In the case of single-species growth we are able to determine some local and global dynamics. In the case of predator-prey systems, we investigate the existence of steady states for a small constant influx of toxicant.On leave from Department of Mathematics, Indian Institute of Technology, Kanpur, IndiaResearch partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A4823     

Competing Effects of Toxin-Producing Phytoplankton on Overall Plankton Populations in the Bay of Bengal

The coexistence of a large number of phytoplankton species on a seemingly limited variety of resources is a classical problem in ecology, known as ‘the paradox of the plankton’. Strong fluctuations in species abundance due to the external factors or competitive interactions leading to oscillations, chaos and short-term equilibria have been cited so far to explain multi-species coexistence and biodiversity of phytoplankton. However, none of the explanations has been universally accepted. The qualitative view and statistical analysis of our field data establish two distinct roles of toxin-producing phytoplankton (TPP): toxin allelopathy weakens the interspecific competition among phytoplankton groups and the inhibition due to ingestion of toxic substances reduces the abundance of the grazer zooplankton. Structuring the overall plankton population as a combination of nontoxic phytoplankton (NTP), toxic phytoplankton, and zooplankton, here we offer a novel solution to the plankton paradox governed by the activity of TPP. We demonstrate our findings through qualitative analysis of our sample data followed by analysis of a mathematical model.     

Comparative studies of light leafspot (Pyrenopeziza brassicae) epidemics on the growth and yield of winter oilseed rape

Comparisons of epidemics of light leafspot of differing duration and time of initiation were made in two experiments using a single cultivar of Brassica napus. Fungicide was applied before introduction of disease to prevent infection or some time after inoculation to stop further disease development. In the first experiment, substantial reductions in green leaf area and total plant dry-matter were found at flowering when disease was introduced in the autumn or in January. Plant dry weight at maturity was also greatly reduced in these treatments. The detrimental effect of an epidemic initiated in the autumn was avoided to a large extent if fungicide application began in February. Epidemics initiated in March had only small effects on final dry-matter yield. Seed yield was negatively correlated with the length of the epidemic. In a second experiment, early epidemics initiated in the autumn were halted after different time intervals. Commencing fungicide application even as early as December failed to prevent some loss of dry weight at flowering. At maturity, however, dry weight and seed yield were reduced significantly when fungicide application was delayed until February. Failure to control the disease resulted in a 46% loss of seed yield.     

Development of cercospora blight epidemics and effect on the summer annual Heliotropium europaeum in the field

Heliotropium europaeum, common heliotrope, is a serious economic weed in southern Australia. Cercospora blight occurs on H. europaeum in both its native (Mediterranean) and non-native (Australian) range. The causal agents are genetically different forms of an asexual pathogen in the form-genus Cercospora in each region. Natural epidemics of cercospora blight killed weed infestations in both Australia and France. The epidemiology of the disease did not differ between the two regions once differences in rainfall had been considered. Rainfall was important for disease spread. In France, field inoculation experiments were conducted using different concentrations of conidia prepared from a monospore isolate. Cercospora blight reduced seed production 89%, but did not reduce viability. Increased inoculum concentration and inoculation of younger plants encouraged an early epidemic, but did not affect the rate of disease development once the epidemic took hold. In order to be effective at controlling this weed in Australia, these pathogens need to attack young hosts and have regular rain-splash to facilitate spread to new growth.     

The human nucleus accumbens is highly susceptible to G protein down-regulation by methamphetamine and heroin

Although the nucleus accumbens is assumed to be a critical brain "pleasure center," its function in humans is unknown. As animal data suggest that a unique feature of this small brain area is its high sensitivity to down-regulation of an inhibitory G protein by drugs of abuse, we compared G protein levels in postmortem nucleus accumbens with those in seven other brain regions of chronic users of cocaine, methamphetamine, and heroin, and of matched controls. Biochemical changes were restricted to the nucleus accumbens in which concentrations of G(alpha)1 and/or G(alpha)2 were reduced by 32-49% in the methamphetamine and heroin users. This selective responsiveness to these abused drugs implies a special role for the human nucleus accumbens in mechanisms of drug reinforcement and suggests that some features of the drug-dependent state (e.g., tolerance) might be related to inhibition of G(alpha)1-linked receptor activity.     

Heroin epidemics, treatment and ODE modelling

The UN [United Nations Office on Drugs and Crime (UNODC): World Drug Report, 2005, vol. 1: Analysis. UNODC, 2005.], EU [European Monitoring Centre for Drugs and Drug Addiction (EMCDDA): Annual Report, 2005.http://annualreport.emcdda.eu.int/en/home-en.html.] and WHO [World Health Organisation (WHO): Biregional Strategy for Harm Reduction, 2005-2009. HIV and Injecting Drug Use. WHO, 2005.] have consistently highlighted in recent years the ongoing and persistent nature of opiate and particularly heroin use on a global scale. While this is a global phenomenon, authors have emphasised the significant impact such an epidemic has on individual lives and on society. National prevalence studies have indicated the scale of the problem, but the drug-using career, typically consisting of initiation, habitual use, a treatment-relapse cycle and eventual recovery, is not well understood. This paper presents one of the first ODE models of opiate addiction, based on the principles of mathematical epidemiology. The aim of this model is to identify parameters of interest for further study, with a view to informing and assisting policy-makers in targeting prevention and treatment resources for maximum effectiveness. An epidemic threshold value, R(0), is proposed for the drug-using career. Sensitivity analysis is performed on R(0) and it is then used to examine the stability of the system. A condition under which a backward bifurcation may exist is found, as are conditions that permit the existence of one or more endemic equilibria. A key result arising from this model is that prevention is indeed better than cure.