James Watmough

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

A precise definition of the basic reproduction number, R0, is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if R0<1, then the disease free equilibrium is locally asymptotically stable; whereas if R0>1, then it is unstable. Thus, R0 is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for R0 near one. This criterion, together with the definition of R0, is illustrated by treatment, multigroup, staged progression, multistrain and vector-host models and can be applied to more complex models. The results are significant for disease control.

Modelling strategies for controlling SARS outbreaks

Severe acute respiratory syndrome (SARS), a new, highly contagious, viral disease, emerged in China late in 2002 and quickly spread to 32 countries and regions causing in excess of 774 deaths and 8098 infections worldwide. In the absence of a rapid diagnostic test, therapy or vaccine, isolation of individuals diagnosed with SARS and quarantine of individuals feared exposed to SARS virus were used to control the spread of infection. We examine mathematically the impact of isolation and quarantine on the control of SARS during the outbreaks in Toronto, Hong Kong, Singapore and Beijing using a deterministic model that closely mimics the data for cumulative infected cases and SARS–related deaths in the first three regions but not in Beijing until mid–April, when China started to report data more accurately. The results reveal that achieving a reduction in the contact rate between susceptible and diseased individuals by isolating the latter is a critically important strategy that can control SARS outbreaks with or without quarantine. An optimal isolation programme entails timely implementation under stringent hygienic precautions defined by a critical threshold value. Values below this threshold lead to control, but those above are associated with the incidence of new community outbreaks or nosocomial infections, a known cause for the spread of SARS in each region. Allocation of resources to implement optimal isolation is more effective than to implement sub–optimal isolation and quarantine together. A community–wide eradication of SARS is feasible if optimal isolation is combined with a highly effective screening programme at the points of entry.

Simple models for containment of a pandemic

Stochastic simulations of network models have become the standard approach to studying epidemics. We show that many of the predictions of these models can also be obtained from simple classical deterministic compartmental models. We suggest that simple models may be a better way to plan for a threatening pandemic with location and parameters as yet unknown, reserving more detailed network models for disease outbreaks already underway in localities where the social networks are well identified. We formulate compartmental models to describe outbreaks of influenza and attempt to manage a disease outbreak by vaccination or antiviral treatment. The models give an important prediction that may not have been noticed in other models, namely that the number of doses of antiviral treatment required is extremely sensitive to the number of initial infectives. This suggests that the actual number of doses needed cannot be estimated with any degree of reliability. The model is applicable to pre-epidemic vaccination, such as annual vaccination programs in anticipation of an ‘ordinary’ influenza outbreak with limited drift, and as a combination of treatment both before and during an epidemic.

A model for influenza with vaccination and antiviral treatment

Compartmental models for influenza that include control by vaccination and antiviral treatment are formulated. Analytic expressions for the basic reproduction number, control reproduction number and the final size of the epidemic are derived for this general class of disease transmission models. Sensitivity and uncertainty analyses of the dependence of the control reproduction number on the parameters of the model give a comparison of the various intervention strategies. Numerical computations of the deterministic models are compared with those of recent stochastic simulation influenza models. Predictions of the deterministic compartmental models are in general agreement with those of the stochastic simulation models.

Further Notes on the Basic Reproduction Number

The basic reproduction number, R0 is a measure of the potential for disease spread in a population. Mathematically, R0 is a threshold for stability of a disease-free equilibrium and is related to the peak and final size of an epidemic. The purpose of these notes is to give a precise definition and algorithm for obtaining R0 for a general compartmental ordinary differential equation model of disease transmission. Several examples of calculating R0 are included, and the epidemiological interpretation of this threshold parameter is connected to the local and global stability of a disease-free equilibrium.

GLOBAL STABILITY OF A NONLINEAR VIRAL INFECTION MODEL WITH INFINITELY DISTRIBUTED INTRACELLULAR DELAYS AND CTL IMMUNE RESPONSES

Determining sharp conditions for the global stability of equilibria remains one of the most challenging problems in the analysis of models for the management and control of biological systems. Yet ...

Trail following in ants: individual properties determine population behaviour

This paper deals with the purposeful marking of trails as a mechanism for coordinating movement. Patterns of motion are adapted to the environmental conditions, the functions to be carried out, and the condition of the organism; therefore, the networks of trails must change both quantitatively and qualitatively over time. The nature of such changes, and how they are controlled at the individual level are discussed. In particular, we show that slight modulations in individual traits, in the trail marker, or in the size of the group can account for major changes in movement patterns at the population level such as abrupt transitions from diffuse area-covering networks to focused trunk trails. Using a mathematical model and computer (cellular automata) simulation we show that trunk trails carrying a high density of traffic can form spontaneously under suitable conditions from an initially randomly distributed group. The key to this self-organizing property stems from interactions between individuals that lead to a “collective effect” in recruitment to trails: the influence of small groups of individuals increases rapidly with group size. The dichotomy between high traffic (strong) trunk trails versus diffuse (weak) networks is discussed.

When Eradication is not an Option: Modeling Strategies for Electrofishing Suppression of Nonnative Brook Trout to Foster Persistence of Sympatric Native Cutthroat Trout in Small Streams

Abstract Subspecies of inland cutthroat trout Oncorhynchus clarkii occupy a fraction of their historic ranges, and displacement by nonnative brook trout Salvelinus fontinalis is among the greatest threats to existing populations in small, headwater streams. Electrofishing is often used to suppress brook trout and enhance cutthroat trout populations, but these efforts are labor intensive and costly. To help managers more effectively plan and implement brook trout control programs, we used survival estimates from a field experiment to construct matrix population models for both species and linked the models by making the vital rates of young cutthroat trout a function of brook trout density to represent the effect of invasion. We then explored the response of cutthroat trout populations (growth rate [λ] and probability of persistence for 50 years) to brook trout suppression across various levels of electrofishing frequency, sampling intensity, capture efficiency, and brook trout immigration rate. Cutthroat ...

Sustained and transient oscillations and chaos induced by delayed antiviral immune response in an immunosuppressive infection model

Sustained and transient oscillations are frequently observed in clinical data for immune responses in viral infections such as human immunodeficiency virus, hepatitis B virus, and hepatitis C virus. To account for these oscillations, we incorporate the time lag needed for the expansion of immune cells into an immunosuppressive infection model. It is shown that the delayed antiviral immune response can induce sustained periodic oscillations, transient oscillations and even sustained aperiodic oscillations (chaos). Both local and global Hopf bifurcation theorems are applied to show the existence of periodic solutions, which are illustrated by bifurcation diagrams and numerical simulations. Two types of bistability are shown to be possible: (i) a stable equilibrium can coexist with another stable equilibrium, and (ii) a stable equilibrium can coexist with a stable periodic solution.

Non-destructive estimation of wheat leaf chlorophyll content from hyperspectral measurements through analytical model inversion

Optimizing nitrogen (N) fertilization in crop production by in-season measurements of crop N status may improve fertilizer N use efficiency. Hyperspectral measurements may be used to assess crop N status indirectly by estimating leaf and canopy chlorophyll content. This study evaluated the ability of the PROSAIL canopy-level reflectance model to predict leaf chlorophyll content of spring wheat (Triticum aestivum L.) during the growth stages between pre-tillering (Zadoks Growth Stage (ZGS 15)) to booting (ZGS50). Spring wheat was grown under different N fertility rates (0–200 kg N ha−1) in 2002. Canopy reflectance, leaf chlorophyll content, N content and leaf area index (LAI) values were measured. There was a weakly significant trend for the PROSAIL model to over-estimate LAI and under-estimate leaf chlorophyll content. To compensate for this interdependency by the model, a canopy chlorophyll content parameter (the product of leaf chlorophyll content and LAI) was calculated. The estimation accuracy for canopy chlorophyll content was generally low earlier in the growing season. This failure of the PROSAIL model to estimate leaf and canopy variables could be attributed to model sensitivity to canopy architecture. Earlier in the growing season, full canopy closure was not yet achieved, resulting in a non-homogenous canopy and strong soil background interference. The canopy chlorophyll content parameter was predicted more accurately than leaf chlorophyll content alone at booting (ZGS 45). A strong relationship between canopy chlorophyll content and canopy N content at ZGS 45 indicates that the PROSAIL model may be used as a tool to predict wheat N status from canopy reflectance measurements at booting or later.