A Contact-Network-Based Formulation of a Preferential Mixing Model

Heterogeneity in the number of potentially infectious contacts and connectivity correlations (“like attaches to like” i.e., assortatively mixed or “opposites attract” i.e., disassortatively mixed) have important implications for the value of the basic reproduction ratio R ₀ and final epidemic size. In this paper, we present a contact-network-based derivation of a simple differential equation model that accounts for preferential mixing based on the number of contacts. We show that results based on this model are in good qualitative agreement with results obtained from preferential mixing models used in the context of sexually transmitted diseases (STDs). This simple model can accommodate any mixing pattern ranging from completely disassortative to completely assortative and allows the derivation of a series of analytical results.     

On the Final Size of Epidemics with Seasonality

We first study an SIR system of differential equations with periodic coefficients describing an epidemic in a seasonal environment. Unlike in a constant environment, the final epidemic size may not be an increasing function of the basic reproduction number ℛ₀ or of the initial fraction of infected people. Moreover, large epidemics can happen even if ℛ₀<1. But like in a constant environment, the final epidemic size tends to 0 when ℛ₀<1 and the initial fraction of infected people tends to 0. When ℛ₀>1, the final epidemic size is bigger than the fraction 1−1/ℛ₀ of the initially nonimmune population. In summary, the basic reproduction number ℛ₀ keeps its classical threshold property but many other properties are no longer true in a seasonal environment. These theoretical results should be kept in mind when analyzing data for emerging vector-borne diseases (West-Nile, dengue, chikungunya) or air-borne diseases (SARS, pandemic influenza); all these diseases being influenced by seasonality.     

A note on a paper by Erik Volz: SIR dynamics in random networks

Recent work by Volz (J Math Biol 56:293–310, 2008) has shown how to calculate the growth and eventual decay of an SIR epidemic on a static random network, assuming infection and recovery each happen at constant rates. This calculation allows us to account for effects due to heterogeneity and finiteness of degree that are neglected in the standard mass-action SIR equations. In this note we offer an alternate derivation which arrives at a simpler—though equivalent—system of governing equations to that of Volz. This new derivation is more closely connected to the underlying physical processes, and the resulting equations are of comparable complexity to the mass-action SIR equations. We further show that earlier derivations of the final size of epidemics on networks can be reproduced using the same approach, thereby providing a common framework for calculating both the dynamics and the final size of an epidemic spreading on a random network. Under appropriate assumptions these equations reduce to the standard SIR equations, and we are able to estimate the magnitude of the error introduced by assuming the SIR equations.     

Two-step concerted mechanism for alkane hydroxylation on the ferryl active site of methane monooxygenase

 A two-step concerted mechanism for the conversion of methane to methanol catalyzed by soluble methane monooxygenase (sMMO) is discussed. We propose that the enzymatic reaction mechanism is essentially the same as that of the gas-phase methane-methanol conversion by the bare FeO⁺ complex. In the initial stage of our mechanism, the ferryl (Fe—O) "iron" active site of intermediate Q and substrate methane come into contact to form the initial Q (CH₄) complex with an OFe—CH₄ bond. The C—H bonds of methane are significantly weakened by the formation of a five-coordinate carbon species, through orbital interactions between a C _3v - or D _2d -distorted methane and the Fe—O active site. The important transition state for an H atom abstraction exhibits a four-centered structure. The generated intermediate involves an HO—Fe—CH₃ moiety, and it is then converted into the final product complex including methanol as a ligand through a methyl migration that occurs via a three-centered transition state. The two-step concerted mechanism is consistent with recent experiments on regioselectivity of enzyme-catalyzed alkane hydroxylations. Received: 15 September 1997 / Accepted: 20 December 1997     

Discussion: The Kermack-McKendrick epidemic threshold theorem

The paper reviews the work of Kermack and McKendrick on the development of simple mathematical models of the transmission dynamics of viral and bacterial infectious agents within population of hosts. The focus of attention is centred on the notion of a threshold density of susceptible hosts to trigger an epidemic and recent extensions of this idea as expressed in the definition of a basic or case reproductive rate of infection. The main body of the paper examines recent developments of the basic Kermack-McKendrick model with an emphasis on deterministic models that describe various types of heterogeneity in the processes that determine transmission between infected and susceptible persons. Particular attention is given to the role of behavioural heterogeneity within the framework of a contact or mixing matrix which defines “who acquires infection from whom”.     

Temporal genetic samples indicate small effective population size of the endangered yellow-eyed penguin

There is an increasing awareness that the long-term viability of endemic island populations is negatively affected by genetic factors associated with population bottlenecks and/or persistence at small population size. Here we use contemporary samples and historic museum specimens (collected 1888–1938) to estimate the effective population size (N _e) for the endangered yellow-eyed penguin (Megadyptes antipodes) in South Island, New Zealand, and evaluate the genetic concern for this iconic species. The South Island population of M. antipodes—constituting almost half of the species’ census size—is thought to be descended from a small number of founders that reached New Zealand just a few hundred years ago. Despite intensive conservation measures, this population has shown dramatic fluctuations in size over recent decades. We compare estimates of the harmonic mean N _e for this population, obtained using one moment and three likelihood based-temporal methods, including one method that simultaneously estimates migration rate. Evaluation of the N _e estimates reveals a harmonic mean N _e in the low hundreds. Additionally, the inferred low immigration rates (m = 0.003) agree well with contemporary migration rate estimates between the South Island and subantarctic populations of M. antipodes. The low N _e of South Island M. antipodes is likely affected by strong fluctuations in population size, and high variance in reproductive success. These results show that genetic concerns for this population are valid and that the long-term viability of this species may be compromised by reduced adaptive potential.     

Generalization of the variance-covariance method for microdosimetric measurements

The variance method of microdosimetric measurements and its extension, the variance-covariance method, permit the determination of an essential parameter of radiation quality, the dose mean event size,y _d. The methods have — among other advantages — the feature that they permit measurements for smaller simulated sites than the conventional single-event technique. It is, therefore, desirable to employ them also for the determination of further moments of the distribution ofy. The formulae for the first three moments are here derived both for the case of constant dose rate and of fluctuating dose rates. A second article will use the same mathematical approach to deduce formulae that remain valid even if there are slow changes of the ratio of dose rates in the two detectors for the variance-covariance method. A third article will explore — in terms of microdosimetric data — the applicability of the formulae.     

Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing

In this paper, we outline the theory of epidemic percolation networks and their use in the analysis of stochastic susceptible-infectious-removed (SIR) epidemic models on undirected contact networks. We then show how the same theory can be used to analyze stochastic SIR models with random and proportionate mixing. The epidemic percolation networks for these models are purely directed because undirected edges disappear in the limit of a large population. In a series of simulations, we show that epidemic percolation networks accurately predict the mean outbreak size and probability and final size of an epidemic for a variety of epidemic models in homogeneous and heterogeneous populations. Finally, we show that epidemic percolation networks can be used to re-derive classical results from several different areas of infectious disease epidemiology. In an Appendix, we show that an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model in a closed population and prove that the distribution of outbreak sizes given the infection of any given node in the SIR model is identical to the distribution of its out-component sizes in the corresponding probability space of epidemic percolation networks. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR models in closed populations.     

Contrasting growth strategies of pond versus marine populations of nine-spined stickleback (Pungitius pungitius): a combined effect of predation and competition?

Gigantism in isolated ponds in the absence of sympatric fish species has previously been observed in nine-spined sticklebacks (Pungitius pungitius). Patterns in sexual size dimorphism suggested that fecundity selection acting on females might be responsible for the phenomenon. However, the growth strategy behind gigantism in pond sticklebacks has not been studied yet. Here, we compared von Bertalanffy growth parameters of four independent nine-spined stickleback populations reared in a common laboratory environment: two coastal marine (typical size) and two pond (giant size) populations. We found that both pond populations had larger estimated final size than marine populations, which in turn exhibited higher intrinsic growth rates than the pond populations. Female growth strategies were more divergent among marine and pond populations than those of males. Asymptotic body size and intrinsic growth rate were strongly negatively correlated. Hence, pond versus marine populations exhibited different growth strategies along a continuum. Our data suggest that quick maturation—even with the cost of being small (low fecundity)—is favoured in marine environments. On the contrary, growth to a giant final size (high fecundity)—even if it entails extended growth period—is favoured in ponds. We suggest that the absence (ponds) versus presence (marine environment) of sympatric predatory fish species, and the consequent change in the importance of intraspecific competition are responsible for the divergence in growth strategies. The sex-dependence of the patterns further emphasizes the role of females in the body size divergence in the species. Possible alternative hypotheses are also discussed.     

Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model

Multiple transmission pathways exist for many waterborne diseases, including cholera, Giardia, Cryptosporidium, and Campylobacter. Theoretical work exploring the effects of multiple transmission pathways on disease dynamics is incomplete. Here, we consider a simple ODE model that extends the classical SIR framework by adding a compartment (W) that tracks pathogen concentration in the water. Infected individuals shed pathogen into the water compartment, and new infections arise both through exposure to contaminated water, as well as by the classical SIR person–person transmission pathway. We compute the basic reproductive number (ℛ₀), epidemic growth rate, and final outbreak size for the resulting “SIWR” model, and examine how these fundamental quantities depend upon the transmission parameters for the different pathways. We prove that the endemic disease equilibrium for the SIWR model is globally stable. We identify the pathogen decay rate in the water compartment as a key parameter determining when the distinction between the different transmission routes in the SIWR model is important. When the decay rate is slow, using an SIR model rather than the SIWR model can lead to under-estimates of the basic reproductive number and over-estimates of the infectious period.     

The Final Size of an Epidemic and Its Relation to the Basic Reproduction Number

We study the final size equation for an epidemic in a subdivided population with general mixing patterns among subgroups. The equation is determined by a matrix with the same spectrum as the next generation matrix and it exhibits a threshold controlled by the common dominant eigenvalue, the basic reproduction number R₀{\mathcal{R}_{0}}: There is a unique positive solution giving the size of the epidemic if and only if R₀{\mathcal{R}_{0}} exceeds unity. When mixing heterogeneities arise only from variation in contact rates and proportionate mixing, the final size of the epidemic in a heterogeneously mixing population is always smaller than that in a homogeneously mixing population with the same basic reproduction number R₀{\mathcal{R}_{0}}. For other mixing patterns, the relation may be reversed.     

Impulsive Vaccination of an SEIRS Model with Time Delay and Varying Total Population Size

Pulse vaccination is an effective and important strategy for the elimination of infectious diseases. A delayed SEIRS epidemic model with pulse vaccination and varying total population size is proposed in this paper. We point out, if R* < 1, the infectious population disappear so the disease dies out, while if R _*; > 1, the infectious population persist. Our results indicate that a long period of pulsing or a small pulse vaccination rate is sufficient condition for the permanence of the model.     

The basic reproduction number and the probability of extinction for a dynamic epidemic model

We consider the spread of an epidemic through a population divided into n sub-populations, in which individuals move between populations according to a Markov transition matrix Σ and infectives can only make infectious contacts with members of their current population. Expressions for the basic reproduction number, R₀, and the probability of extinction of the epidemic are derived. It is shown that in contrast to contact distribution models, the distribution of the infectious period effects both the basic reproduction number and the probability of extinction of the epidemic in the limit as the total population size N → ∞. The interactions between the infectious period distribution and the transition matrix Σ mean that it is not possible to draw general conclusions about the effects on R₀ and the probability of extinction. However, it is shown that for n = 2, the basic reproduction number, R₀, is maximised by a constant length infectious period and is decreasing in ?, the speed of movement between the two populations.     

The model of Kermack and McKendrick for the plague epidemic in Bombay and the type reproduction number with seasonality

The figure showing how the model of Kermack and McKendrick fits the data from the 1906 plague epidemic in Bombay is the most reproduced figure in books discussing mathematical epidemiology. In this paper we show that the assumption of constant parameters in the model leads to quite unrealistic numerical values for these parameters. Moreover the reports published at the time show that plague epidemics in Bombay occurred in fact with a remarkable seasonal pattern every year since 1897 and at least until 1911. So the 1906 epidemic is clearly not a good example of epidemic stopping because the number of susceptible humans has decreased under a threshold, as suggested by Kermack and McKendrick, but an example of epidemic driven by seasonality. We present a seasonal model for the plague in Bombay and compute the type reproduction numbers associated with rats and fleas, thereby extending to periodic models the notion introduced by Roberts and Heesterbeek.     

Does the badge of status influence parental care and investment in house sparrows? An experimental test

Theory predicts that traits which signal parental quality might evolve in males of species with biparental care. In avian species, male ornaments may be the most likely candidates for such signals. Male house sparrows (Passer domesticus) possess a black throat patch often referred to as a “badge” or a “badge of status”. By assuming a trade-off between male attractiveness (reflected in male ornaments) and parental care under the differential allocation hypothesis, we predicted that badge size would be negatively correlated with male parental investment. An experiment in which the badge was enlarged in one group and unchanged in a control group was conducted. Our manipulation was predicted to affect female as well as male parental investment. However, we found that eight variables associated with parental investment—the start date for breeding, clutch size, male and female incubation time, male and female food provisioning rate, and average chick weight and the number of fledglings—barely differed between treatments. Also, little evidence for correlations between natural variation in badge size and any of these eight variables was found. Instead, the start date for breeding and the number of fledglings were significantly correlated with both male and female age, while clutch size increased with female age. Female condition was a positive predictor of clutch size and number of fledglings. Female tarsus length, unexpectedly, is related to both male and female incubation time. Badge size was also positively correlated with male age. However, parental age (male or female) was not related to parental care. We conclude that badge size does not signal parental quality, but that the ages of both sexes and the condition of the female play significant roles in the reproductive performance of this species.     

Epidemic models with uncertainty in the reproduction number

One of the first quantities to be estimated at the start of an epidemic is the basic reproduction number, ${\mathcal{R}_0}$ . The progress of an epidemic is sensitive to the value of ${\mathcal{R}_0}$ , hence we need methods for exploring the consequences of uncertainty in the estimate. We begin with an analysis of the SIR model, with ${\mathcal{R}_0}$ specified by a probability distribution instead of a single value. We derive probability distributions for the prevalence and incidence of infection during the initial exponential phase, the peaks in prevalence and incidence and their timing, and the final size of the epidemic. Then, by expanding the state variables in orthogonal polynomials in uncertainty space, we construct a set of deterministic equations for the distribution of the solution throughout the time-course of the epidemic. The resulting dynamical system need only be solved once to produce a deterministic stochastic solution. The method is illustrated with ${\mathcal{R}_0}$ specified by uniform, beta and normal distributions. We then apply the method to data from the New Zealand epidemic of H1N1 influenza in 2009. We apply the polynomial expansion method to a Kermack–McKendrick model, to simulate a forecasting system that could be used in real time. The results demonstrate the level of uncertainty when making parameter estimates and projections based on a limited amount of data, as would be the case during the initial stages of an epidemic. In solving both problems we demonstrate how the dynamical system is derived automatically via recurrence relationships, then solved numerically.     

The uptake of zinc by erythrocytes under near-physiological conditions

The in vitro uptake of zinc by erythrocytes was measured under near-physiological conditions, using⁶⁵Zn as a radioactive tracer. Because of the presence of serum albumin—a strong zinc ligand—a low concentration of medium free zinc was maintained. Under these conditions a high-affinity carrier for zinc transport over the cell membrane was identified. With human erythrocytes, a Michaelis constant (K _m ) of 0.2 nM with respect to free medium zinc was measured and aV _max of 4.5 nmoles Zn transported per h/g dry wt. TheK _m for medium Zn increases when the size of the internal erythrocytic Zn pool is augmented, whereasV _max remains virtually unchanged. A model to explain this phenomenon is proposed. It is suggested that this phenomenon could underlie observations, confirmed here, that the in vitro uptake of Zn by animal erythrocytes depends on the Zn status of the animal.