Separable mixing: The general formulation and a particular example focusing on mask efficiency.

The aim of this short note is twofold. First, we formulate the general Kermack-McKendrick epidemic model incorporating static heterogeneity and show how it simplifies to a scalar Renewal Equation (RE) when separable mixing is assumed. A key general feature is that all information about the heterogeneity is encoded in one nonlinear real valued function of a real variable. Next, we specialize the model ingredients so that we can study the efficiency of mask wearing as a non-pharmaceutical intervention to reduce the spread of an infectious disease. Our main result affirms that the best way to protect the population as a whole is to protect yourself. This qualitative insight was recently derived in the context of an SIR network model. Here, we extend the conclusion to proportionate mixing models incorporating a general function describing expected infectiousness as a function of time since infection.

Waning and boosting: on the dynamics of immune status.

The aim is to describe the distribution of immune status (as captured by antibody level) on the basis of a within-host submodel for continuous waning and occasional boosting. Inspired by Feller's fundamental work and the more recent delay equation formulation of models for the dynamics of physiologically structured populations, we derive, for given force of infection, a linear renewal equation. The solution is obtained by generation expansion, with the generation number corresponding to the number of times the individual became infected. Our main result provides a precise characterization of the stable distribution of immune status.

Shallow genetic divergence and species delineations in the endemic Labeobarbus species flock of Lake Tana, Ethiopia.

To assess whether the species distinctions of Lake Tana's Labeobarbus spp. are supported by genetic information, microsatellite markers were used. A total of 376 Labeobarbus spp., belonging to 24 populations of 11 species from three regions of the lake (north, south and east), were sampled. Eight microsatellite markers were analysed. In general, differences between conspecific populations were smaller than differences between populations of different species. For six species, conspecific populations from different regions in the lake were consistently more similar than populations of other species from the same region. For four species this was not the case, while for one species two populations were similar, but different from the third population. River-spawning species appeared to be more distinct than presumed lake spawners. On the species level, there was a significant correlation between genetic and morphological differentiation, especially in morphological aspects associated with ecological functioning. This suggests that genetic differentiation arose together with adaptive radiation, although the overall genetic differentiation among the Lake Tana Labeobarbus spp. is small.

A two-phase within-host model for immune response and its application to serological profiles of pertussis.

We present a simple phenomenological within-host model describing both the interaction between a pathogen and the immune system and the waning of immunity after clearing of the pathogen. We implement the model into a Bayesian hierarchical framework to estimate its parameters for pertussis using Markov chain Monte Carlo methods. We show that the model captures some essential features of the kinetics of titers of IgG against pertussis toxin. We identify a threshold antibody level that separates a large increase in antibody level upon infection from a small increase and accordingly might be interpreted as a threshold separating clinical from subclinical infections. We contrast predictions of the model with observations reported in the literature and based on independent data and find a remarkable correspondence.

A didactical note on the advantage of using two parameters in Hopf bifurcation studies.

In order to maximize the information that a linearized stability analysis provides, one should work with two free parameters rather than one. Moreover, it is recommended to first consider coefficients in the characteristic equation as parameters and in a second step (try to) invert the map that defines the coefficients in terms of the parameters as they occur in the original equation. Our aim is to substantiate these claims by way of a delay equation example taken from the literature.

Dynamic concurrent partnership networks incorporating demography.

We introduce a population model that incorporates From a mathematical point of view we deal with continuous-time Markov chains at the individual level, with the interaction between individuals captured by a global variable describing opportunities for new partnerships. We show that for large time a stationary distribution is attained and we deduce various statistical features of that distribution, with particular attention for concurrency, i.e. the overlap in time of multiple partnerships of one and the same individual. Our ultimate motivation is to model the spread of sexually transmitted infections in the population, for which the present paper serves as a prelude.

On the formulation of epidemic models (an appraisal of Kermack and McKendrick).

The aim of this paper is to show that a large class of epidemic models, with both demography and non-permanent immunity incorporated in a rather general manner, can be mathematically formulated as a scalar renewal equation for the force of infection.

Modeling non-inherited antibiotic resistance.

A mathematical model is presented for the increase and decrease of non-inherited antibiotic resistance levels in bacteria. The model is applied to experimental data on E. coli exposed to amoxicillin or tetracyclin in different concentrations. The parameters of the model are estimated using a Monte Carlo Markov Chain method. The model accurately describes build-up and decline of antibiotic resistance caused by physiological adaptations as long as no genetic changes have occurred. The main conclusion of the analysis is that short time periods are sufficient to re-obtain low MIC-values after long-lasting exposure to these antibiotics.

The protective effects of temporary immunity under imposed infection pressure.

The aim of this paper is to show in explicit detail that, due to the effects of waning and boosting of immunity, an increasing force of infection does not necessarily lead to an increase in the incidence of disease. Under certain conditions, a decrease of the force of infection may in fact lead to an increase of the incidence of disease. Thus we confirm and reinforce the conclusions from Águas et al. (2006), concerning pertussis. We do so, however, in the context of Campylobacter infections in humans deriving from animal reservoirs. For such an externally 'driven' epidemic we can ignore the transmission feedback cycle and treat the force of infection as a parameter. As this parameter is, to a certain extent, under public health control, our findings constitute an important warning: reducing exposure may not necessarily lead to a reduction in the occurrence of clinical illness. In a second part of the paper we relate the model parameters to the available data concerning campylobacteriosis.

How to lift a model for individual behaviour to the population level?

The quick answer to the title question is: by bookkeeping; introduce as p(opulation)-state a measure telling how the individuals are distributed over their common i(ndividual)-state space, and track how the various i-processes change this measure. Unfortunately, this answer leads to a mathematical theory that is technically complicated as well as immature. Alternatively, one may describe a population in terms of the history of the population birth rate together with the history of any environmental variables affecting i-state changes, reproduction and survival. Thus, a population model leads to delay equations. This delay formulation corresponds to a restriction of the p-dynamics to a forward invariant attracting set, so that no information is lost that is relevant for long-term dynamics. For such equations there exists a well-developed theory. In particular, numerical bifurcation tools work essentially the same as for ordinary differential equations. However, the available tools still need considerable adaptation before they can be practically applied to the dynamic energy budget (DEB) model. For the time being we recommend simplifying the i-dynamics before embarking on a systematic mathematical exploration of the associated p-behaviour. The long-term aim is to extend the tools, with the DEB model as a relevant goal post.

The concept of population in clonal organisms: mosaics of temporally colonized patches are forming highly diverse meadows of Zostera marina in Brittany.

Seagrasses structure some of the world's key coastal ecosystems presently in decline due to human activities and global change. The ability to cope with environmental changes and the possibilities for shifts in distribution range depend largely on their evolvability and dispersal potential. As large-scale data usually show strong genetic structure for seagrasses, finer-grained work is needed to understand the local processes of dispersal, recruitment and colonization that could explain the apparent lack of exchange across large distances. We aimed to assess the fine-grained genetic structure of one of the most important and widely distributed seagrasses, Zostera marina, from seven meadows in Brittany, France. Both classic population genetics and network analysis confirmed a pattern of spatial segregation of polymorphism at both regional and local scales. One location exhibiting exclusively the variety 'angustifolia' did not appear more differentiated than the others, but instead showed a central position in the network analysis, confirming the status of this variety as an ecotype. This phenotypic diversity and the high allelic richness at nine microsatellites (2.33-9.67 alleles/locus) compared to levels previously reported across the distribution range, points to Brittany as a centre of diversity for Z. marina at both genetic and phenotypic levels. Despite dispersal potential of several 100 m, a significant pattern of genetic differentiation, even at fine-grained scale, revealed 'genetic patchiness'. Meadows seem to be composed of a mosaic of clones with distinct origins in space and time, a result that calls into question the accuracy of the concept of populations for such partially clonal species.

The construction of next-generation matrices for compartmental epidemic models.

The basic reproduction number (0) is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation of (0) where finitely many different categories of individuals are recognized. We clear up confusion that has been around in the literature concerning the construction of this matrix, specifically for the most frequently used so-called compartmental models. We present a detailed easy recipe for the construction of the NGM from basic ingredients derived directly from the specifications of the model. We show that two related matrices exist which we define to be the NGM with large domain and the NGM with small domain. The three matrices together reflect the range of possibilities encountered in the literature for the characterization of (0). We show how they are connected and how their construction follows from the basic model ingredients, and establish that they have the same non-zero eigenvalues, the largest of which is the basic reproduction number (0). Although we present formal recipes based on linear algebra, we encourage the construction of the NGM by way of direct epidemiological reasoning, using the clear interpretation of the elements of the NGM and of the model ingredients. We present a selection of examples as a practical guide to our methods. In the appendix we present an elementary but complete proof that (0) defined as the dominant eigenvalue of the NGM for compartmental systems and the Malthusian parameter r, the real-time exponential growth rate in the early phase of an outbreak, are connected by the properties that (0) > 1 if and only if r > 0, and (0) = 1 if and only if r = 0.

Numerical equilibrium analysis for structured consumer resource models.

In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for "Daphnia consuming algae" models in C-code. The results obtained by way of this implementation are shown in the form of graphs.

Can a species keep pace with a shifting climate?

Consider a patch of favorable habitat surrounded by unfavorable habitat and assume that due to a shifting climate, the patch moves with a fixed speed in a one-dimensional universe. Let the patch be inhabited by a population of individuals that reproduce, disperse, and die. Will the population persist? How does the answer depend on the length of the patch, the speed of movement of the patch, the net population growth rate under constant conditions, and the mobility of the individuals? We will answer these questions in the context of a simple dynamic profile model that incorporates climate shift, population dynamics, and migration. The model takes the form of a growth-diffusion equation. We first consider a special case and derive an explicit condition by glueing phase portraits. Then we establish a strict qualitative dichotomy for a large class of models by way of rigorous PDE methods, in particular the maximum principle. The results show that mobility can both reduce and enhance the ability to track climate change that a narrow range can severely reduce this ability and that population range and total population size can both increase and decrease under a moving climate. It is also shown that range shift may be easier to detect at the expanding front, simply because it is considerably steeper than the retreating back.

Calculating the time to extinction of a reactivating virus, in particular bovine herpes virus.

The expected time to extinction of a herpes virus is calculated from a rather simple population-dynamical model that incorporates transmission, reactivation and fade-out of the infectious agent. We also derive the second and higher moments of the distribution of the time to extinction. These quantities help to assess the possibilities to eradicate a reactivating infection. The key assumption underlying our calculations is that epidemic outbreaks are fast relative to the time scale of demographic turnover. Four parameters influence the expected time to extinction: the reproduction ratio, the reactivation rate, the population size, and the demographic turn-over in the host population. We find that the expected time till extinction is very long when the reactivation rate is high (reactivation is expected more than once in a life time). Furthermore, the infectious agent will go extinct much more quickly in small populations. This method is applied to bovine herpes virus (BHV) in a cattle herd. The results indicate that without vaccination, BHV will persist in large herds. The use of a good vaccine can induce eradication of the infection from a herd within a few decades. Additional measures are needed to eradicate the virus from a whole region within a similar time-span.

An algorithm to estimate the importance of bacterial acquisition routes in hospital settings.

An algorithm is presented to calculate likelihoods of acquisition routes using only individual patient data concerning period of stay and microbiologic surveillance (without genotyping). The algorithm also produces estimates for the prevalence and the number of acquisitions by each route. The algorithm is applied to colonization data of third-generation cephalosporin-resistant Enterobacteriaceae (CRE) from September 2001 to May 2002 in two intensive care units (ICUs) (n = 277 and n = 180, respectively) of Utrecht, Kingdom of the Netherlands. Genotyping and epidemiologic linkage are used as the reference standard. Surveillance cultures were obtained on admission and twice weekly thereafter. All CREs were genotyped. According to the reference standard, the daily prevalence of CRE in ICU-1 and ICU-2 was 26.1% (standard deviation: 15.4) and 15.1% (standard deviation: 13.4), respectively, with five of 23 (21.7%) and six of 21 (28.6%) cases of acquired colonization being of exogenous origin, respectively. On the basis of the algorithm, the endogenous route was responsible for more acquisitions than the exogenous route (p = 0.003 and p < 0.001 for ICU-1 and ICU-2, respectively). The estimated number of acquisitions is 30 and 27, and the estimated prevalence is 27.6% and 17.6% for ICU-1 and ICU-2, respectively. By use of longitudinal colonization data only, the algorithm determines the relative importance of acquisition routes taking patient dependency into account.

Relative effects of barrier precautions and topical antibiotics on nosocomial bacterial transmission: results of multi-compartment models.

Nosocomial bacterial infections in critically ill patients are generally preceded by asymptomatic carriage (i.e. colonization) at one, or even several, body sites such as the skin, the gastro-intestinal and the respiratory tract. Different routes of transmission between the colonized sites create a complex epidemiology, which is additionally complicated by the smallness of the patient population size and the rapid patient turnover, characteristic for intensive care units (ICUs). Naturally occurring large fluctuations in the prevalence of colonization make it very difficult to determine the efficacy of control measures that aim to reduce the prevalence of antibiotic-resistant bacteria in ICUs. Theoretical models can sharpen our intuition through carefully designed thought experiments. In this spirit, we introduce and investigate two models that incorporate the fact that patients may be colonized at multiple body sites. Our study can be applied to several pathogens commonly found in ICUs, such Pseudomonas Aeruginosa, enteric Gram-negative bacteria, MRSA and enterococci. We evaluate the effects of barrier precautions (improved hygiene, use of gloves and gowns, etc.) and of administration of nonabsorbable antibiotics on the prevalence of colonization in ICUs and find that the effect of the controversial, though widely used, antibiotic prophylaxis can only be substantial if the patient-to-patient transmission has already been reduced to a subcritical level by barrier precautions. Taking into account that the very use of antibiotics may increase the selection for resistant strains and may thereby only add to the ever increasing problem of antibiotic resistance, our findings hence represent a firm theoretical argument against the routine use of topical antimicrobial prophylaxis for infection control.

Controlling methicillin-resistant Staphylococcus aureus: quantifying the effects of interventions and rapid diagnostic testing.

Control of nosocomial transmission of methicillin-resistant Staphylococcus aureus (MRSA) has been unsuccessful in most countries. Yet, some countries have maintained low endemic levels by implementing nationwide MRSA-specific infection control measures, such as "search & destroy" (S&D). These strategies, however, are not based on well designed studies, and their use in countries with high levels of endemicity is controversial. We present a stochastic three-hospital model and an analytical one-hospital model to quantify the effectiveness of different infection control measures and to predict the effects of rapid diagnostic testing (RDT) on isolation needs. Isolation of MRSA carriers identified by clinical cultures is insufficient to control MRSA. However, combined with proactive search (of high-risk patients on admission and/or contacts of index patients), it will maintain prevalence levels <1%. Concerted implementation of S&D in countries with high nosocomial endemicity reduces nosocomial prevalence to <1% within 6 years. Stepwise implementation of control measures can reduce isolation capacities needed. RDT can reduce isolation needs by >90% in low-endemic settings and by 20% in high-endemic settings. Surveillance of colonization and improved hand hygiene can markedly increase control efficacy. These findings strongly suggest that: (i) causality exists between S&D and low MRSA prevalence; (ii) isolating MRSA carriers identified by clinical cultures as a single measure is insufficient for control; (iii) a combined approach of isolation and screening confers efficacy; and (iv) MRSA-prevalence levels can be reduced to <1% in high-endemic settings by S&D or a stepwise approach to interventions. RDT can markedly enhance feasibility.

On the (dis) advantages of cannibalism.

Cannibalism is an interaction between individuals that can produce counter- intuitive effects at the population level. A striking effect is that a population may persist under food conditions such that the non-cannibalistic variant is doomed to go extinct. This so-called life boat mechanism has received considerable attention. Implicitly, such studies sometimes suggest, that the life boat mechanism procures an evolutionary advantage to the cannibalistic trait. Here we compare, in the context of a size structured population model, the conditions under which the life boat mechanism works, with those that guarantee, that a cannibalistic mutant can invade successfully under the steady environmental conditions as set by a non-cannibalistic resident. We find qualitative agreement and quantitative difference. In particular, we find that a prerequisite for the life boat mechanism is, that cannibalistic mutants are successful invaders. Roughly speaking, our results show that cannibalism brings advantages to both the individuals and the population when adult food is limiting.