Calculation of Disease Dynamics in a Population of Households

Early mathematical representations of infectious disease dynamics assumed a single, large, homogeneously mixing population. Over the past decade there has been growing interest in models consisting of multiple smaller subpopulations (households, workplaces, schools, communities), with the natural assumption of strong homogeneous mixing within each subpopulation, and weaker transmission between subpopulations. Here we consider a model of SIRS (susceptible-infectious-recovered-susceptible) infection dynamics in a very large (assumed infinite) population of households, with the simplifying assumption that each household is of the same size (although all methods may be extended to a population with a heterogeneous distribution of household sizes). For this households model we present efficient methods for studying several quantities of epidemiological interest: (i) the threshold for invasion; (ii) the early growth rate; (iii) the household offspring distribution; (iv) the endemic prevalence of infection; and (v) the transient dynamics of the process. We utilize these methods to explore a wide region of parameter space appropriate for human infectious diseases. We then extend these results to consider the effects of more realistic gamma-distributed infectious periods. We discuss how all these results differ from standard homogeneous-mixing models and assess the implications for the invasion, transmission and persistence of infection. The computational efficiency of the methodology presented here will hopefully aid in the parameterisation of structured models and in the evaluation of appropriate responses for future disease outbreaks.     

The theoretical possibility of reverse translation of proteins into genes

It is shown on a theoretical basis that the existence of a “power law” relationship between body mass M and total metabolic heat generation rate Q of the form Q = kM^α does not uniquely determine the dependence of metabolic rate on body temperature. However, it is shown that a particular assumption for this temperature dependence, successful in other problems, does predict a “power law” similar to the empirical one. At the same time it also accounts satisfactorily for the linear dependence of metabolic rate on ambient temperature.     

A formula for the mean infection success per miracidium and a method of proving the homogeneous susceptibility of snail populations to trematode infection

A formula for computing the mean infection success per miracidium (“infection index”) is given for every permutation of the numbers of snails and miracidia in miracidial exposure experiments. This “infection index” allows the comparison of the results from quantitatively different exposure methods and may be important for characterizing special trematode- intermediate host systems. By means of the “infection index” it is shown that a laboratory population of Lymnaea stagnalis is homogeneous with regard to its susceptibility to Isthmiophora melis miracidia, i.e. that there is no appreciable rate of resistant individuals. Some experimental results which have been published by other authors and have been interpreted by them as indicating the resistance of a portion of snail populations can be numerically explained by means of the formula of the “infection index” and without necessarily assuming resistance.     

Arguments against the significance of the Fenton reaction contributing to signal pathways under in vivo conditions

One of the common explanations for oxidative stress in the physiological milieu is based on the Fenton reaction, i.e. the assumption that radical chain reactions are initiated by metal-catalyzed electron transfer to hydrogen peroxide yielding hydroxyl radicals. On the other hand — especially in the context of so-called “iron switches” — it is postulated that cellular signaling pathways originate from the interaction of reduced iron with hydrogen peroxide.

Using fluorescence detection and EPR for identification of radical intermediates, we determined the rate of iron complexation by physiological buffer together with the reaction rate of concomitant hydroxylations of aromatic compounds under aerobic and anaerobic conditions. With the obtained overall reaction rate of 1,700 M^-1s^-1 for the buffer-dependent reactions and the known rates for Fenton reactions, we derive estimates for the relative reaction probabilities of both processes.

As a consequence we suggest that under in vivo conditions initiation of chain reactions by hydroxyl radicals generated by the Fenton reaction is of minor importance and hence metal-dependent oxidative stress must be rather independent of the so-called “peroxide tone”. Furthermore, it is proposed that — in the low (subtoxic) concentration range — hydroxylated compounds derived from reactions of “non-free” (crypto) OH radicals are better candidates for iron-dependent sensing of redox-states and for explaining the origin of cellular signals than the generation of “free” hydroxyl radicals.     

On the effects of the spatial distribution in an epidemic model based on cellular automaton

Several theoretical studies on disease propagation assume that individuals belonging to different groups regarding their health conditions are homogeneously distributed over the space. This is the well-known homogenous mixing assumption, which supports epidemiological models written in terms of ordinary differential or difference equations. Here, we consider that the host population infected by a contagious pathogen is composed by two groups with distinct traits and habits, which can be homogeneously mixed or not. The pathogen propagation is modeled by using an asynchronous probabilistic cellular automaton. Our main goal is to examine how a heterogeneous spatial distribution of these groups affects the endemic state. We noted that homogeneous distribution favors the occurrence of oscillations in the population composition. Surprisingly, we found out that the propagation dynamics of the heterogeneous distribution can also be described by a set of ordinary difference equations.     

Vector borne diseases on an urban environment: The effects of heterogeneity and human circulation

We study the effect of human circulation and host/vector heterogeneities on the onset of epidemics of arboviruses. From a meta-population dynamics based on the classical Bailey–Dietz model, we derive a multi-group model under three assumptions: (i) fast host sojourn time-scale; (ii) mosquitoes do not move; (iii) time homogeneity and strong connectivity of human circulation. Within this modelling framework, three different kinds of R₀ appear: (i) the “true” or “global” R₀—derived from the corresponding next generation matrix; (ii) the uniform R₀—obtained if the patches are taken homogeneous; (iii) the local R₀s—obtained if the patches are disconnected. We show that there is relevant epidemiological information associated to all of them. In particular, they can be used to understand the effects of changing the circulation on the value of the global R₀. We also present additional results on the effects on R₀ of different vector control policies, and a simulation with data from the city of Rio de Janeiro, Brazil.     

A test of heterogeneous mixing as a mechanism for ecological persistence in a disturbed environment

Persistence is a central issue in population ecology with important implications for population management. Most theoretical studies have focused on continually interacting populations, even though many systems are subject to ecological disturbances which confound analysis of persistence. In this paper, we use a combination of a simple parasite–hyperparasite model with disturbances and field data to investigate the factors contributing to the observed persistence of the parasite population. The field data are taken from a two-year experiment (including five growing seasons) investigating the use of the mycoparasite Sporidesmium sclerotivorum as a persistent biological control agent of Sclerotinia minor, an economically important fungal parasite of lettuce. We show that the standard assumption of homogeneous mixing fails to predict the observed persistence of the parasite population. We demonstrate that allowing for heterogeneous mixing prevents the fade-out predicted in the homogeneous mixing case. The implications of the results for broad classes of host–parasite systems are discussed.     

The genetic mating structure of subdivided populations I. Open-mating model

The genetic mating structure of a subdivided population can describe how parental genotypes gave rise to zygotes. When parents of the same genotype are considered together as one class (“open-mating”), three independent parameters of inbreeding and mating structure are needed to describe this structure at a diallelic locus. One is Wright's fixation index F. The other two are mating structure parameters, derived herein and termed the “effective selfing” rate E and the “inbreeding assortative selfing” rate D. E is the genetically equivalent proportion of self-fertilization at a single locus, and is given by standardized second and third central moments of gene frequencies of mates. E is a summary measure of inbreeding that includes effects due to self-fertilization and mating to relatives, as well as correlations between mates induced by Wahlund effects and/or selective diversification among neighborhoods. The second parameter D measures the tendency of inbred or more homozygous individuals to effectively self more (or less) than outbred or more heterozygous individuals. D is related to the maintenance of variation of inbreeding among individuals and/or to the prevalence of spatial variation of selection. D is independent of E, but together with E controls the generational change of inbreeding, ΔF. Extensions of the model to unequal allele frequencies in male vs female mates, and to multi-allelic loci, are also examined.     

Phenoptosis as genetically determined aging influenced by signals from the environment

Aging is a complex and not well understood process. Two opposite concepts try to explain its causes and mechanisms — programmed aging and aging of “wear and tear” (stochastic aging). To date, much evidence has been obtained that contradicts the theories of aging as being due to accumulation of various damages. For example, creation of adequate conditions for the functioning of the organism’s components (appropriate microenvironment, humoral background, etc.) has been shown to cause partial or complete reversibility of signs of its aging. Programmed aging and death of an organism can be termed phenoptosis by analogy to the term apoptosis for programmed cell death (this term was first suggested by V. P. Skulachev). The necessity of this phenomenon, since A. Weismann, has been justified by the need for population renewal according to ecological and evolutionary requirements. Species-specific lifespan, age-dependent changes in expression pattern of genes, etc. are compatible with the concept of phenoptosis. However, the intraspecific rate of aging was shown to vary over of a wide range depending on living conditions. This means that the “aging program” is not set rigidly; it sensitively adjusts an individual to the specific realities of its habitat. Moreover, there are indications that in rather severe conditions of natural habitat the aging program can be completely cancelled, as the need for it disappears because of the raised mortality from external causes (high extrinsic mortality), providing fast turnover of the population.     

Inbreeding: One word, several meanings

The way geneticists use the word “inbreeding” can be somewhat puzzling. How can the same word be applied to situations as different as the presence of commong ancestors in the pedigrees of two individuals, on one hand, and the attitude towards marriage in a population, on the other? How can the same parameter, i.e., the “inbreeding coefficient F,” be used both, to measure the degree of genetic resemblance between individuals within a single population and also to measure the degree of genetic differences between groups of individuals.It will be useful for the sake of translation as well as for that of clarity, to delineate the various situations in which geneticists are led to refer to “inbreeding.”We shall see that this one word is used in, at least, five concepts: relationship between relatives; genetic drift; departure from panmixia of mating behaviour; subdivision of a population into several isolated groups; and divergence between the actual genotypic structure of a population and the reference “Hardy-Weinberg Structure.”     

Reiterated transfer RNA genes of Xenopus laevis

The proportion of the Xenopus laevis genome complementary to “7 S” RNA, unfractionated transfer RNA and some selected aminoacyl-tRNAs, and the sequence complexity of these RNA species, have been determined by following the kinetics of RNA-DNA hybridization on filters under conditions of RNA excess at optimum rate temperature. For hybridization of aminoacyl-labelled tRNAs, conditions for optimum aminoacylation were first determined and, where necessary, aminoacyl-tRNAs were treated with nitrous acid to prevent discharge during annealing. Neither the extent nor rate of hybridization was affected by this treatment.“7 S” RNA, coded for by 580 genes per haploid complement of chromosomes, reacts like a single family of nucleotide sequences, whereas about 43 basic tRNA sequences are coded for by at least 7800 genes. If hybrids are not treated with RNase A, the apparent tDNA redundancy is some 23% greater but no more nucleotide sequences are detectable. Taken together, the results suggest that each tRNA sequence is, on average, 200-fold reiterated.The reiteration varies, however, for different aminoacyl-tRNAs. Thus, hybridization resolves only one valyl-tRNA which is coded for by 240 genes, but at least four leucyl-tRNA sequences can be distinguished by hybridization, each of which is on average 90-fold reiterated. Reiteration also varies for the two methionyl-tRNAs detectable both by hybridization and by reversed phase chromatography: tRNA₁^Met and tRNA₂^Met are 310- and 170-fold reiterated, respectively, and each is kinetically homogeneous. These saturation values are almost exactly additive and are not influenced by the presence of other tRNA species. Thus the results suggest that Xenopus tRNAs are no more heterogeneous than would be predicted by the genetic code, despite the high but variable multiplicity of tRNA cistrons.     

Isolation and characterization of temperature-sensitive mutants in a Chinese hamster cell line

A replica plating method was used for the isolation of temperature-sensitive (ts) mutants after treatment of Chinese hamster cells with ethyl methanesulfonate (EMS). No significant increase in ts mutants was found after this treatment. The limitations and advantages of the replicating procedure to detect such differences, as well as an alternative method, are discussed.Mutants isolated were classified into two general groups—density-dependent and clear-cut—as measured by survival at low and high cell densities at the restrictive temperature. The density-dependent mutants may be truly “leaky”, losing a metabolite to the medium at an excessive rate at the restrictive temperature. On the other hand, the one clear-cut mutant analyzed extensively dies at a rate determined by its ability to utilize one or more components from the medium. It shows an inverse density relationship in rate of death, as inferred from rates of macromolecular synthesis, as opposed to its growth rate at the permissive temperature.     

Genetic variation and differentiation in striped field mouse <Emphasis Type="Italic">Apodemus agrarius</Emphasis> inferred from RAPD-PCR analysis

Genetic variation and differentiation of the trans-Palearctic species Apodemus agrarius (striped field mouse), whose range consists of two large isolates—European-Siberian and Far Eastern-Chinese, were examined using RAPD-PCR analysis. The material from the both parts of the range was examined (41 individual of A. agrarius from 18 localities of Russia, Ukraine, Moldova, and Kazakhstan); the Far Eastern-Chinese part was represented by samples from the Amur region, Khabarovsk krai, and Primorye (Russia). Differences in frequencies of polymorphic RAPD loci were found between the European-Siberian and the Far Eastern population groups of striped field mouse. No “fixed” differences between them in RAPD spectra were found, and none of the used statistical methods permitted to distinguish with absolute certainty animals from the two range parts. Thus, genetic isolation of the European-Siberian and the Far Eastern population groups of A. agrarius is not strict. These results support the hypothesis on recent dispersal of striped field mouse from East to West Palearctics (during the Holocene climatic optimum, 7000 to 4500 years ago) and subsequent disjunction of the species range (not earlier than 4000–4500 years ago). The Far Eastern population group is more polymorphic than the European-Siberian one, while genetic heterogeneity is more uniformly distributed within it. This is probably explained by both historical events that happened during the species dispersal in the past, and different environmental conditions for the species in different parts of its range. The Far Eastern population group inhabits the area close to the distribution center of A. agrarius. It is likely that this group preserved genetic variation of the formerly integral ancestral form, while some amount of genetic polymorphism could be lost during the species colonization of the Siberian and European areas. To date, the settlement density and population number in general are higher than within the European-Siberian isolate, which seems to account for closer interpopulation associations, intense genetic exchange, and “smoothing” of polymorphism within the Far Eastern population group of A. agrarius.     

Sucrose density gradient sedimentation of E. coli ribosomes.

The behavior of E. coli ribosomes during sedimentation on sucrose gradients is predicted under a variety of conditions by computer simulations. Since numerous recent kinetic studies indicate equilibration in times short compared to the time of sedimentation, these simulations assume that the system attains local reaction equilibrium at every point in the gradient at all times. For any type of homogeneous equilibrating ribosome population, governed by a single formation constant at one atmosphere pressure for 70S couples, no more than two clearly defined zones will be resolved, although the presence of large dissociating effects due to pressure gradients in high speed experiments will spread the subunit zone. Normally the pattern will consist of a 30S zone and a so-called “70S” zone, which is in reality a mixture of 70S couples and 30S and 50S subunits in local equilibrium. The greater the dissociation into subunits, the more the “70S” zone will be slowed below the nominal rate of 70 Svedberg units. If ribosomes have been collected from the “70S” zone in several successive cycles of purification, the repeated deletion of resolved 30S subunits can result in a preparation with so large a molar excess of 50S subunits that the ensuing sucrose density gradient sedimentation pattern may exhibit a “70S” zone followed by zone of 50S subunits, insteadof a zone of 30S subunits. Our most important conclusion is that whenever a well-resolved 50S zone is present in a sucrose density gradient sedimentation experiment on E. coli ribosomes, in addition to a 30S and a “70S” zone, under conditions where ribosomes and subunits should be in reversible equilibrium, the preparation must be microheterogeneous, containing a mixture of “tight” and “loose” couples. Moreover in such cases the content of large subunits in the 50S zone must be derived entirely from “loose” couples whereas the 30S zone must contain small subunits derived from both “tight” and “loose” couples. Sedimentation patterns predicted for various mixtures of “tight” and “loose” couples display all the major characteristics of published experimental patterns for E. coli ribosomes, including the partial or complete resolution into three zones, depending on rotor velocity and level of Mg²⁺.     

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.     

On a formal definition of organization

A mathematic definition is proposed to account for the intuitive features of what is usually meant by organization. To account for both functional and structural aspects of organization the rate at which information content of a system changes in time is examined. It can be shown that Shannon's expression for ambiguity in a channel has two different meanings according to whether one is interested in the information transmitted in the channel or in the information transmitted to the observer from a whole system in which the channel is a part of a redundant communication network. This was applied in a previous work to show that the effects of noise on the information content H of a system result in two kinds of ambiguities, “autonomy-producing” and “destructive” leading to increase and decrease in H, respectively. By making use of this observation and Shannon's definition of redundancy R, a single equation for $\text{d}\text{H}\text{dt}$ is proposed to define organization on the basis of a kinetics of change of information content of a system under the effects of environmental noise-producing factors accumulated in time. It is shown how these factors, obviously responsible for a decrease in H, i.e. a “disorganizing” effect, can be responsible also—under certain conditions and up to a certain time or “dose” of noise—for an initial increase in H interpreted as a process of “self”-organization. The autonomy-producing ambiguity is expressed by a term of decrease in redundancy, while a second term in the equation, of decrease in maximum-non-redundant-information content expresses the destructive ambiguity. A given organization is defined at least by three parameters, which determine the main features of its characteristic function H(t). One of them is the initial information content H₀ and has a structural meaning. A second parameter, with a dimension of time, has the meaning of a functional reliability, related to the overall resistance of the system to noise-producing factors. The third parameter, namely the initial redundancy R₀, is both structural and functional in character, since structural redundancy is known to help insure reliability. Various conditions on these parameters lead to various kinds of organizations, with and without self-organizing properties.