A net reproductive number for periodic matrix models

We give a definition of a net reproductive number R (0) for periodic matrix models of the type used to describe the dynamics of a structured population with periodic parameters. The definition is based on the familiar method of studying a periodic map by means of its (period-length) composite. This composite has an additive decomposition that permits a generalization of the Cushing-Zhou definition of R (0) in the autonomous case. The value of R (0) determines whether the population goes extinct (R (0)<1) or persists (R (0)>1). We discuss the biological interpretation of this definition and derive formulas for R (0) for two cases: scalar periodic maps of arbitrary period and periodic Leslie models of period 2. We illustrate the use of the definition by means of several examples and by applications to case studies found in the literature. We also make some comparisons of this definition of R (0) with another definition given recently by Baca?r.     

On a new perspective of the basic reproduction number in heterogeneous environments

Although its usefulness and possibility of the well-known definition of the basic reproduction number R0 for structured populations by Diekmann, Heesterbeek and Metz (J Math Biol 28:365-382, 1990) (the DHM definition) have been widely recognized mainly in the context of epidemic models, originally it deals with population dynamics in a constant environment, so it cannot be applied to formulate the threshold principle for population growth in time-heterogeneous environments. Since the mid-1990s, several authors proposed some ideas to extend the definition of R0 to the case of a periodic environment. In particular, the definition of R0 in a periodic environment by Baca?r and Guernaoui (J Math Biol 53:421-436, 2006) (the BG definition) is most important, because their definition of periodic R0 can be interpreted as the asymptotic per generation growth rate, which is an essential feature of the DHM definition. In this paper, we introduce a new definition of R0 based on the generation evolution operator (GEO), which has intuitively clear biological meaning and can be applied to structured populations in any heterogeneous environment. Using the generation evolution operator, we show that the DHM definition and the BG definition completely allow the generational interpretation and, in those two cases, the spectral radius of GEO equals the spectral radius of the next generation operator, so it gives the basic reproduction number. Hence the new definition is an extension of the DHM definition and the BG definition. Finally we prove a weak sign relation that if the average Malthusian parameter exists, it is nonnegative when R0>1 and it is nonpositive when R0<1.     

On the biological interpretation of a definition for the parameter R₀ in periodic population models

An adaptation of the definition of the basic reproduction number R (0) to time-periodic seasonal models was suggested a few years ago. However, its biological interpretation remained unclear. The present paper shows that in demography, this R (0) is the asymptotic ratio of total births in two successive generations of the family tree. In epidemiology, it is the asymptotic ratio of total infections in two successive generations of the infection tree. This result is compared with other recent work.     

Epidemic growth rate and household reproduction number in communities of households,schools and workplaces

In this paper we present a novel and coherent modelling framework for the characterisation of the real-time growth rate in SIR models of epidemic spread in populations with social structures of increasing complexity. Known results about homogeneous mixing and multitype models are included in the framework, which is then extended to models with households and models with households and schools/workplaces. Efficient methods for the exact computation of the real-time growth rate are presented for the standard SIR model with constant infection and recovery rates (Markovian case). Approximate methods are described for a large class of models with time-varying infection rates (non-Markovian case). The quality of the approximation is assessed via comparison with results from individual-based stochastic simulations. The methodology is then applied to the case of influenza in models with households and schools/workplaces, to provide an estimate of a household-to-household reproduction number and thus asses the effort required to prevent an outbreak by targeting control policies at the level of households. The results highlight the risk of underestimating such effort when the additional presence of schools/workplaces is neglected. Our framework increases the applicability of models of epidemic spread in socially structured population by linking earlier theoretical results, mainly focused on time-independent key epidemiological parameters (e.g. reproduction numbers, critical vaccination coverage, epidemic final size) to new results on the epidemic dynamics.     

Stochastic and deterministic models for SIS epidemics among a population partitioned into households

Models for the spread of an SIS epidemic among a population consisting of m households, each containing n individuals, are considered and their behaviour is analysed under the practically relevant situation when m is large and n small. A threshold parameter R* is determined. For the stochastic model it is shown that the epidemic has a non-zero probability of taking off if and only if R* > 1, and the extension to unequal household sizes is also considered. For the deterministic model, with households of size 2, it is shown that if R* < or = 1 then the epidemic dies out, whilst if R* > 1 the epidemic settles down to an endemic equilibrium. The usual basic reproductive ratio R0 does not provide a good indicator for the behaviour of these household epidemic models unless the household size n is large.     

Extending the range of additivity in using inclusive fitness

Inclusive fitness is a concept widely utilized by social biologists as the quantity organisms appear designed to maximize. However, inclusive fitness theory has long been criticized on the (uncontested) grounds that other quantities, such as offspring number, predict gene frequency changes accurately in a wider range of mathematical models. Here, we articulate a set of modeling assumptions that extend the range of scenarios in which inclusive fitness can be applied. We reanalyze recent formal analyses that searched for, but did not find, inclusive fitness maximization. We show (a) that previous models have not used Hamilton''s definition of inclusive fitness, (b) a reinterpretation of Hamilton''s definition that makes it usable in this context, and (c) that under the assumption of probabilistic mixing of phenotypes, inclusive fitness is indeed maximized in these models. We also show how to understand mathematically, and at an individual level, the definition of inclusive fitness, in an explicit population genetic model in which exact additivity is not assumed. We hope that in articulating these modeling assumptions and providing formal support for inclusive fitness maximization, we help bridge the gap between empiricists and theoreticians, which in some ways has been widening, demonstrating to mathematicians why biologists are content to use inclusive fitness, and offering one way to utilize inclusive fitness in general models of social behavior.     

A markov chain approach to calculate r(0) in stochastic epidemic models

R(0) has been defined as "The expected number of secondary infections originated by a "typical" infective individual when introduced into a population of susceptibles", and it is perhaps the single most important parameter in epidemic models. A general framework to calculate R(0) that can be applied to complicated stochastic epidemic models that may include demography, several strains, latent or carrier-like states, with or without density-dependent parameters is introduced. This framework helps us to understand the concept of a "typical" infective individual used in the deterministic definition of R(0). The method is illustrated with applications to several epidemic models, including some in which it has been found that the disease may persist even if R(0)<1. It is shown that although the probability of extinction is difficult to calculate in these latter cases, it is possible to give general conditions on the parameters under which eventual extinction is certain.     

Towards a marxist demography: A comparison of Puerto Rican landowners, peasants and rural proletarians

Conclusion The basic axis of the analysis presented in this inquiry has been the comparison of three barrios and three social classes. The following generalizations can be made:The landowning class was older and not rapidly replacing itself; the heads were united with their spouses in legal unions within simple family households that were larger than the households of the other social classes. Landowners were primarily whites. When they reached old age, most had mature children living with them to provide support and continuity.The rural proletarians, who were most rapidly reproducing themselves, as an apparent function of lower life-expectancy, typically lived in simple family households united by legal or consensual unions, or headed by widowers. They tended to be mulattos. Fewer proletarian households headed by elderly persons had their own mature male children at home. The peasantry was the class most rapidly reproducing itself, and had the largest average number of children in the household, as well as the largest proportion of households with agregados present. The flexibility of social organization among peasants is indicated by the incidence of consensual unions and of extended family households. Landless peasants tended to live in legal unions and in simple family households, similar to the rural proletarians. Only half of the peasant households headed by older persons had resolved the social security problem through family structures that included mature male children; agregados may have served the function of providing support and continuity.The fourth group discussed here, although not properly speaking a social class, is the significant number of households headed by women-domestics. A few were widows, but most had not married, or perhaps had not maintained consensual unions. These households appear to have existed in a more or less direct proportion to rural proletarian households It is not clear whether they were consensual unions in transition, obscured polygyny, represented a stable adaptation of women heads, or some other, unrecognized adaptation. The characteristics of the households headed by women-domestics indicate that if they stood by themselves without direct links to employment or to subsistence; life for them must have been very difficult.The demographic and agrarian patterns analyzed, clearly establish that the three barrios represent various degrees of articulation to the evolving plantation system, ranging from the peasant-proletarian community of Sabana Eneas, through the proletarian community of Maresúa, to the landowner community of Rosario Alto. Stated in different terms, these barrios reveal the differential effects of proletarianization, which probably affected Maresúa more than the other two barrios, because of its linkage with the plantation system.The significance of this interpretation is that the differing circumstances presented contrasting bases for change and stability in preinvasion Puerto Rico. Rosario Alto's lack of articulation with the plantation system probably meant that its characteristics would persist longer than those of the other two barrios, while the demographic and agrarian structures of Maresúa would change most rapidly. The peasant orientation of Sabana Eneas provided some elasticity in response to the demands of the plantation system.Some summary remarks may be added about the problems cited by Mintz for the materialist historiography of rural populations. While basic characteristics, including the hierarchy of social relationships, are discernible from historical documents, the most important aspect of peasant and rural proletariat society is not. I refer to the dialectical movement of persons between peasant and proletarian adaptations throughout their life cycles. There were instances of peasants with proletarian sons, as well as of landowners with peasant and proletarian sons, but not a sufficient number in either case to establish a generalization. In order to understand this process, the anthropologist must be prepared to dissect the stages of individual and family life cycles, just as the economist deploys marxist economic theory in order to disaggregate the circuits of capital, and thus come to understand the generation of surplus value . This in turn requires a familiarity with, and willingness to use, appropriate quantitative techniques. In future analyses of these Puerto Rican census materials and related sources, I hope to demonstrate the utility of such an approach in a marxist framework.James Wessman teaches in the Department of Sociology, Saint Olaf College, Northfield, Minnesota.

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Threshold conditions for infection persistence in complex host-vectors interactions

As classically defined by Macdonald in the early 1950s, for the case of diseases with one vector and one host, the Basic Reproduction Number, R0, is defined as the number of secondary infections caused by a single infective of the same type (vector or host) during its infectiousness period in an entirely susceptible population. In the case of a disease which has one vector and one host, it is easy to show that R0 coincides with the threshold for the establishment of an endemic state: if R0 > 1 (< 1), the disease can invade (cannot invade) the host population. In this paper we examine various epidemic situations in which there are more than one vector and/or host. We show that in those more complex systems it is not possible to deduce a single R0 but rather a threshold for infection persistence which is a composite of several quantities closely related to the classical expression of R0. Another definition of R0 given by Diekmann, Heesterbeek and Metz, and denoted in this paper R0NGO is discussed and applied as an alternative to calculate the thresholds for infection establishment.     

Socioeconomic disparities and the familial coexistence of child stunting and maternal overweight in guatemala

The double burden of malnutrition, defined here as households with a stunted child and an overweight mother (SCOM), is a growing problem in Guatemala. We explored the magnitude of SCOM and the identification of socio-economic factors associated with this malnutrition duality. From the 2000 Living Standards Measurement Study from Guatemala, we obtained a sample of 2492 households with pairs of children 6–60 months and their mothers (18–49 years) and estimated the prevalence of SCOM. Economic characteristics of this sample were assessed with the Concentration Index (CI). Results revealed higher prevalence of child stunting, but a lower prevalence of maternal overweight among the poor compared to the rich households. Economic inequality in child stunting was greater than economic inequality in maternal overweight (CI = ?0.22 vs. +0.14). SCOM pairs were more prevalent among the poor and middle SES groups as compared to the rich households. A multivariate logistic regression model showed that SCOM was more likely to occur in households from the middle consumption quintile than in those from the first quintile (odds ratio = 1.7). The findings reported here add new insights into the complex phenomenon observed in households with both extremes of the malnutrition continuum, and support the need for the identification of economic, social and biological interventions aimed at, on the one hand, the prevention of this duality of the malnutrition in those households where it is still non-existent, and on the other hand, to deter or correct the economic, social and biological environments where those mother–child dyads are already affected by such phenomena.     

The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.     

Concentration of Access to Information and Communication Technologies in the Municipalities of the Brazilian Legal Amazon

This study fills demand for data on access and use of information and communication technologies (ICT) in the Brazilian legal Amazon, a region of localities with identical economic, political, and social problems. We use the 2010 Brazilian Demographic Census to compile data on urban and rural households (i) with computers and Internet access, (ii) with mobile phones, and (iii) with fixed phones. To compare the concentration of access to ICT in the municipalities of the Brazilian Amazon with other regions of Brazil, we use a concentration index to quantify the concentration of households in the following classes: with computers and Internet access, with mobile phones, with fixed phones, and no access. These data are analyzed along with municipal indicators on income, education, electricity, and population size. The results show that for urban households, the average concentration in the municipalities of the Amazon for computers and Internet access and for fixed phones is lower than in other regions of the country; meanwhile, that for no access and mobile phones is higher than in any other region. For rural households, the average concentration in the municipalities of the Amazon for computers and Internet access, mobile phones, and fixed phones is lower than in any other region of the country; meanwhile, that for no access is higher than in any other region. In addition, the study shows that education and income are determinants of inequality in accessing ICT in Brazilian municipalities and that the existence of electricity in rural households is directly associated with the ownership of ICT resources.     

Genealogy with seasonality,the basic reproduction number,and the influenza pandemic

The basic reproduction number R ₀ has been used in population biology, especially in epidemiology, for several decades. But a suitable definition in the case of models with periodic coefficients was given only in recent years. The definition involves the spectral radius of an integral operator. As in the study of structured epidemic models in a constant environment, there is a need to emphasize the biological meaning of this spectral radius. In this paper we show that R ₀ for periodic models is still an asymptotic per generation growth rate. We also emphasize the difference between this theoretical R ₀ for periodic models and the “reproduction number” obtained by fitting an exponential to the beginning of an epidemic curve. This difference has been overlooked in recent studies of the H1N1 influenza pandemic.     

Transmission and dynamics of tuberculosis on generalized households

Tuberculosis (TB) transmission is enhanced by systematic exposure to an infectious individual. This enhancement usually takes place at either the home, workplace, and/or school (generalized household). Typical epidemiological models do not incorporate the impact of generalized households on the study of disease dynamics. Models that incorporate cluster (generalized household) effects and focus on their impact on TB's transmission dynamics are developed. Detailed models that consider the effect of casual infections, that is, those generated outside a cluster, are also presented. We find expressions for the Basic Reproductive Number as a function of cluster size. The formula for R0 separates the contributions of cluster and casual infections in the generation of secondary TB infections. Relationships between cluster and classical epidemic models are discussed as well as the concept of critical cluster size.     

Food-Sharing Networks in Lamalera, Indonesia: Status, Sharing, and Signaling

Costly signaling has been proposed as a possible mechanism to explain food sharing in foraging populations. This sharing-as-signaling hypothesis predicts an association between sharing and status. Using exponential random graph modeling (ERGM), this prediction is tested on a social network of between-household food-sharing relationships in the fishing and sea-hunting village of Lamalera, Indonesia. Previous analyses (Nolin 2010) have shown that most sharing in Lamalera is consistent with reciprocal altruism. The question addressed here is whether any additional variation may be explained as sharing-as-signaling by high-status households. The results show that high-status households both give and receive more than other households, a pattern more consistent with reciprocal altruism than costly signaling. However, once the propensity to reciprocate and household productivity are controlled, households of men holding leadership positions show greater odds of unreciprocated giving when compared to households of non-leaders. This pattern of excessive giving by leaders is consistent with the sharing-as-signaling hypothesis. Wealthy households show the opposite pattern, giving less and receiving more than other households. These households may reciprocate in a currency other than food or their wealth may attract favor-seeking behavior from others. Overall, status covariates explain little variation in the sharing network as a whole, and much of the sharing observed by high-status households is best explained by the same factors that explain sharing by other households. This pattern suggests that multiple mechanisms may operate simultaneously to promote sharing in Lamalera and that signaling may motivate some sharing by some individuals even within sharing regimes primarily maintained by other mechanisms.     

Endemic threshold results in an age-duration-structured population model for HIV infection

In this paper we consider an age-duration-structured population model for HIV infection in a homosexual community. First we investigate the invasion problem to establish the basic reproduction ratio R(0) for the HIV/AIDS epidemic by which we can state the threshold criteria: The disease can invade into the completely susceptible population if R(0)>1, whereas it cannot if R(0)<1. Subsequently, we examine existence and uniqueness of endemic steady states. We will show sufficient conditions for a backward or a forward bifurcation to occur when the basic reproduction ratio crosses unity. That is, in contrast with classical epidemic models, for our HIV model there could exist multiple endemic steady states even if R(0) is less than one. Finally, we show sufficient conditions for the local stability of the endemic steady states.     

Reproductive numbers, epidemic spread and control in a community of households

Many of the studies on emerging epidemics (such as SARS and pandemic flu) use mass action models to estimate reproductive numbers and the needed control measures. In reality, transmission patterns are more complex due to the presence of various social networks. One level of complexity can be accommodated by considering a community of households. Our study of transmission dynamics in a community of households emphasizes five types of reproductive numbers for the epidemic spread: household-to-household reproductive number, leaky vaccine-associated reproductive numbers, perfect vaccine reproductive number, growth rate reproductive number, and the individual reproductive number. Each of those carries different information about the transmission dynamics and the required control measures, and often some of those can be estimated from the data while others cannot. Simulations have shown that under certain scenarios there is an ordering for those reproductive numbers. We have proven a number of ordering inequalities under general assumptions about the individual infectiousness profiles. Those inequalities allow, for instance, to estimate the needed vaccine coverage and other control measures without knowing the various transmission parameters in the models. Along the way, we have also shown that in choosing between increasing vaccine efficacy and increasing coverage levels by the same factor, preference should go to efficacy.     

The physical and radiobiological basis of the local effect model: a response to the commentary by R. Katz

The physical and biological basis of our model to calculate the biological effects of charged particles, termed the local effect model (LEM), has recently been questioned in a commentary by R. Katz. Major objections were related to the definition of the target size and the use of the term cross section. Here we show that the objections raised against our approach are unjustified and are largely based on serious misunderstandings of the conceptual basis of the local effect model. Furthermore, we show that the approach developed by Katz and coworkers itself suffers from exactly those deficiencies for which Katz criticizes our model. The essential conceptual differences between the two models are discussed by means of some illustrative examples, based on a comparison with experimental data. For these examples, the predictions of the local effect model are fully consistent with the experimental data. In contrast, e.g. for very heavy ions, there are significant discrepancies observed for the Katz approach. These discrepancies can be attributed to the inadequate definition of the target size in this model. Experimental data are thus clearly in favor of the definition of the target as used in the local effect model. Agreement with experimental data is achieved for protons within the Katz approach but at the cost of questionable approximations in combination with the violation of the fundamental physical principle of energy conservation.     

Drivers and socioeconomic impacts of tourism participation in protected areas

Nature-based tourism has the potential to enhance global biodiversity conservation by providing alternative livelihood strategies for local people, which may alleviate poverty in and around protected areas. Despite the popularity of the concept of nature-based tourism as an integrated conservation and development tool, empirical research on its actual socioeconomic benefits, on the distributional pattern of these benefits, and on its direct driving factors is lacking, because relevant long-term data are rarely available. In a multi-year study in Wolong Nature Reserve, China, we followed a representative sample of 220 local households from 1999 to 2007 to investigate the diverse benefits that these households received from recent development of nature-based tourism in the area. Within eight years, the number of households directly participating in tourism activities increased from nine to sixty. In addition, about two-thirds of the other households received indirect financial benefits from tourism. We constructed an empirical household economic model to identify the factors that led to household-level participation in tourism. The results reveal the effects of local households' livelihood assets (i.e., financial, human, natural, physical, and social capitals) on the likelihood to participate directly in tourism. In general, households with greater financial (e.g., income), physical (e.g., access to key tourism sites), human (e.g., education), and social (e.g., kinship with local government officials) capitals and less natural capital (e.g., cropland) were more likely to participate in tourism activities. We found that residents in households participating in tourism tended to perceive more non-financial benefits in addition to more negative environmental impacts of tourism compared with households not participating in tourism. These findings suggest that socioeconomic impact analysis and change monitoring should be included in nature-based tourism management systems for long-term sustainability of protected areas.     

Stability analysis of within-host parasite models with delays

We provide a global analysis of systems of within-host parasitic infections. The systems studied have parallel classes of different length of latently infected target cells. These systems can also be thought as systems arising from within-host parasitic systems with distributed continuous delays. We compute the basic reproduction ratio R0 for the systems under consideration. If R0< or =1 the parasite is cleared, if R0>1 and if a sufficient condition is satisfied we conclude to the global asymptotic stability (GAS) of the endemic equilibrium. For some generic class of models this condition reduces to R0>1. These results make possible to revisit some parasitic models including intracellular delays and to study their global stability.