A new method for estimating the effort required to control an infectious disease

We propose a new threshold quantity for the analysis of the epidemiology of infectious diseases. The quantity is similar in concept to the familiar basic reproduction ratio, R0, but it singles out particular host types instead of providing a criterion that is uniform for all host types. Using this methodology we are able to identify the long-term effects of disease-control strategies for particular subgroups of the population, to estimate the level of control necessary when targeting control effort at a subset of host types, and to identify host types that constitute a reservoir of infection. These insights cannot be obtained by using R0 alone.     

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.     

The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model

In this paper, we develop the theory of a state-reproduction number for a multistate class age structured epidemic system and apply it to examine the asymptomatic transmission model. We formulate a renewal integral equation system to describe the invasion of infectious diseases into a multistate class age structured host population. We define the state-reproduction number for a class age structured system, which is the net reproduction number of a specific host type and which plays an analogous role to the type-reproduction number [M.G. Roberts, J.A.P. Heesterbeek, A new method for estimating the effort required to control an infectious disease, Proc. R. Soc. Lond. B 270 (2003) 1359; J.A.P. Heesterbeek, M.G. Roberts, The type-reproduction number T in models for infectious disease control, Math. Biosci. 206 (2007) 3] in discussing the critical level of public health intervention. The renewal equation formulation permits computations not only of the state-reproduction number, but also of the generation time and the intrinsic growth rate of infectious diseases.Subsequently, the basic theory is applied to capture the dynamics of a directly transmitted disease within two types of infected populations, i.e., asymptomatic and symptomatic individuals, in which the symptomatic class is observable and hence a target host of the majority of interventions. The state-reproduction number of the symptomatic host is derived and expressed as a measurable quantity, leading to discussion on the critical level of case isolation. The serial interval and other epidemiologic indices are computed, clarifying the parameters on which these indices depend. As a practical example, we illustrate the eradication threshold for case isolation of smallpox. The generation time and serial interval are comparatively examined for pandemic influenza.     

On the definition and the computation of the type-reproduction number T for structured populations in heterogeneous environments

In the context of mathematical epidemiology, the type-reproduction number (TRN) for a specific host type is interpreted as the average number of secondary cases of that type produced by the primary cases of the same host type during the entire course of infection. Here, it must be noted that T takes into account not only the secondary cases directly transmitted from the specific host but also the cases indirectly transmitted by way of other types, who were infected from the primary cases of the specific host with no intermediate cases of the target host. Roberts and Heesterbeek (Proc R Soc Lond B 270:1359–1364, 2003) have shown that T is a useful measure when a particular single host type is targeted in the disease control effort in a community with various types of host, based on the fact that the sign relation sign(R ₀ ? 1) = sign(T ? 1) holds between the basic reproduction number R ₀ and T. In fact, T can be seen as an extension of R ₀ in a sense that the threshold condition of the total population growth can be formulated by the reproduction process of the target type only. However, the original formulation is limited to populations with discrete state space in constant environments. In this paper, based on a new perspective of R ₀ in heterogeneous environments (Inaba in J Math Biol 2011), we give a general definition of the TRN for continuously structured populations in heterogeneous environments and show some examples of its computation and applications.     

Interval estimates for epidemic thresholds in two-sex network models

Epidemic thresholds in network models of heterogeneous populations characterized by highly right-skewed contact distributions can be very small. When the population is above the threshold, an epidemic is inevitable and conventional control measures to reduce the transmissibility of a pathogen will fail to eradicate it. We consider a two-sex network model for a sexually transmitted disease which assumes random mixing conditional on the degree distribution. We derive expressions for the basic reproductive number (R(0)) for one and heterogeneous two-population in terms of characteristics of the degree distributions and transmissibility. We calculate interval estimates for the epidemic thresholds for stochastic process models in three human populations based on representative surveys of sexual behavior (Uganda, Sweden, USA). For Uganda and Sweden, the epidemic threshold is greater than zero with high confidence. For the USA, the interval includes zero. We discuss the implications of these findings along with the limitations of epidemic models which assume random mixing.     

Reproduction numbers and thresholds in stochastic epidemic models. I. Homogeneous populations.

We compare threshold results for the deterministic and stochastic versions of the homogeneous SI model with recruitment, death due to the disease, a background death rate, and transmission rate beta cXY/N. If an infective is introduced into a population of susceptibles, the basic reproduction number, R0, plays a fundamental role for both, though the threshold results differ somewhat. For the deterministic model, no epidemic can occur if R0 less than or equal to 1 and an epidemic occurs if R0 greater than 1. For the stochastic model we find that on average, no epidemic will occur if R0 less than or equal to 1. If R0 greater than 1, there is a finite probability, but less than 1, that an epidemic will develop and eventuate in an endemic quasi-equilibrium. However, there is also a finite probability of extinction of the infection, and the probability of extinction decreases as R0 increases above 1.     

Population-level responses to nutrient heterogeneity and density by Abutilon theophrasti (Malvaceae): an experimental neighborhood approach

An experimental approach was used to examine the effects of spatial nutrient heterogeneity and planting density on the sizes of plants within populations of Abutilon theophrasti. Planting locations were generated using random numbers and replicated among populations growing on two different scales of heterogeneity and homogeneous soils. The same quantity of nutrients (dehydrated cow manure) was added to each population, regardless of the spatial nutrient distribution. The higher density was achieved by adding additional planting locations to those present at the lower density. Plant biomass was compared among ten planting locations present in all populations. Plants in seven locations were smaller at the higher density, but the spatial distribution of nutrients affected plant size in only two locations. At the population level, the higher density reduced mean plant biomass and increased both total biomass and the coefficient of variation in biomass, a measure of size inequality. Only when populations on both scales of heterogeneity were together compared with those on homogeneous soils were population-level measurements found to be significantly affected by soil treatment; heterogeneity resulted in decreased total biomass and an increase in the coefficient of variation, apparently due to an increase in the number of small plants in the population. These results, together with the finding that fine root biomass increased in nutrient-enriched patches, suggest that on heterogeneous soils most plants were able to access nutrient patches.     

Assessing biodiversity with species accumulation curves; inventories of small reptiles by pit‐trapping in Western Australia

Abstract We examined 11 non‐linear regression models to determine which of them best fitted curvilinear species accumulation curves based on pit‐trapping data for reptiles in a range of heterogeneous and homogenous sites in mesic, semi‐arid and arid regions of Western Australia. A well‐defined plateau in a species accumulation curve is required for any of the models accurately to estimate species richness. Two different measures of effort (pit‐trapping days and number of individuals caught) were used to determine if the measure of effort influenced the choice of the best model(s). We used species accumulation curves to predict species richness, determined the trapping effort required to catch a nominated percentage (e.g. 95%) of the predicted number of species in an area, and examined the relationship between species accumulation curves with diversity and rarity. Species richness, diversity and the proportion of rare species in a community influenced the shape of species accumulation curves. The Beta‐P model provided the best overall fit (highest r²) for heterogeneous and homogeneous sites. For heterogeneous sites, Hill, Rational, Clench, Exponential and Weibull models were the next best. For homogeneous habitats, Hill, Weibull and Chapman–Richards were the next best models. There was very little difference between Beta‐P and Hill models in fitting the data to accumulation curves, although the Hill model generally over‐estimated species richness. Most models worked equally well for both measures of trapping effort. Because the number of individuals caught was influenced by both pit‐trapping effort and the abundance of individuals, both measures of effort must be considered if species accumulation curves are to be used as a planning tool. Trapping effort to catch a nominated percentage of the total predicted species in homogeneous and heterogeneous habitats varied among sites, but even for only 75% of the predicted number of species it was generally much higher than the typical effort currently being used for terrestrial vertebrate fauna surveys in Australia. It was not possible to provide a general indication of the effort required to predict species richness for a site, or to capture a nominated proportion of species at a site, because species accumulation curves are heavily influenced by the characteristics of particular sites.     

Revisiting the basic reproductive number for malaria and its implications for malaria control

The prospects for the success of malaria control depend, in part, on the basic reproductive number for malaria, R0. Here, we estimate R0 in a novel way for 121 African populations, and thereby increase the number of R0 estimates for malaria by an order of magnitude. The estimates range from around one to more than 3,000. We also consider malaria transmission and control in finite human populations, of size H. We show that classic formulas approximate the expected number of mosquitoes that could trace infection back to one mosquito after one parasite generation, Z0(H), but they overestimate the expected number of infected humans per infected human, R0(H). Heterogeneous biting increases R0 and, as we show, Z0(H), but we also show that it sometimes reduces R0(H); those who are bitten most both infect many vectors and absorb infectious bites. The large range of R0 estimates strongly supports the long-held notion that malaria control presents variable challenges across its transmission spectrum. In populations where R0 is highest, malaria control will require multiple, integrated methods that target those who are bitten most. Therefore, strategic planning for malaria control should consider R0, the spatial scale of transmission, human population density, and heterogeneous biting.     

Population size and the risk of local extinction: empirical evidence from rare plants

Due to habitat fragmentation many plant species today occur mainly in small and isolated populations. Modeling studies predict that small populations will be threatened more strongly by stochastic processes than large populations, but there is little empirical evidence to support this prediction for plants. We studied the relationship between size of local populations (number of flowering plants) and survival over ten years for 359 populations of eight short-lived, threatened plants in northern Germany ( Lepidium campestre , Thlaspi perfoliatum , Rhinanthus minor , R . serotinus , Melampyrum arvense , M . nemorosum , Gentianella ciliata and G . germanica ). Overall, 27% of the populations became extinct during the study period. Probability of survival of a local population increased significantly with its size in all but one species ( R. minor ). However, estimated population sizes required for 90% probability of survival over 10 years varied widely among species. Survival probability increased with decreasing distance to the nearest conspecific population in R . serotinus , but not in the other species. The mean annual growth rate of surviving populations differed greatly between species, but was only for G . germanica significantly lower than 1, suggesting that there was no general deterministic decline in the number of plants due to deteriorating habitat conditions. We conclude that the extinction of populations was at least partly due to stochastic processes. This is supported by the fact that in all species a considerable proportion of small populations survived and developed into large populations.     

Network theory and SARS: predicting outbreak diversity

Many infectious diseases spread through populations via the networks formed by physical contacts among individuals. The patterns of these contacts tend to be highly heterogeneous. Traditional "compartmental" modeling in epidemiology, however, assumes that population groups are fully mixed, that is, every individual has an equal chance of spreading the disease to every other. Applications of compartmental models to Severe Acute Respiratory Syndrome (SARS) resulted in estimates of the fundamental quantity called the basic reproductive number R0--the number of new cases of SARS resulting from a single initial case--above one, implying that, without public health intervention, most outbreaks should spark large-scale epidemics. Here we compare these predictions to the early epidemiology of SARS. We apply the methods of contact network epidemiology to illustrate that for a single value of R0, any two outbreaks, even in the same setting, may have very different epidemiological outcomes. We offer quantitative insight into the heterogeneity of SARS outbreaks worldwide, and illustrate the utility of this approach for assessing public health strategies.     

A qualitative analysis of a model for the transmission of varicella-zoster virus

Varicella-zoster virus (VZV) is a herpesvirus which is the known agent for causing varicella (chickenpox) in its initial manifestation and zoster (shingles) in a reactivated state. The standard SEIR compartmental model is modified to include the cycle of shingles acquisition, recovery, and possible reacquisition. The basic reproduction number R(0) shows the influence of the zoster cycle and how shingles can be important in maintaining VZV in populations. The model has the typical threshold behavior in the sense that when R(0)1, the virus persists over time and so chickenpox and shingles remain endemic.     

Dynamical analysis of a multi-strain model of HIV in the presence of anti-retroviral drugs

One major drawback associated with the use of anti-retroviral drugs in curtailing HIV spread in a population is the emergence and transmission of HIV strains that are resistant to these drugs. This paper presents a deterministic HIV treatment model, which incorporates a wild (drug sensitive) and a drug-resistant strain, for gaining insights into the dynamical features of the two strains, and determining effective ways to control HIV spread under this situation. Rigorous qualitative analysis of the model reveals that it has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold (R t 0) is less than unity and that the disease will persist in the population when this threshold exceeds unity. Further, for the case where R t 0 > 1, it is shown that the model can have two co-existing endemic equilibria, and competitive exclusion phenomenon occurs whenever the associated reproduction number of the resistant strain (R t r) is greater than that of the wild strain (R t w). Unlike in the treatment model, it is shown that the model without treatment can have a family of infinitely many endemic equilibria when its associated epidemiological threshold (R(0)) exceeds unity. For the case when [Formula in text], it is shown that the widespread use of treatment against the wild strain can lead to its elimination from the community if the associated reduction in infectiousness of infected individuals (treated for the wild strain) does not exceed a certain threshold value (in this case, the use of treatment is expected to make R t w < R t r.     

Control of invasive American bullfrog Lithobates catesbeianus in small shallow water bodies

Setting up cost-efficient control programs for alien invasive species requires the development of adequate removal methods in combination with insights in population size and dynamics. American bullfrog Lithobates catesbeianus is an alien invasive species, which is suspected to cause substantial ecological damage around the globe. However, control of bullfrog populations is difficult, as no conclusive management measures have yet been determined. We investigated how double fyke nets could contribute to bullfrog management by assessing the tadpole population size in 10 permanent small shallow water bodies. Two population size estimate methods were applied, being the catch–depletion and mark–recapture method. Catchability of bullfrog tadpoles proved to be very consistent over ponds and methods, with one catch per unit of effort (one double fyke net for 24 h) retaining on average 6 % of the tadpole population. Population density varied considerably among ponds, ranging from 950 to 120,804 larger tadpole individuals/ha. Using these insights in developing a cost-efficient eradication program for the species, we projected the number of catch efforts needed to reduce tadpole numbers to a threshold that more than likely affects final bullfrog population size. Predictions indicated that for the specified thresholds the use of eight double fyke nets at a time is most cost-efficient in high abundance populations, while using five double fyke nets seems most suitable in low abundance populations. What the exact threshold number of remaining tadpole individuals should be is uncertain, but forecasts demonstrate that only half of the budget would be needed when aiming at a drop to fewer than 100 remaining tadpoles than when a decrease to fewer than 10 remaining tadpoles is pursued. Given the fairly limited cost of bullfrog management with double fyke nets, however, it may be worthwhile to fully reduce the tadpole population.     

Parasite strain coexistence in a heterogeneous host population

Heterogeneity in host susceptibility and transmissibility to parasite attack allows a lower transmission rate to sustain an epidemic than is required in homogeneous host populations. However, this heterogeneity can leave some hosts with little susceptibility to disease, and at high transmission rates, epidemic size can be smaller than for diseases where the host population is homogeneous. In a heterogeneous host population, we model natural selection in a parasite population where host heterogeneity is exploited by different strains to varying degrees. This partitioning of the host population allows coexistence of competing parasite strains, with the heterogeneity-exploiting strains infecting the more susceptible hosts, in the absence of physiological tradeoffs and spatial heterogeneity, and even for markedly different transmission rates. In our model, intermediate-strategy parasites were selected against: should coexistence occur, an equilibrium is reached where strains occupied only the extreme ends of trait space, under appropriate conditions selecting for lower R₀.     

Invasion and persistence of infectious agents in fragmented host populations

One of the important questions in understanding infectious diseases and their prevention and control is how infectious agents can invade and become endemic in a host population. A ubiquitous feature of natural populations is that they are spatially fragmented, resulting in relatively homogeneous local populations inhabiting patches connected by the migration of hosts. Such fragmented population structures are studied extensively with metapopulation models. Being able to define and calculate an indicator for the success of invasion and persistence of an infectious agent is essential for obtaining general qualitative insights into infection dynamics, for the comparison of prevention and control scenarios, and for quantitative insights into specific systems. For homogeneous populations, the basic reproduction ratio R(0) plays this role. For metapopulations, defining such an 'invasion indicator' is not straightforward. Some indicators have been defined for specific situations, e.g., the household reproduction number R*. However, these existing indicators often fail to account for host demography and especially host migration. Here we show how to calculate a more broadly applicable indicator R(m) for the invasion and persistence of infectious agents in a host metapopulation of equally connected patches, for a wide range of possible epidemiological models. A strong feature of our method is that it explicitly accounts for host demography and host migration. Using a simple compartmental system as an example, we illustrate how R(m) can be calculated and expressed in terms of the key determinants of epidemiological dynamics.     

The effect of spatial heterogeneity and parasites on the evolution of host diversity

Both spatial heterogeneity and exploiters (parasites and predators) have been implicated as key ecological factors driving population diversification. However, it is unclear how these factors interact. We addressed this question using the common plant-colonizing bacterium Pseudomonas fluorescens, which has been shown to diversify rapidly into spatial niche-specialist genotypes when propagated in laboratory microcosms. Replicate populations were evolved in spatially homogeneous and heterogeneous environments (shaken and static microcosms, respectively) with and without viral parasites (bacteriophage) for approximately 60 bacterial generations. Consistent with previous findings, exploiters reduced diversity in heterogeneous environments by relaxing the intensity of resource competition. By contrast, exploiters increased diversity in homogeneous environments where there was little diversification through resource competition. Competition experiments revealed this increase in diversity to be the result of fitness trade-offs between exploiter resistance and competitive ability. In both environments, exploiters increased allopatric diversity, presumably as a result of divergent selection for resistance between populations. Phage increased total diversity in homogeneous environments, but had no net effect in heterogeneous environments. Such interactions between key ecological variables need to be considered when addressing diversification and coexistence in future studies.     

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.     

Areawide models comparing synchronous versus asynchronous treatments for control of dispersing insect pests

Integrated pest management (IPM) has the goal of combining several control methods that reduce populations of pest insects and their damage to tolerable levels and thereby reduce the use of costly pesticides that may harm the environment. Insect populations can be monitored during the season to determine when the densities exceed an economic threshold that requires treatment, often as an insecticide application. We developed a simulation model where insect populations varied in exponential growth in fields and dispersed to adjacent fields each day of a season. The first model monitored populations of individual fields in a grid of fields and treated any field with insecticide if it exceeded a threshold population (asynchronous model) as done in traditional IPM. The second model treated the entire grid of fields with insecticide when the average population of all fields exceeded the threshold (synchronous model). We found that the synchronous model at all growth and dispersal rates tested had average field populations during a season that were significantly lower and required fewer treatments than the asynchronous method. Parameters such as percentage of fallow fields, number of fields, and treatment threshold had little affect on relative differences between the two models. The simulations indicate that cooperation among growers in areawide monitoring of fields to obtain an average population estimate for use in treatment thresholds would result in significantly less insect damage and fewer insecticide treatments. The synchronous method is more efficient because population refugia are precluded from which dispersal could reintroduce insects.     

Mapping genes for polygenic disorders: considerations for study design in the complex trait of inflammatory bowel disease

While the methodology for the mapping of Mendelian disorders is well established, the practical and theoretical steps required for successful gene identification in a complex trait are still difficult to predict. A number of analytical models and simulations based on repetitive drawings from predefined statistical distributions are available. To supplement these analytical models, we developed an integrated simulation approach by directly simulating entire populations under a disease model based on epidemiological data. Random mating, nonoverlapping populations and the absence of differential fitness were assumed. Samples were drawn from these homogeneous and heterogeneous populations and analyzed with established analysis tools. We investigated the properties of linkage and association studies in inflammatory bowel disease - modeled as a six-locus polygenic disorder - as an example of this approach. In nonparametric linkage studies, lod scores varied widely, with the median required sample size depending on the locus-specific relative sibling risk. A fine mapping resolution <4 cM was found to require nonparametric lod scores >10. Family-based association studies (TDT test) and case-control studies showed a similar sensitivity and can identify risk loci in populations with moderate levels of linkage disequilibrium in sample sizes of 500-800 triplets. Case-control association studies were prone to false-positive results if applied in heterogeneous populations, with the false-positive rate increasing with sample size because population heterogeneity is detected with increasing power.