The Paradigm Formula: Your 2012 Survival Guide
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For decreasing, U-shaped, and unimodal mortality models, other parameters are also important.
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This is particularly true for the unimodal mortality curve where vectorial capacity is sensitive to changes in the parameter determining the location of the mortality maximum. Figure 2 shows vectorial capacity in a stable population in dependence on intrinsic rate of grows r for three mortality models: exponential, Gompertz and logistic.
The largest differences in vectorial capacity between these models occurs when value of r is intermediate or low. For age-independent mortality, lower r values correspond to lower birth rates because the mortality rate is constant.
In this case, decreasing r by decreasing birth rates causes an increase in the proportion of the vector population in older age classes that are capable of transmitting a pathogen, which in turn increases vectorial capacity. Conversely, with age-dependent mortality age-specific mortality and fecundity rates vary non-linearly with different values of r.
The complex interaction between nonlinear birth and death rates and r lead to a decrease in vectorial capacity as r decreases, which can occur because mean age and thus transmission potential decrease as r decreases.
This reinforces the need for new data that can be used to refine and test predictions regarding associations between intrinsic rate of growth and the complexity of age-dependent mortality. Depending on mosquito population density, there are epidemiological important differences in dynamics of human and mosquito infections with age-dependent versus age-independent vector mortality Fig.
For such high values of R o , epidemic levels of transmission would be expected for both age-independent and -dependent mortality and stationary values would be similar for infectious host and vector populations Fig. With age-dependent mortality and the proportion of infected vector and host populations declines with time Fig.
For age-independent mortality and the infected proportion of the population increases over time. Solid line denotes humans and dashed line mosquitoes. The new mathematical models derived and investigated in this paper extend the theoretical and empirical understanding of age-dependent vector mortality in ways that can add to the conceptual basis of vector-borne disease prevention. The formula for dependence of vectorial capacity from the vector survival is general enough and can be used in calculation of vectorial capacity for any of age-dependent mortality of vectors.
One immediate utility of the formula is in the case, when a vector is exposed to two causes of death: exogenous, which does not depend on the vector age, such as predation, swatting, weather conditions, and endogenous, related to ageing and vector senescence. It is realistic to consider these two causes as independent hazards, which mathematically means that the resulting mortality is a sum of the exogenous and endogenous mortalities.
The presented formula allows calculations of vectorial capacity to find the limits of control, when measures, aimed to increase the exogenous mortality, can be effective against the background of endogenous age-dependent mortality. In this case the value of age-independent mortality component can be considered as a parameter for construction of optimal control strategy under given resource limitations.
Introduction of age-dependent mortality broadens and refines the vectorial capacity paradigm by introducing an age-structured extension to the model. It encourages further research on the actuarial dynamics of vector populations and the contribution of dynamics in vector mortality to dynamics in pathogen transmission. It provides a quantitative basis for understanding the relative impact of reductions in vector longevity compared to other strategies for prevention of vector-borne disease; i.
Our analysis indicates that different age-dependent patterns of mortality can influence vectorial capacity differently. Although mean lifespan remains an important determinant of vectorial capacity, other parameters that affect the shape of a mortality curve are also important. We used demographic entropy to illustrate such influence in one value, which is free from the specific parameterization of age-dependent mortality.
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In addition to mean life span, our analysis of entropy illustrates how survival curves can differ from the original assumption of exponential, or age-independent, mortality profile. The shift from age-independent to age-dependent mortality can be viewed as conceptually advantageous because it captures transmission dynamics in a more biological relevant way . It also increases the mathematical complexity of the models, which raises questions about their general applicability in applied, epidemiologic contexts.
A key issue in this regard concerns the strength of the insights gained from the increased complexity, biological and mathematical. That is, does including the additional age-dependent elements substantially improve the power of the models for guiding disease surveillance and prevention? Results from our analyses indicate that there are important differences in the epidemiologic output from age-dependent vs -independent vector mortality models.
Understanding the nature of output differences is challenging because it depends on variation in complex characteristics of the system being examined. Examples include novel vector control strategies that target older vectors or aim to shorten vector lifespan  ,  or prevent disease by pathogen elimination . Our vectorial capacity model has several limitations. It assumes no emigration or immigration of vectors or hosts, which can be particularly important if rates are not equal. It is deterministic and does not account for individual variatiability hidden heterogeneity in chances of survival.
In some cases it may be difficult to fit simple mortality functions e. More research is needed to examine the impact on vectorial capacity when vector age-structure and population growth rate vary over time. Estimates of vectorial capacity, and thus R 0 , for mosquito-borne pathogens differ most between cases of age-independent and age-dependent mortality when: 1 vector densities are relatively low, 2 vector population growth rate is relatively low and there are differences in the complex interactions between birth and death; and 3 vectors exhibit complex patterns of age-dependent mortality.
In many parts of the world vector control is an integral or even primary component of vector-borne disease prevention. Our analyses indicate that when vector control programs are successful and mosquito densities are reduced to low levels, the mortality model used to predict sustainability of that success or the effort needed for the final push to eliminate the pathogen can lead to strikingly different conclusions.
An age-independent model may overestimate the effort needed to meet public health goals at low vector densities. Our conclusions support earlier results indicating that age-dependent vector mortality can influence transmission dynamics and the success of disease prevention strategies in meaningful ways  ,  , . In a similar study, Bellan  demonstrated that age-dependent vector mortality has important effects on vectorial capacity and vector control.
By focusing only on the logistic mortality model and possible effects of two control measures decreased survival and decreased recruitment that is equal across all ages, with each intervention affecting a single parameter and simplifying the equation for vectorial capacity, he demonstrated that the effects of interventions may be over- or under-estimated when assuming age-independent survival. Our study, however, goes further by providing a single complete formula for vectorial capacity that can be used for any vector, mortality model, and control scenario.
Our formula allows researchers to calculate vectorial capacity and assess the effects of various control measures for their own particular system. Importantly, this equation may be easily expanded to include more complex functions affecting vectorial capacity including density-dependent mortality in addition to age-dependent mortality or factors affecting biting rate.
It is important to note that despite growing evidence showing notable effects of age-dependent mortality on estimates of vectorial capacity and effects of control, we still know very little about the prevalence and pattern of age-dependent mortality in natural mosquito populations  ,  ,  , .
Age-dependent mortality and the diversity of its patterns has been examined for non-vector insects  , . Most of those studies lacked sufficiently large sample sizes to accurately estimate mortality rates of relatively rare, older-aged individuals .
We similarly know little about what factors govern disease vector mortality patterns or how mortality patterns vary through space and time. Theoretically, variation in patterns of age-dependent mortality could cause frequent and dramatic fluctuations in vectorial capacity and entomological thresholds below which epidemic pathogen transmission will cease.
New techniques to better estimate patterns of age-dependent vector mortality, how mortality patterns vary in space and time, and the factors determining those patterns are needed to better understand when and how age-dependent vector mortality has its greatest affects on transmission dynamics and disease intervention campaigns. The field of aging research can make substantial contributions to improved understanding and more efficient prevention of vector-borne disease because it deals with factors and mechanisms affecting age-specific patterns of mortality among different species  , .
Identifying concepts and interventions capable of accelerating vector aging processes, and understanding how such manipulations affect pathogen transmission parameters can stimulate investigation of new approaches for vector-borne disease control. Consideration of more realistic situations will require more sophisticated models and more comprehensive computational analyses of alternative scenarios.
For example, undefined heterogeneities in vector mortality patterns are likely to be important determinants in the success or failure of vector control programs. Due to variation within and between mortality patterns, a strategy that works well at one place and time may not work at another. Our analysis indicates that a more sophisticated analytical framework, which is mathematically and computationally plausible, will stimulate increasingly insightful thinking about age-dependent vector mortality and prevention of vector-borne disease.
Although our analyses use data from a single mosquito species, Ae, aegypti , our models are intended to have broad application for a wide range of vector species and vector-borne diseases. Vectorial capacity in stable and stationary vector population.
Proportion of the host population that is infected or infectious when the vector population exhibits age-dependent mortality. Browse Subject Areas? Click through the PLOS taxonomy to find articles in your field. Abstract Background Vectorial capacity and the basic reproductive number R 0 have been instrumental in structuring thinking about vector-borne pathogen transmission and how best to prevent the diseases they cause.
Introduction The basic reproductive number R 0 and vectorial capacity are integral parts of the language, science, and control of vector-borne disease . Download: PPT. Materials and Methods Background Styer et.
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Vectorial Capacity in Vector Population Based on equation 1 , we modeled total vectorial capacity of a population of mosquitoes, C. Table 2.
Formulas for vectorial capacity in stable and stationary vector populations. Role of Age-dependent Mortality in Determining R 0 Age-dependent mortality can have important implications for the spread of vector-borne pathogens through host populations. The general relationship between the basic reproductive number and the vector capacity is given de from which the basic reproductive number in stable population with age-dependent mortality is given by: We use the same parameter values as above and the survival function for logistic mortality to explore the dynamics of these models.
Results Vectorial Capacity for Different Patterns of Age-dependent Mortality To examine the behavior of the model, we calculated vectorial capacity for Ae. Figure 1. Illustration of mortality models examined with different parameter values. Figure 2. Vectorial capacity in a stable population for three mortality models see Table 3 for functions. Figure 3. Hypothetical survivorship curves for different values of H, entropy.
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Figure 4. Vectorial capacity versus entropy H for different values of mean life span e 0. Role of Intrinsic Growth Rate r Figure 2 shows vectorial capacity in a stable population in dependence on intrinsic rate of grows r for three mortality models: exponential, Gompertz and logistic. R 0 and Dynamics of Host Infection Depending on mosquito population density, there are epidemiological important differences in dynamics of human and mosquito infections with age-dependent versus age-independent vector mortality Fig.
Figure 5. Discussion The new mathematical models derived and investigated in this paper extend the theoretical and empirical understanding of age-dependent vector mortality in ways that can add to the conceptual basis of vector-borne disease prevention. Supporting Information. Supplement S1. Supplement S2. References 1. Accessed Jan Ross R The Prevention of Malaria. John Murray, London. Oxford University Press, Oxford. Nature — View Article Google Scholar 5. View Article Google Scholar 6. View Article Google Scholar 7.