Heterogeneity across scales.

<p>Heterogeneity of outgoing secondary infections from subsets of blood-feeding habitats under different assumptions about their spatial arrangement. Phylograms are structured from bottom (depiction of spatial arrangement of blood feeding habitats) to top (by spatial extent and patch size) such that nearby blood-feeding habitats are grouped together, nearby groups combine to form larger groups, and so on, until all blood-feeding habitats are grouped together. Colors on the branches of the phylograms show the coefficient of variation of the sums across the rows of the diagonal submatrix of corresponding to blood-feeding habitats that comprise each cluster.</p

Heterogeneity, Mixing, and the Spatial Scales of Mosquito-Borne Pathogen Transmission

<div><p>The Ross-Macdonald model has dominated theory for mosquito-borne pathogen transmission dynamics and control for over a century. The model, like many other basic population models, makes the mathematically convenient assumption that populations are well mixed; <i>i.e.</i>, that each mosquito is equally likely to bite any vertebrate host. This assumption raises questions about the validity and utility of current theory because it is in conflict with preponderant empirical evidence that transmission is heterogeneous. Here, we propose a new dynamic framework that is realistic enough to describe biological causes of heterogeneous transmission of mosquito-borne pathogens of humans, yet tractable enough to provide a basis for developing and improving general theory. The framework is based on the ecological context of mosquito blood meals and the fine-scale movements of individual mosquitoes and human hosts that give rise to heterogeneous transmission. Using this framework, we describe pathogen dispersion in terms of individual-level analogues of two classical quantities: vectorial capacity and the basic reproductive number, . Importantly, this framework explicitly accounts for three key components of overall heterogeneity in transmission: heterogeneous exposure, poor mixing, and finite host numbers. Using these tools, we propose two ways of characterizing the spatial scales of transmission—pathogen dispersion kernels and the evenness of mixing across scales of aggregation—and demonstrate the consequences of a model's choice of spatial scale for epidemic dynamics and for estimation of , both by a priori model formulas and by inference of the force of infection from time-series data.</p></div

Spatial kernels.

<p>Panels correspond to the matrices that summarize pathogen dispersion by mosquitoes (a), pathogen dispersion by vertebrate hosts (b), pathogen dispersion through both species (c), and the spread of secondary host infections (d). Gray histograms show the empirical densities of each matrix's weighting at different distances, and black curves show a smoothed version of these data. Dashed lines show the average distance at which the events described by each matrix take place and therefore represent one way of defining the spatial scales of transmission with a single number. For example, the dashed line in (d) indicates that, on average, mosquito bites conferring a secondary host infection occur a distance of E away from where the corresponding primary host transmitted the pathogen to a mosquito.</p

Basic reproductive numbers () and vaccination thresholds () calculated by methods that variously account for heterogeneity (H), poor mixing (M), and finite host numbers (F), in both low- and high-transmission contexts in a randomly simulated population.

<p>Because the value of based on the matrix (denoted here as ) is the most complete description of the underlying transmission process, we also list the ratio of all other values to it (). All methods compute based solely on parameter values of the model, except Favier <i>et al.</i> 2006, which use both a subset of model parameters and an empirical estimate of the force of infection based on simulated epidemic data.</p

Mixing across scales.

<p>Evenness of mixing of secondary infections within subsets of blood-feeding habitats under different assumptions about their spatial arrangement. Phylograms are structured from bottom (depiction of spatial arrangement of blood feeding habitats) to top (by spatial extent and patch size) such that nearby blood-feeding habitats are grouped together, nearby groups combine to form larger groups, and so on, until all blood-feeding habitats are grouped together. Colors on the branches of the phylograms show the evenness of mixing in the diagonal submatrix of corresponding to blood-feeding habitats that comprise each cluster.</p

Epidemic dynamics across scales.

<p>When mosquito and host movement are both well mixed (a), each infectious bite originating from a single primary host is made on a unique secondary host. When mosquito and host movement are both poorly mixed (b), some hosts receive multiple infectious bites. Under these different scenarios about movement, epidemics originating in hosts with equal unfold much differently. Pathogen spread through a well-mixed population is quick, consistent, and complete (red in c), whereas pathogen spread through poorly-mixed populations is slower, more variable, and does not infect the entire host population (blue in c).</p

Model schematic.

<p>The model is specified on a continuous landscape with a point set of blood-feeding habitats, , and a point set of aquatic habitats, . The model is discrete in time with a time step equal to the length of the mosquito feeding cycle, in which mosquitoes take a blood meal, search for aquatic habitat (, red arrow), lay eggs, and repeat the search for another blood meal (, green arrow). Each host in allocates its time proportionally at multiple blood-feeding habitats (, brown arrows). During a single feeding cycle, each mosquito present at a given blood-feeding habitat takes a single blood meal, the collection of which are distributed differentially on hosts according to the proportion of time each spends there and a quantity describing each host's biting suitability (). This model structure allows for the derivation of weighted, bidirectional networks that summarize pathogen dispersion among blood-feeding habitats (houses) or among hosts (circles). From this process-based description of transmission, it is possible to derive network summaries of pathogen dispersion. Pathogen dispersion by mosquitoes: describes how mosquitoes taking an infective blood meal at one blood-feeding habitat distribute secondary, potentially infectious bites at other blood-feeding habitats. Pathogen dispersion by hosts: specifies the probability that a secondary bite on a human infected at one blood-feeding habitat takes place at some other blood-feeding habitat. Pathogen amplification: gives the total number of secondary bites on a host arising from primary bites on another host in a single feeding cycle. Host infection: contains the probabilities that a primary infection in one host will result in a secondary infection in some other host.</p

Understanding productivity (<i>i.e.</i>, the emergence rate of adults ) in heterogeneous habitats depends upon understanding the relationship between egg laying, carrying capacity (), and crowding.

<p><b>a</b>) The functional relationship between the rate of egg-laying and productivity depends on the functional response to crowding. In this model, the relationship is sensitive to the power-law scaling relationship ( blue; red; purple). Carrying capacity is given for a single value of egg laying rates, given at the steady state if that pool had existed in isolation. <b>b</b>) In a system with 2 pools linked by egg-laying, where the carrying capacity of pool 1 is approximately 90% of the total (dashed blue line) and pool 2 has the rest (dashed red line), the population totals overall (solid black) are generally below the maximum, unless egg laying is fine-tuned such that the proportion of eggs laid was equal to that pool’s proportion of carrying capacity (vertical grey). <b>c</b>) A comparison of productivity (red) and carrying capacity (black line) for a typical set of heterogeneous aquatic habitats. Productivity equals carrying capacity when the distribution of eggs laid is finely tuned to match the distribution of carrying capacities (i.e. ). <b>d</b>) The ratio of productivity to carrying capacity was computed for 100 sets of heterogeneous aquatic habitat. The green line plots the 1∶1 ratio, when productivity equals carrying capacity. These distributions, plotted here as the median (solid line) and the 10^th and 90^th quantiles (dashed lines), shows the robust pattern that the habitats with the lowest productivity tend to be under capacity and the few highly productive habitats tend to be over capacity.</p

The scaling between egg-laying and productivity is only apparent after normalizing both productivity and egg laying by carrying capacity.

<p>In completely heterogeneous environments, there may be a poor correlation between <b>a</b>) carrying capacity and productivity; <b>b</b>) egg laying and productivity; and <b>c</b>) egg laying and carrying capacity. <b>d</b>) The crowding law governing density dependence is found by plotting the ratio of eggs laid to carrying capacity against the ratio of productivity to carrying capacity (i.e. ). The constant was used to scale the <i>x</i>-axis.</p

The “effect size” of LSM in relation to coverage tend to be either linear or quadratic depending on whether eggs are laid in “treated” habitats and how well LSM is targeted.

<p><b>a</b>) Holding the total number of productive pools constant, adult mosquito population density declines as the number of unproductive pools increases and absorb eggs. <b>b</b>) The “egg sink” effect gives a non-linear effect to LSM if adult mosquitoes continue to lay eggs in the treated pools, so that treating 50% of the pools reduces adult density by 75%, and treating 75% of the pools reduces adult density by 95% (red). If adult mosquitoes do not lay eggs in the treated pools, however, then reductions in mosquito density are proportional to the % of habitat treated (blue). <b>c</b>) The change in adult mosquito density due to LSM in highly heterogeneous habitat as a function of the proportion of habitats treated depending on whether the adults lay eggs in treated pools (red) or avoid treated pools (blue), and depending on whether LSM was done in one particular random order (grey spikes), perfectly efficiently targeted (dashed lines), or perfectly inefficiently targeted (dotted lines). The black line represents a linear response with respect to coverage. <b>d</b>) For the same graphs as 3c, the effect sizes are plotted on a semi-log scale to highlight the benefits of LSM at high coverage. The best case for this system, with efficient targeting and egg-sink effects, predicts a hundred-fold (99%) reduction in mosquito density for 60% coverage. These benefits also get larger for higher coverage and show that there is enormous potential for LSM to reduce transmission through targeted repeated application of modern larvicides.</p