The risk of a mosquito-borne infection in a heterogeneous environment

Abstract

A common assumption about malaria, dengue, and other mosquito-borne infections is that the two main components of the risk of human infection--the rate at which people are bitten (human biting rate) and the proportion of mosquitoes that are infectious--are positively correlated. In fact, these two risk factors are generated by different processes and may be negatively correlated across space and time in heterogeneous environments. Uneven distribution of blood-meal hosts and larval habitat creates a spatial mosaic of demograPhic sources and sinks. Moreover, mosquito populations fluctuate temporally, forced by environmental variables such as rainfall, temperature, and humidity. These sources of spatial and temporal heterogeneity in the distribution of mosquito populations generate variability in the human biting rate, in the proportion of mosquitoes that are infectious, and in the risk of human infection. To understand how heterogeneity affects the epidemiology of mosquito-borne infections, we developed a set of simple models that incorporate heterogeneity in a stepwise fashion. These models predict that the human biting rate is highest shortly after the mosquito densities peak, near breeding sites where adult mosquitoes emerge, and around the edges of areas where humans are aggregated. In contrast, the proportion of mosquitoes that are infectious reflects the age structure of mosquito populations; it peaks where old mosquitoes are found, far from mosquito breeding habitat, and when mosquito population density is declining. Finally, we show that estimates for the average risk of infection that are based on the average entomological inoculation rate are strongly biased in heterogeneous environments.

Figure 1. Dynamics with Temporal Heterogeneity

The components of EIR follow different trends when mosquito populations vary temporally. Mosquito density (solid black) forms the dominant component of HBR. The density of infected mosquitoes (solid gray) peaks shortly after the density of mosquitoes (dotted vertical lines align the peaks). In contrast, the proportion of infectious mosquitoes (dashed) peaks while the mosquito population is declining. Seasonal mosquito emergence was modeled to have an long-term average M/H ≈ 2 (K = 2 and H = 1). The ticks on the x-axis mark the peaks of the wet and dry seasons.

Figure 2. Statics with Homogeneous Humans and Heterogeneous Mosquitoes

The components of EIR follow different trends when larval habitat is distributed at a single point and humans are uniformly distributed (the gray background illustrates the human distribution). (A) Mosquito density (solid) declines monotonically, but PIM (dashed) increases monotonically. The density of infected mosquitoes (Z, dotted) also declines monotonically. (B) HBR (solid) and EIR (dashed) both decline monotonically away from the source, reflecting the steep gradient in mosquito density. (C) The density of infected humans (dashed) and prevalence of infection in humans (solid) also decline monotonically (the curves coincide).

Figure 3. Statics with Heterogeneous Humans and Homogeneous Mosquitoes

HBR and EIR reflect mosquito movement and human distribution patterns when larval habitat is evenly distributed but humans have a low–high–medium distribution, such as a town with rural and suburban populations on either side (the gray background illustrates the human distribution). (A) Mosquito density (solid) is highest in town, peaks at the edges of town, and dips just outside of town. PIM (dashed) and the density of infected mosquitoes (Z, dotted) follow similar patterns. (B) HBR (solid) and EIR (dashed) are both high on the low-density side of town and lowest on the medium-density side of town, with peaks just inside town and troughs just outside of town. (C) The density of infected humans (dashed) and the prevalence of infection in humans (solid) peak at the edge of town, but prevalence of infection in humans is less variable than HBR or EIR.

Figure 4. Statics with Heterogeneous Humans and Mosquitoes

When human density increases smoothly away from a larval habitat (the gray background illustrates the human distribution), the patterns of EIR components reflect heterogeneity in the distribution of larval habitat and human populations. (A) Mosquito density (solid) peaks an intermediate distance away from the source. The peak density of infected mosquitoes (Z, dotted) is further from the source because PIM (dashed) increases monotonically away from the source. (B) HBR (solid) decreases monotonically away from the source, reflecting mosquito density, but EIR (dashed) has a minor peak away from the source. (C) The density of infected humans (dashed) peaks away from the source, but the prevalence of infection in humans (solid) remains relatively constant near the source, dropping off sharply further away.

Figure 5. The Relationship between EIR and Human Prevalence, with Heterogeneity

(A) EIR and the prevalence of infection in humans have a tidy relationship among patches; each small symbol is from a single patch in Figures 2–4. The relationship, given by equation 1, is plotted in gray. The phase plane of the dynamic relationship over time from Figure 1 is plotted with dashed lines. Average EIR is plotted against the average prevalence (large symbols). Predicting the average prevalence of human infection from average EIR leads to underestimates. (B–D) The density of infectious mosquitoes (Z) (B), HBR (HBR = aM/H) (C), and PIM (PIM = Z/M) (D) are plotted against the proportion of humans who are infected and infectious. PIM is a particularly bad measure of the risk of infection; in heterogeneous habitats, it peaks far from larval habitat, where mosquito density and prevalence of infection in humans is lowest. This accounts for the large number of points where PIM is high, but the proportion of infectious humans is low.