The unexpected importance of mosquito oviposition behaviour for malaria: non-productive larval habitats can be sources for malaria transmission

Here, a recent spatial model for malaria epidemiology on heterogeneous landscapes was modified by incorporating a more detailed description of the gonotrophic cycle into models for mosquito infection dynamics [32] (Figure 1). Let S denote the density of susceptible (i.e. uninfected) mosquitoes, L denote the density of latent (i.e. in the incubation period: from the onset of infection to the beginning of the infective period) and I the density of infective mosquitoes. Whatever her state of infection, the mosquito alternates between the activities of blood-meal feeding and ovipositing. Fed, gravid mosquitoes, denoted with subscript f, have taken a blood-meal and seek a place to oviposit, while unfed mosquitoes, denoted with subscript u, have recently oviposited and seek a blood-meal host.

Spatial heterogeneity was incorporated by subdividing the landscape into i patches in an array or grid (see below), with subscripts denoting the mosquito location and state. Thus, I _{i, f} describes the density of fed, infective mosquitoes in patch i (Eq. 1). Let A_i,u, denote the rate (speed of an event over time) at which unfed mosquitoes feed on a human host in patch i, if one is available; upon biting, the mosquito changes state from "unfed" (u) to "fed" (f). Similarly, let A_i,fdenote the rate at which fed mosquitoes oviposit in patch i, if water is available; upon ovipositing, mosquitoes change state from fed to unfed. Thus, if hosts and water are available in patch i, the expected time to find a host is A_i,u^-1, and the expected time to oviposit is A_i,f^-1, so the duration of one gonotrophic cycle in patch i is A_i,f^-1 + A_i,u^-1. Note that the human biting rate (HBR) in each patch includes biting only by unfed mosquitoes: HBR _i = A _i,u (S _i,u + L _i,u + I _i,u ) / H _i , where H _i denotes human density in the i ^th patch. In this model, the entomological inoculation rate (EIR) in each patch includes biting only by infective, unfed mosquitoes, . Similarly, the sporozoite rate in unfed mosquitoes is defined as ratio of unfed infective mosquitoes to total number of unfed mosquitoes .

Let ε_i denote the local emergence rate of adults. If ε_i > 0, a patch was considered to be a productive source for mosquitoes. Some patches might have water where mosquitoes can oviposit, but no adults emerge. Thus, if A_i,f> 0, but ε_i = 0, the patch was called a non-productive water source. Obviously ε_i = 0 if no water was available. Thus, it is implicitly assumed that the emergence rate of adult mosquitoes is regulated at the pre-adult stages, not limited by the availability of eggs.

It is assumed that mosquitoes move among patches [33], depending on their gonotrophic state and the availability of hosts and water. Since water or hosts might not be available in some patches, A_i,fand A_i,udenote the local biting rates, subject to host availability [34] and local oviposition rates subject to the availability of oviposition habitat [35]. If humans were not available, A_i,u = 0, and if water was not available, A_i,f = 0: a mosquito in an unfed or fed state, respectively, would migrate to another patch in search of a blood-meal host or larval habitat (see below). Thus, the residence time in each disease state is longer if no host or no water is available. Otherwise, the parameters assume a positive value (Table 1). It is assumed that emigration of mosquitoes depends on the presence of water for fed mosquitoes and on human density for unfed mosquitoes. Let w denote the per-capita emigration rate of a fed mosquito, which depends on water availability: it describes the expected number of patches a mosquito would cross in one day if no water were available w = 0 if water is available). Similarly, it is assumed that the migration of unfed mosquitoes is a function of local human density, H _i . Let γ denote the per capita emigration rate of an unfed mosquito; thus γ = ςe^-ψHi, where ς corresponds to the maximum daily number of patches a mosquito would visit in a day if no humans were available and ψ describes her responsiveness to human density. Another parameterω _{i, j} describes the proportion of mosquitoes leaving patch j that fly into patch i; thus, . It is assumed that ω_i,j= 0 unless two patches are adjacent and that mosquitoes move in a random direction. The same migration rates were applied for mosquitoes without regard for infection status. Let Ω(C _ij ) denote the total migration rate for mosquitoes in state C and location i; for example, represents the net migration of fed susceptible mosquitoes moving from patch i, and the net migration of unfed susceptible mosquitoes from patch i.

Finally, it is assumed that a susceptible mosquito biting an infective human becomes infected with probability c, that latent mosquitoes become infective at the rate θ and that mosquitoes die at rate μ. Let X _i denote the proportion of humans in patch i who are infective, assuming no latency (i.e. no pre-patent period) and no delay between the appearance of merozoites and gametocytes. Let b denote the probability that a bite by an infective mosquito produces a human infection, and r denote the rate at which a human infection is cleared; in other words, it is assumed that the recovery period is exponentially distributed with average duration r^-1. The infection dynamics of malaria in mosquitoes and humans over time and space, including the mosquito gonotrophic cycle, are described by the following equations:

Parameter values are listed in Table 1. The emergence rate was manipulated such that the overall ratio of mosquitoes to humans was 2. The equations were numerically solved over a period of four years; by that date, the system of equations was at the equilibrium . The resulting static spatial distributions of the variables were plotted on two kinds of hypothetical landscapes.

First, simulations of malaria dynamics were performed on a linear array of 17 patches. Humans were uniformly distributed in patches 2 to 16 and absent from the edges. For mosquitoes, three scenarios were considered:

This landscape is similar to that in a recent study in Tanzania that evaluated mosquito dispersion within three hamlets and showed that marked vectors dispersed differently in relation to the distribution of breeding sites [36].

Second, simulations of malaria dynamics were performed on a square grid of 100 patches (10 × 10). All 10 patches on the left side of the grid were productive water sources (e.g. a stream or pond edge). A non-productive water source was located near the centre of the grid and the human population was uniformly distributed on all the patches except on the left side of the grid. Among the total number of mosquitoes leaving a patch, it was assumed that 80% were randomly flying to patches that shared a side (up, down, left and right patches) and 20% to patches that shared a corner (on the diagonal). This scenario is based on a real-world example: we simulate a village away from a river where the breeding sites are mainly located [37] and with a water source in its centre that is non-productive because it was sprayed with larvicide or prone to desiccation.

A sensitivity analysis on the linear array from the scenario (c) above was performed. The evaluation of the influence of water on mosquito migration and distribution focused on the segment of the mosquito population that was fed and seeking a site to oviposit. A multivariate sensitivity analysis [38] to assess the impact of the parameters that describe the behaviour of fed mosquitoes on the proportion of infected humans was performed; parameters examined were the migration parameter w (the expected number of patches a mosquito would fly without water), the oviposition parameter A _f ^-1 (the time between taking a blood-meal and oviposition) and mortality μ^-1 (mosquito lifespan). These three parameters were sampled in accord with a Latin Hypercube Sampling (LHS) scheme [39]. The distribution of malarial infections in humans was computed using 10,000 sets of parameters drawn from a uniform distribution, with bounds described in Table 1. To evaluate the impact of uncertainty in these parameters we calculated the 5^th and 95^th percentile of the resulting distribution of the human malaria prevalence for each patch. The Partial Rank Correlation Coefficients (PRCCs) were also calculated using the 10,000 values for each parameter and the 10,000 predicted values in malaria prevalence over each patch. A univariate sensitivity analysis on the three parameters described above was performed using the extreme values.