Mosquito Ecology
Overview
Mosquito behavior and ecology plays a core role in malaria epidemiology, but it has played only a minor role in malaria transmission dynamics and control.
In George Macdonald’s model, mosquito infection dynamics were summarized by a few core parameters: the ecology part was never spelled out [1,2]. The classic parameter set was codified in formulas for \(R_0\) and vectorial capacity (VC) [2–4]. While the original formala for the VC was derived from a very specific set of assumptions, the formula for VC has taken on a life of its own [5–7].
In developing ramp.xds, we recognized several major types of models describing adult mosquito ecology. The MY component modules fall into two categories:
In basic ecology models, the compartments for adult mosquito populations represent infection states. Macdonald’s mosquito models were not extensible, so the base model for
ramp.xdsdevelopment was published by Joan Aron and Bob May [8].In behavioral state models, the compartments subdivide a population by behavioral states as well as infection states [9].
A complete framework for mosquito ecology needs to explicitly define egg laying and then couple it to the dynamics of immature mosquito populations in aquatic habitats, and emergence [10].
- To model
In mosquito ecology, there has been a long-standing interest in using parity to measure the reproductive age of a mosquito.
Basic Ecology
Adult Mosquitoes
In 1982, Joan Aron and Bob May published a model that made that implicit model explicit and then extended it. In that model \(M\) is a variable describing adult mosquito density, \(\Lambda(t)\) the emergence rate of adult mosquitoes, and \(g\) the death rate [8]. In this model, an equation described mosquito population dynamics: \[ \frac{dM}{dt} = \Lambda(t) - g M \]
Basic Infection Dynamics
To model infection dynamics, we let \(f\) be the overall feeding rate, \(q\) the human fraction, and \(\kappa\) the net infectiousness (NI). Letting \(Y\) denote the density of infected mosquitoes, we get: \[\frac{dY}{dt}= fq\kappa(M-Y)-g Y\] To get the density of infecious mosquitoes, \(Z,\) Aron & May use a fixed delay to model the latent period, commonly known as the extrinsic incubation period (EIP). If mosquitoes become infected \(\tau\) days after they became infected, then they survive the EIP with probability \(e^{-g\tau}\), and we introduce the subscript \(\tau\) to denote the lagged value of a term of variable: \[\frac{dZ}{dt}= fq\kappa_\tau(M_\tau-Y_\tau) e^{-g\tau} -g Z\] The net infectious biting rate is \(fqZ\)
Aquatic Ecology
A model for adult mosquito population dynamics can be coupled to a model for aquatic mosquito population dynamics if we model egg laying by adult mosquitoes. Software developed around a basic model by Smith et al. (2013) [10]: adults lay eggs in aquatic habitats, where larvae compete for resources.
Spatial Dynamics
In ramp.xds, we established a convention for modeling mosquito spatial ecology using a demographic matrix for both survival and dipsersal, \(\Omega.\) In a spatial model with \(N_p\) patches, \(\Lambda(t)\) is a vector of length \(N_p\), \(M\) a vector of state variables of length \(N_p\), \(\Omega\) a \(N_p \times N_p\) matrix, and \[
\frac{dM}{dt} = \Lambda(t) - \Omega \cdot M
\] If \(\Lambda\) is a constant, then mosquito population density will come to a steady state \[ \bar M = \Omega^{-1} \cdot \Lambda.\] and we can expand the definition of vectorial capacity for spatial models. Vectorial capacity is the number of infectious bites arising from all the mosquitoes blood feeding on a single available human in a single patch on a single day.
The classic formula is problematic, for various reasons, and software development was based on two formulas (using either \(\Lambda\) or \(\bar M\)) that are equivalent (under Macdonald’s assumptions): \[V = \frac {\bar M} H \frac{f^2 q^2}{g} e^{-g\tau} = \frac \Lambda H \frac{f^2 q^2}{g^2} e^{-g\tau}\]
Aquatic Ecology
Non-Autonomous Dynamics
To get a model that is equivalent to the one used by George Macdonald, we set \(N_p=1\), and if we let \(m=\bar M/H,\) \(a=fq\), \(y = Y/M,\) and \(z=Z/M.\)
Behavioral States
Behavioral state models for mosquito ecology consider the physiological status of a mosquito and its associated behaviors [9,11]. Gravid mosquitoes will tend to be seeking water to lay eggs, but after laying eggs, the mosquito will seek blood or perhaps sugar. Compartmental models consider both behavioral states and infection states, so they are more complex than basic models. In ramp.library