Table 1.
The five parameters comprising the classical formula for vectorial capacity (VC or denoted V), describing the total number of infectious bites arising from all the mosquitoes feeding on a single human on a single day under the assumptions of the Ross-Macdonald model [4, 6, 11]. The expected number of blood meals on the pathogen’s hosts, summed over a mosquito lifespan, is given by the term S = fQ/g. The probability of surviving the EIP is P = e−gn. Mosquito population density is m = λ/g. Under the assumptions, the formula for VC is V = λS2 P. In the Ross-Macdonald model, the entomological inoculation rate, , is related to VC by a formula:
, where κ is the proportion of bites on humans that infects a mosquito; the approximation holds when κ is small, such that mosquito super-infection is rare. In MBITES, the same quantity can be computed directly by Monte Carlo simulation.
Fig 1.
Structure of an activity bout.
Top) MBITES and MBDETES model mosquito behavioral states and state transitions required for the gonotrophic cycle. The first two columnns list the behavioral states, and the last two columns describe the potential state transitions. A mosquito is either searching for a blood host (F) or attempting to blood feed (B), searching for aquatic habitat (L) or attempting to oviposit (O), or resting (R). Transitions depend on whether the last bout was a success or failure, and optionally on refeeding behavior† or laying a partial egg batch and skip oviposit‡. The next activity bout is also affected by whether a mosquito decides to make an attempt or initiate a search*. Bottom) In MBITES models, each behavioral state has an associated activity bout that has a common structure, as illustrated in the diagram. The activity bout involves a sequence of four phases: launch, do an activity (either a search or an attempt), land, and rest. The type of activity is determined both by its behavioral state and by presence and availability of resources. A mosquito will stay (S) unless there are no resources present or if the mosquito has become frustrated (*), in which case it will initiate a search. If the mosquito decides to stay, it makes a choice and an approach that may or may not succeed at what it was trying to do. When a mosquito lands, it selects a micro-site for a resting spot from the set of possibilities at that site. During the resting period, data from the last bout are logged, the behavioral state is updated, and the waiting time to the next launch is determined. A mosquito enters the bout either after emerging from aquatic habitat or after exiting its previous bout and surviving.
Table 2.
State transitions & waiting times.
In MBITES, it is possible to compute the expected state transitions and waiting times from any state to the next state. In MBDETES and limiting cases of MBITES, these single-state transition expectations can be used to estimate the state transition probabilities and waiting times from one state to every other state, including the length of a gonotrophic cycle, from resting to resting (i.e., from R → R). The table gives formulas for the probability of surviving to reach the behavioral state Y starting from another state X, ΨX,Y, where X, Y ∈ {L, O, F, B, R}. Note that PX,Y denotes the single activity bout probability of a state transition. It also gives the expected waiting time to Y from X is TX,Y. These formulae are expressed in terms of the single bout state transitions and waiting times, PX,Y and (or they can be by making a simple substitution from one of the formulas appearing in the table above it).
Fig 2.
The structure of a blood feeding attempt bout in MBITES.
The flowchart follows the progression of a mosquito through simulated events, from the launch (dark grey oval), choosing a host from the atRiskQ (aquamarine diamond), and the events that follow depending on what sort of host was chosen (yellow rectangles). If a human is chosen (or more generally, a blood host that is also a host for the pathogen), then each mosquito must approach and attempt to probe (salmon rectangle) and then blood feed (red rectangle). If a non-human host is, probing is ignored. Traps mimicking a blood host can also be chosen. After a blood meal (red rectangle), a mosquito must land and choose a resting spot (yellow diamonds). A post-prandial resting period follows a successful blood meal which has its own hazards (purple oval), including additional hazards associated with a flight laden with blood, which may be followed by decision to feed again (dark red diamond). Similarly, after failing the attempt (green pentagons to green rectangle), a mosquito must land and choose a resting spot (yellow diamonds). At each step, it is possible to die (light grey ovals). At any point when failure occurs or during landing, a mosquito could choose to leave the haunt and initiate a search on the next bout. This condition is checked after completing the bout (pink diamond). At the end of a bout, the mosquito’s behavioral state and other state variables are updated. The endpoint of each bout is either death (grey ovals), a repeated blood feeding attempt (dark red oval) or a state transition to either a blood feeding search (pink oval) or to oviposit (blue ovals).
Fig 3.
The structure of an egg laying attempt bout in MBITES.
The flowchart follows the progression of a mosquito through simulated events, from the launch, choosing a habitat or trap from the eggQ (aquamarine diamond), and the events that follow depending on whether the mosquito chose a habitat or a trap (yellow rectangles). If a mosquito approaches the habitat, it could lay eggs. Alternatively, a mosquito could approach a trap and fail in the approach (thus surviving) or die (light grey ovals). If a mosquito is deterred in the approach to its habitat or the trap, it fails (green pentagons to green rectangle). After a successful approach to a habitat, a mosquito lays eggs (blue rectangle). After laying eggs or failing, a mosquito must land and survive (yellow diamonds). If not all eggs were laid, a mosquito can choose another habitat to lay (light blue diamond). At any point when failure occurs or during landing, a mosquito could choose to leave the haunt and initiate a search on the next bout. This condition is checked after completing the bout (pink diamonds). At the end of a bout, the mosquito’s behavioral state and other state variables are updated. The outcome of each bout is either death (light grey ovals), a repeated egg laying attempt bout (dark blue ovals) or a state transition to either an egg laying search (light blue ovals) or a blood feeding attempt (red or pink ovals).
Fig 4.
Comparison of results from MBITES and MBDETES under restricted (Markovian) assumptions on waiting times and state transition probabilities.
B: Egg laying rate is the number of eggs laid, per female, per day. C: Blood feeding by age is the age distribution of mosquitoes taking bloodmeals. D: Feeding cycle duration is the time between post-prandial resting periods. In each panel, MBITES is summarized as a red histogram overlaid against the smooth density (in blue) predicted by MBDETES. All cases see excellent agreement, with MBITES fluctuating around MBDETES due to finite sample size of mosquitoes in the stochastic simulation.
Fig 5.
Measures of mosquito dispersion.
Smoothed distribution (red line) and density (blue area) functions are displayed for summary statistics calculated for one particular landscape (50% peri-domestic habitats). A: The spatially averaged movement kernel is simply the probability of movement by distance, averaged over all haunts on the landscape. B: Cumulative movement, gives the distribution of total distance traveled by mosquitoes over their entire lifetime, and has a long right tail. C: Lifetime displacement is the absolute displacement of a mosquito, that is, the distance between the natal aquatic habitat they emerged from and the site at which their died. D: Dispersion of VC shows the distribution of secondary bites by distance, and follows closely absolute displacement of mosquitoes. All plots are calibrated to the same x-axis for comparison.
Fig 6.
3 simulated landscapes at A: 0%, B: 50%, and C: 100% peri-domestic habitats. Haunts that contain only blood feeding haunts are plotted as red circles, haunts that contain only aquatic habitats are plotted as green triangles, and those haunts that contain both types of resources are shown as blue squares (i.e., peri-domestic habitats). Dispersal kernels were calibrated as if this was an area of about 100 km2.
Fig 7.
In MBITES, vectorial capacity (VC) is computed directly as the average number of infectious bites (i.e., probing) arising from all the mosquitoes blood feeding on a single human on a single day; it is effectively the number of pairs of events where a blood meal by a mosquito is followed at least EIP days later by that same mosquito probing in attempt to feed on a human, measured per human, per day. Summary VC A,B,C: and number of human blood meals per mosquito over its lifespan (D,E,F; referred to as the stability index by Macdonald) are shown by column for 0%, 50%, and 100% peri-domestic habitats. Each histogram gives the distribution of VC or the number of human blood hosts across mosquitoes for that percent peri-domestic habitats.
Fig 8.
Behavioral state distribution.
Chord diagrams showing the empirical state transition matrices for three of the 26 experiments: A: 0%, B: 50%, and C: 100% peri-domestic habitats. These were calculated for each experiment by summing transitions for each mosquito between two states and then averaging to produce a Markov transition matrix. The width of the directed edges between each behavioral state is proportional to the probability of that transition, and the area on the perimeter of the circle labeled for each state is proportional to the mean time spent in that state. The three chord diagrams are accompanied below (D-F) by quasi-stationary probability distributions which give the asymptotic distribution of how a mosquito spends time across behavioral states conditional on survival.
Fig 9.
Simulations in MBITES illustrate that all of the bionomic parameters are sensitive to the proportion of peri-domestic habitats, which gives a measure of how frequently a mosquito must search. The x-axis of each plot ranges from 0% to 100%, and each summary bionomic parameter is plotted as mean (solid line), median (dashed line), and the shaded area covers the 20-80% quantile range of the data. The distribution of number of blood hosts B: exhibits significant right skew, such that the mean exceeds the 80% quantile at low proportion peri-domestic breeding habitats. Because the simulations are stochastic, the exact number of mosquitoes from which Monte Carlo estimates of the bionomic parameters were computed varied somewhat over the 26 landscapes, the mean was 456,579 mosquitoes with a standard deviation of 754 mosquitoes.
Fig 10.
Dispersion and movement parameters.
In MBITES, vectorial capacity (VC) and its dispersion are highly sensitive to the proportion of peri-domestic habitats. Interpretation of axes follows Fig 9, and each summary bionomic parameter is plotted as mean (solid line), median (dashed line), and the shaded area covers the 20-80% quantile range of the data. A: Number of secondary bites produced increases dramatically as a function of peri-domestic habitats. B: Spatial dispersion shows no strong trend however, due to the strong clustering of haunts in the landscape (it largely follows the trend of absolute lifetime displacement (C), as opposed to cumulative movement (D). At low percent peri-domestic breeding habitats, significant right skew in the distribution of VC pulls the mean above the 80% quantile.