An analytically tractable, age-structured model of the impact of vector control on mosquito-transmitted infections

Vector control is a vital tool utilised by malaria control and elimination programmes worldwide, and as such it is important that we can accurately quantify the expected public health impact of these methods. There are very few previous models that consider vector-control-induced changes in the age-structure of the vector population and the resulting impact on transmission. We analytically derive the steady-state solution of a novel age-structured deterministic compartmental model describing the mosquito feeding cycle, with mosquito age represented discretely by parity—the number of cycles (or successful bloodmeals) completed. Our key model output comprises an explicit, analytically tractable solution that can be used to directly quantify key transmission statistics, such as the effective reproductive ratio under control, Rc, and investigate the age-structured impact of vector control. Application of this model reinforces current knowledge that adult-acting interventions, such as indoor residual spraying of insecticides (IRS) or long-lasting insecticidal nets (LLINs), can be highly effective at reducing transmission, due to the dual effects of repelling and killing mosquitoes. We also demonstrate how larval measures can be implemented in addition to adult-acting measures to reduce Rc and mitigate the impact of waning insecticidal efficacy, as well as how mid-ranges of LLIN coverage are likely to experience the largest effect of reduced net integrity on transmission. We conclude that whilst well-maintained adult-acting vector control measures are substantially more effective than larval-based interventions, incorporating larval control in existing LLIN or IRS programmes could substantially reduce transmission and help mitigate any waning effects of adult-acting measures.

In order to consider interventions in terms of coverage, we define the action of these three measures as described in Table A. We consider coverage of LLINs to describe the percentage of indoor-sleeping individuals who sleep daily under bednets; coverage of IRS describes the percentage of people who have had their bedrooms sprayed in the previous 6 months; coverage of larvicides is taken to be the percentage coverage (by area) of larval sites with weekly larvicidal treatment.
An experimental hut trial in Benin tested the efficacy of LLIN and IRS interventions using a pyrethroid-impregnated polyester LLIN, untreated nets and chlorfenapyr IRS [1] against Anopheles gambiae and Culex quinquefasciatus.The LLINs used were deliberately provided with either 6 holes (4cm 2 each) or 80 holes (2cm 2 each) to simulate different levels of integrity.The results for Anopheles gambiae vectors are shown in Tables B and C. LLINs were found to have the highest repelling effect, with only 12.1% of vectors successfully feeding in the presence of a bednet.In comparison, untreated nets allowed 56.4% of vectors to successfully feed (see Table B).The insecticide used was assumed to have a half-life of 2 years [2], allowing variation of repelling and mortality parameters from those observed for new LLINs (6 holes, insecticide-treated) to those observed for untreated nets.IRS parameters for deterrence (pre-entering), feeding inhibition and death (postfeed) were taken from a review study looking at the efficacy of IRS in Africa (see Table C).
Use of the biological larvicide Bacillus thuringiensis israelensis (Bti) to treat Anopheles breeding sites has been tested in a study in Peru and Ecuador [4].The larvicide was found to be effective, but due to the surface feeding habits of Anopheles larvae, it was found to be only effective for the first 7-10 days following spraying, after which it had sank sufficiently below the surface to have no further impact.The study saw an average adult density reduction (measured in bites per person per hour) of 50 -70% in the 7 days following treatment across all identified larval breeding sites in a 2km radius.In the model we assume sustained larvicidal coverage provides up to a 60% reduction in the adult population -included as a reduction in the emergence rate and scaled by the coverage, or proportion of larval sites treated, where 100% coverage results in a 60% reduction.This is likely to be a conservative estimate, as this assumes the reported reductions of 50-70% are linked to perfect coverage, but the magnitude of larvicidal impact will be setting-and species-dependent.
Table D contains a summary of the vector-specific parameters, including feeding and biological parameters as well as vector control intervention assumptions.The parameters π i for i = 1, ..., 4 are chosen to give a 3 day feeding cycle length with 0.68 day mean blood-seeking duration [5].Lardeux et al. observed a minimum 2 day period for gestation, matching up to an approximate 3 day gonotrophic cycle [7], hence the blood-seeking and gestating stages are assumed to take up the majority of the cycle duration.Feeding and ovipositing periods are therefore short in comparison and are set to be equal to make up the remainder of the cycle period in the absence of further evidence (1/π 1 = 1/π 3 = 0.16).

Disease parameters
Table E shows the values used for disease-specific parameters, such as the intrinsic and extrinsic incubation periods and the mean human infection duration.

Section B: Derivations of vector control dependent transmission measures
Consider the age-structured gonotrophic cycle model at equilibrium, such that the number of blood-seeking vectors in generation i is K i B 0 , where and It is sufficient to consider the blood-seeking class as this is the stage of the feeding cycle where vectors have potential to pick up or transmit disease through biting.If we define the binomial probability, κ = xc, of a successful feed leading to a new vector infection, then the probability a mosquito becomes exposed during generation i is κ (i) = (1 − κ) i−1 κ.Hence the probability a mosquito has been exposed before generation i is [1−(1−κ) i ].Combining these gives the number of vectors in generation n that are already infected: (C) The total number of diseased, D (exposed, Y , or infectious, Z), vectors is given by summation across the generations; assuming n is the maximum number of generations a vector may live for.
If the incubation period is assumed to be equivalent to N generations (or cycles), then the probability of surviving until infectious is given by K N and can be treated as a multiplicative factor when calculating the numbers of infectious and exposed vectors: and hence directly calculate transmission measures as discussed in the main text.
In the absence of interventions the mean feeding cycle length is, ) where 1/π 2 is the average time to hunt and take a blood meal.IRS and larvicides won't impact hunting time, but the repelling effect of bednets will result in some vectors taking longer to move from emerged to fed.
If we assume a repelled vector begins the hunting process from scratch, then the expected time taken to successfully feed will be equal to the time taken to feed given a successful first attempt plus the expected time taken to feed scaled by the proportion of vector that repeat on any given attempt.
Now we can express overall feeding cycle length, δ, in terms of bednet parameters: and the human blood feeding rate is given by: The ratio of vectors to humans m, can be scaled by changes in the mosquito population (m = M/H), where and K describes the probability of surviving each feeding cycle, with dependence on vector control parameters included in B 0 and K.
The death rate will depend on IRS and bednet usage.We can relate the probability of a vector surviving one feeding cycle, K, to a per cycle death rate −ln(K), then we have as the instanenous daily death rate.Now that we have vector control dependent expressions for all relevant parameters, these can be substituted into our equations to calculate key transmission measures, such as R 0 .In the presence of vector control measures, we relabel R 0 as the basic reproductive number under control, R c .

Entomological inoculation rate
Using above expressions for a, m and D, we have:

Vectorial capacity
Using above expressions for a, m and g, we have:

Reproductive number under control
Using above expression for V we have that

E
[Time to feed] = E[Time | Successful attempt] (I) +P[Repeat]E[Time to feed] (J)

Table A
Vector control coverage definitions

Table B
[1]lt vector control: outcome probabilities from feeding attempts in the presence of LLINs[1].

Table C
[3]lt vector control: outcome probabilities from feeding attempts in the presence of IRS, taken from a systematic review of IRS efficacy in Africa[3].

Table D
Parameters for mosquito biology and vector control (Anopheles gambiae).

Table E
Disease parameters for malaria (Plasmodium falciparum).