An elaborated feeding cycle model for reductions in vectorial capacity of night-biting mosquitoes by insecticide-treated nets

Insecticide-treated nets (ITNs) are now regarded as standard tools for malaria control [1]. ITNs reduce malaria transmission by vectors that bite during the night; they work by reducing the intensity of malaria transmission, the average number of infectious bites received by a person over some time period, called the entomological inoculation rate (EIR). In one study, ITNs reduced EIR by a factor of 10 [2]. Previously published reports have demonstrated that the use of ITNs reduced mortality attributable to malaria by 30% in young children [3] and reduced severe, life-threatening malaria among children by 44% [4]. The benefits of ITNs may vary from place to place depending on both the baseline EIR and the total reduction in EIR [5, 6].

ITNs are effective at reducing EIR because they simultaneously affect several different aspects of the mosquito feeding cycle. ITNs kill mosquitoes that land on treated nets, with mortality reaching up to 90% [7]. By increasing adult mortality, ITNs reduce transmission in at least four ways: a reduced probability that a mosquito will become infected, a reduced probability that an infected mosquito survives sporogony to become infectious, fewer expected bites by a vector after becoming infectious, and fewer eggs laid. ITNs also repel mosquitoes – in quantitative terms, mosquitoes are less likely to enter houses with ITNs and they leave sooner. The mean number of Anopheles mosquitoes per house was 77% lower in an area using ITNs than in a comparable one without ITNs [2]. Other effects have also been reported, including a shift to outdoor biting, a shift in time of biting, or diversion to feed on other blood meal sources [8].

The combined effects of ITNs, including both adult killing and repelling effects, are likely to be different, depending on the intrinsic biting preferences of the vector. Classical descriptions of malaria transmission are based on a quantitative description of the feeding cycle [9]. The mosquito feeding cycle involves host seeking, feeding, resting, oviposition-site seeking, and oviposition. Most formulas that describe malaria transmission by mosquitoes reduce this feeding cycle to a single term that describes the human feeding rate, a, where 1/a is the average interval between two successive human bites. The human feeding rate can be further broken down into two terms, the average time to complete one feeding cycle, 1/f, and the proportion of bites on humans, Q; the human feeding rate is a = fQ. The capacity of a single vector to transmit malaria is also affected by its lifespan and the duration of parasite sporogony; let p denote the probability of surviving one day and n the number of days required to complete sporogony. The total capacity for a vector population increases with the density of the vector population; let m be the population density of vectors divided by the population density of humans. One index of malaria transmission intensity is vectorial capacity, the number of infectious bites that would arise from all bites on a single infected human on a single day:C = m Q 2 f 2 p n − ln p MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqGH9aqpdaWcaaqaaiabd2gaTjabdgfarnaaCaaaleqabaGaeGOmaidaaOGaemOzay2aaWbaaSqabeaacqaIYaGmaaGccqWGWbaCdaahaaWcbeqaaiabd6gaUbaaaOqaaiabgkHiTiGbcYgaSjabc6gaUjabdchaWbaaaaa@3D27@[10]. Vectorial capacity is closely related to other indices of malaria transmission, including EIR and R₀ [11, 12]. Here, the differences in bed-net effects are assessed using a mathematical model for the change in transmission intensity produced by ITN use, based on a detailed quantitative description of the mosquito feeding cycle, extending previous work on mosquitoes and malaria transmission [9, 13, 14]. The study aims to improve understanding of how host-feeding by mosquitoes would be influenced by the use of ITNs, and to show how ITNs can have similar reductions in EIR for different reasons, despite innate differences in mosquito ecology.

Here, the mosquito feeding cycle is described by 2 stages: host-seeking through to successful feeding, and resting through to oviposition (Figure 1). To understand ITNs, all the events that are changed by ITNs when a mosquito bites or attempts to bite a host are examined in details (Figure 1). First, during the host-seeking period, a proportion Q(0) of mosquitoes find a human, and 1-Q(0) find some other vertebrate host. With ITN use, a fraction φ of the humans are protected by ITNs. If a mosquito finds a protected human, one of three things can happen: a) the mosquito successfully feeds, regardless of the net, with probability s. Successful feeding on humans protected by ITNs may occur if the net has holes, if the net is not properly deployed, or if the human is unprotected and bitten for short stretches during the night b) the mosquito lands on the net to bite a human, but dies from contact with the insecticide, with probability d c) the mosquito leaves the vicinity to search for another host, effectively restarting the host-seeking cycle, with probability r = 1 - s - d. In summary, at the end of one biting attempt, some mosquitoes have successfully fed on a non-human host, some have fed on a human, some have died from the insecticide in the net, and some have been frustrated and begin again.

Since the study is interested in understanding how ITNs affect malaria transmission changes to vectorial capacity are assessed in a controlled population relative to a baseline population without control. Without ITNs, host-seeking takes τ₁(0) days and mosquitoes survive this process with a probability p₁(0). Following a successful blood meal, mosquitoes rest, find larval habitat, and oviposit. This process lasts τ₂ days and the mosquito survives with probability p₂ regardless of species biting preferences. This elaborated description is related to classic formulations: the time to complete one feeding cycle with no ITN use is 1/f(0) = τ₁(0) + τ₂, and the daily probability of surviving one day is p(0) = [p₁(0) p₂]^f(0). The proportion of blood meals that are taken on a human is also affected by ITNs; let Q(0) denote the proportion of blood meals taken on a human in the absence of ITNs, i.e. the human visiting rate. Functions were elaborated to describe vectorial capacity in a population where a fraction of humans, φ, are protected by ITNs; in other words, what are f(φ), p(φ), and Q(φ)?

A surviving, diverted mosquito can successfully feed after several attempts. To derive the following formulas, we assume that they may repeat the attempt as many times as necessary to complete their feeding cycle. A mosquito successfully feeds by finding a non-human host, by finding an unprotected human host, or by successfully feeding on a protected human host. Altogether, during a single attempt, a surviving mosquito succeeds with probability W = 1 - Q(0) + Q(0) (1-φ) + Q(0) φs = 1 - Q(0) φ (1-s). A mosquito resets and begins a new search with probability Z = Q(0) φr, assuming that the mosquito survived the ordinary hazards of feeding and seeking, that it survived the ITN, and that it failed to feed successfully.

First, a formula was elaborated for the expected delay. By assumptions, after a failed first attempt, the mosquito starts over, with probability Z, so the expected time to complete the life-cycle is given by:

τ₁(φ) = τ₁(0) + Zτ₁(φ) = τ₁(0)/(1-Z);     (Equation 1)

Intuitively, this expression says that the expected time to feed in a population where ITNs are used is the baseline time multiplied by the number of attempts required to complete a feeding cycle, 1/(1-Z). Thus, the time to complete one feeding cycle is

1/f(φ) = τ₁(0)/(1-Z) + τ₂;     (Equation 2)

Second, a formula was elaborated for the expected probability that a mosquito survives through completion of feeding. At each attempt, mosquitoes may die, succeed, or fail and try again. Thus the probability of surviving through a complete feeding cycle is the probability of surviving and succeeding the first attempt, plus the probability of surviving the first attempt and succeeding the second attempt if mosquitoes failed the first attempt, and so on. Importantly, we assume that the baseline hazards apply each time a mosquito attempts to feed:

p₁(φ) = p₁(0) [W + Z p₁(φ)] = p₁(0)W/[1 - Z p₁(0)];     (Equation 3)

Thus, the probability of surviving one day is given by:

p(φ) = [p₁(0) p₂W/(1-Z p₁(0)]^f(φ);     (Equation 4).

Finally, with ITN use, a larger fraction of bites is taken on non-human hosts. Overall, the probability that a feeding attempt ends with a blood meal on a human host is:

q(φ) = p₁(0) [Q(0)(1 - φ + φ s) + Z q(φ)]

= p₁(0) Q(0)(1 - φ + φ s)/[1-Z p₁(0)]     (Equation 5)

The proportion of blood meals taken on a human host is:

Q(φ) = q(φ)/p₁(φ) = Q(0)(1 - φ + φ s)/W.     (Equation 6)

Thus, three formulas have been derived (Equation 2, 4 and 6) relating the probability of surviving the feeding cycle, the average length of a feeding cycle, and the proportion of bites taken on a human to the analogous baseline rates in the absence of ITNs as a function of ITN coverage, φ. The functions rely on only two additional parameters: the probability that the average mosquito dies, d, or that it feeds successfully, s, during a single feeding attempt.

Finally, ITNs can have other effects on vector population density. By killing adult mosquitoes, the total rate of oviposition goes down in the surrounding area. More importantly, there are direct reductions in adult vector density. The rate at which mosquitoes emerge, per-human, per day, is a constant λ; it follows that mosquito density is reduced slightly because of the higher death rates, to m(φ) = λ/-ln p(φ) Thus, the total effect of ITNs on vectorial capacity can be rewritten as a function of each one of these separate effects:

C ( φ ) = m ( φ ) Q ( φ ) 2 f ( φ ) 2 p ( φ ) n − ln p ( φ ) = λ Q ( φ ) 2 f ( φ ) 2 p ( φ ) n ( − ln p ( φ ) ) 2 ( E q u a t i o n 7 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqcqGGOaakiiGacqWFgpGzcqGGPaqkcqGH9aqpdaWcaaqaaiabd2gaTjabcIcaOiab=z8aMjabcMcaPiabdgfarjabcIcaOiab=z8aMjabcMcaPmaaCaaaleqabaGaeGOmaidaaOGaemOzayMaeiikaGIae8NXdyMaeiykaKYaaWbaaSqabeaacqaIYaGmaaGccqWGWbaCcqGGOaakcqWFgpGzcqGGPaqkdaahaaWcbeqaaiabd6gaUbaaaOqaaiabgkHiTiGbcYgaSjabc6gaUjabdchaWjabcIcaOiab=z8aMjabcMcaPaaacqGH9aqpcqWF7oaBdaWcaaqaaiabdgfarjabcIcaOiab=z8aMjabcMcaPmaaCaaaleqabaGaeGOmaidaaOGaemOzayMaeiikaGIae8NXdyMaeiykaKYaaWbaaSqabeaacqaIYaGmaaGccqWGWbaCcqGGOaakcqWFgpGzcqGGPaqkdaahaaWcbeqaaiabd6gaUbaaaOqaaiabcIcaOiabgkHiTiGbcYgaSjabc6gaUjabdchaWjabcIcaOiab=z8aMjabcMcaPiabcMcaPmaaCaaaleqabaGaeGOmaidaaaaakiaaxMaacaWLjaacbiGae4hkaGIaemyrauKaemyCaeNaemyDauNaemyyaeMaemiDaqNaemyAaKMaem4Ba8MaemOBa4MaeeiiaaIae43naCJae4xkaKcaaa@8148@

Using the classical assumptions, changes in vectorial capacity are related to changes in the entomological inoculation rate by the formula:

where X(φ) denotes the parasite rate, the fraction of humans that carry parasites, c is the infectivity, the fraction of bites on infected humans that would infect a mosquito (i.e the fraction of parasitemic humans that transmit gametocytes to mosquitoes, per bite) and b is the probability a human gets infected from mosquitoes bites.

Sample parameters describing the baseline feeding cycle process result from epidemiological surveys [15‐17] and are listed in Table 1.

Using an elaborated description of the classic feeding cycle model, simple formulas have been derived to describe how ITNs change the intensity of malaria transmission, as summarized by the vectorial capacity. ITNs accomplish these changes by reducing daily survival (and by extension, mosquito density), increasing the duration of the mosquito feeding cycle, and decreasing the proportion of blood meals taken on a human host. ITNs affect all these aspects of transmission for all vectors, but the dominant effects differ for anthropophilic and zoophilic vectors. ITNs reduce vectorial capacity mainly through reductions in vector lifespan; the effects are largest for species with high baseline survival, but these effects are enhanced by lengthening the feeding cycle and diverting bites onto non-human hosts. The model also predicts that ITNs delay feeding more for anthropophilic vectors. For zoophilic vectors, ITNs increase the proportion of bites that are taken on non-human hosts. Despite these differences, the total reduction in transmission is similar, and is strongly affected by the baseline mosquito ecology. As a result, the risk of malaria would be reduced more and the vectorial capacity would decrease by more than 90% in any species, regardless of host biting preferences.

The risk of an infection is related to changes in EIR, but such changes will differ enormously depending on the absolute value of EIR at the baseline, not just the proportional reductions in vectorial capacity. Baseline EIR and vectorial capacity can greatly vary because of mosquito density, a quantity that is related to mosquito larval ecology. While ITNs can reduce the number of eggs laid by adult mosquitoes, this may not reduce the number of emerging adults, especially if larval populations are regulated in a density dependent way by other factors, especially competition for resources and predation.

When baseline EIR is high, a large proportional reduction in vectorial capacity (i.e. by a factor of 100) may simply reduce EIR from 700 to 7, but an annual EIR of 7 still represents a high level of risk. Reductions from extremely high EIR with ITNs can produce short-term reductions in morbidity and mortality that reflect residual levels of immunity acquired under baseline transmission. The long-term effects, all else equal, are expected to adjust such that morbidity and mortality in the controlled populations would be expected to resemble other places with that baseline EIR. The real unanswered question is whether the total burden of malaria is importantly different in populations with high baseline EIR. How different is the total burden of malaria in a population where annual EIR is constantly 7 compared to one where it is 70 or 700 [5]?

These models also suggest that complete ITN use can generate reductions in transmission intensity [20] that are high enough to locally eliminate malaria from some areas where the intensity of transmission is low or moderate (i.e. by a factor of a hundred or more) if all transmission is by night-biting mosquitoes and if ITN use is high (i.e. in excess of 50%). At low EIR, large proportional changes in vectorial capacity might lead to significant reductions in parasite rate, and consequently the proportion of mosquitoes that become infected per bite. The added reductions in human prevalence might happen after a delay, but the combined effects of reductions in vectorial capacity and reductions in the parasite rate will produce even larger changes in EIR when baseline EIR is low.

The effects of ITNs might have different important effects in areas with perennial transmission compared with acutely seasonal transmission, and this raises important questions about the timing of insecticide treatments and public health messages to promote ITN use [21]. Should ITNs be emphasized and treated during times of high or low transmission? In some places, ITN use during the dry season combined with mass drug administration could be used to knock out the cryptic human infections that spark transmission during the following season [22].

Here, ITN use is considered in terms of the proportion of mosquito-human encounters with an ITN-protected human; this is the relevant quantity for malaria transmission, but it is obviously a different definition than ITN coverage, the proportion of households that own a ITN. Achieving 100% ITN coverage may be unrealistic. In the Gambia, where the use of ITNs is widely accepted, ITN coverage evaluated during the National Insecticide Impregnated Bed-net (ITN) Programme ranged from 46.4% to 78.6% in five different villages [23]. Furthermore, ITNs may become inefficient if not re-impregnated regularly, and the rate of re-treatment may vary from 7% to 62% after introduction of cost-retrieval [24]. Thus, ITN evaluation through mathematical models should take into account realistic use of ITNs, as is done here; mosquitoes are allowed to bite through inefficient nets.

The cycling model does not take into account heterogeneous distributions of mosquitoes or hosts [27], but it has been shown that mosquitoes' demographic characteristics influence malaria risk [14]. For instance, villages close to larval habitats are at higher risk of malaria infection [25]. Nevertheless, the model results are in agreement with epidemiological studies. In this model, decreases in daily survival rate range from 9 to 26%, similar to the 10 to 40% decreases observed in field surveys [26]. To mimic real contexts, the input and model parameters used are developed from studies carried out in Africa, in Papua New Guinea [16, 17] or in Latin America [15]. Finally, the model provides simple formulae for mosquito survival rate, duration of the feeding cycle and proportion of bites taken on humans.

This approach provides a framework for predicting ITN efficacy based on mosquito characteristics and behavioural response in relation to ITN use. Such methods provide tools for planning interventions and evaluating other measures such as the use of untreated ITNs [28]. The results of the model suggest that malaria risk would be greatly reduced by medium ITN coverage regardless of species biting preference, thereby providing important observations with clear public health implications to achieve sufficient protection and control of malaria.