Defining biological coverage based on the example of blood resources
Biological coverage (
C
A,p
) of all the blood resources upon which mosquitoes rely, with long-lasting insecticidal nets (LLINs) or any other personal protection measure, has been previously defined as the proportion of all mosquito attacks upon all available hosts for which those hosts were covered with a protective (
p) intervention at that time and place[
11,
24]:
where the total attack availabilities of the all hosts (
A), and covered hosts at times and places when they are actually protected (
A
p
), are defined kinetically[
25,
42] as the rates per night at which an individual host-seeking mosquito respectively encounters and attacks[
26] either all hosts or all hosts that are protected at the time of the encounter and attack events.
However, to allow simplified notation for generalization of this approach to a greater diversity of distinct resources, here the symbol
A is replaced by
R to denote the total kinetic availability rate of a specific given resource, which may be specified as blood (
v), resting sites (
r), sugar (
s) or any other resource (
R ∈
v,
r,
s …). Also, the terms
attack and
protected which were previously used to define availability in kinetic terms for models of blood resource seeking and acquisition[
26,
27] are not entirely appropriate for non-blood resources, so these are replaced with more generally applicable terms
attempt to utilize and
covered, respectively. Furthermore, now that biological coverage has been defined to explicitly consider only protection that is in place at the times and places when that resource is utilized,
de facto coverage (
c) and protection (
p) are equivalent to each other (
c ≈ p), so the former is used to simplify and harmonize the notation. Expressing Equation
1 in terms of this revised notation yields:
where
C
v
is the proportion of all mosquito attacks upon real (live vertebrate hosts) or perceived (artificial odor-baited traps, sometimes referred to as pseudo-hosts[
27]) blood resources to which effective coverage with a vector control intervention applies at that time and place,
v is the total rate at which individual mosquitoes encounter and attack all hosts and pseudo hosts, and
v
c
is the total rate at which individual mosquitoes encounter and attack all hosts and pseudo hosts at times and places when they are effectively covered with a vector control intervention.
In the case of interventions such as LLINs, which only protect humans while they use them indoors, biological coverage can be calculated as the product of the proportion of all bloodmeals (
v) that originate from the human (
h) host species subset
, the proportion of human exposure to mosquito bites that would otherwise occur indoors (
i) without an LLIN
, and the proportional
demographic coverage of humans, measured as the proportion of humans who reported using a net while asleep indoors the previous night
[
11]:
(3)
where all three terms on the right hand side of Equation
3 are defined as sequentially nested fractions and sub-fractions of the total availability of all blood hosts (
v) that are represented by humans (
h), those humans while indoors (
i), and those humans while indoors and protected by coverage with LLIN use at the time (
c):
(5)
(6)
Adapting the concept of biological coverage to rationalize vector control impact based on utilization rates of diverse resource targets
Expressing Equation
2 in more general terms that may be applied to any given resource (
R), rather than blood specifically (
v), yields the following formula:
LLINs that directly kill mosquitoes when they encounter and attack protected human blood sources are the best established[
43] and easiest resource targets to conceptualize and model, but previous formulations predicted their impact upon vector survival by assuming mosquitoes feed once and only once per gonotrophic cycle[
26]. However, resting sites, oviposition sites, and even blood resources themselves, may be utilized more than once per gonotrophic cycle[
44], while sugar sources may be used less than once[
21,
45]. To adapt the concept of biological coverage to more diverse resource targets which are used more than once per gonotrophic cycle, the term
resource utilization is defined as the mean rate at which mosquitoes utilize any given resource (
R) per gonotrophic cycle (
α
R
). This definition of resource utilization rate can be expressed mathematically as the product of the duration of the gonotrophic cycle, expressed as nights per gonotrophic cycle (
g), and the rate per night at which a mosquito population utilizes that resource (
m
R
), divided by the size of the mosquito population (
M):
where
g is expressed as units of nights,
m
R
in units of utilization attempt events per night, and
M as the number of individual adult mosquitoes present in the population. For any targetable, intervention-covered (
c) proportion of that resource (
R
c
), the corresponding rate at which mosquitoes encounter and attempt to utilize that covered fraction
, by definition, varies in proportion to the fraction of the kinetic availability of that resource that it represents:
(9)
Hence the quotient of the availability or utilization rates for the total resource, divided into those for the intervention-covered fraction, are equivalent to biological coverage of that resource:
(10)
Most vector control strategies only target a specific subset (
x) of the resource that they are delivered to, which is practically definable, identifiable, accessible, and treatable in the field. Similarly to resource coverage (Equation
10), the proportion of all available forms of a specific resource (
R) that is accounted for by a given subset (
R
x
) of that resource
, can be defined and measured in terms of the rate at which mosquitoes encounter and attempt to utilize it[
26,
27,
42] by generalizing Equation
4 for subsets of any possible resource, rather than specifying blood:
(11)
Similarly, the proportion of that subset (
x) that is effectively protected at times when it is utilized by mosquitoes (
y) can be expressed in terms of the proportion of resource utilization attempt events it accounts for in that resource subset:
(12)
Hence, the biological coverage of all forms of that resource (
C
R
) can be expressed more explicitly than in Equation
10 as the product of the proportion of that resource represented by that subset
, the proportion of utilization attempt events for that subset to which protection effectively and conditionally applies
and measured intervention coverage of that resource subset at times and places when it may be effectively protected
:
(13)
Note that
is the utilization rate of the covered fraction of the targeted resource subset, equivalent to
in Equation
10, because all covered forms of the resource occur amongst the intervention-targeted subset of that resource (
x) at the times and places at which they mosquitoes actually attempt to utilize them (
y):
(14)
Predicting intervention impact based on resource subset coverage and utilization rates
Interventions targeting adult mosquitoes may have quite complex modes of action, repelling mosquitoes away from humans[
46] or contaminating them with agents that affect their longevity[
47,
48], competence[
47,
48] or fecundity[
49,
50]. Biological agents may be transmitted horizontally or vertically through the population[
47,
48,
50], while coverage amplification of chemicals may be achieved by mosquito-mediated transfer between resources[
49]. Regardless of their complexity, these diverse strategies can all be enhanced by maximizing biological coverage of the resource targeted to ensure maximum contact with the mosquito population, and this is a critically important determinant of success in its own right. Previous formulations describing mosquito survival and mortality as a function of exposure to LLINs or IRS[
26] are therefore adapted and simplified as follows to allow for a range of utilization rates ranging from zero to several times per gonotrophic cycle, rather than the previously assumed utilization rate of once per gonotrophic cycle for all blood resources (
α
v
=1). All predictions of impact upon mosquito survival, and the entomologic inoculation rates they mediate, were implemented and parameterized exactly as previously described[
51], except that Equation
14 of the original formulation[
26] was adapted to incorporate the mortality risks of utilizing all covered and uncovered resources in a more generally applicable manner:
(15)
where
P
γ
is the probability of surviving all utilization attempt events for all resources per gonotrophic cycle,
Pγ,0 is the probability of surviving all utilization attempt events for all resources per gonotrophic cycle in the absence of any intervention, and
is the probability of surviving all attempts to utilize intervention-covered forms of the targeted resources per gonotrophic cycle. The probability of surviving all attempts to utilize intervention-covered forms of the targeted resource per gonotrophic cycle
can be calculated as an exponential decay function of the product of the mortality probability associated with exposure to a covered form of the resource through a single utilization attempt event
, and the mean utilization rate for all covered forms of that resource
:
(16)
By substituting rearranged forms of Equation
10 and then Equation
13 into Equation
16, a solution with two field measurable parameters for the targetable, quantifiable, surveyable, subset is derived:
(17)
It is therefore not necessary to know the proportion of that total resource which the targeted subset represents, or the coverage (C
R
) or utilization rate (α
R
) for all available forms of a resource. Impact can be predicted directly so long as the coverage of the targeted subset itself, and utilization rates for that subset under conditions that enable the intervention to protect it against safe utilization by the mosquito, can be measured.
This approach to predicting the survival probability assumes that utilization attempt events are randomly, and independently, distributed across all resource units and mosquitoes. Specifically, the number of times one mosquito utilizes a resource (or resource subset) in one gonotrophic cycle is assumed to be a non-negative integer valued random variable (0, 1, 2, 3…) since the mosquito may not necessarily use the resource or, alternatively, may access it multiple times. Hence, the utilization rate of these resources should be understood as an expected value depending on random events that may be expressed as a mean. This is clearly not the case in relation to obligate utilization of blood from one of
all available blood resources (
R =
v). Each mosquito must utilize one of these available resources to complete the gonotrophic cycle, so complete coverage of all blood resources (
C
R
= 1) that are utilized at a mean rate of once per gonotrophic cycle (
α
R
= 1) with an insecticide which induces comprehensive fatality
would deterministically result in reduction of survival probability to zero
, rather than merely reduced to the minor proportion of mosquitoes that are inaccurately assumed by Equation
16 to have completed a gonotrophic cycle without taking any bloodmeal. However, for a covered
subset of a resource (Equation
17), rather than for all available forms of that resource (Equation
16), it is realistic to assume that the number of utilization attempt events per gonotrophic cycle is a random variable for individual mosquitoes and utilization rates per gonotrophic cycle are expected values (expressed as a mean), even for obligate blood resource utilization behaviours.
Measuring utilization rates for subsets of undefined resources by comparison with those for quantifiable blood resources
Adult mosquitoes use many distinct resources during their lifetimes, including several that they need afresh every time they complete a gonotrophic cycle: blood, resting sites, and oviposition sites. Most of these resources are difficult to quantify directly, so the same is true of the rates at which mosquitoes utilize them, thereby making contact with them. However, measurements of feeding rates upon humans or livestock allow ready quantification of absolute mosquito population size or recruitment rate[
31]. This is because the size of these mammalian host populations can be conveniently measured by direct census, and blood acquisition occurs at a measurable rate per host[
52] with a measurable probability for a given blood host species[
22,
23]. Also, blood acquisition usually occurs at a utilization rate of only once per gonotrophic cycle (
α
v
≈ 1) where sugar availability is not limiting[
45], except for the first gonotrophic cycle where two bloodmeals may be required[
53,
54]. For example, the emergence or recruitment rate of mosquitoes (
E) in a given setting can be calculated as function of the measured mean biting rate experienced by individual humans (
B
h
), the number of humans living there (
N
h
), the proportion of bloodmeals obtained from humans
, and survival probability per feeding cycle (
P
f
)[
31]:
where the mean lifetime number of bloodmeals per emerging mosquito (
b) is calculated as the sum of the probabilities of surviving to all plausible gonotrophic ages, expressed as the number of gonotrophic cycles completed (
j)[
26,
31]:
Similarly, for a very zoophagic (predominantly animal-feeding) vector with a strong preference for a known, accessible, manageable non-human host species such as cattle, goats, sheep, pigs or other livestock (
l), it may be easier to accurately measure biting rates upon such livestock (
B
l
) so the equivalent calculation can be made if the proportion of blood obtained from that host species
, and the population size of that host species (
N
l
) can be determined:
The key to applying Equations
18 and
20 to estimate absolute vector population sizes is the assumption that the fraction of all available sources of blood that each entomologically surveyed host represents
can be readily estimated by host census and bloodmeal identification from a sample of resting, blood-fed mosquitoes, so it is not necessary to directly detect all biting events on all hosts. Generalizing this principle, the emergence rate of mosquitoes (
E) can be estimated based on the rate at which mosquitoes are trapped or observed utilizing
a surveyed sample subset (
z) of any targetable subset (
x) of a given resource (
R
x,z
) if the proportion of all available forms of that resource which that surveyed subset represents (
R
x,z
/R), and the rate at which individual mosquitoes utilize all available forms of that resource per gonotrophic cycle (
α
R
), are both known. Note also that equation
18 and
20 both also implicitly include a term in the denominator for the utilization rate of all blood sources, that was negated by assumed a value approximating unity (
α
v
≈ 1) but can be explicitly reintroduced for the purposes of generalization. Substituting
for
B
h
or
B
l
,
R
x,z
/R for
or
, and
α
R
for
α
v
in Equations
18 and
20, respectively, yields the following general formula:
(21)
Blood resources can be readily identified as discrete units and their total numbers can be quantified by head-count census. However, units of sugar, resting site, oviposition site, and mating site resources are difficult to define unambiguously, except where these are introduced artificially (sugar baits, houses, boxes, pots, barrier screens, water containers, or swarming markers), and the total quantity of these resources available in the environment is even more difficult, if not impossible, to ascertain. Therefore, it is not obvious how
R
x,z
/R can be estimated for these resources with existing field survey methods. However, this is not necessary to know, because intervention impact can be conveniently rationalized in terms of coverage and utilization rates for definable, targetable subsets of those resources (Equation
17), and it is possible to calculate their relative rates of utilization compared with those for blood from a known proportion of all available blood resources. Here the quantifiable total blood resource, and a surveyed sample (
z) of an identifiable subset (
x) of that blood resource, is specified (
R = v) and distinguished from equivalent terms for other resources, such as resting sites (
R = r) or sugar (
R = s), with the specific terms
v and
v
x,z
. Also, the per gonotrophic rates at which mosquitoes attempt to utilize (
α) all blood resources (
v) or a distinct, identifiable subset of blood resources (
v
x
), as well as the per night rate at which utilization attempt events occur (
m) at a surveyed sample (
z) of that blood resource subset (
v
x
), are distinguished from those for other resources with the specific terms
, and
, respectively. Given that resource utilization rate per gonotrophic cycle is proportional to the total rate at which utilization attempt events occur in the overall population (Equations 8, 9, 10 and 11), the relative rate of utilization of such a resource subset compared to all blood resources can be expressed by dividing Equation
21, which is specified for a given non-blood resource subset and a surveyed sample thereof (
R
x
/
Rx,z), by an equivalent formulation specified for all blood resources, and a surveyed sample of hosts from a subset for which bloodmeals recovered from the midguts of recently fed specimens can be identified and distinguished from other sources (
v/
vx,z), and rearranging:
(22)
Note that the emergence and mean longevity terms cancel each other out so that estimates of these parameters are not required to estimate the relative rate of utilization of a resource compared with blood, as described below.
The most obvious vertebrate blood resource subsets (
v
x
) which are readily surveyed, including detection of blood feeding events and identification of blood source in specimens of fed mosquitoes, are humans (
x = h) and livestock (
x = l)[
52]. The proportion of all available blood resources sampled by the field survey can be quantified as the product of the proportion of bloodmeals obtained from humans
or livestock
and the number of humans (
N
h,z
) or cattle (
N
l,z
) sampled by the host attack survey, divided by the total number of humans (
N
h
) or livestock (
N
l
) present:
(23)
Fortunately, while gonotrophic discordance beyond the first feeding cycle does occur in
Anopheles, it is unusual and can be quantified[
45]. In most cases, it is therefore reasonable to explicitly assume that mosquitoes predominantly utilize blood approximately once per feeding cycle (
α
v
≈ 1), so substituting the host-specified (
v
x
= v
h
or
v
l
) formula of Equation
23 into Equation
22, and replacing
α
v
with unity, yields a solution for
for which all the terms are measurable in the field:
(24a)
or
(24b)
where R
x,z
/R
x
is the proportion of all available forms of the targeted non-blood resource subset that was surveyed entomologically to measure the rate per night at which the entire mosquito population attempts to utilize it, where N
h,z
/N
h
and N
l,z
/N
l
are the proportions of all humans or livestock that were respectively surveyed to measure the rate at which mosquitoes attempted to utilize their blood, and where and are the proportions of bloodmeals the vector population obtains from all available humans and livestock, respectively.
Where two resources co-occur and overlap completely with each other, specifically the example of resting sites (
R =
r
i
) and human blood indoors within houses (
v
h,i
), the proportion of each resource subset that is sampled is no longer required because these cancel each other out. All that is required is an estimate of the number of persons or person nights sampled by the host attack survey (
N
h,z
), and the total number people staying in those sampled houses (
N
h,Ω
), or even their ratio, which is commonly referred to as the mean number of occupants per house (
Nh,Ω/
Nh,z):
(25a)
In some experiments, however, resting events following more than one bloodmeal are represented in surveys of resting sites because those events may last two or more days. Recent standardized trials to compare various techniques for catching host-seeking and resting mosquitoes[
55‐
57] placed these in or immediately outside of different houses within a defined sampling frame each night, so that the former would not compete with the latter by trapping mosquitoes before they can feed and rest. Also, spray catches must be spaced by intervals of several days to allow residual pyrethrum to dissipate. In both cases, mosquitoes gestating over two or more preceding nights are allowed to accumulate from multiple nights of blood feeding (
τ) in the surveyed houses, and this must be accounted for when estimating utilization rates:
(25b)
Literature review and utilization rate estimate extraction
Studies, or sets of studies, were identified which presented sufficient parameter estimate data for utilization rates of specific, intervention-targetable resource subsets to be calculated for specific malaria vector species in specific, distinct locations. In addition to the authors’ archives of literature and unpublished data, the Pubmed database was also queried with the search term ‘Anopheles AND ((pyrethrum spray OR aspirator) OR (insecticide AND (cattle OR livestock)) OR odour-baited OR sugar)’. For utilization of livestock blood and sugar, consideration was limited to studies in settings where trials of insecticide-treated livestock or sugar baits, respectively, have been either implemented or specifically suggested. To avoid cluttering of the second figure presented in the Results section, consideration of studies enabling estimation of indoor resting site utilization was restricted to recent unpublished studies of our own and those published in the last decade. Where results for a species complex or group were reported, these are attributed to the most common sibling species identified in that population.
Utilization rates for blood from humans while indoors, when they can be protected with LLINs, was calculated as the product of the proportion of human exposure to mosquito bites occurring indoors
and the human blood index
, by assuming a single bloodmeal per gonotrophic cycle:
(26)
Where local estimates for the proportion of bloodmeals obtained from humans
were not available for the vector in question, the median values for that species from a previous review[
39] were applied. Where direct estimates of the proportion of bloodmeals obtained from cattle and other treated livestock
were not available, utilization of blood from other non-human sources was assumed to be negligible, so that this quantity could be calculated as the complement of the proportion obtained from humans
:
(27)
Utilization rates for odour-baited traps are calculated by assuming that the probability of a mosquito attacking a trap, rather than a natural host, per gonotrophic cycle is equivalent to the proportion of all available human hosts, animal hosts, and pseudo-hosts, that they represent[
27]. This is calculated on the basis of the ratio of traps to people (
N
t
/N
h
), the relative availability of those traps (
λ
t
), and the proportion of bloodmeals obtained from non-human hosts
[
27]:
(28)
Utilization rates for sugar resource subsets
were calculated as follows, based on direct field measurements of utilization rates per day for dye-labelled sugar baits
[
18‐
20], and an assumed mean gonotrophic cycle duration of three days (
g = 3):
Utilization rates for resting site subsets
, such as the insides of houses or artificial shelters and netting barriers placed in or around them, were estimated using Equation
25a or 25b based on the quotient of the mean rate at which mosquitoes were caught resting inside a sample of them (
m
r,x,z
) adjusted, where necessary, for an assumed indoor resting period of 2 days (
τ = 2), divided by the rate at which they were caught attacking human hosts indoors
, the mean number of exposed occupants per house or room (
N
h,Ω
/
N
h,z
) adjusted for reported usage rates of LLINs
which were assumed to confer complete protection, the proportion of bloodmeals obtained from humans
, and the proportion of human bloodmeals obtained indoors
, with the latter two parameters usually assumed from mean literature values reported for that species[
32,
39]:
(30)