While essentially identical results were also obtained by using far more complex, detailed models of mosquito behaviour and malaria transmission [14, 17, 36], a far simpler form of these models [37] is applied here to illustrate key processes at hand as clearly as possible to a broad epidemiological audience.
A simple kinetic model of mosquito foraging for blood [37] is extended slightly, to distinguish between the availability of blood hosts to encounter and attempt attack upon (A) and the availability of blood per se (Z) [36]. Note that both of these distinctive availability terms are defined as the total rates at which the relevant attack (A) or feeding (Z) events occur on all hosts, or a specified subset of hosts, per individual foraging, host-seeking mosquito [36]. Note that this model assumes a single, homogenously-mixed mosquito population that is equally likely to attempt to attack humans when they are encountered indoors or outdoors, but that the distribution of human exposure across indoor and outdoor spaces is determined by where and when human and mosquito activities overlap [14, 20]. All cattle are assumed to always be outdoors, but this has no impact upon the model because they are also assumed to be all unprotected and equally vulnerable to attack indoors and outdoors. All parameter definitions and symbols are detailed in Table 1.
Assuming the same input values for baseline human (Z
h,0
) and cattle (Z
c,0
) host availability as recent simulations of the impacts of LLINs and IRS combinations upon An. arabiensis in the Kilombero Valley [17], there are assumed to be 140 cattle for every 1000 humans (I
c
/I
h
= 0.14) and each head of cattle is assumed to have 61 % greater availability to foraging mosquitoes than an average human (λ
c
= 1.61) [38], presumably because of their larger size and greater attractiveness to this vector species. To enable simplification of the equations for proportional output parameters, specifically the proportion of all blood meals obtained from cattle (Q
c
) or human (Q
h
) host subsets and the proportions of human biting exposure occurring indoors (π
h,i
) or outdoors (π
h,o
), all blood availability terms for cattle (Z
c
) and protected human populations (Z
h,N
, Z
h,i,N
and Z
h,o,N
) are calculated relative to Z
h,0
. The proportion of exposure to An. arabiensis occurring indoors for unprotected humans lacking a bed net (π
h,i
) is assumed to match field measurements of 82 % [25], representing the de facto limit of the level of personal protection that even a net with very high protective efficacy (ρ = 90 %) [39] can provide. Assuming that cattle and humans are the only major sources of blood for this vector [38], the proportions of blood meals obtained from humans, protected or otherwise (Q
h
), and cattle (Q
c
) is equivalent to the proportion of all available sources of blood that these host species account for (Z
h
and Z
c
, respectively). For a scenario with no LLINs (Ω = 0), these human and cattle blood indices can be predicted (Q
h,0
and Q
c,0
) as a simple function of the relative population sizes and relative availability of individuals from these two host populations as sources of blood (λ
c
for one head of cattle relative one human):
$$Q_{h,0} = \frac{{Z_{h,0} }}{{Z_{h,0} + Z_{c} }} = \frac{1}{{1 + \lambda_{c} (I_{c} /I_{h} )}}$$
(1)
$$Q_{c,0} = 1 - Q_{h,0} = \frac{{Z_{c} }}{{Z_{h,0} + Z_{c} }} = \frac{{\lambda_{c} (I_{c} /I_{h} )}}{{1 + \lambda_{c} (I_{c} /I_{h} )}}$$
(2)
Furthermore, in the absence of bed nets (Ω = 0) the proportion of all blood meals obtained from humans while indoors (Q
h,i,0
) and outdoors (Q
h,o,0
) can be separately calculated based on the measured proportion of human exposure that occurs indoors (π
h,i,0
) and outdoors (π
h,o,0
= 1 − π
h,i,0
):
$$Q_{h,i,0} = \frac{{Z_{h,i,0} }}{{Z_{h,0} + Z_{c} }} = \frac{{\pi_{h,i,0} }}{{1 + \lambda_{c} (I_{c} /I_{h} )}}$$
(3)
$$Q_{h,o,0} = \frac{{Z_{h,o,0} }}{{Z_{h,0} + Z_{c} }} = \frac{{1 - \pi_{h,i,0} }}{{1 + \lambda_{c} (I_{c} /I_{h} )}}$$
(4)
Predictions of these same proportional contributions of various blood sources (Q
h,N
, Q
h,i,N
, Q
h,o,N
and Q
c,N
) can also be made for a scenario with high bed net use (Ω = N), specified as four fifths of residents using bed nets (U
h
= 80 %). The availability of human blood is reduced in proportion to the de facto mean level of personal protection provided by those nets at population level when limitations of human net usage rates (U
h
) and the proportion of bites that would otherwise occur indoors (π
h,i,0
) are accounted for:
$$Q_{h,N} = \frac{{Z_{h,N} }}{{Z_{h,N} + Z_{c} }} = \frac{{1 - \rho \pi_{h,i,0} U_{h} }}{{(1 - \rho \pi_{h,i,0} U_{h} ) + \lambda_{c} (I_{c} /I_{h} )}}$$
(5)
$$Q_{c,N} = 1 - Q_{h,N} = \frac{{Z_{c} }}{{Z_{c} + Z_{h,N} }} = \frac{{\lambda_{c} (I_{c} /I_{h} )}}{{(1 - \rho \pi_{h,i,0} U_{h} ) + \lambda_{c} (I_{c} /I_{h} )}}$$
(6)
$$Q_{h,i,N} = \frac{{Z_{h,i,N} }}{{Z_{h,N} + Z_{c} }} = \frac{{\pi_{h,i,0} (1 - \rho U_{h} )}}{{(1 - \rho \pi_{h,i,0} U_{h} ) + \lambda_{c} (I_{c} /I_{h} )}}$$
(7)
$$Q_{h,o,N} = \frac{{Z_{h,o,N} }}{{Z_{h,N} + Z_{c} }} = \frac{{1 - \pi_{h,i,0} }}{{(1 - \rho \pi_{h,i,0} U_{h} ) + \lambda_{c} (I_{c} /I_{h} )}}$$
(8)
The proportion of all human biting exposure that occurs outdoors can then be readily estimated as the outdoor fraction of all blood meals obtained from humans:
$$\pi_{h,o,N} = \frac{{Z_{h,o,N} }}{{Z_{h,N} }} = \frac{{1 - \pi_{h,i,0} }}{{1 - \rho \pi_{h,i,0} U_{h} }}$$
(9)
Total blood availability is defined as the rate at which individual mosquitoes encounter, attack and successfully feed upon all hosts, so it is inversely proportional to the length of time spent foraging and number of host encounters required to do so [36]. So, for example, if high coverage with protective measures like bed nets prevent successful feeding following half all host encounter events, total blood availability will be halved and each successful blood meal will require twice as many host encounters. Assuming that essentially all encounters with hosts in the absence of nets result in a successful feeding event, the mean number of encounters required for a mosquito to feed each at the end of each host-seeking interval in the presence of nets (E
N
) can be estimated as the total blood availability in the absence of nets (Z
0
), divided by the total blood availability in the presence of nets (Z
N
):
$$E_{N} = \frac{{Z_{0} }}{{Z_{N} }} = \frac{{1 + \lambda_{c} (I_{c} /I_{h} )}}{{(1 - \rho \pi_{i} U_{h} ) + \lambda_{c} (I_{c} /I_{h} )}}$$
(10)
Obviously, the longer a mosquito lives and has to spend foraging for hosts, the greater the number of host encounters it experiences, and the lower the probability that it has never been inside a house. The proportion of mosquitoes that have been inside a house at least once therefore increases to saturation with age, at a rate that increases as protective measures like nets necessitate more host encounters per successful blood meal. The proportion of mosquitoes that have been inside house but failed to feed because the occupants were protected by LLINs at least once (F
N
), can therefore be calculated as an exponential decay function of the number of failed encounters per gonotropic cycle (E
N
−1) and the age of the mosquito in terms of the number of gonotrophic cycles (x) it has completed:
$$F_{x,N} = 1 - e^{{ - x (E_{N} - 1)}}$$
(11)
Note that while insecticide treatment of nets causes negligible additional mortality of An. arabiensis in this setting [18] and another recently described setting in Ethiopia [19], this phenomenon does need to be assumed because this model only predicts the choices of those mosquitoes that actually fed successfully and, therefore, survived any hazards associated with foraging and all attempted host attacks. The parameters for mean proportional bed net usage (U
h
) and mean protective efficacy while in use (ρ) therefore represent those for all treated and untreated nets present in the community.
