Vectorial Capacity
The Original Formula
Vectorial capacity (VC) was developed to measure potential malaria transmission by mosquitoes and the impact of vector control. The original formula traces back to a mathematical formula for the daily entomological inoculation rate (EIR). Here, we rewrite the original formula and show that it is a concise description of transmission: to transmit malaria parasites, a mosquito must blood feed on a human to get infected, survive long enough to become infectious, and then blood feed on humans again to transmit.
See Related for links to closely related vignettes.
Quantitative medical entomology was a half-centry old when George Macdonald published a synthesis around a mathematical formula for the sporozoite rate (SR) [1]. Later that year, using a formula for the EIR, he published a formula for the basic reproductive number for malaria parasites, \(R_0\) [2]. A dozen years later, Garrett-Jones named vectorial capcity (VC):
I propose that “vector control” should refer, in this context, to reduction of the mosquito population’s vectorial capacity [3].
The formula for VC was extracted from the formula for \(R_0.\) The mosquito-specific parameters in the formula for \(R_0\) were derived from a formula for the daily EIR, which was the product of the human biting rate (HBR) and the SR.
The two concepts — EIR and VC — are thus very closely related, which is apparent from their definitions:
daily EIR (\(E\)) is the average number of infective bites received by a single person on a single day; and
VC (\(V\)) is the expected number of infective bites arising from all the mosquitoes blood feeding on a single fully infectious human on a single day.
The EIR describes bites received (exposure) and VC bites arising (potential transmission). The main difference is that VC is computed as if the hosts were fully infectious, while the EIR only counts mosquitoes that actually got infected.
In models where the parameters remain constant for all time and no mosquitoes leave or enter the system, the number of bites arising will, after a lag, be equal to the number of bites received. If we knew the fraction of mosquitoes that would become infected after blood feeding on a human \((\kappa),\) or the net infectiousness (NI) of humans, then it should be true that \[E \approx V\kappa.\] The formula for VC (building on Macdonald’s analysis of sensitivity to mosquito mortality) emphasized the importance of understanding how vector control changed mosquito bionomic parameters. In most cases, the EIR is a more practical metric to use for entomological surveillance and evaluation [4], so long as we take into account changes in NI [5].
In the following, we
present the formula VC and explain the parameters and some of the assumptions that went into it;
update the formula;
present a version of the dynamical system for mosquito infection dynamics and derive a formula for the EIR;
discuss parity.
VC: The Original Formula
The original formula for vectorial capacity was extracted from Macdonald’s papers:
\[V = \frac{ma^2}{-\ln p} p^n.\] To understand the formula and extend or apply it, we will need to deconstruct and rebuild it.
\(p\) - daily mosquito survival
We start with the parameter \(p,\) the fraction of mosquitoes that survive one day. The original model was probably formulated as a differential equation with a parameter describing the per-capita mosquito mortality rate, here called \(g.\) The probability of surviving one day is \(p = e^{-g}\). The model assumes mosquitoes survive at the same rate regardless of age; the lifepan is thus exponentially distributed with mean \(1/g\) (for a longer discussion, see [6]). The expected lifespan of a mosquito is thus \[\frac{1}{g} = \frac{1}{-\ln p}.\]
\(p^n\) - survival through the EIP
The extrinsic incubation period (EIP) was the time elapsed between the blood meal that infected a mosquito and the point when sporozoites had reached the salivary glands, such that the mosquito was infective (\(n\) days). In the formula, the propbability of surviving through the EIP is \[p^n = e^{-gn}.\]
\(a\) - the human blood feeding rate
The parameter \(a\) is the human blood feeding rate. In 1964, Garrett-Jones published The human blood index of malaria vectors in relation to epidemiological assessment [7], a simple way to measure the fraction of mosquitoes that had blood fed on humans. The paper drew attention to a studies of anthropophagy, the propensity to blood feed on humans [8]. We have thus split the parameter into an overall blood feeding rate (\(f\)), and the human fraction (\(q\)), so \[a=fq.\]
\(ma\) - the human biting rate
The human biting rate (HBR) in Macdonald’s analysis was \(ma:\) it is the number of mosquito bites, per human, per day. Here, we define mosquito population density (\(M\)) and human population density (\(H\)).
If there are \(M\) mosquitoes, and each mosquito takes \(a=fq\) human bloodmeals in a day, then the number of blood meals per person per day (i.e. the HBR) is: \[B = ma = \frac {fqM}{H}.\] This is a useful formula, but in some cases, we might want to understand mosquito ecology, the factors that determine \(M,\) and how mosquito population density is modified by vector control.
In writing a basic equation describing adult mosquito ecology, we follow Aron and May [9]. Let \(\Lambda(t)\) denote the number of adult females emerging from aquatic habitats each day. \[\frac{dM}{dt} = \Lambda - g M.\] If \(\Lambda\) is constant, the system reaches a steady state: \[M = \frac{\Lambda}{g}\] Now we can write the HBR as: \[B = \frac{\Lambda}{H} \frac{fq}{g} \] In Macdonald’s analysis of the sporozoite rate, he drew attention to the importance of mosquito mortality [1]. In his analysis of equilibrium in malaria, where he presented the formula for \(R_0,\) he overlooked the fact that the HBR is also affected by adult mosquito mortality [6,10].
An Updated Formula
In the new notation, the original formula for VC can be rewritten:
\[V = \frac{\Lambda}{H} \frac{f^2 q^2}{g^2} e^{-gn}\]
In this form, there are six quantities affecting transmission, but some of these parameters always appear together. In particular, the term:
\[s = \frac{fq}{g}\]
is the expected number of human blood meals per mosquito, summed over its lifespan. (When Macdonald derived the formula for \(R_0,\) he named the stability index.)
The fraction of mosquitoes surviving the EIP is:
\[\Upsilon = e^{-gn}\]
Under these assumptions, we can rewrite the classical formula:
\[V = \frac{\Lambda}{H} s \Upsilon s \]
In this form, it tells the story of transmission: after emergence \((\Lambda)\), a mosquito must blood feed on a human \((1/H)\) to get infected \((s),\) then survive the EIP \((\Upsilon)\), and then blood feed on humans to transmit \((s).\)
Macdonald
Macdonald was not a mathematician, and his papers present formulas with only minimal derivation (see the Ross-Macdonald vignette). He was presenting mathematical analysis that had been done by others [11]. Here, we show how VC was derived from Macdonald’s formulas. Here, we use the the updated notation and variables.
The Sporozoite Rate
Macdonald’s formula for the sporozoite rate can be dervied from a pair of equations describing infection dynamics in mosquitoes.
Let \(Y\) denote the density of infected mosquitoes. We need to know the probability a mosquito becomes infected after blood feeding on a human, \((\kappa)\):
\[\frac{dY}{dt} = fq\kappa(M-Y)-gY.\] At the steady state, \[y = \frac{Y}{M} = \frac{fq\kappa}{g+fq\kappa} = s \frac{\kappa}{1+s\kappa}.\]
Macdonald’s formula assumed a mosquito would become infectious after a fixed delay of \(n\) days. We introduce the subscript \(n\) to mean the value of a variable at time \(t-n.\) The denssity of infective mosquitoes (\(Z\)) is
\[\frac{dZ}{dt} = e^{-gn} fq\kappa(M_\tau-Y_\tau) -gY.\] At the steady state, the sporozoite rate is: \[z = \frac{Z}{M} = e^{-gn} y = \frac{fq\kappa}{g+fq\kappa} e^{-gn} = s \Upsilon \frac{\kappa}{1+s\kappa}.\]
EIR
Macdonald’s formula for \(R_0\) began with the formula for the EIR. It was the product of the human biting rate and the sporozoite rate. In this model, the human biting rate is \[ma = \frac{fqM}{H} = B.\] So the eir would be \[E = Bz = V \frac{\kappa}{1 + s\kappa}\]
The formula for \(R_0\) describes malaria transmission in a population where the parasite has been absent, so \(\kappa=0.\) If we write the EIR as a function of the NI, then then in the limit as \(\kappa\) gets small: \[\lim_{\kappa \rightarrow 0} \frac{d E(\kappa)}{d\kappa} = V.\] such that \[E \approx V \kappa.\] At steady state, the approximation is off by the factor \((1 + s \kappa)^{-1},\) which describes the expected number of times an infectious mosquito has been infected. This difference is, in simpler terms, a way of approximating mosquito superinfection.
In the Ross-Macdonald Model vignette, we presented the original formula for \(R_0.\) Here, we can rewrite it as: \[R_0 = V \frac{b}{r}\] If we knew \(\kappa,\) (and \(s\) and \(b\) and \(r\)) then we could estimate VC from the EIR [12]:
\[R_0 = E \; \frac{b}{r} \; \frac{1+s\kappa}{\kappa} \] We emphasize that this is model-dependent. If we wanted a robust estimate of \(R_0,\) we would need to consider many other factors. We can follow the same logic in complex models, but the logic is easier to follow in these simple ones.
Parity & Bionomics
Macdonald’s analysis of the sporozoite rate had drawn attention to the importance of measuring mosquito survival and parity [13,14]. The concept of parity has played an important role in malaria, so it is worth deriving the formula for it. If we let \(P\) denote the fraction of mosquitoes that has laid at least one batch of eggs, and if we assume that blood feeding is equivalent to egg laying (a dubious assumption for several reasons), then:
\[\frac{dP}{dt} = f(M-P)-gP.\] such that at the steady state:
\[\frac{P}{M} = \frac{f}{f+g}\]
The ratio of parous to non-parous mosquitoes is thus: \[\frac{f}{g}\] Parity is thus, potentially, an ideal way of measuring blood feeding. If we also measured the human blood index (HBI) [7], then we could estimate the human fraction \((q).\)