Vectorial Capacity: A Framework
Potential Malaria Transmission by Mosquitoes
Vectorial capacity (VC) describes potential malaria transmission by mosquitoes: after emergence, 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. The concept is so concise and compelling that if we didn’t have this formula, we would reinvent something just like it. In fact, the formula is model-dependent, and wew might want to relax some of the highly restrictive assumptions of that model to use it. Here, we present a generalized framework for understanding VC as a concept that can be used to compute VC.
See Related for links to closely related vignettes.
The formulas for \(R_0\) and VC are sometimes used as if they expressed some universal truths about transmission, not as formulas that were derived from a specific model. Macdonald was not a mathematician, and his papaers present formulas with only minimal derivation (see the Ross-Macdonald vignette). He was presenting mathematical analysis that had been done by others [1]. The formulas are often used out of context, as if they expressed some universal truths about transmission. In fact, the formulas for \(R_0\) and VC are model dependent. While the formulas are based on some compelling quantitative logic, that same logic also works for other mathematical models. In the next vignette, we present a generic framework.
The formula for vectorial capacity (VC) traces back to Macdonald’s formula for \(R_0\) [2,3]. Garrett-Jones named it, and argued it was the proper bases for developing theory for vector control:
I propose that “vector control” should refer, in this context, to reduction of the msoquito population’s vectorial capacity [4].
If we want to use VC as a basis for theory for vector control, we must acknowledge its limitations. The original formula was derived from Macdonald’s model; as written, it is model-dependent. Here, we present a generalized formula that we can use to compute VC in other models, including models where mosquitoes senesce, in models with spatial dynamics [5], in non-autonomous models with exogenous forcing on bionomic parameters, in models with vector control, and in agent-based models [6]. VC is a useful way of benchmarking models.
The formula for VC is embedded in Macdonald’s formula for \(R_0,\) which was derived from a formula for the EIR. Macdonald’s 1952 papers on sporozoite rate and steady states in malaria had identified four parameters: the ratio of mosquitoes to humans (\(m\)), the human blood feeding rate (\(a\)), the fraction of mosquitoes surviving one day (\(p\)), and the duration of the extrinsic incubation period (EIP, \(n\)) [2,3]. In the body of this vignette, we show that the formula for VC combines these parameters into a coherent summary of potential transmission.
Macdonald’s analysis of the sporozoite rate had demonstrated that transmission was highly sensitive to mosquito mortality, and several studies pointed out that the fraction of mosquitoes that were parous (i.e. that had laid at least one egg batch) could be used to measure a mosquito’s reproductive age [7,8].
In 1952, Macdonald published a formula for the sporozoite rate [2]: \[z = \frac{a\kappa}{-\ln p + a\kappa} p^n\]
Later in 1952, Macdonald published formulas for the endemic equilibrium and \(R_0\) [3] that was based on the EIR, which was the product of the the human biting rate (HBR) and the sporozoite rate (SR). In Macdonald’s notation, the HBR was \(ma\) so his forrmula for the EIR was:
\[E = \frac{ma^2\kappa}{-\ln p + a\kappa} p^n.\]
The formula for \(R_0\) contains the formula for VC. Note that in the limit as \(\kappa\) becomes small:
\[\lim_{\kappa \rightarrow 0} \frac{d E(\kappa)}{d\kappa } = \frac{ma^2}{-\ln p } p^n = V.\]
VC and EIR
The formula for VC was extracted from the formula for \(R_0.\) The entomological parameters in the formula for \(R_0\) were derived from the formula for EIR. The two concepts are thus very closely related:
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.
One describes bites received (exposure) the other bites arising (potential transmission). In models where the parameters remain constant for all time and no mosquitoes leave or enter the system, the number of bites arising will be equal to the number of bites received.
The biggest difference between the two formulas is that EIR accounts for partially infectious humans. If we define net infectiousness (NI or \(\kappa\)) as the fraction of mosquitoes that become infected after blood feeding on a human, then \[E \approx V \kappa\]
Starting from these definitions, we will present a new general formula for VC. After that, we will revisit the original formulas.
The HBR
The definitions for the EIR and VC both include the human biting rate (HBR). The HBR is a real quantity, but it difficult to measure accurately. Here, we deal with the mathematical formulas that describe the expectation of that process in a highly simplified way.
The EIR is the product of the HBR and the sporozoite rate (SR):
\[E = Bz\]
where
the HBR (\(B\)) is the average or expected number of bites by a single vector species, received by a single person, on a single day; and
the SR (\(z\)) is the fraction of infective mosquitoes, those sporozoites in their salivary glands and thus presumed to be infectious.
To write the formula for VC, we define two terms that appear in any formula for the SR:
let \(\Upsilon\) denote the fraction of mosquitoes surviving the EIP; and
let \(s_z\) denote the number of human blood meals after becoming infectious.
Using these terms, vectorial capacity is the HBR, times the probability of surviviving the EIP, times the expected number of human bloodmeals that a mosquito would give after becoming infectious:
\[V = B \Upsilon s_z\] The two formulas emphasize, once again, the important similarities between the two terms. While the EIR is looking at events happening now, integrating processes from the past, VC is making a projection into the future, from the moment when a mosquito became infected.
The HBR, the EIR, and VC all have the same units: human population density is in the denominator. To define the HBR, we need four terms:
let \(M\) denote the population density of adult female mosquitoes;
let \(H\) denote human population density;
let \(f\) denote the overall blood feeding rate;
let \(q\) denote the fraction of blood meals that are taken on a human or the human fraction.
\[B = \frac{fqM}{H}\]
This is the formula that we will use to compute VC in most models.
A New Formula
It is sometimes useful to understand the human biting rate in terms of mosquito ecology. The HBR is affected by some of the same processes affecting the sporozoite rate [9].
Let \(\Lambda\) denote the number of adult, female mosquitoes emerging from aquatic habitats, per day
Let \(s\) denote the expected number of human blood meals a msoquito would take over its lifespan.
The HBR includes all the adult mosquitoes that emerged in the past, survived and blood feed on humans \((s).\)
\[B = \frac{\Lambda}{H} s \]
We are now in a position to rewrite VC in a new way that elucidates the processes affecting transmission. In its most general form, we could make a distinction between the blood meal that infects a mosquito (\(s_u\)) and the bites that transmit \((s_z).\)
\[V = \frac{\Lambda}H s_u \Upsilon s_z\]
In this form, vectorial capacity tells the story of transmission: after emergence \((\Lambda),\) a mosquito must blood feed on a human \((1/H)\) to get infected \((s_u),\) then survive the EIP \((\Upsilon)\), and then blood feed on humans to transmit \((s_z).\)
We think of this formula as general template that can be applied in various contexts. It serves as a benchmark for comparing models that make different assumptions about mosquito ecology, the EIP, mosquito survival, or spatial or temporal heterogeneity.
Extending VC
In computing basic reproductive numbers for malaria in models with space and seasonality, we need to compute VC and its dispersion over time and space:
If we want to take mosquito senescence into account (the death rate of mosquitoes increases with age), the formulas for the EIR and VC must be modified to account for the differences in a mosquito lifespan after it becomes infectious.
If we want to understand mosquito spatial ecology in patch-based models, then we will find it useful to compute a VC matrix: the number of infective bites arising in every patch from the number of mosquitoes blood feeding in each patch [5].
In models with seasonal forcing, we can compute the temporal dispersion of VC, the number of infective bites arising each day from all the mosquitoes blood feeding each day.
In models with vector control, we can treat mass distribution as a perturbation to the system and show how vector control modifies the temporal distribution of the VC.
SimBA includes algorithms to compute VC for patch-based meta-population models.