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 logic motivating the concept of vectorial capacity is so compelling that if we didn’t have this formula, we would reinvent it. In fact, the formula is model-dependent. To use it in other contexts, we must relax some of the highly restrictive assumptions of that model. 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 were derived from a model of malaria transmission by mosquitoes, but they are sometimes used as if they expressed some universal truths about transmission. While the formula is based on some compelling quantitative logic (for example, see [1]), it is based on some highly unrealistic simplifying assumptions. There is a great potential to apply the formula in other contexts, but to avoid misusing the formula, we must relax the underlying assumptions and develop new formulas that are fit for purpose. These formulas are derived from other models that have been developed to describe malaria transmission by mosquitoes [1,2]. Here, we present a generic framework and notation:
to describe VC for models with spatial dynamics
for models with forcing by weather or vector control; and
generalized vectorical capacity that describes potential transmission over space and time.
To review a bit, Garrett-Jones named the formula for vectorial capacity (VC):
I propose that “vector control” should refer, in this context, to reduction of the msoquito population’s vectorial capacity [3].
The formula traces back to Macdonald’s formula for \(R_0\) [4,5]. Macdonald was not a mathematician, and his papers present formulas with only minimal derivation (see the Ross-Macdonald vignette). Macdonald’s mathematical analysis was partly done by others [6].
Macdonald’s formulas for \(R_0\) and VC are, in fact, model dependent. That model assumes all the bionomic parameters are constant, and transmission occurs in a single well-mixed population. If we want to use VC, we must acknowledge these limitations.
Given the compelling logic, the reformulated formula for VC provides a sound basis for benchmarking models. The formula could be used as a basis for spatial targeting, but it would need to be modified to describe mosquito dispersal, and differences in potential transmission across space [2]. The formula could be used to describe potential transmission in models with seasonal forcing or vector control, but the formulas should then emphasize how VC is different on every day.
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, \(n\)) [4,5]. 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 [4]: \[z = \frac{a\kappa}{-\ln p + a\kappa} p^n\]
Later in 1952, Macdonald published formulas for the endemic equilibrium and \(R_0\) [5] 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 Macdonald’s formula for \(R_0\) [5]. The entomological parameters in the formula for \(R_0\) were derived from the formula for EIR [4,5]. The two concepts are 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 and notions, we will present a new general formula for VC.
VC, Generically
The definitions for the EIR and VC both include the human biting rate (HBR). The HBR describes something real, but it difficult to measure accurately. Here, we deal with the mathematical formulas that describe it as the expectation of a process.
First, we define a quantity \(B,\) the total daily blood feeding rate by a mosquito population in a place. We must define 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 human fraction: the fraction of all blood meals that are taken from humans.
Using this notation, the total biting rate \((B),\) defined as the total number of bites by a single vector species in a population is: \[B=fqM\] The human biting rate is defined as the number of bites, per human, per day. Letting \(H\) denote the number of humans, the HBR is: \[\mbox{HBR} = \frac{B}{H}\] Later, we will use \(W,\) the total availability of humans, for the denominator.
The SR (\(z\)) is the fraction of infective mosquitoes, those sporozoites in their salivary glands and thus presumed to be infectious. The EIR is the product of the HBR and the sporozoite rate (SR):
\[E = \frac{B}{H}z\]
To write the formula for VC, we define two terms to describe the number of infectious bites arising. These terms are also used to compute the SR. To transmit, a mosquito must survive the EIP and then bite other humans:
let \(\Upsilon\) denote the fraction of mosquitoes surviving the EIP; and
let \(s_z\) denote the number of human blood meals after becoming infectious.
In some models, it is much easier to compute infectious bites arising without computing both terms separately.
Using these terms, vectorial capacity is the HBR, times the probability of surviving the EIP, times the expected number of human bloodmeals that a mosquito would give after becoming infectious:
\[V = \frac B H \Upsilon s_z\] The two formulas emphasize, once again, the important similarities between the two terms. While the EIR is a measure of infectious biting now, integrating processes from the past, VC starts with blood feeding now and projects infective biting into the future, from the moment when a mosquito became infected.
VC, with Mosquito Ecology
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.
In our simple model, \[\frac{dM}{dt} = \Lambda - g M,\] so \[M = \frac{\Lambda}{g}.\] Later, we will replace \(g\) with the mosquito demographic matrix that computes survival and dispersal.
The HBR includes all the adult mosquitoes that emerged in the past, survived and blood feed on humans \((s_u).\)
\[B = fq \frac{\Lambda}{g} = \Lambda s_u \]
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. Vectorial capacity can serve 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 [2].
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.
Recommended
In Vector bionomics and vectorial capacity as emergent properties of mosquito behaviors and ecology, Sean Wu et al. present an individual-based simulation model for adult mosquito behavior, and they study vectorial capacity [1].
In Spatial dynamics of malaria transmission, Sean Wu et al. present the mathematical framework for modeling malaria spatial dynamics adoped by this website, including a formula for the VC matrix [2]