Environmental Heterogeneity

Heterogeneous blood feeding is a basic feature of malaria transmission (see Heterogeneous Transmission). In exDE the term environmental heterogeneity is used to describe the distribution of the expected number of bites within a homogenous human population stratum: biting is extremely heterogeneous even for individuals who have the same expectation. The approach is motivated by a study of heterogeneous exposure by (Cooper L, et al., 2019)¹.

In the following, we derive formulas for the force of infection (FoI) from the model for the attack rate (AR) under the Poisson and Negative Binomial families of models.

Attack Rates and the Force of Infection

In mechanistic models of malaria, the hazard rate for exposure is generally assumed to be a linear function of the entomological inoculation rate. In the following, we assume that the number of bites per person over a day (or over some longer interval, \(\tau\)), is a random variable, and we formulate approximating models for attack rates and hazard rates.

Poisson Hazard Rates

We let \(E\) denote the EIR, the expected number of bites per person over a day. If we assume that the distribution of the daily EIR is Poisson, and if a fraction \(b\) of infective bites cause an infection, then the relationship between the between EIR and the FoI is a Poisson compounded with a binomial, which is also Poisson:

\[ Z \sim F_E(z) = \mbox{Poisson}(z, \mbox{mu} = bE(t)) \]

Over a day, the daily attack rate, \(\alpha\), is the fraction of individuals who received at least one infection, or:

\[ \begin{array}{rl} \alpha &= 1-F_E(0) \\ &= 1-\mbox{Poisson}(0, \mbox{mu} = bE(t)) \\ &= 1- e^{-bE(t)} \\ \end{array} \]

The daily FoI, \(h\), is given by a generic formula:

\[ \alpha = 1 - e^{-h} \mbox{ or equivalently } h = -\ln (1-\alpha) \]

In this case, the relationship between the FoI and the EIR is:

\[ h(t) = b E(t) \]

It is highly mathematically convenient that the relationship is invariant with respect to the sampling period.

Negative Binomial Daily Hazards

If we assume the number of infective bites, per person, per day, has a Gamma distribution in a population, then we could model the number of infective bites as a Gamma - Poisson mixture process, or a negative binomial distribution. Under this model, the counts for bites by sporozoite positive mosquitoes over one day, \(Z\), would be a negative binomial random variable with mean \(E\):

\[ Z \sim F_E(z) = \mbox{NB}(z, \mbox{mu} = bE(t), \mbox{size} = 1/\phi) \]

Assuming an infectious bite causes an infection with probability \(b\), the daily attack rate is:

\[ \begin{array}{rl} \alpha &= 1-F_E(0) \\ &= 1-\mbox{NB}(0, \mbox{mu} = b E(t), \mbox{size} = 1/\phi) \\ &= 1- \left(1+b E(t)\phi \right)^{-1/\phi} \end{array} \]

This is consistent with a formula that has a continuous daily FoI:

\[ h = \frac{\ln \left(1 + bE(t)\phi \right)} {\phi} \]