Exposure Trace Functions

To model exposure, ramp.falciparum uses the trace function library in ramp.func

Exposure to malaria is defined as the number of bites by infective mosquitoes, per person, per day, called the daily entomological inoculation rate (abbreviated EIR and denoted $E(t)$ ). Since the daily EIR (dEIR) is often seasonal, we also discuss the annual EIR (aEIR), average dEIR over a year. The force of infection is the hazard rate for infection for susceptible individuals (abbreviated FoI and denoted $h(t)$ ).

ramp.falciparum is focused on malaria epidemiology defined in a narrow sense to include anything happening in the human population: exposure, infection, disease, immunity, infectiousness, treatment with anti-malarial drugs, vaccines, detection (by various diagnostic methods), heterogeneity, and demography. It has avoided models of malaria transmission dynamics and control: for that, we developed ramp.xds and SimBA.

Here, we use trace functions to model exposure. If we model malaria transmission dynamics, the EIR and the FoI would be the outcome of parasite transmission among humans and mosquitoes through blood feeding. Mosquito ecology and infection dynamics could be seasonally forced or highly variable, and the models would output the EIR and the FoI as intermediate terms. If we wanted to remain focused on malaria epidemiology in the narrow sense, we could take the essential information about the EIR and the FoI — the trace — and use it to force exposure.

Composed Time Series Functions

To model complex patterns of exposure, we use composed time series functions from the ramp.func a function library. Each trace function is the product of four components:

$E(t) = \bar E \times F_S(t) \times F_T(t) \times F_K(t)$

where

$\bar E$ is the mean value of an unperturbed system.
$F_S(t)>0$ is a seasonal pattern function, normalized over the interval such that $\int_{t_0}^{t_1} F_S(t) \;dt = t_1 - t_0$
$F_T(t)>0$ is a trend pattern function, normalized over the interval such that $\int_{t_0}^{t_1} F_T(t) \; dt = t_1 - t_0$
$F_K(t)>0$ is a shock function, a perturbation to the system

Heterogeneous Exposure

The EIR and the FoI are related, under some model for heterogeneous exposure and pre-erythrocytic immunity. In developing models of heterogeneous exposure, we recognize several kinds of heterogeneity that we handle very differently (see Cooper L, et al., 2019)¹: heterogeneous biting, environmental heterogeneity, and complex exposure.

Heterogeneous Biting

By heterogeneous biting, we mean that the expected rate of exposure differs for population strata: biting is a kind of frailty: if the population average dEIR is $E$ , and if $w$ is the frailty term for one stratum, then the expected dEIR is $wE.$ A frailty term is a multiplicative modifier of risk. If $E$ is the population average, and if $D$ describes the distribution of frailty over all strata then: $\int_0^\infty D(w) dw = 1$ and $\int_0^\infty w E D(w) \; dw = E$ In this case, $w$ is the relative biting rate.

Environmental Heterogeneity

Exposure can be heterogeneous in a homogenous population stratum (i.e. there are no differences in the expected rate). Consider two humans with the same expectation, but on one day, one of them happens to spend their time in a house with glass windows and a door that closes, while the other one is outside and walks into a swarm of mosquitoes. From a mosquito perspective, mosquitoes populations move around and aggregate around resources. Those mosquito swarms can be moved around by wind.

To handle environmental heterogeneity, we let the expectation have a distribution. If, for example, the expectation is gamma distributed, then the number of bites per human is a mixture process: a $gamma$ -Poisson gives us a negative binomial distribution of bites per person. Notably, for most mosquito catch counts data, the variance is much higher than the mean, and the distributions can be modeled with negative binomial distributions.

Complex Exposure

In models for superinfection, a single infection event can infect a human with more than one distinct parasite genotype. The distribution of the number of distinct parasite types, per bite, is called the multiplicity of exposure (MoE).

Exposure and Infection

Exposure is heterogeneous, but so is the risk of infection. In some cases, we want to consider models with pre-erythrocytic immunity: the risk of infection per infectious bite is denoted $b$ . To handle this, various mathematical models will represent infection states — denoted $\mathbf{X}$ — in different ways. Generically, the FoI is related to the EIR by a function:

$h = F_b(E, \mathbf{X})$

Cohort Dynamics

Since malaria epidemiology is related to age and exposure, all the mathematical models developed in ramp.falciparum are formulated for cohorts as they age. To explore these models, we must simulate arbitrary patterns of exposure in cohorts as they age, with the ability to compare two cohorts of different ages in the same population. If $E(t)$ denotes the average daily EIR in a population, we want to model exposure in some particular cohort, born on day $d.$ For every cohort, age and time are related by the identity $t = a + d.$ It follows that exposure at age $a$ is

$E(a|d) = E(t-d).$

Age

Motivated by the evidence for age-dependent exposure, the rate of exposure in a cohort is assumed to change with age. We use the age-weighting function, $\omega(a)$ , consistent with the available evidence from both anopheline and aedine mosquitoes (Port GR, et al., 1980;², Carnevale P, et al., 1978;³, Liebman KA, et al., 2014⁴) . Since biting varies with age, we let $F_\omega(a)$ be a function that computes the relative biting rate by age, a kind of frailty. The EIR for a cohort is:

$E(a|d) = F_\omega(a) E(t-d) = \bar E \times F_\omega(a) \times F_S(t-d) \times F_T(t-d) \times F_K(t-d)$ We can thus formulate models for exposure in a population and compare outcomes for two different cohorts.

In some cases, we can simply skip the model for pre-erythrocytic immunity. We $\bar h$ be a scaling parameter and:

$h(a|d) = \bar h \times F_\omega(a) \times F_S(t-d) \times F_T(t-d) \times F_K(t-d)$

Trace Functions