ScenarioTransmission

Levels of Transmission

OpenMalaria simulations of malaria transmission require specification of:

• The level and seasonality of exposure (measured by the Entomological Inoculation Rate, EIR) to malaria at the start of the simulation

• The model for malaria transmission from humans to mosquitoes

• The dynamics of malaria parasite cycle within humans and also the model for transmission from mosquitoes to humans (the entomological model). There are three different variants of the entomological model:

• The "Non-vector" variant does not consider mosquito dynamics and hence does not allow the user to modify the vectorial capacity. It is appropriate for modeling situations where interventions (such as chemotherapy or vaccines that only act on humans) and is described in: Smith et al, 2006.. Specification of this variant is described below

• The "Vector" variant comprises discrete-time population models that simulate how many mosquitoes belong in each of several categories at each time. The models assume that the infectious (sporozoite positive) mosquitoes act to distribute infections at random to the human population (with human exposure proportionate to availability). Entomological interventions modify the vectorial capacity and require the "vector" transmission model variant. The simulations that include non-periodic changes in the vectorial capacity use a seasonally forced version of the difference equation model for vector dynamics of Chitnis et al (2008)Journal of Biological Dynamics Vol. 2, No. 3, July 2008, 259–285, further described in Chitnis et al (2010) American Journal of Tropical Medicine and Hygiene Vol. 83, No. 2, 230--240.

The vector transmission model is required for modeling interventions that have effects on mosquitoes, and hence change the vectorial capacity. In addition to the specification described below additional XML parameters for specifying this sub-model are described here.

The default use of both the "non-Vector" and "Vector" model variants follows the original Ross-Macdonald model in assuming that intervention-induced reductions in adult mosquitoes do not affect the numbers of emerging females, (which depend on local carrying-capacity of the breeding sites). OpenMalaria also supports an extension of the "Vector" model variant that incorporates a full-life cycle model that can capture effects of adulticiding on emergence, described here.

Non-vector transmission

The "non vector" model assumes a fixed seasonal vectorial capacity, and either forces infection rates (EIR) or makes EIR dependent on human infectiousness to vectors while forcing vectorial capacity. Initial exposure of humans to infectious mosquito bites is input and any intervention effects on transmission to the mosquito translate into proportionate effects on transmission back to the human. This model is valid when the only interventions are ones that do not affect the vectorial capacity (e.g. vaccines or chemotherapeutic interventions). When vector control interventions are applied, the "Vector" model must be used.

The nonVector element primarily consists of a list of daily EIR (Entomological Inoculation Rate) parameters (EIRDaily elements) specifying the annual EIR (thus 365 values are expected) (see example above). Assuming the first value is the EIR for January 1st, time 0 corresponds to the beginning of the year (since this is the only input affecting seasonality it can be rotated as desired).

Values in this list are averaged per timestep to calculate the EIR per timestep of the year for the pre-intervention equilibrium state. Where data for more than one year are provided, the data is assumed to wrap into the next year and all values for the same timestep of the year are averaged.

nonVector also has an eipDuration attribute: the extrinsic incubation period (sporozoite development time, in days), which determines the delay before changes in human infectiousness affect the EIR (in dynamic mode only).

Initial level and seasonality of EIR

Entomological data are described by the entomology element, containing either a vector or a nonVector sub-element. For example:

<entomology mode="4" name="a name">
  <nonVector eipDuration="10">
    <EIRDaily origin="monthly">0.0738</EIRDaily>
    <EIRDaily origin="monthly">0.0738</EIRDaily>
    <EIRDaily origin="monthly">0.0738</EIRDaily>
    <EIRDaily origin="monthly">0.0738</EIRDaily>
    <EIRDaily origin="monthly">0.0738</EIRDaily>
    <EIRDaily origin="monthly">0.0738</EIRDaily>
    ...
  </nonVector>
</entomology>

Attributes of the entomology element:

name	versions	type	description
name	versions	type	description
name	all	text	A user-friendly name for the transmission settings

Note that prior to schema 24, the vector model used EIR in units of infectious bites per person per time-period, averaged across the population, while the non-vector model used units of infectious bites per adult per time-period. From schema version 24 both use units of infectious bites per adult per day/timestep/month/year. (The difference being that children receive fewer bites than adults.)

Entomological Inoculation Rate (EIR)

The level and seasonality of transmission are input via a description of the approximate seasonal pattern of the EIR. EIR can be specified either via 5 Fourier coefficients and a rotation factor (EIR element) or via 12 monthly values plus an annual level (monthlyEIR element). Exactly one of these elements must appear. The models require as input, data on the overall average transmission level in the absence of interventions (measured by the entomological inoculation rate, EIR).

Data on seasonality of malaria transmission for driving models might be available in the form of seasonality in any of a number of malariological indices (see Table below). The models, however, expect the seasonality in the EIR as the input. If the data are available in the form of a different measure of seasonality, they need to be transformed before being used to drive the models. The easiest approximation is to introduce a fixed lag period, depending on which measure is used. While this is a considerable simplification, because it assumes proportionality between different measures, this may be reasonable, especially if the data relate to mosquito densities or emergence rates.

The table contains suggestions for what might be the approximate lag periods between different measures of seasonality. A positive value (Lx) for the lag for measure x implies that the EIR seasonality reflects the value of x, Lx days previously. This table provides only a very approximate guide with values rounded to multiples of 5 days. The actual average lag periods in the simulations are model dependent and will vary somewhat from these. The lag periods in the field also vary and, in the case of quantities measured in the vector population, are dependent on the environmental temperature.

Table: Approximate lag periods between different seasonality measures

Transmission measure	Lag period (Lx) (days)
Rainfall	+30
Emergence rate of vector	+20
Density of host-seeking vectors	+10
Entomological inoculation rate	0
Incidence of infection	-5
Incidence of patent infection	-10
Incidence of clinical malaria	-15
Incidence of severe malaria	-20

Ways of specifying level and seasonality in EIR

EIR can be specified either via 5 Fourier coefficients and a rotation factor (EIR element) or via 12 monthly values plus an annual level (monthlyEIR element). Exactly one of these elements must appear.

EIR via Fourier coefficients

This is the older method (the only available method before schema version 22).

The EIR element describes the Entomological Infection Rate for this mosquito species, which is used as a target when fitting the emergence rate of adult mosquitoes. The EIR is given via a Fourier series and a rotation offset; more accurately, the exposed EIR, in units of bites per day, is:

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Here, Ξ_t is the number of innoculations per person per day, where t is the day of year (from 0 to 364), w = 2π / 365, θ is the EIRRotateAngle attribute and a₀ to a₂, b₁ and b₂ are the corresponding attributes of the EIR element.

If we introduce the function f(t) dependent on a_n and b_n for n≥1, we can reformulate Ξ_t as

and thus show that the annual EIR is scaled by :

This is configured using a section in the XML, per species, similar to the following:

<seasonality annualEIR="178.60558666831946" input="EIR">
          <fourierSeries EIRRotateAngle="0">
            <coeffic a="-0.692164" b="0.002098"/>
            <coeffic a="0.401189" b="-0.375356"/>
          </fourierSeries>
        </seasonality>

EIR via monthly values

As of schema version 22 this method of entering EIR was added to simplify entry of field data. Fourier coefficients are still used within the code (for smoothing), but are calculated internally.

The monthlyEIR element requires one attribute: annualEIR, specifying the total annual EIR as infectious bites per person per year. It should have a sequence of 12 child elements, named item, specifying the relative in terms of infectious bites per person per month (which might be approximated by densities of mosquito densities, with an appropriate lag), in the months of January through December. The overall EIR is scaled to the annualEIR value indicated at the top of the element. An (unrealistic) example:

        <seasonality annualEIR="20" input="EIR">
          <monthlyValues smoothing="fourier">
            <value>10</value>
            <value>1</value>
            <value>1</value>
            <value>1</value>
            <value>1</value>
            <value>1</value>
            <value>1</value>
            <value>1</value>
            <value>1</value>
            <value>1</value>
            <value>1</value>
            <value>1</value>
          </monthlyValues>
        </seasonality>

Seasonality in the static transmission mode (mode="2")

With static transmission the seasonality pattern during the intervention phase of the simulation, reproduces the input pattern of seasonality.

Seasonality in the dynamic transmission mode (mode="4")

If the simulation is run using the dynamic transmission mode (mode="4") then the simulated EIR is not constrained to repeat the same seasonal pattern as the input EIR after the warm-up period.

The simulated EIR will vary because of the effects of interventions. In addition, there may be stochastic variations or fluctuations even in the absence of interventions. This is particularly the case with small simulated populations or when the simulation is close to the limit of sustainable transmission (e.g. with high case-management coverage) the simulated EIR will usually be somewhat below the level input.

During the warm-up period, the entomological parameters are adjusted to try to ensure that the output EIR corresponds as closely as possible to that input. An exact match, consistent with the emergence rate (N_v0) being positive, is in general impossible. The seasonality of the simulated EIR (which is close to that of S_v plus the delay due to the lag) may differ from the seasonality of the input EIR substantially

In versions before 30 of OpenMalaria the change from the static to dynamic generation of the EIR is at the end of the warm-up period. In these simulations, even if the annualEIR is equal before and after the end of the warm-up period, the switch in seasonality may alter the annual number of new infections. Running scenarios without interventions shows the dynamics of EIR and human infectiousness (the continuous output mode with duringInit="true" is convenient for viewing this).