Page 255 - DCAP601_SIMULATION_AND_MODELING

Unit 13: Simulation Languages (I)

tracks an individual’s progress along an infection timeline also sketches the life cycle of the Notes
parasite. Because the fundamental question for two-party interactions is whether one participant
is infectious and the other susceptible with respect to a parasite, the state of each host and vector
thereby represents the presence or absence of a parasite life cycle stage appropriate for infectivity
or susceptibility. Unit interactions simply mimic host-vector contact – i.e., the mosquito taking
a blood meal – and each corresponding potential state transition the potential transmission of a
parasite between host and vector. Actual transmission of a parasite in any given host-vector
interaction may involve chance as well as additional biological factors; our representations here
include a host immune component and a vector mortality function.

Infection timelines express the dynamic individual-level states that lead to the compartments of
traditional population-level models; their single-valued state representations allow compact,
efficient data structures that permit representations of very large interacting populations. Simple
binning can translate the individual state descriptions into familiar population-level classes
such as “infected” and “infectious,” and allows ready calculation of prevalence and other
epidemiological measures.
The full infection timeline in our malaria models corresponds to the “incubation interval” in
Macdonald’s model, i.e., “the complete period from the occurrence of infective gametocytes in
one case to the development of infective gametocytes in the secondary cases derived from it,”
comprising “the period of extrinsic development of the parasite in the mosquito; the pre-patent
period, or incubation period as it is normally known, in man; and any interval between the
patency of asexual parasites and the development of fully infective gametocytes.” Our malaria
models represent this full interval, and the parasite life cycle, as a circuit from position “0” on
the host timeline through position “0” on the vector timeline and back to the host “0,” by way
of two blood meals.

Three of the five temporal parameters of our basic model correspond directly to those in
Macdonald’s:
1. Vector Delay (DV) is the length of the interval between infection (gametocyte ingestion)
and the onset of infectivity (sporozoite migration) in a vector, i.e., Macdonald’s “period of
extrinsic development,” above;
2. Host Delay (DH) is the length of the interval between infection (sporozoite inoculation)
and the onset of infectivity (gametocyte maturation) in a host, i.e., Macdonald’s “pre-
patent period” and subsequent “interval,” above;
3. Vector Survivorship (VS) is the daily probability of a mosquito’s survival, i.e., synonymous
with Macdonald’s expression for the “probability of survival through one day.”

4. Host Window (WN) is the duration of a host’s infectivity to vectors, from the first to the
final presence of infective gametocytes. This factor was addressed by Macdonald at most
indirectly, in terms of recovery rates and infective proportions; WN is closely analogous
to the “loss rate” a in the Dietz et al. extension of Macdonald’s model.
1
5. Host Immunity (IM) defines a host’s susceptibility to re-infection through the daily decay
of a blocking immunity. This was addressed at most indirectly in Macdonald’s model; IM
closely resembles the “loss rate” ã in the Aron and May extension.

Implementation

Populations of hosts and vectors are represented by arrays that contain the state of each constituent
host or vector with respect to its stage of infection, with these states represented on the
corresponding infection timeline by the variables h and v, respectively. Thus a value 0 < h < D
H
indicates that the host is infected but not yet infectious, –WN < h <= 0 that the host is (infected

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