Page 256 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes and) infectious, and any other value of h that the host is not currently infected. For h < –WN, the
host’s immunity to re-infection is declining, as explained below. Similarly, –D < v < 0 indicates
V
a vector that is infected but not infectious, v <= –D a vector that is (infected and) infectious, and
V
any other value of v a vector that is not currently infected. The basic model does not encompass
the possibility of mosquito immunity. At each successive day of a simulation run, all h and v
values are decremented by 1, advancing them through the appropriate delay, window or decay
periods.
Parasite transmission is represented as a single state-altering interaction between an individual
host and an individual vector. At each time step the model selects a mosquito at random and a
human at random for an interaction. An initial parameter, V , the total number of daily host-
B
vector interactions, sets the number of times this random pair selection is repeated each day.
Only two pairs of host-vector states permit interactions that may directly induce state transitions:
1. If the mosquito is infectious, then with a probability based on the immune state of the
human, that human changes state and becomes infected, setting h to D ;
H
2. If the human is infectious, then with some probability, taken to be 1 in the basic model,
that mosquito changes state and becomes infected, setting v to 0.

In the other possible state combinations there are no transitions: super infection of a human
already infected with a given strain does not reset h, and super infection of a mosquito already
infected with a given strain does not reset v. Hence, in humans, super infections with identical
strains affect neither immune responses nor gametocyte production, and in mosquitoes they
simply accumulate, neither accelerating nor delaying the onset of infectiousness. This follows
Macdonald’s conventions, but an advantage of our modeling approach is that virtually any
super infection scheme can be implemented. Host and vector populations are sampled with
replacement, so multiple feedings by a single mosquito as well as multiple bites on a single
human may occur within a single day; both phenomena occur in nature.
Any host not previously infected is considered wholly susceptible. The decay of immunity in a
previously-infected host is represented by a changing probability of re-infection of that host
when bitten by an infectious mosquito. During an infection, i.e., for h greater than –WN, this
probability is considered 0, consistent with the absence of state change noted above; after an
infection is cleared, i.e., when the host state reaches –WN, this probability exponentially,
asymptotically approaches 1. For h less than –WN, we model the probability of infection as 1–
a im (hewn) ; note that with h always more negative than –WN, the exponent is always negative. The
immune half-life, (ln2)/IM, is the number of days required for this probability to reach 0.5.
When an infectious mosquito is paired with any human, that human’s probability of acquiring
an infection is calculated, as above, and a uniform random number between 0 and 1 is compared
with that probability. If the random number is less than the calculated probability, then the
human becomes infected, setting the host state h to D ; otherwise the host state remains
H
unchanged.
At each time step for each vector, a uniform random number between 0 and 1 is compared with
the probability of death, (1–VS). If the random number is less than that probability, then the
mosquito is removed from the population and replaced with a new mosquito, maintaining the
vector population at constant size. The vector half-life, (ln2)/ (1–VS), is the number of days over
which mortality reduces any given cohort of new mosquitoes by half.
The initial parameters N and N fix the host and vector population sizes, respectively. Initially
H o
no mosquitoes are infected, and an initial fraction of the host population is infected by setting
each h to a random value between D and 0. The initial state of each uninfected host is set to a
H
large negative value, implying no initial immunity. We coded the models in C++ and ran them
on an IBM-compatible PC under Windows 95.

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