Page 264 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes is greater than that of a or n, which are in turn greater than that of b, m, or r, and hence that vector
survivorship is the single most important element in the basic reproduction rate of malaria.
Macdonald’s “affected” proportions do not distinguish between infected and infectious states,
but his conclusion with respect to host infectivity was that: “Transmission can be altered by
reduction of the mean period of infectivity of a case of malaria. The influence is, however,
relatively small; the reproduction rate varies directly with the mean duration of infectivity,
very great changes in which would be necessary to reduce the high’ rates common in Africa and
some other places below the critical level.”
In terms of our model the “basic reproduction rate” is:

Z = – k[V /(N N )][VS /In(VS)] = kC
2
DV
0 B H V
where “k” is an equivalent to the ratio b/r (see below).
Our parameters VS and D correspond directly to Macdonald’s “p” and “n,” and the ratios of our
V
initial parameters N /N and V /N translate his “m” and “a,” respectively (substituting “bites”
V H B V
2
2
for “men bitten”), such that for our model “ma ” = V /(N N ). Macdonald’s “b” is a measure of
B H V
incidence (e.g. by its role in expressions for “inoculation rate” and “force of infection”), and “r”
the reciprocal of the average duration of the “affected” state. Macdonald wrote that “in nature
the value of the reproduction rate is greatly influenced by immunity altering the values of r and
b,” and in our model these proportions actually do vary dynamically with distributions of host
immune states and infection histories, in a convolved, partly stochastic manner.
Therefore it is difficult to interpret “b” and “r” in terms of our model, particularly in terms of
our parameters WN and IM. However, the C values considered here range from 3.7 to 29.2, in
accord with field estimates, but see, so if we consider 1/(D +WN) roughly equivalent to r, then
H
k ranges from 30b to 50b. Even if we consider the b-equivalent values in our model as ranging
from 0.1 to 1 (k values from 3 to 50), field estimates span the resulting range of Z values.
0

Did u know? What is equilibrium?
Equilibrium is the condition of a system in which competing influences are balanced.

Discussion

There are many useful approaches to modeling human malaria, and many differences among
models constructed for different purposes, but some forms that are analogous are not equivalent:
analogous classic differential-equation and differential-delay-equation models have different
properties, and each has properties very different from those of the discrete-event models
developed here, including different dynamics leading to the same equilibrium.
Among the factors in malaria epidemiology most difficult to represent in compartment models
is the immunity of individuals. In the discrete-event simulation models, complex population-
level dynamics emerge from a simple representation of individual-level malaria infections.
Accordingly, we can readily represent probabilities that a human becomes infected if bitten by
an infectious mosquito, even if those probabilities depend on that host’s prior infection history
and waning immunity to reinfection. We can represent the decay of immunity within an
individual to one “strain,” even if that decay depends on the interval since that individual
cleared another “strain,” and so forth. The decay of immunity within an individual may have a
very complex relation to the decay of the immune component of a population, and in fact the
time scales involved at the individual and the population levels may differ by orders of magnitude.

Any individual character that changes based on calculations with respect to that individual can
be represented by aggregates in a priori population-level models only if the aggregates either

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