Page 263 - DCAP601_SIMULATION_AND_MODELING

Unit 13: Simulation Languages (I)

rate in an exponential decay process. As above, assuming the system at equilbrium, we set p = k Notes
= (ln2)/WN = (ln2)/D , and c = (ln2)/D . With respect to vector mortality, because we assume
H V
that each mosquito has the same probability of dying each day we do not need to know the
distribution, and we take this directly as the aggregate, exponential decay, i.e., d = (1–VS).
The second, differential-delay form replaces each of the parameters k, p and c with an explicit
time lag corresponding to the host delay, the host window and the vector delay parameters in
our simulation model, respectively, such that:
dS/dt = qR – hFS,

dM/dt = hFS – [hFS]
t–k
dG/dt = [hFS] – [hFS]
t–k t–(k+p)
dR/dt = [hFS] – qR,
t–(k+p)
dU/dt = b – hGU – dU,
–cd
dL/dt = hGU – [1 – e ][hGU] – dL, and
t–c
dF/dt = [1 – e ][hGU] – dF,
–cd
t–c
where the dynamic variables and other parameters are as above.
Here the system is underdetermined, i.e., no equilibrium can be calculated, and its dynamics
depend on the initial conditions.
We obtained numerical approximations to solutions (Figure 13.9) by translating each of the
models into a BASIC program, typically setting 25% of the hosts and none of the vectors initially
infected, as in the discrete-event simulation model.

Macdonald’s model can be expressed by the equations:
1. dX/dt = abm Y – X(abm Y + r), and
2. dY/dt = aX – Y(aX – Inp),
where the dynamic variable X represents, according to Macdonald, “the proportion of people
affected,” the dynamic variable Y its (implicit) counterpart in the vector population, and the
parameters as follows, also quoted from Macdonald:

m the anopheline density in relation to man,
a the average number of men bitten by one mosquito in one day,
b the proportion of those anophelines with sporozoites in their glands which are actually infective,
p the probability of a mosquito surviving through one whole day, and
the proportion of affected people, who have received one infective inoculum only, who revert to
r
the unaffected state in one day.

The crucial aspects of Macdonald’s model are summarized in his formula for Z , the “basic
0
reproduction rate” of malaria:
2
n
Z = – (ma b)p /[r(lnp)] = b/r)C,
0
where the parameter n represents “the time taken for completion of the extrinsic cycle,” and C
{= “(ma )p /(Inp)} summarizes the “vectorial capacity” of malaria.
n
2
Macdonald derived Z as an estimate of the average number of secondary cases arising in a very
0
large population of completely susceptible humans following the introduction of a single primary
case, and Z = 1 as the transmission threshold, i.e., the value above which cases propagate and
0
below which they recede. This formula for Z holds that the influence of vector survivorship, p,
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