In this chapter we are primarily
concerned with the basic concept of control; how can control measures
be efficiently applied so as to minimise prevelance, minimise the
incidence of disease or even to erradicate infection all together.
We begin with a simple extension to the standard SIR model, including
prophylactic vaccination of new borne individuals. This form of control
simply 'moves' a proportion p
of new-bornes into the recovered (vaccinated) class rather than into
the susceptible class:
To allow a greater appreciation for
the impact of vaccination, the model is first integrated until time tV without vaccination (p=0) after which time the
vaccination campaign is begun.
All rates are
the proportion of new-borne individuals who are vaccinated.
|| is the transmission
rate and incorporates the encounter rate between susceptible and
infectious individuals together with the probability of transmission.
called the removal
or recovery rate, though often we are more interested in its reciprocal
(1/γ) which determines the average infectious period.
|is the over-all birth set. We
set ν=μ to keep the population size
|is the per captia death rate.
||is the time at which the
vaccination program is begun.
proportion of the population that are susceptible.
proportion of the population that are infectious.
All parameters must be positive, S(0)+I(0) ≤ 1 and 0 ≤ p ≤ 1.
Python Program, MATLAB Code.