Modeling Infectious Diseases in Humans and Animals
Matt J. Keeling & Pejman Rohani

SIR model with pulsed vaccination (page 302)

An alternative approach that has been tried in a variety of locations is pulse vaccination, where children in certain age cohorts are periodically immunized. The principle aim of pulse vaccination is to ensure the susceptible fraction is maintained below this level by periodically immunizing a fraction of the susceptible population. Pulse vaccination has gained in prominence as a result of its highly successful application in the field. Compared to “continual” pediatric vaccination, it has the additional advantage that it is often logistically simpler to implement. A well-publicized example is the spectacular control of poliomyelitis and measles in Central and South America.
Here we assume that a proportion pV of susceptible individuals are vaccinated every T time units (days):

Again we allow for a delay, such that vaccination is begun after time tV.

is the proportion of the susceptible individuals that are vaccinated with each pulse.
is the time between vaccination pulses.
β is the transmission rate and incorporates the encounter rate between susceptible and infectious individuals together with the probability of transmission.
γ is 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 constant
is the per captia death rate.
tV is the time at which the vaccination program is begun.
S(0) is the initial proportion of the population that are susceptible.
I(0) is the initial proportion of the population that are infectious.
All rates are specified in days.

All parameters must be positive, S(0)+I(0) ≤ 1 and 0 ≤ pV ≤ 1.

Python ProgramMATLAB Code.

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