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

SIR model with paediatric vaccination (page 293)

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.

is the proportion of new-borne individuals who are vaccinated.
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 ≤ p ≤ 1.

Python ProgramMATLAB Code.

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