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

SIR model with wildlife vaccination (page 296)

We now consider a second extension to the standard SIR model, introducing random vaccination at a rate v. Obviously vaccination only has an impact on susceptible individuals (vaccination of infecteds and recovereds is ineffective) and moves them into the recovered class. This is an ideal model of the random vaccinaton of wildlife populations where it is impossible to target the vaccination towards susceptible individuals.

To allow a greater appreciation for the impact of vaccination, the model is first integrated until time tV without vaccination (v=0) after which time the vaccination campaign is begun.

is the rate of random vaccination.
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.

Python ProgramMATLAB Code.

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