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

SIR model with two types of imports (page 210)

We now utilise the approach given in programs 6.3 and 6.4 to implement event-driven stochasticity into the standard SIR equations when there is the potential for stochastic imports of infection. Although disease incidence in real populations may frequently undergo stochastically driven fade-outs, imports of pathogen from outside the population can prevent permanent extinction. In human populations, such imports usually take the form of visitors who remain for a limited period but, in doing so, re-introduce the pathogen. Here we model imports in two distinct ways:
1) Imports due to an external force of infection, which occur at rate εX,
2) Imports due to an infected individual moving into the population, which occur at rate δ.

Once again we assume that the population size N is constant which prevents the permanent extinction of the host population.
Note that we are using numbers (X,Y,Z) throughout this chapter for greater clarity.

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.
μ is the per capita death rate.
ε is the import rate due to an external force of infection
δ is the import rate due to infectious individuals immigrating into the population
X(0) is the initial number or density of susceptible individuals.
Y(0) is the initial number or density of infectious individuals.
is the population size -- assumed to be constant. We assume Z(0)=N-X(0)-Y(0)
All rates are specified in days.

All parameters must be positive. Remember, X, Y, Z and N all refer to integer numbers.

Python ProgramMATLAB Code.

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