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

SIR model with Constant additive noise (page 194)

An intuitive way to incorporate noise is to introduce it directly into the deterministic equations. As such, the dynamics at each point in time are subject to some random variability and this variability is propagated forward in time by the underlying equations. We are, therefore, concerned with the interplay between deterministic and stochastic forces—how they cancel out or amplify each other.
As a starting point, let us incorporate noise into only the transmission term. Let ξ(t) be a time series of random deviates derived from the normal distribution with mean zero and unit variance (see Box 6.2). The basic equations, assuming frequency-dependent (mass-action) transmission, are transformed to:

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.
amount of noise experienced in the transmission term.
noise term which is generated as a function of the time step.
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 birth and death rate.
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.
All rates are specified in days.

All parameters must be positive. Remember, X, Y and N all refer to numbers.
A time step δt also has to be defined and this sets both the integration step and scales the noise term ξ.

Python ProgramMATLAB Code.

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