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

SIR model with Scaled additive noise (page 197)

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.
In the previous example (program 6.1) noise was only applied to the transmission term and its magnitude was set as an external parameter. We now seek a more realistic formulation by making two changes to this basic formulation:
1) a noise term is included for each of the six possible processes (birth, infections, death of susceptibles, recovery, death of infecteds, death of recovereds).
2) the magnitude of this noise term is a function of the rate of each process. This is implemented by assuming events are Poisson distributed such that the standard deviation of the noise scales with the square-root of the mean.
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.
a set of six noise terms which are 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 death rate.
is the birth rate, which is assumed to be constant and equal to μN; therefore preventing extinction of the host 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.
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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