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

SIR metapopulation model for animals (page 241)




Metapopulations are one of the simplest spatial models, but are also one of the most applicable to modeling many human diseases. The metapopulation concept is to subdivide the entire population into distinct “subpopulations”, each of which has independent epidemiological dynamics, together with limited interaction between the subpopulations.
For many animal populations, it is plausible to assume that the spread of disease is due to the migration or permanent movement of individuals. The simplest means of modeling this is to allow animals to randomly move between subpopulations, although other assumptions based on known dispersal behavior of specific species leading to different spatio-temporal dynamics may be more appropriate. The metapopulation SIR-type model is then:

Here, coupling is governed by the parameter mij, which measures the rate at which hosts migrate to subpopulation i from j -- and therefore captures both emigration and immigration.
Note that we are using numbers (X,Y,Z) for greater clarity and assuming density dependent transmission.

Parameters
n
is the number of sub-populations. Note that all parameters are vectors of size n, or matrices of size n × n
βi is the transmission rate for each subpopulation; β is a vector of length n
γi is called the removal or recovery rate for each subpopulation; γ is a vector of length n
νi is the total birth rate for each subpopulation; ν is a vector of length n
μi is the per capita death rate for each subpopulation; μ is a vector of length n
mij is the rate at which hosts migrate to subpopulation i from subpopulation j. m is a matrix of size n × n
Xi(0) is the initial number or density of susceptible individuals in each subpopulation; X(0) is a vector of length n.
Yi(0) is the initial number or density of infectious individuals in each subpopulation; Y(0) is a vector of length n.
All rates are specified in days.

Requirements.
All parameters must be positive.


Files
C++ ProgramPython ProgramParametersMATLAB Code.



Questions and comments to: M.J.Keeling@warwick.ac.uk or rohani@uga.edu
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Matt Keeling      Pejman Rohani