One interesting facet of multi-strain
models is their potential to sustain large-amplitude complex
oscillations. Here we investigate one of the simplest in which
oscillations can persist with just four strains arranged in a circle.
It is assumed that infection (and recovery) from strain i provides partial immunity to
neighbouring strains, where partial immunity is modelled as a decrease
in transmission but no change in susceptiblity. This leads to the
following general equations:
where S, P and R refer to all individuals that are
susceptible to, partially susceptible to, or recovered (or infected
with) a given strain. We note that in this formulation Si+Pi+Ri=1
but Σi Si
etc could be greater than one. In this simplest of formultions it is
also assumed that all strains are identical and so have the same
transmission and recovery rates.
All rates are
|is the number of strains.
is the transmission
rate (the same for all strains).
recovery rate (the same for all strains).
|is the per capita death rate.
|is the modified
rate due to partial immunity.
the localised conferance of immunity. It is assumed that cij=1
if i=j or i and j are neighbours, otherwise cij=0.
proportion of the population that are susceptible to strain i.
proportion of the population that are partially immune to strain i.
the initial force of infection due to strain i.
All parameters must be positive and it is generally assumed that a<1.
Python Program, MATLAB Code.