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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.
n
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is the number of strains.
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| β |
is the transmission
rate (the same for all strains).
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| γ |
is the
recovery rate (the same for all strains).
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μ
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is the per capita death rate.
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a
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is the modified
transmission
rate due to partial immunity.
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cij
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is
the localised conferance of immunity. It is assumed that cij=1
if i=j or i and j are neighbours, otherwise cij=0.
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| Si(0) |
is
the initial
proportion of the population that are susceptible to strain i.
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| Pi(0) |
is
the initial
proportion of the population that are partially immune to strain i. |
| λi(0) |
is
the initial force of infection due to strain i. |
All rates are
specified
in days.
Requirements.
All parameters must be positive and it is generally assumed that a<1.
Files
Python Program, MATLAB Code.
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