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

Partial immunity model that cycles. (page 123)

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.

is the number of strains.
β is the transmission rate (the same for all strains).
γ is the recovery rate (the same for all strains).
is the per capita death rate.
is the modified transmission rate due to partial immunity.
is the localised conferance of immunity. It is assumed that cij=1 if i=j or i and j are neighbours, otherwise cij=0.
Si(0) is the initial proportion of the population that are susceptible to strain i.
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.

All parameters must be positive and it is generally assumed that a<1.

Python ProgramMATLAB Code.

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