In order to examine the relative dynamical consequences of different forms of seasonality, we now focus on wildlife diseases more closely and explore model dynamics using parameters estimated for Rabbit Hemorrhagic Disease in the United Kingdom. In contrast to the above examples of seasonal variation in wildlife populations, we now allow both the transmission rate and the per capita birth rate to be seasonally varying. In addition, following the generally observed behavior of natural populations, we include :
- density-dependent regulation of the hosts (in terms of an increased death rate),
- mortality due to infection,
- density-dependent transmission of infection.
This necessitates us to construct the differential equations in terms of numbers of susceptible and infected hosts, as well as keeping track of the total population size: We note that for the default parameters, Rabbit Hemorrhagic Disease has a devastating impact on the host population density.
Parameters
| β0 |
is the mean transmission rate |
| β1 |
is the amplitude of sinuoidal forcing |
| α0 |
is the mean birth rate |
| α1 |
is the amplitude of sinuoidal forcing for the birth rate |
| ω |
is the frequency of the oscillations. We set ω=2π/365, such that oscillations are annual. |
| 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, and the population level birth rate |
| m |
is the mortality rate due to infection (note, for the default parameters mortality is very high) |
| K |
is the carrying capacity associated with the host populations |
| X(0) |
is the initial number of susceptible hosts (rabbits) |
| Y(0) |
is the initial number infectious hosts (rabbits) |
| N(0) |
is the intital total population size |
All rates are specified in days.
Requirements.
All parameters must be positive, α1, β1 ≤ 1, and X(0)+Y(0) ≤ N(0).
Files
MATLAB Code, Python Program, R Code, C++ Program, Fortran Program, Parameters.
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