A large number of species experience
seasonal forcing, either due to host aggregating a certain times of the
year, or due to transmission rates being influenced by climatic
conditions. We begin to analyse such seasonally forced systems,
by making the transmission rate β
vary sinusoidally about its mean value. This system of equations was
first studied by Klaus Dietz in 1976, when he showed the complex
dynamics that could arise from including a small amount of sinusoidal
forcing. The basic equations are as follows:
It should be noted that large values
of β1 can lead to
large-amplitude fluctuations and numerical errors.
All rates are
the mean transmission
the amplitude of sinuoidal forcing
the frequency of the oscillations. We set ω=2π/365, such that
oscillations are annual.
the per capita death
rate, and the population level birth rate.
called the removal
or recovery rate, though often we are more interested in its reciprocal
(1/γ) which determines the average infectious period.
proportion of the population that are susceptible.
proportion of the population that are infectious.
The programs can return either standard time-series, or bifurcation
plots. Bifurcation plots are achieved by setting β1 to be a vector in the
Matlab code, or by setting Num_Bif_Steps in the parameter file for the
C and Fortran code.
All parameters must be positive, β1
≤ 1, and S(0)+I(0) ≤ 1
C++ Program, Python Program, Fortran Program, Parameters, MATLAB Code.