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| We study the spread of disease in an SIS model of the form
\[ \dot{x}=(-D_{\sigma(t)}+B_{\sigma(t)}-\diag(x)B_{\sigma(t)})x \,.\]
Here $D_i$ is a diagonal matrix, representing the recovery coefficients of the different (groups of) individuals and $B_i$ is a weighted adjecency matrix representing the infection graph. In order to model changing circumstances for the spread of the disease
we have a set of scenarios $\{ D_1,\ldots,D_m \}$, $\{B_1,\ldots,B_m\}$. The time-dependent switching signal $\sigma$ represents the time-varying change of scenario.
The model considered
is a time-varying, switched model, in which the parameters of the SIS model are
subject to abrupt change. We show that the joint spectral radius can be used as a
threshold parameter for this model in the spirit of the basic reproduction number for time-invariant
models. We also present conditions for persistence and the existence of periodic orbits for the switched model and results for a stochastic switched model.
The results extend in a fairly straightforward manner to switched SIR or SIRS models and we briefly comment on this. |
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