Recent work has produced examples where models of the spread of infectious disease with immigration of infected hosts are shown to be globally asymptotically stable through the use of Lyapunov functions. In each case, the Lyapunov function was similar to a Lyapunov function that worked for the corresponding model without immigration of infected hosts.
We distill the calculations from the individual examples into a general result, finding algebraic conditions under which the Lyapunov function for a model without immigration of infected hosts extends to be a valid Lyapunov function for the corresponding system with immigration of infected hosts.
Finally, the method is applied to a multi-group $ SIR $ model.
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