Discrete and continuous SIS epidemic models: A unifying approach

Abstract The susceptible-infective-susceptible (SIS) epidemiological scheme is the simplest description of the dynamics of a disease that is contact-transmitted, and that does not lead to immunity. Two by now classical approaches to such a description are: (i) the use of a mass-action compartmental model that leads to a single ordinary differential equation (SIS-ODE); (ii) the use of a discrete-time Markov chain model (SIS-DTMC). While the former can be seen as a mean-field approximation of the latter under certain conditions, it is also known that their dynamics can be significantly different, if the basic reproduction number is greater than one. The goal of this work is to introduce a continuous model, based on a partial differential equation (SIS-PDE), that retains the finite populations effects present in the SIS-DTMC model, and that allows the use of analytical techniques for its study. In particular, it will reduce itself to the SIS-ODE model in many circumstances. This is accomplished by deriving a diffusion-drift approximation to the probability density of the SIS-DTMC model. Such a diffusion is degenerated at the origin, and must conserve probability. These two features then lead to an interesting consequence: the biologically correct solution is a measure solution. We then provide a convenient representation of such a measure solution that allows the use of classical techniques for its computation, and that also provides a tool for obtaining information about several dynamical features of the model. In particular, we show that the SIS-ODE gives the most likely state, conditional on non-absorption. As a further application of such representation, we show how to define the disease-outbreak probability in terms of the SIS-PDE model, and show that this definition can be used both for certain and uncertain initial presence of infected individuals. As a final application, we compute an approximation for the extinction time of the disease. In addition, we present many numerical examples that confirm the good approximation of the SIS-DTMC by the SIS-PDE.

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