A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique

• Random nonlinear SIS-epidemiological models under general hypotheses are proposed.
• Random Variable Transformation method provides probabilistic solution of SIS-model.
• First probability density of the solution of random SIS-model is determined.
• Time until a certain proportion of susceptibles remains in the population is given.
• A probabilistic meaning of the reproductive number R0 is provided.

This paper provides a complete probabilistic description of SIS-type epidemiological models where all the input parameters (contagion rate, recovery rate and initial conditions) are assumed to be random variables. By applying the Random Variable Transformation technique, the first probability density function, the mean and the variance functions, as well as confidence intervals associated with the solution of SIS-type epidemiological models, are determined. It is done under the general hypothesis that model random inputs have any joint probability density function. The distributions to describe the time until a given proportion of the population remains susceptible and infected are also determined. Finally, a probabilistic description of the so-called basic reproductive number is included. The theoretical results are applied to an illustrative example showing good fitting.