Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique

• Random nonlinear SI-epidemiological models under general hypotheses are proposed.
• Random Variable Transformation method provides a probabilistic solution of SI-model.
• First probability density function of the solution of random SI-model is determined.
• Time until a certain proportion of susceptibles remain in the population is given.
• Punctual and confidence estimates to model a real problem is shown.

This paper presents a full probabilistic description of the solution of random SI-type epidemiological models which are based on nonlinear differential equations. This description consists of determining: the first probability density function of the solution in terms of the density functions of the diffusion coefficient and the initial condition, which are assumed to be independent random variables; the expectation and variance functions of the solution as well as confidence intervals and, finally, the distribution of time until a given proportion of susceptibles remains in the population. The obtained formulas are general since they are valid regardless the probability distributions assigned to the random inputs. We also present a pair of illustrative examples including in one of them the application of the theoretical results to model the diffusion of a technology using real data.