Markov‐Associated Quadratic Stochastic Families and the Susceptible‐Infected‐Recovered (SIR) Model

Source abstract

ABSTRACT We introduce a Markov‐associated construction for quadratic stochastic families generated by a symmetric cubic stochastic matrix and a finite‐state homogeneous Markov semigroup. The constructed family is not assumed to be a classical quadratic stochastic process of type or type ; rather, it is a Markov‐associated family whose output index is evolved by the given Markov semigroup. In the continuous‐time finite‐dimensional setting, continuity at zero of the semigroup yields a generator. We derive the corresponding differential equations for the constructed coefficients and show that these equations characterize the Markov‐associated construction once the initial coefficients form a symmetric cubic stochastic matrix. We also define the associated averaged stochastic process, prove its semigroup and differential properties, and formulate conditions under which the averaged process reconstructs the original family. Dobrushin ergodicity coefficients are then used to obtain sufficient conditions for weak ergodicity, coefficient‐level convergence, and weak ergodicity of related non‐homogeneous Markov chains. Finally, we apply the construction to a quadratic stochastic operator arising from a discrete SIR model with an arbitrary averaging vector and compute the limiting behavior of the corresponding discrete‐time Markov‐associated family.