Modeling transmission intensity in SI epidemics via CIR and Jacobi processes: Asymptotic results and preliminary...

This paper studies an SI epidemic model with stochastic transmission rates of the form $ (\beta_t = \varphi(t)P_t:t\geq0) $, where $ \varphi(t) $ is a deterministic modulation function and $ P_t $ is a positive stochastic process. We show that the asymptotic behavior of the epidemic is determined by the integrated intensity process $ (H_t = \int_0^t \beta_s\, ds:t\geq0) $. We consider two stochastic models for $ (P_t:t\geq0) $: the bounded Jacobi process and the Cox-Ingersoll-Ross (CIR) process. Both preserve positivity, but differ in the support of their sample paths. In the non-modulated regime $ (\varphi\equiv1) $, the CIR framework allows explicit expressions for Laplace transforms and probabilistic bounds associated with the integrated intensity process. Additionally, we present numerical simulations in two regimes: the non-modulated case $ (\varphi(t) = 1) $ and the exponentially damped case $ (\varphi(t) = e^{-\alpha t}) $. The simulations show that the bounded and unbounded structures of the stochastic transmission processes produce different tail behaviors, particularly in high-volatility regimes.