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.
Citation: León A. Valencia, Raúl Alejandro Morán-Vásquez, Duván H. Cataño Salazar. Modeling transmission intensity in SI epidemics via CIR and Jacobi processes: Asymptotic results and preliminary intervention strategies[J]. Mathematical Biosciences and Engineering, 2026, 23(7): 1995-2017. doi: 10.3934/mbe.2026073
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.
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León A. Valencia, Raúl Alejandro Morán-Vásquez, Duván H. Cataño Salazar. Modeling transmission intensity in SI epidemics via CIR and Jacobi processes: Asymptotic results and preliminary intervention strategies[J]. Mathematical Biosciences and Engineering, 2026, 23(7): 1995-2017. doi: 10.3934/mbe.2026073
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