Evaluating the Impact of Epidemic Control via State-Dependent Markovian Switching Modeling

Stochastic epidemic models with pulse vaccination, varying infectivity and waning immunity

We introduce a fully stochastic, non-Markovian SIRS-type epidemic model that incorporates varying infectivity, waning immunity and a pulse vaccination strategy that may not confer permanent immunity. The model is constructed at the individual level, where each person is characterized by random infectivity and susceptibility functions, and vaccination campaigns occur at the jump times of a Poisson random measure with arbitrary intensity. We rigorously derive the epidemic dynamics as the large-population limit of an interacting stochastic particle system, leading to a system of nonlinear Volterra-type integral equations governing the average susceptibility and total force of infection. We establish a functional law of large numbers(FLLN) for the empirical processes and provide explicit expressions for the limiting compartmental proportions. The long-term behavior of the system is analyzed: we prove that the infection-free solution is globally asymptotically stable when the basic reproduction number falls below a critical threshold, and that the disease persists when this threshold is exceeded. The threshold is given by the harmonic mean of the maximal susceptibility across individuals and generalizes previous results by incorporating vaccination and memory effects. Our framework provides a probabilistically grounded extension of classical deterministic pulse vaccination models and offers new insights into the control of epidemics through scheduled immunization policies.

Thresholds for Stochastic SIRS with Switching Exponents.

This work examines a nonlinear stochastic SIRS epidemic model evolving in a randomly changing environment described by a finite-state Markov chain. The transmission mechanism incorporates regime-dependent nonlinear incidence rates, where the contact interaction between susceptible and infectious individuals is modeled by the term [Formula: see text] . This switching nonlinearity allows the model to capture varying environmental effects and heterogeneous transmission patterns more accurately, thereby providing a more realistic description of epidemic dynamics. To the best of our knowledge, the sufficient criteria governing the persistence and extinction of stochastic SIRS models with transmission rate exponents governed by Markovian switching have not yet been established in the existing literature. The principal contribution of the present study is the derivation of the rigorous sufficient conditions that characterize both the extinction and the long-term persistence of disease dynamics. Specifically, a threshold parameter Λ, expressed in terms of the switching exponents ρξ(t) and ζξ(t), is derived. That is, if Λ > 0, the disease exhibits strong stochastic persistence; conversely, if Λ < 0, the disease-free equilibrium state becomes globally asymptotically stable in probability, leading to eventual disease extinction. In the special case where there is no regime switching and ρξ(t)=ζξ(t)=1, our model recovers the classical threshold found in the literature. To support and validate the theoretical findings, numerical simulations are provided to demonstrate the dynamical behavior of the model under different environmental regimes.

From Individual-Based Stochastic Epidemics to Heterogeneous SIR Equations

We develop a stochastic framework for a broad class of heterogeneous SIR epidemic models. In the finite-population construction, each initially susceptible individual is assigned a fixed nonnegative susceptibility, and infection occurs when the accumulated population-level infection pressure exceeds an individual random threshold. Infectious periods are independent and exponentially distributed with a common recovery rate. For any susceptibility distribution with finite mean, we prove a uniform-on-compact law of large numbers for the susceptible, infectious, removed, and cumulative infection-pressure processes. In the limit, an individual with susceptibility lambda remains susceptible under cumulative pressure x with probability exp(-x lambda). It follows that the susceptible fraction is given by the Laplace transform of the initial susceptibility distribution, while the incidence rate is governed by the mean susceptibility among those who remain susceptible. The resulting limits recover several familiar heterogeneous SIR systems, including the classical power-law model, and also yield other closed nonlinear incidence forms. The framework therefore provides a unified probabilistic foundation for deterministic epidemic models with persistent individual heterogeneity.

Markov SIRD Epidemic Model with Semi-Markov Sojourn-Time Analysis of COVID-19

Infection-age structured transmission model highlights the efficacy of case-finding operations.

Early epidemic control often relies on case-finding operations. To assess their theoretical efficacy, we develop an infection-age structured early growth transmission model that incorporates symptom-based diagnosis, mass screening, and both forward and backward contact tracing. By deriving integral equations for the forward and backward tracing rates and coupling them with a Lotka-Euler-type equation for the early growth rate, we obtain a closed deterministic system. This system enables the computation of early growth rates and reproduction numbers under different case-finding scenarios. Using parameter values calibrated from the 2021 Yangzhou outbreak, the theoretical scenario comparisons show that diagnosis alone provides limited control of early epidemic growth, whereas the integrated case-finding scenario substantially suppresses transmission. The results further show that reducing the effective reproduction number below one requires stronger mass screening and contact tracing when infected individuals generate more secondary infections on average. Together, these findings indicate that the theoretical efficacy of case-finding operations depends jointly on intervention intensity and the effective transmission structure.

Discrete state-structured epidemic model with varying susceptibility and infectivity and its application to the measles outbreak in the United States

Host individuals with vaccinations for infectious diseases often transition among different states of susceptibility and infectivity. Thus, we derive a class of epidemic models in which susceptible and infected individuals have a discrete set of susceptibility and infectivity states, respectively. Our model is based on double-dose vaccination and accounts for all possible state transitions between states. A comprehensive mathematical analysis of the proposed model is conducted, including assessing the control reproduction number and the global dynamical behaviors of the solutions. Using an improved affine-invariant ensemble Markov Chain Monte Carlo method, we also fit the model with measles case data in the United States. The results indicate that despite high vaccination coverage and efficacy rates, sporadic or small-scale measles outbreaks still occur. This suggests that reliance solely on vaccination policies is insufficient to fully interrupt its transmission chain. Consequently, enhancing the diagnosis rate of measles cases is another crucial measure for controlling measles transmission and outbreaks, building upon existing vaccination efforts.