Stochastic Modeling of a Vaccine-Structured Epidemic Model Using Data from South Africa

Abstract

Introduction: Understanding the initial dynamics of an epidemic, especially whether it will establish itself or die out, is critical for public health policy. Deterministic models provide insight into average population behavior but cannot capture the random chance, or demographic stochasticity, that governs the fate of an outbreak when infectious case numbers are low. This is particularly relevant for COVID-19, where population heterogeneity due to vaccination significantly influences transmission. In this paper, we develop and analyze a vaccine-structured epidemic model to quantify the probability of disease extinction and understand how vaccination status impacts these early, uncertain dynamics.Materials and Methods: We formulated a deterministic model using a system of eight ordinary differential equations (ODEs) to represent non-vaccinated and vaccinated populations, incorporating waning immunity. A corresponding Continuous-Time Markov Chain (CTMC) model was developed to capture stochastic effects. The basic reproduction number, R_0, was derived using the next-generation matrix method. We applied multitype branching process theory to analytically calculate the probability of disease extinction (P_0) and used Gillespie's Stochastic Simulation Algorithm to run 10,000 CTMC sample paths to numerically approximate this probability (P_A) and the finite time to extinction. The model was grounded using parameters fitted to COVID-19 data from South Africa during the period from March 5, 2020 to March 21, 2022.Results: The basic reproduction number was calculated as R_0 is approximately equals to 1.41, indicating the potential for sustained transmission. The extinction probability derived from the branching process (P_0) showed excellent agreement with the simulated approximation (P_A). A key finding is that an infection introduced by a vaccinated individual has a significantly higher chance of extinction (P_A approximately equals 0.90-0.93) compared to one from a non-vaccinated individual (P_A approximately equals 0.75-0.78). Furthermore, outbreaks initiated by an infectious vaccinated person that do go extinct resolve the fastest (T approximately equals 35 days), while those from an infectious non-vaccinated person persist the longest (T approximately equals 62 days).Conclusion: This study demonstrates that vaccination provides a dual benefit in containing new disease introductions: it substantially increases the probability of stochastic fade-out and shortens the duration of abortive outbreaks. These findings highlight the limitations of relying solely on deterministic thresholds like R}_0 and underscore the importance of stochastic models in providing a more nuanced risk assessment for public health planning, emphasizing that vaccination is a powerful tool for preventing new sparks from becoming major epidemics.