Temporal Dynamics and Interplay of Transmission Rate, Vaccination, and Mutation in Epidemic Modeling: A Poisson Point Process Approach

One of the significant challenges when a new virus circulates in a host population is to detect the outbreak as it arises in a timely fashion and implement the appropriate preventive policies to halt the spread of the disease effectively. The conventional computational epidemic models provide a state-space representation of the dynamic changes of various sub-clusters of a society based on their exposure to the virus and are primarily developed for small-size epidemics. In this work, we reformulate the conventional computational epidemic modeling approach inspired by the complex temporal dynamics observed during the COVID-19 pandemic. We utilize the Poisson point process to delineate transitions between various states, enabling us to track the exposed population effectively. The proposed model, based on random event-based Poisson arrivals, offers a comprehensive framework for understanding disease spread when the exposed state is intermediate between susceptibility and infectiousness and delays in implementing mitigation strategies are inevitable. Moreover, our newly proposed framework allows the construction of the transmission probability (p) as a probabilistic function of contributing factors such as virus mutation, immunity waning, and immunity resilience. Our results unravel the interplay between delays, transmission probability, vaccination, virus mutation, immunity loss, and their indirect impacts on the endemic states and waves of the spread. The proposed model provides a mathematical framework that allows policy-makers to improve preparedness for curtailing a lingering infectious disease spreading and unfolds the optimal time frame for vaccination given the available resources and the probability of virus mutation for the current and unforeseen outbreaks.