Simulating the COVID-19 Epidemic: A Numerical Examination of SIR, SIRID, and SIRVI Models

Abstract

A comprehensive understanding of the propagation dynamics of COVID-19 is the paramount goal of this study. Two innovative mathematical models, namely the susceptible, infected, recovered, infected, dead (SIRID) model and the susceptible, infected, recovered, vaccinated, infected (SIRVI) model, are introduced. These models extend the conventional susceptible, infected, recovered (SIR) model by contemplating two pivotal factors: reinfections and the impact of vaccination. The SIRID model encapsulates the potential for a previously infected and recovered population to experience a secondary infection leading to death. The model forecasts crucial phases in this intricate progression: initial infection, recovery, reinfection, and subsequent fatality. Reinfections are underscored as a potentially significant driver of mortality in the SIRID model. The SIRVI model, however, integrates vaccination into the analysis, evaluating how immunization may modulate the virus's spread amid post-vaccination reinfections. In essence, the SIRVI model estimates the key stages: initial infection, recovery, vaccination, and reinfection notwithstanding immunization. This model underscores the potential for vaccination to mitigate the pandemic's severity, while also highlighting the ongoing challenges associated with reinfections. The methodologies employed to construct the SIR, SIRID, and SIRVI models stem from an adaptation of the classic SIR model to incorporate reinfections and vaccination. Each model was built using a comparable approach, albeit with additional compartments to capture the intricate interplay among various pandemic dynamics. The models' compartments (S, I, R, etc.) represent distinct population states based on disease status. The transitions between compartments illustrate the flux of individuals from one state to another. For the SIRID and SIRVI models, an innovative approach was adopted: every compartment accounts for incoming and outgoing fluxes as additions and subtractions, respectively. This allows infections, recoveries, reinfections, and deaths to be represented as dynamic variables, each with specific equations. The interactions between compartments were regulated according to inflows and outflows, capturing the complexity of viral spread, potential reinfections, and vaccination impact. Once the equations were formulated, numerical methods were employed to solve these differential equations. The model parameters were adjusted to align with real-world pandemic data, and iterations were conducted to observe various possible scenarios. This permitted detailed predictions about the pandemic's progression, considering potential reinfections and vaccination, thereby providing valuable insights for public health decision-making.