Abstract

EULER ITERATIVE METHOD FOR SOLVING SEIRV MODEL EQUATIONS FOR HUMAN INFECTIOUS DISEASES

Mohemid Maddallah Al-Jebouri*, Mohammed Nokhas Murad Kaki

ABSTRACT

Background: The SEIRV model introduces an additional compartment—vaccinated individuals—allowing for a more realistic and policy-relevant analysis of disease spread and control strategies. Investigations have been focused on the numerical simulation of the SEIRV model using the Euler method, a straightforward yet effective approach for approximating solutions to differential equations. The present study is an attempt to investigate the existence and uniqueness of disease-free and endemic equilibrium points in the SEIRV model. Methodology: This study employs a matrix-based formulation of the SEIRV model alongside Euler’s method to compute numerical solutions. The approach allows for comparison with alternative numerical techniques. Special emphasis is placed on the use of Euler’s method for small time-step intervals, highlighting its computational simplicity and effectiveness in short-term simulations. Further analysis explores the implications of step size on accuracy and model behavior, supporting the method’s practical relevance in epidemic modeling. This study presents a numerical analysis of an extended SEIR model that incorporates vaccination, known as the SEIRV model. Using the Euler method, we simulate the temporal dynamics of a closed population exposed to an infectious disease under specific epidemiological parameters, including infection rate (β =0.03), incubation rate (σ = 0.2), recovery rate (γ = 0.1), vaccination rate (ν = 0.5), and natural death rate (μ = 0.1). Initial conditions are set for a population of 1000 individuals distributed across Susceptible, Exposed, Infectious, Recovered, and Vaccinated compartments. Results: The results demonstrate a rapid decline in the susceptible population and a substantial increase in the vaccinated group within a few time steps, reflecting the significant effect of early vaccination. Both the exposed and infectious compartments show a steady decrease, suggesting that high vaccination coverage can effectively drive the system toward disease elimination. These findings align with established theoretical expectations and support the importance of vaccination in epidemic control. Conclusions: This study highlights the practical utility of the SEIRV framework in modeling and analyzing infectious disease dynamics, particularly in scenarios where vaccination plays a central role in public health strategy. By extending the classical SEIR model to include a vaccinated compartment, the SEIRV model offers a more comprehensive and policy-relevant representation of disease transmission and control efforts.

Keywords: SEIRV model, Infectious disease modeling, Epidemiological simulation, Vaccination dynamics, Numerical analysis.