# Fractional epidemic model of coronavirus disease with vaccination and crowding effects

Sci Rep. 2024 Apr 8;14(1):8157. doi: 10.1038/s41598-024-58192-7.

## Abstract

Most of the countries in the world are affected by the coronavirus epidemic that put people in danger, with many infected cases and deaths. The crowding factor plays a significant role in the transmission of coronavirus disease. On the other hand, the vaccines of the covid-19 played a decisive role in the control of coronavirus infection. In this paper, a fractional order epidemic model (SIVR) of coronavirus disease is proposed by considering the effects of crowding and vaccination because the transmission of this infection is highly influenced by these two factors. The nonlinear incidence rate with the inclusion of these effects is a better approach to understand and analyse the dynamics of the model. The positivity and boundedness of the fractional order model is ensured by applying some standard results of Mittag Leffler function and Laplace transformation. The equilibrium points are described analytically. The existence and uniqueness of the non-integer order model is also confirmed by using results of the fixed-point theory. Stability analysis is carried out for the system at both the steady states by using Jacobian matrix theory, Routh-Hurwitz criterion and Volterra-type Lyapunov functions. Basic reproductive number is calculated by using next generation matrix. It is verified that disease-free equilibrium is locally asymptotically stable if ${R}_{0}<1$ and endemic equilibrium is locally asymptotically stable if ${R}_{0}>1$ . Moreover, the disease-free equilibrium is globally asymptotically stable if ${R}_{0}<1$ and endemic equilibrium is globally asymptotically stable if ${R}_{0}>1$ . The non-standard finite difference (NSFD) scheme is developed to approximate the solutions of the system. The simulated graphs are presented to show the key features of the NSFD approach. It is proved that non-standard finite difference approach preserves the positivity and boundedness properties of model. The simulated graphs show that the implementation of control strategies reduced the infected population and increase the recovered population. The impact of fractional order parameter $\alpha$ is described by the graphical templates. The future trends of the virus transmission are predicted under some control measures. The current work will be a value addition in the literature. The article is closed by some useful concluding remarks.

Keywords: Basic reproductive number; Existence and uniqueness; Infection dynamics; Non-linear differential equation; Non-standard finite difference approach; Precautionary measures; SIVR model; Stability.

## MeSH terms

• Basic Reproduction Number
• COVID-19* / epidemiology
• COVID-19* / prevention & control
• Epidemics* / prevention & control
• Humans
• Upper Extremity
• Vaccination