Review of fractional epidemic models

Abstract

The global impact of corona virus (COVID-19) has been profound, and the public health threat it represents is the most serious seen in a respiratory virus since the 1918 influenza A(H1N1) pandemic. In this paper, we have focused on reviewing the results of epidemiological modelling especially the fractional epidemic model and summarized different types of fractional epidemic models including fractional Susceptible-Infective-Recovered (SIR), Susceptible-Exposed-Infective-Recovered (SEIR), Susceptible-Exposed-Infective-Asymptomatic-Recovered (SEIAR) models and so on. Furthermore, we propose a general fractional SEIAR model in the case of single-term and multi-term fractional differential equations. A feasible and reliable parameter estimation method based on modified hybrid Nelder-Mead simplex search and particle swarm optimisation is also presented to fit the real data using fractional SEIAR model. The effective methods to solve the fractional epidemic models we introduced construct a simple and effective analytical technique that can be easily extended and applied to other fractional models, and can help guide the concerned bodies in preventing or controlling, even predicting the infectious disease outbreaks.

Keywords: 26A33; 37M05; 37N30; 97M60; Epidemic models; Fractional order differential equations; Hybrid simplex search and particle swarm optimisation; Implicit numerical method; Multi-term epidemic models; Parameter estimation.

Fig. 1

Number of infected humans $I\left(t\right)$ in a middle school in the 2007: Comparison of numerical results of one-term fractional SEIAR model with the real data with the estimated parameters obtained by MH-NMSS-PSO method. The root-mean-square error is rMSE $=4.0792$.

Fig. 2

Number of infected humans $I\left(t\right)$ in a middle school in the 2007: Comparison of numerical results of three-term fractional SEIAR model with the real data with the estimated parameters obtained by MH-NMSS-PSO method. The root-mean-square error is rMSE $=2.6041$.