Abstract
In the paper, we use the Caputo fractional derivative to consider general single-term and multiterm fractional-order SEIAR models for the outbreak of Norovirus. Then, the inverse problem about parameter estimation for these fractional-order SEIAR models of the Norovirus outbreak is studied firstly. To provide the numerical solution of the single-term (or multiterm) fractional-order nonlinear differential equation, the GMMP scheme and Newton method are introduced. Then, we make use of the modified hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) algorithm to obtain the fractional orders and parameters for these fractional-order SEIAR models of Norovirus outbreak. To guarantee the correctness and effectiveness of the methods, the data of a 2007 Norovirus outbreak in a middle school in one city is used as the real data to solve the inverse problem of the parameter estimation. With the new parameters, all numerical studies illustrate that the numerical solutions fit very well with the real data, which reveals that the single-term and multiterm fractional-order SEIAR models of Norovirus outbreak all can predict the number of the infectious people accurately. And it also shows that the GMMP scheme and the MH-NMSS-PSO method are efficient and valid for estimating the parameters of the single-term (or multiterm) fractional-order nonlinear equations. Then, we research the impact of changes in each parameter on the amount of infected humans I t when the remaining parameters are unchanged. All results of numerical simulation reveal that the single-term and multiterm fractional-order SEIAR model of Norovirus can provide better results than other models. And we also study the effect of the isolation on different days. The conclusion is obtained that the earlier the isolation is taken, the less the infected people are. Hence, for a fractional-order application in the SEIAR model of Norovirus outbreak, we establish the effective parameter estimation methods.
Highlights
Norovirus is one of the most important pathogens of infectious diarrhea and outbreaks of all ages [1,2,3]
In this paper, we need to find the optimal parameters to make the numerical solution of the fractional-order Norovirus system as close as possible to the number of people infected with Norovirus adopting the MH-Nelder-Mead simplex search (NMSS)-particle swarm optimization (PSO) algorithm
We will make use of the MH-NMSS-PSO algorithm to obtain the ideal parameters to make the numerical results of the Norovirus system (16) as close as possible to the number of people infected with Norovirus
Summary
Norovirus is one of the most important pathogens of infectious diarrhea and outbreaks of all ages [1,2,3]. We mainly use the single-term and multiterm fractional-order SEIAR model to describe the outbreak of Norovirus explained by the Caputo fractional derivatives. Using the statistics from the Norovirus outbreak in a middle school in 2007 [42], the fractional-order SEIAR model we proposed can be determined Based on these new parameters and orders, the numerical results provided by two fractional-order SEIAR models are very closer to the real data.
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