# Parameter Estimation for the One-Term (Multiterm) Fractional-Order SEIAR Models of Norovirus Outbreak

Authors: Tianzeng Li; Yu Wang; Weiqiu Pan
Journal: Advances in Mathematical Physics
 (2021)

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## 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 humansIt 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.


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## Body

## 1. Introduction

Norovirus is one of the most important pathogens of infectious diarrhea and outbreaks of all ages [1–3]. In the United States, Norovirus gives rise to approximately 21 million cases each year [2], 71 thousand hospitalizations [4], and 8 hundred deaths [2, 5]. In developing countries, there are frequent outbreaks of medical institutions and schools [6], which have a great impact on the health of residents. The disease is mainly transmitted by means of the fecal-oral route, and the infection dose is very low. Ingestion of 18 viruses at a time can cause infection [7]. Therefore, it is easy to cause transmission, usually by human contact and water or food spread. School outbreaks can also lead to absenteeism or even suspension of classes, affecting normal teaching order and increasing the burden of family care for children. Therefore, in-depth study of the dynamic characteristics of Norovirus infectious diarrhea outbreaks and the evaluation of the effects of various types of prevention and control measures have important public health significance.These are difficult to analyze through traditional epidemiological methods, and mathematical modeling has become a crucial research tool for studying the dynamics of infectious diseases [8, 9]. Based on the natural history of Norovirus-infected diarrhea, the authors established the integer-order SEIAR model of Norovirus transmission in schools. Taking an outbreak event in a city in 2007 as an example, the dynamic characteristics of Norovirus were studied and the key prevention and control was quantitatively evaluated. But the results indicate that the match between the number of infections derived by the model and the real data needs to be upgraded further to give expression to the spread of Norovirus, especially the transmission speed before the intervention.During the last three decades or so, fractional calculus is the focus of almost all dynamic researchers as we often see it in the fields of chemistry physics [10], engineering [11], biology [12, 13], and image processing [14]. Fractional-order derivatives can be called a superb tool which depicts the memory and hereditary properties of all kinds of materials and processes [14, 15]. Fractional differential equations are more adequate than integer-order derivatives to describe some phenomena associated with nonlocality. Hence, the fractional-order systems are also used to deal with some problems of infectious disease [16–18]. The fractional-order derivative can offer a better infectious disease model compared to the integer-order derivative [19]. There are many advantages of fractional differential equation for describing infectious disease. One is that the fractional derivatives are nonlocal operations, i.e., the fractional-order differential equation can establish a powerful tool for integration of memory and hereditary properties of the systems, while the integer-order derivatives are local operations, i.e., the effects are ignored or tough to incorporate in the integer-order differential equation. Another one is that it is precisely because the fractional-order model has one more degree of freedom compared to the integer order when we fit the model to the real data; the fractional-order model is more suitable for studying infectious diseases.As we know, some parameters are unknown in the fractional-order model, which need to be determined from the real data. Thus, the fractional inverse problem [20] is generated which has aroused worldwide attention on account of the extensive applications [21]. And there are many researches for classical problems [22, 23]. The two-dimensional nonlinear reaction-subdiffusion equation is solved via the novel compact numerical method [24]. Utilizing a Haar wavelet operational matrix, literature [25] considered an identification method aimed at the fractional-order linear system. In [26], aiming at the one-dimensional fractional diffusion equation, the authors discussed the uniqueness result with regard to deciding the fractional order and the space-dependent diffusion coefficient. Literature [27] proposed a feasible parameter estimation method based on hybrid Nelder-Mead simplex search and particle swarm optimization to perform the curve fitting for the multiterm time-fractional Bloch equations. In [10], the authors applied a parameter fitting approach to the classical monoexponential model, a previously developed anomalous relaxation model, and the extended time-fractional relaxation model. They find that the extended time-fractional model is able to fit the experimental data with smaller mean-squared error than the classical monoexponential relaxation model and the anomalous relaxation model. Aside from the parameter estimation method we used in the article, there are also the Bayesian method put forward by Fan et al. [28], the modified quasiboundary value method [29], and so on. Nowadays, coronavirus is a pandemic that has become a concern for the whole world. Scholars at home and abroad predict [30], analyze [31, 32], and intervene [33] the spread of the coronavirus according to the infectious disease models of coronavirus outbreak. It can be seen that the analysis of infectious diseases based on infectious disease models is of great significance for understanding the trend of infectious diseases, reducing the number of infections, and preventing recurrence.In this paper, we mainly use the single-term and multiterm fractional-order SEIAR model to describe the outbreak of Norovirus explained by the Caputo fractional derivatives. These SEIAR models are general models with different fractional orders. For the sake of maintaining the same units on both sides of the models, some new parameters are introduced which have different dimensions. This makes certain that both sides of the models have the same dimension. The GMMP schemes are modified to acquire an implicit difference equation which can be identified as nonlinear equations solved by the Newton method. Thus, the problem of numerical solution is derived. There are some other numerical methods, such as q-Homotopy Analysis Transform Method (q-HATM) [34–37], Fractional Euler Method (FEM) [38], Discretized Collocation Method [39], Variational Iteration Method (VIM) [40], and Fractional Natural Decomposition Method (FNDM) [41]. Then, the corresponding inverse problem of parameter estimation is studied by the MH-NMSS-PSO. 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.The content of this paper is introduced as follows: the integer-order Susceptible-Exposed-Infectious/Asymptomatic-Removed (SEIAR) Model of Norovirus outbreak is introduced in Section2. The fractional derivatives and the single-term fractional-order SEIAR model and its numerical method are introduced in Section 3. In Section 4, the MH-NMSS-PSO algorithm is described. In Section 5, the application of the parameter estimation in the single-term fractional SEIAR model is presented. The multiterm fractional-order SEIAR model is introduced, and the application of the parameter estimation is presented in Section 6. In Section 7, the conclusion is derived.


## 2. The Classical SEIAR Model of Norovirus Outbreak

The SEIAR model is the most common method for studying the dynamic characteristics of some infectious disease, such as Norovirus, influenza, and worm propagation [42, 43]. It is better to divide the total human population N into five subpopulations: St susceptible humans, Et exposed humans (infected but not yet fully contagious), It infected humans, At asymptomatic humans, and Rt recovered (or removed) humans. Hence, the Norovirus classical differential equation is as follows:
(1)dSdt=−βSI+kA,(2)dEdt=βSI+kA−μω′E−1−μωE,(3)dIdt=1−μωE−γI,(4)dAdt=μω′E−γ′A,(5)dRdt=γI+γ′A.In this model (1), the parameters denote different meanings, respectively:
(i)
β is the probability of transmission per contact(ii)
k is the relative transmissibility of asymptomatic to symptomatic individuals(iii)
μ is the proportion of asymptomatic individuals(iv)
1/ω is the incubation period of the disease(v)
1/ω′ is the latent period of the disease(vi)
γ is the removal rate of the infectious individuals(vii)
γ′ is the removal rate of the asymptomatic individualsIn practice, the quarantine measures were applied to all patientsI, that is, all patients I were quarantined when the health department intervened on the first day. And one patient was quarantined in the subsequent treatment, that is, all patients were quarantined once they were found. Because patient I is quarantined, quarantined patients cannot transmit the disease because they are not in contact with susceptible persons. During isolation, the disease progression of the patient was the same as that of the unquarantined, and the removal rate coefficient from the quarantined remained. But people with latent infections can still contract the disease through person-to-person contact. The differential equation of the quarantine patient model is as follows:
(6)dSdt=−βkSA,dEdt=βkSA−μω′E−1−μωE,dIdt=1−μωE−γI,dAdt=μω′E−γ′A,dRdt=γI+γ′A,where the parameters of this model are the same as the model (1).In reference [42], they study the data about a Norovirus infectious diarrhea incident reported in a middle school in a city. The information includes the number of people affected, the onset time of all cases, the intervening time of the department of the centers for disease control and prevention, and the preventive and control measures. The details of the outbreak are as follows: on March 8, the department of the centers for disease control and prevention of a city received a telephone report from a middle school, saying that there were more than ten cases of vomiting, abdominal pain, and diarrhea in the school recently. The following case definitions were established: vomiting or diarrhea and other symptoms such as abdominal pain, fever, headache, and dizziness have occurred among the students and staff of the school since March 5. There are 93 classes in 5 grades in the school, with 5225 students and 430 teachers. The number of cases reached a peak on 8 March. After the intervention on 8 March, isolation measures were taken. The epidemic situation began to decline gradually.The authors use the integer-order SEIAR model to predict the number of the infected people. The parameters were selected asβ=8.3452×10−4, k=3.9065×10−11, ω=1, ω′=1, μ=0.3, γ=0.3333, and γ′=0.03846; the number of infected humans is shown in Figure 1. Based on the results and Figure 1, we find that the root-mean-square error among the numerical solution and the real data is g=7.9138, which identifies that the numerical solutions we obtained should be closer to the real data to reflect the spread of Norovirus, especially the speed of the spread before the intervention. Hence, we should find some new model to make the numerical results fit well with the real data. We could predict the change in the number of people infected more correctly.Figure 1
Quantity of the Norovirus infectionsIt compare with the numerical results of integer-order SEIAR model (1) obtained by the MGAM method and g=7.9138.


## 3. Fractional-Order SEIAR Model of Norovirus Outbreak and Its Numerical Method

### 3.1. The Definitions and Properties of the Fractional Derivatives

Fractional calculus has a great influence on the development of scientific research [14, 44]. The most influential definitions for fractional-order derivatives include Riemann-Liouville (R-L), Caputo, and Grünwald-Letnikov (G-L) definition [15].Definition 1.
The fractional integral is introduced as(7)aDt−αft=1Γα∫att−τα−1fτdτ,where the order is α>0 and Γz=∫0∞tz−1e−tdt is the gamma function.Definition 2 (see [24]).
The expression of Caputo derivative definition is explained in this way:(8)aCDtαft=0Dt−n−αdndtnft=1Γn−α∫att−τn−α−1fnτdτ,where n−1<α<n,n∈Z+.Definition 3.
In 1847, the German mathematician Riemann made further additions on this basis and formed the first more complete definition of fractional calculus—the Riemann-Liouville definition.(9)aRLDtαft=dndtnaDt−n−αft=1Γn−αdndtn∫att−τn−α−1fτdτ,where n−1<α<n,n∈Z+.Definition 4.
The Grünwald-Letnikov derivative is considered as(10)aGLDtαft=limh⟶0mh=th−α∑r=0m−1rαrft−rh=∑k=0n−1fk0tk−αΓk+1−α+1Γn−α·∫att−τn−α−1fnτdτ,where n−1<α<n.
Depending on the above definitions, we know that the G-L and R-L fractional derivatives can be transformed into each other; in other words, they are equivalent while the R-L and Caputo definitions are not, in which the difference can be shown as(11)aCDtαft=aRLDtαft−∑k=0n−1t−ak−αfkaΓk−α+1.If the initial condition is chose asfka=0,k=0,1,⋯,n−1, depending on (11), the Caputo and R-L derivatives are equivalent.Lemma 5.
Assumed functionft∈Cna,T, then
(12)aCDtαft=aRLDtαft−∑k=0n−1t−akfkak!,where n−1<α≤n.Proof.
Suppose(13)ht=ft−∑k=0n−1t−akfkak!,where hka=0,k=0,1,⋯,n−1. On the basis of (11), aCDtαht=aRLDtαht, i.e.,
(14)aCDtαht=aRLDtαht=aRLDtαft−∑k=0n−1t−akfkak!.
We can obtainaCDtαt−ak=0 under the condition of 0≤k<α by the definition of the Caputo derivative. The left side of equation (14) can be presented as
(15)aCDtαht=aCDtαft.
Hence, the result (12) is found.☐


### 3.2. Single-Term Fractional-Order SEIAR Model of the Norovirus Outbreak

On the basis of the interorder SEIAR model, the fractional-order SEIAR model and quarantine SEIAR model are as follows:(16)λα10CDtα1S=−βSI+kA,(17)λα20CDtα2E=βSI+kA−μω′E−1−μωE,(18)λα30CDtα3I=1−μωE−γI,(19)λα40CDtα4A=μω′E−γ′A,(20)λα50CDtα5R=γI+γ′A,(21)λα10CDtα1S=−βkSA,(22)λα20CDtα2E=βkSA−μω′E−1−μωE,(23)λα30CDtα3I=1−μωE−γI,(24)λα40CDtα4A=μω′E−γ′A,(25)λα50CDtα5R=γI+γ′A,where Dtαi0C represents the Caputo fractional derivative with order αi,i=1,2,⋯,5. The parameters β,k,μ, ω,ω′, γ,γ′ are defined in model (1). In these fractional-order models, we can see that the units on the left of system (16) and (21) maintaining the dimensions of days−αi are not same as the right, in which the dimension is days−1. If we take the left-hand side of system (16) and (21) by λαii=1,2,⋯,5 which has the dimension of daysαi−1, then we can have the same units days−1 on both sides of this system.


### 3.3. Numerical Result of the Fractional-Order Differential Equation

There are all kinds of numerical methods which have been confirmed to deal with equations (21), covering the Power Series Method [14], the Predictor-Corrector Method [14, 15, 45], the Mellin Transform Method [14], and others. In this article, we adopted the GMMP scheme [46] and Newton method to carry out the equations (21), which are much more efficient than other numerical methods.Next, we converted equations (21) to the form of
(26)λ⊙aCDtαxt=ft,xt,where xt=St,Et,It,At,RtT and λ=λα1,λα2,λα3,λα4,λα5T. For the sake of establishing the numerical solution of (21), the uniform grids with 0<α<1 are applied to discretize in time, that is, tj=a+jh,j=0,1,2,⋯,N, Nh=t−a. As we know, the R-L and G-L fractional derivatives can be approximated applying this equation:
(27)aRLDtαxt=aGLDtαxt=limh⟶01hα∑k=0NckαxtN−k≈1hα∑k=0NckαxtN−k.Similarly, the Caputo fractional derivatives can be explained as(28)aCDtαxt≈1hα∑k=0NckαxtN−k−∑j=0n−1t−ajxjaj!,where,
(29)ckα=−1kαj,are binomial coefficients.This method appears for the first time in [46] and is widely used by people. In [47], researchers call it the GMMP scheme. Based on it, the numerical solutions of the fractional-order differential equations are revealed. For introducing this method, equation (21) can be written as
(30)λ⊙DtαaCxt=ft,xt,0≤t≤T,xka=x0k,k=0,1,⋯,n−1,where aCDtα represents the Caputo fractional derivative.It follows from (28) that
(31)λ⊙∑k=0NckαxtN−k−∑j=0n−1t−ajxjaj!=hαftN,xtN,i.e.,
(32)xtN=hα⊘λ⊙ftN,xtN+∑j=0n−1t−ajxjaj!−∑k=1NckαxtN−k−∑j=0n−1t−ajxjaj!.In particular, if0<α≤1, (32) should be written as
(33)xtN=hα⊘λ⊙ftN,xtN+xa−∑k=1NckαxtN−k−xa.In line with the Grünwald-Letnikov formula, the solution form (33) of the fractional-order nonlinear equation has been obtained which can be considered as an equation regarding an unknown variable xtN. We choose the Newton algorithm to gain the value of xtN via equation (33) which is expressed as
(34)xn+1=xn−JFxn−1Fxn,n=0,1,2,⋯,where JFxn is the Jacobian matrix at xn. The LU factorization of JFxn can be used to solve the above equations. It is worth mentioning that any other factorization can be presented as well such as QR or Cholesky. Firstly, we should input the initial iterate x0, the nonlinear map F, and termination tolerances ε into the program.Step 1.
Compute and factor the Jacobian matrixJFx0=LU.Step 2.
Solve the linear equationLUs=−Fx0, x=s+x0.Step 3.
Whilex satisfies the condition x−x0>ε, then
(3a) Letx0=x, then factor the Jacobian matrix JFx0=LU
(3b) Solve the linear equationLUs=−Fx0, x=s+x0
(3c) Compute and evaluatex−x0. Ifx−x0>ε, goto step (3a)Then, we can get the outputx=xtN, i.e., the solution of equation (33).


## 3.1. The Definitions and Properties of the Fractional Derivatives

Fractional calculus has a great influence on the development of scientific research [14, 44]. The most influential definitions for fractional-order derivatives include Riemann-Liouville (R-L), Caputo, and Grünwald-Letnikov (G-L) definition [15].Definition 1.
The fractional integral is introduced as(7)aDt−αft=1Γα∫att−τα−1fτdτ,where the order is α>0 and Γz=∫0∞tz−1e−tdt is the gamma function.Definition 2 (see [24]).
The expression of Caputo derivative definition is explained in this way:(8)aCDtαft=0Dt−n−αdndtnft=1Γn−α∫att−τn−α−1fnτdτ,where n−1<α<n,n∈Z+.Definition 3.
In 1847, the German mathematician Riemann made further additions on this basis and formed the first more complete definition of fractional calculus—the Riemann-Liouville definition.(9)aRLDtαft=dndtnaDt−n−αft=1Γn−αdndtn∫att−τn−α−1fτdτ,where n−1<α<n,n∈Z+.Definition 4.
The Grünwald-Letnikov derivative is considered as(10)aGLDtαft=limh⟶0mh=th−α∑r=0m−1rαrft−rh=∑k=0n−1fk0tk−αΓk+1−α+1Γn−α·∫att−τn−α−1fnτdτ,where n−1<α<n.
Depending on the above definitions, we know that the G-L and R-L fractional derivatives can be transformed into each other; in other words, they are equivalent while the R-L and Caputo definitions are not, in which the difference can be shown as(11)aCDtαft=aRLDtαft−∑k=0n−1t−ak−αfkaΓk−α+1.If the initial condition is chose asfka=0,k=0,1,⋯,n−1, depending on (11), the Caputo and R-L derivatives are equivalent.Lemma 5.
Assumed functionft∈Cna,T, then
(12)aCDtαft=aRLDtαft−∑k=0n−1t−akfkak!,where n−1<α≤n.Proof.
Suppose(13)ht=ft−∑k=0n−1t−akfkak!,where hka=0,k=0,1,⋯,n−1. On the basis of (11), aCDtαht=aRLDtαht, i.e.,
(14)aCDtαht=aRLDtαht=aRLDtαft−∑k=0n−1t−akfkak!.
We can obtainaCDtαt−ak=0 under the condition of 0≤k<α by the definition of the Caputo derivative. The left side of equation (14) can be presented as
(15)aCDtαht=aCDtαft.
Hence, the result (12) is found.☐


## 3.2. Single-Term Fractional-Order SEIAR Model of the Norovirus Outbreak

On the basis of the interorder SEIAR model, the fractional-order SEIAR model and quarantine SEIAR model are as follows:(16)λα10CDtα1S=−βSI+kA,(17)λα20CDtα2E=βSI+kA−μω′E−1−μωE,(18)λα30CDtα3I=1−μωE−γI,(19)λα40CDtα4A=μω′E−γ′A,(20)λα50CDtα5R=γI+γ′A,(21)λα10CDtα1S=−βkSA,(22)λα20CDtα2E=βkSA−μω′E−1−μωE,(23)λα30CDtα3I=1−μωE−γI,(24)λα40CDtα4A=μω′E−γ′A,(25)λα50CDtα5R=γI+γ′A,where Dtαi0C represents the Caputo fractional derivative with order αi,i=1,2,⋯,5. The parameters β,k,μ, ω,ω′, γ,γ′ are defined in model (1). In these fractional-order models, we can see that the units on the left of system (16) and (21) maintaining the dimensions of days−αi are not same as the right, in which the dimension is days−1. If we take the left-hand side of system (16) and (21) by λαii=1,2,⋯,5 which has the dimension of daysαi−1, then we can have the same units days−1 on both sides of this system.


## 3.3. Numerical Result of the Fractional-Order Differential Equation

There are all kinds of numerical methods which have been confirmed to deal with equations (21), covering the Power Series Method [14], the Predictor-Corrector Method [14, 15, 45], the Mellin Transform Method [14], and others. In this article, we adopted the GMMP scheme [46] and Newton method to carry out the equations (21), which are much more efficient than other numerical methods.Next, we converted equations (21) to the form of
(26)λ⊙aCDtαxt=ft,xt,where xt=St,Et,It,At,RtT and λ=λα1,λα2,λα3,λα4,λα5T. For the sake of establishing the numerical solution of (21), the uniform grids with 0<α<1 are applied to discretize in time, that is, tj=a+jh,j=0,1,2,⋯,N, Nh=t−a. As we know, the R-L and G-L fractional derivatives can be approximated applying this equation:
(27)aRLDtαxt=aGLDtαxt=limh⟶01hα∑k=0NckαxtN−k≈1hα∑k=0NckαxtN−k.Similarly, the Caputo fractional derivatives can be explained as(28)aCDtαxt≈1hα∑k=0NckαxtN−k−∑j=0n−1t−ajxjaj!,where,
(29)ckα=−1kαj,are binomial coefficients.This method appears for the first time in [46] and is widely used by people. In [47], researchers call it the GMMP scheme. Based on it, the numerical solutions of the fractional-order differential equations are revealed. For introducing this method, equation (21) can be written as
(30)λ⊙DtαaCxt=ft,xt,0≤t≤T,xka=x0k,k=0,1,⋯,n−1,where aCDtα represents the Caputo fractional derivative.It follows from (28) that
(31)λ⊙∑k=0NckαxtN−k−∑j=0n−1t−ajxjaj!=hαftN,xtN,i.e.,
(32)xtN=hα⊘λ⊙ftN,xtN+∑j=0n−1t−ajxjaj!−∑k=1NckαxtN−k−∑j=0n−1t−ajxjaj!.In particular, if0<α≤1, (32) should be written as
(33)xtN=hα⊘λ⊙ftN,xtN+xa−∑k=1NckαxtN−k−xa.In line with the Grünwald-Letnikov formula, the solution form (33) of the fractional-order nonlinear equation has been obtained which can be considered as an equation regarding an unknown variable xtN. We choose the Newton algorithm to gain the value of xtN via equation (33) which is expressed as
(34)xn+1=xn−JFxn−1Fxn,n=0,1,2,⋯,where JFxn is the Jacobian matrix at xn. The LU factorization of JFxn can be used to solve the above equations. It is worth mentioning that any other factorization can be presented as well such as QR or Cholesky. Firstly, we should input the initial iterate x0, the nonlinear map F, and termination tolerances ε into the program.Step 1.
Compute and factor the Jacobian matrixJFx0=LU.Step 2.
Solve the linear equationLUs=−Fx0, x=s+x0.Step 3.
Whilex satisfies the condition x−x0>ε, then
(3a) Letx0=x, then factor the Jacobian matrix JFx0=LU
(3b) Solve the linear equationLUs=−Fx0, x=s+x0
(3c) Compute and evaluatex−x0. Ifx−x0>ε, goto step (3a)Then, we can get the outputx=xtN, i.e., the solution of equation (33).


## 4. The Technique for Obtaining the Parameters of Fractional-Order Nonlinear Systems

In order for the numerical solution presented by the fractional-order model to be closer to the real number of infected people, we must estimate and correct the original parameters. Regarding the unknown parameters, this problem will become more difficult in the dilemma of the bounded parameters and highly nonlinear on functionf. Next, a technique is proposed for estimating parameters. The system (26) can be described as the model.
(35)λ⊙DtαaCxt=ft,xt,P,0≤t≤T,(36)xka=x0k,k=0,1,⋯,n−1,where x=x1,x2,⋯,xnT are state variables, f=f1,f2,⋯,fnT are n-dimensional vector functions, and every fii=1,2,⋯,n represent uncertain parameters P=p1,p2,⋯,pmT and m is the amount of parameters.As we know, many researches adopted the NMSS method [48] and the PSO [49] method to identify parameters. The NMSS concentrates on “exploitation” and PSO concentrates on “exploration.” We can distinguish these two methods according to the following two points. One is how to choose the initial point. The initial point of the former is predetermined while the latter is a collection of random points. Second, it depends on the conditions and direction of the proceeding steps. The former moves to points with better function values while the latter moves from points with the worst performance. The MH-NMSS-PSO method has been applied to estimate parameters for integer and fractional models, which fully combines the merits of the NMSS and PSO methods. The idea of combining the NMSS and PSO methods is applied not only to parameter estimation [50–52] but also to engineering design [53–55], image processing [56], and economic [57].In the following, 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-NMSS-PSO algorithm. Its steps are as follows.We suppose thatP=p1,p2,⋯,pmT∈Ω, where Ω=p1min,p1max×p2min,p2d×⋯×pmmin,pmmax is a bounded domain. Let xtj be one of numerical solutions of equation (35) with the given parameters P=p1,p2,⋯,pm∈Ω (such as It); this result cannot be used as our final data. What we want is the approximation of the unknown parameter vector p1∗,p2∗,⋯,pm∗ determined by the root-mean-square error function (RMSEF):
(37)gP∗=minP∈DgP=minp∈D∑j=0Nxtj−xj2N+1,where xj are real data.A potential global minimumgP∗ in equation (37) with the parameters P∈Ω will be confirmed by the MH-NMSS-PSO method. It initiates with 3m+1 initial particles consisting of two parts. First of all, the predetermined points are derived to organize an initial simplex with m+1 particles which is used in the NMSS method, and the 2m particles are optionally created with the help of the PSO method. Then, we sorted the total 3m+1 particles from smallest to largest via the function values gP in equation (37). In the following, the NMSS method will deal with the best m+1 particles while the task of adjusting the last 2m particles is taken care of by the PSO method.Specifically, it can be divided into the following points.Step 1.
Initialization. Produce a population with size3m+1.
For the minimization of the functionsgP of m variables (unknown parameters), create m+1 vertex points Pi=p1,i,p2,i,⋯,pm,i∈D,i=1,2,⋯,m+1 to form an initial m-dimensional simplex, i.e.,
(38)Pi=p1,i,p2,i,⋯,pm,i∈D,i=1,2,⋯,m+1,where
(39)pj,i=pjmin+i−1×pjmax−pjminm+1,j=1,2,⋯,m,i=1,2,⋯,m+1.
Two particles are randomly generated in each dimension for the PSO part:(40)Pi=p1,i,p2,i,⋯,pm,i∈D,i=m+2,⋯,3m+1,where
(41)pj,i=pjmin+Rand×pjmax−pjmin,j=1,2,⋯,m,i=m+2,⋯,3m+1,and Rand is a random number in the range 0,1. The particle’s initial velocities in each dimension are selected by
(42)Vj,i=Vjmax−VjminLj,j=1,2,⋯,m,i=m+2,⋯,3m+1,where Ljj=1,2,⋯,m are selected integers.Step 2.
Fornum=1:Niter do.
(2a)Evaluation ranking: the function value gP of each particle is different. Rank them by evaluating the objective function values
(43)gP1≤gP2≤⋯≤gP3m+1.
(2b) The NMSS method is applied to the bestm+1 particles and the m+1th particle. Calculate PO, the center of gravity of all points except Pm+1:
(44)PO=p1,O,p2,O,⋯,pm,O∈D,where pj,O=∑i=1mpj,i/m,j=1,2,⋯,m. And the reflected point
(45)Pr=1+αPO−αPm+1,where α is the reflection coefficient (α>0). Nelder and Mead suggest using α=1 [48]. Then, there are three cases in the following:
Case 1:gP1≤gPr≤gPm, then Pr replaces Pm+1.
Case 2:gPr≤gP1, then compute the expanded point.
(46)Pe=γPr+1−γPO.
IfgPe≤gP1, Pe replaces Pm+1; otherwise Pr replaces Pm+1. Nelder and Mead suggest using γ=2 [48].
Case 3:gPr≥gPm, if gPr≤gPm+1, Pr replaces Pm+1. Compute the contracted point:
(47)Pc=βPm+1+1−βPc.
IfgPc≤gPm+1, Pr replaces Pm+1; otherwise, let
(48)Pi=σPi+1−σP1,i=1,2,⋯,m+1.
Nelder and Mead suggest usingβ=0.5 and σ=0.5 [48].
(2c) The PSO method is applied to adjust the last2m particles.
(2d)Discriminate the stopping criterion: if Sc<ε, the loop will stop. Let the criterion be defined by
(49)Sc=∑i=1m+1g¯−gi2m+11/2,where g¯=∑i=1m+1gi∗/m+1 and gi∗=gi=gip1,p2,⋯,pm.Step 3.
Output the best estimated parameter valuesP=p1,p2,⋯,pm.Remark 6.
The Nelder-Mead simplex search (NMSS) is likely to be trapped in local optima rather than global optima and the particle swarm optimization (PSO) algorithm has a slow convergence rate and low balance between exploration and exploitation. Interestingly, the modified hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) algorithm has been demonstrated to outperform both in terms of solution quality and convergence rate. Although the optimal parameters obtained by the new method can provide a solution very close to the real data, it takes a long time to run, and sometimes, the number of iterations is relatively high in order to obtain the optimal parameters.


## 5. The Numerical Simulations of the Fractional-Order SEIAR Model

In this chapter, 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. For exhibiting the parameters of the fractional-order SEIAR model, the real data of a 2007 Norovirus outbreak in a middle school given by [42] is used to carry out the inverse problem of parameter estimation. Each of the parameters in the SEIAR model (16) has its particular biological meaning. Then, each parameter has a corresponding value range. The choice of parameter intervals to narrow the target value is crucial for the result. Hence, based on these ranges, the intervals and initial velocities are chose in this way:
(50)0≤λαi=pi≤2,i=1,2,⋯,5,0≤αi=pi+5≤1,i=1,2,⋯,5,1×10−4≤β=p11≤1×10−3,1×10−11≤k=p12≤1×10−10,0.01≤μ=p13≤0.5,0.5≤ω=p14≤2,0.5≤ω′=p14≤2,0.17≤γ=p15≤1,0.01≤γ′=p16≤0.09,Vi=0.02,i=1,2,⋯,5,0.01,i=6,7,⋯,10,1×10−4,i=11,1×10−11,i=12,0.025,i=13,14,0.01,i=15,16.Based on the potential global minimum ofgP, the unknown parameter P∗ is obtained
(51)λa1=0.35201,λa2=0.36513,λa3=1.99264,λa4=1.27217,λa5=1.04215,a1=0.81292,a2=0.99693,a3=0.99981,a4=0.93916,a5=0.91295,β=8.33565×10−4,k=1.02143×10−11,p=0.21034,ω=2.00000,ω=1.99679,γ=0.50863,γ′=0.08663,with g=3.1922. As seen in Figure 2, both the two methods are capable of effectively solving inverse problems for the fractional-order Norovirus system.Figure 2
Quantity of the Norovirus infectionsIt compared with the numerical results of one-term fractional-order SEIAR model (16) obtained by the MGAM method and g=3.1922.By observing how the parameters influence the variety of infected peopleIt when other parameters are unchanged, we can obtain the conclusion when λα4,λα5,α4,α5,k,γ′ changed; the number of infected people It basically did not change while other parameters have a certain impact on It. And the impacts of other orders and parameters are given in Figures 3and 4, which show that the parameter P∗ we estimated is indeed the ideal parameter.Figure 3
The influences ofλαi,αii=1,2,3 on the quantity of infections It when the remaining parameters remain unchanged.Figure 4
The influences ofβ,μ,ω,ω′,γ on the quantity of infections It when the remaining parameters remain unchanged.


## 6. The Numerical Simulations of the Multiterm Fractional-Order SEIAR Model

### 6.1. The Expression of Numerical Solution

As we know, the multiterm fractional-order differential equations have acted with a significative role in revealing the behaviour of real materials, particularly in the area of the viscoelastic mechanics. In the recent years, some studies have shown that it is more suitable for the study of epidemic models than the integer-order derivative [10, 27]. In the following, the three-term fractional-order SEIAR model and quarantine SEIAR model of the Norovirus outbreak are proposed as follows:
(52)0Dtα1,α2,α0St=−βSI+kA,(53)0Dtα1,α2,α0Et=βSI+kA−μω′E−1−μωE,(54)0Dtα1,α2,α0It=1−μωE−γI,(55)0Dtα1,α2,α0At=μω′E−γ′A,(56)0Dtα1,α2,α0Rt=γI+γ′A,(57)0Dtα1,α2,α0St=−βkSA,(58)0Dtα1,α2,α0Et=βkSA−μω′E−1−μωE,(59)0Dtα1,α2,α0It=1−μωE−γI,(60)0Dtα1,α2,α0At=μω′E−γ′A,(61)0Dtα1,α2,α0Rt=γI+γ′A,where 0Dtα1,α2,α0=λ1·0Dtα1+λ2·0Dtα2+λ0·0Dtα0, and the other parameters in the equation are defined as model (1). However, we can see that the units on the left of equations (52) and (55) maintaining the dimensions of days−αi are not same as those on the right in which the dimensions are days−1. If we take the left-hand side of equations (52) and (55), by λii=0,1,2 in which the dimension is daysαi−1i=0,1,2, then we can have the same unit days−1 to the equations.The GMMP scheme [46, 47] is also adopted to express the numerical solutions of systems (52) and (55). For convenience, the uniform grids tj=jh, j=0,1,2,⋯,n, and nh=t are applied to discretize in time. We converted equations (52) and (55) to the form of
(62)aCDtα1,⋯,αr,α0xt=ft,xt,0≤t≤T,where xt=St,Et,It,At,RtT, and 0<α1<⋯<αr<α0=1.Firstly, the fractional derivative can be discretized by equation (52) as follows:
(63)0CDtαixtn=1hαi∑k=0nckαixtn−k−x0,i=1,2,⋯,r,where
(64)ckαi=−1kαij,are binomial coefficients. When α0=1, we obtain
(65)0CDtα0xtn=dxtndt=xtn−xtn−1h+Oh.From equations (62) and (65), we can obtain that
(66)0CDtα1,⋯,αr,α0xtn=∑i=1rλihαi∑k=0nckαixtn−k−x0+1hxtn−xtn−1=∑k=0n∑i=1rλihαickαixtn−k−x0+1hxtn−xtn−1=∑k=0n∑i=1rλihαickαixtn−k−x0+1hxtn−xtn−1=∑k=0nBknxtn−k−x0+1hxtn−xtn−1,where Bkn=∑i=1rλi/hαickαi.The discrete scheme of equations (52) is given in the following:
(67)0CDtα1,⋯,αr,α0xtn=ftn,xtn.It follows from equation (67) that
(68)∑k=0nBknxtn−k−x0+1hxtn−xtn−1=ftn,utn,i.e.,
(69)xtn=ftn,xtnB0n+h−1+B0nx0+h−1xtn−1B0n+h−1−∑k=1nBknB0n+h−1xtn−k−x0.Since there is an unknown variablextn on both sides of (69), we choose the Newton method to gain the value of xtn via equation (69).


### 6.2. The Numerical Simulations of the Three-Term Fractional-Order SEIAR Model

In the following, we use the above GMMP scheme [46, 47] to exhibit the numerical solution for simulating this SEIAR model (52) and also utilize the MH-NMSS-PSO algorithm to find the ideal fractional orders and parameters, with which the three-term fractional SEIAR model (52) can provide numerical results that fit well with the real data.As the one-term SEIAR model (16), each of the parameters in the multiterm SEIAR model (52) has its particular biological meaning. And each parameter has a corresponding value range. Then, each parameter has a corresponding value range. The choice of parameter intervals to narrow the target value is crucial for the result. Hence, based on these ranges, the intervals and initial velocities are chose in this way:
(70)0≤λ1=p1≤2,0≤λ2=p2≤2,0≤λ0=p3≤2,0≤α1=p4≤1,0≤α2=p5≤2,1×10−4≤β=p6≤1×10−3,1×10−11≤k=p7≤1×10−10,0.01≤μ=p8≤0.5,0.5≤ω=p9≤2,0.5≤ω′=p10≤2,16≤γ=p11≤1,0.01≤γ′=p12≤0.092,Vi=0.01,i=1,2,⋯,5,1×10−4,i=6,1×10−11,i=7,0.02,i=8,0.05,i=9,10,0.02,i=11,0.01,i=12.Based on the potential global minimum ofgP, the unknown parameter P∗ is obtained:
(71)λ1=0.32333,λ2=0.000346,λ0=0.27650,α1=0.98135,α2=0.99672,β=4.3321×10‐4,κ=1.0035×10‐11,μ=0.50000,ω=2.00000,ω=2.00000,γ=0.16667,γ′=0.16667,with g=3.7399. As seen in Figure 5, the three-term fractional SEIAR model (52) can provide numerical results that agree well with the real data.Figure 5
Quantity of the Norovirus infectionsIt compared with the numerical results of the SEIAR model (52) obtained by MGAM method and g=3.7399.By observing how the parameters influence the variety of infected peopleIt when the remaining parameters are unchanged, we can obtain the conclusion when the parameters λ2,α2,k,γ′ changed; the number of infected people It basically did not change while other parameters have a certain impact on It. And the influences of other orders and parameters are given in Figures 6 and 7, which show that the parameter P∗ we estimated is indeed the ideal parameter.Figure 6
The influences ofλ1,λ0,α1,β,μ,ω on the quantity of infections It when the remaining parameters remain unchanged.Figure 7
The influences ofω′,γ on the quantity of infections It when the remaining parameters remain fixed.In the following, we will study the effects of the isolation which is taken on different days. In reference [42], they study the data about a Norovirus infectious diarrhea incident reported in a middle school in a city. On March 8th, the department of the centers for disease control and prevention of a city received a telephone report from a middle school, saying that there were more than ten cases of vomiting, abdominal pain, and diarrhea in the school recently. The following case definitions were established: vomiting or diarrhea and other symptoms such as abdominal pain, fever, headache, and dizziness have occurred among the students and staff of the school since March 5th. The number of cases reached a peak on March 8th. After the intervention on March 8th, isolation measures were taken. The epidemic situation began to decline gradually. In the following, we will assess the effects of the isolation on different days. From the numerical simulation and Figure 8, the conclusion about the effects of isolation can be obtained in the following. When the intervention and isolation measures were taken on March 8th, the number of infected people reached a peak on March 8th. When the isolation measure were taken after March 8th, the number of the infected people would increase more than the real data. In particular, when the isolation measures were taken on March 12th, the changes of the number of the infected people would be similar to those with no isolation measure, which indicates that the intervention and isolation are ineffective for controlling the number of the infected people. These conclusions have practical significance for controlling the spread of the Norovirus. The earlier the intervention and isolation are taken, the less people are infected.Figure 8
Changes in the number of the infectionsIt of the multiterm fractional-order SEIAR model when the intervention and isolation are taken on different days.


## 6.1. The Expression of Numerical Solution

As we know, the multiterm fractional-order differential equations have acted with a significative role in revealing the behaviour of real materials, particularly in the area of the viscoelastic mechanics. In the recent years, some studies have shown that it is more suitable for the study of epidemic models than the integer-order derivative [10, 27]. In the following, the three-term fractional-order SEIAR model and quarantine SEIAR model of the Norovirus outbreak are proposed as follows:
(52)0Dtα1,α2,α0St=−βSI+kA,(53)0Dtα1,α2,α0Et=βSI+kA−μω′E−1−μωE,(54)0Dtα1,α2,α0It=1−μωE−γI,(55)0Dtα1,α2,α0At=μω′E−γ′A,(56)0Dtα1,α2,α0Rt=γI+γ′A,(57)0Dtα1,α2,α0St=−βkSA,(58)0Dtα1,α2,α0Et=βkSA−μω′E−1−μωE,(59)0Dtα1,α2,α0It=1−μωE−γI,(60)0Dtα1,α2,α0At=μω′E−γ′A,(61)0Dtα1,α2,α0Rt=γI+γ′A,where 0Dtα1,α2,α0=λ1·0Dtα1+λ2·0Dtα2+λ0·0Dtα0, and the other parameters in the equation are defined as model (1). However, we can see that the units on the left of equations (52) and (55) maintaining the dimensions of days−αi are not same as those on the right in which the dimensions are days−1. If we take the left-hand side of equations (52) and (55), by λii=0,1,2 in which the dimension is daysαi−1i=0,1,2, then we can have the same unit days−1 to the equations.The GMMP scheme [46, 47] is also adopted to express the numerical solutions of systems (52) and (55). For convenience, the uniform grids tj=jh, j=0,1,2,⋯,n, and nh=t are applied to discretize in time. We converted equations (52) and (55) to the form of
(62)aCDtα1,⋯,αr,α0xt=ft,xt,0≤t≤T,where xt=St,Et,It,At,RtT, and 0<α1<⋯<αr<α0=1.Firstly, the fractional derivative can be discretized by equation (52) as follows:
(63)0CDtαixtn=1hαi∑k=0nckαixtn−k−x0,i=1,2,⋯,r,where
(64)ckαi=−1kαij,are binomial coefficients. When α0=1, we obtain
(65)0CDtα0xtn=dxtndt=xtn−xtn−1h+Oh.From equations (62) and (65), we can obtain that
(66)0CDtα1,⋯,αr,α0xtn=∑i=1rλihαi∑k=0nckαixtn−k−x0+1hxtn−xtn−1=∑k=0n∑i=1rλihαickαixtn−k−x0+1hxtn−xtn−1=∑k=0n∑i=1rλihαickαixtn−k−x0+1hxtn−xtn−1=∑k=0nBknxtn−k−x0+1hxtn−xtn−1,where Bkn=∑i=1rλi/hαickαi.The discrete scheme of equations (52) is given in the following:
(67)0CDtα1,⋯,αr,α0xtn=ftn,xtn.It follows from equation (67) that
(68)∑k=0nBknxtn−k−x0+1hxtn−xtn−1=ftn,utn,i.e.,
(69)xtn=ftn,xtnB0n+h−1+B0nx0+h−1xtn−1B0n+h−1−∑k=1nBknB0n+h−1xtn−k−x0.Since there is an unknown variablextn on both sides of (69), we choose the Newton method to gain the value of xtn via equation (69).


## 6.2. The Numerical Simulations of the Three-Term Fractional-Order SEIAR Model

In the following, we use the above GMMP scheme [46, 47] to exhibit the numerical solution for simulating this SEIAR model (52) and also utilize the MH-NMSS-PSO algorithm to find the ideal fractional orders and parameters, with which the three-term fractional SEIAR model (52) can provide numerical results that fit well with the real data.As the one-term SEIAR model (16), each of the parameters in the multiterm SEIAR model (52) has its particular biological meaning. And each parameter has a corresponding value range. Then, each parameter has a corresponding value range. The choice of parameter intervals to narrow the target value is crucial for the result. Hence, based on these ranges, the intervals and initial velocities are chose in this way:
(70)0≤λ1=p1≤2,0≤λ2=p2≤2,0≤λ0=p3≤2,0≤α1=p4≤1,0≤α2=p5≤2,1×10−4≤β=p6≤1×10−3,1×10−11≤k=p7≤1×10−10,0.01≤μ=p8≤0.5,0.5≤ω=p9≤2,0.5≤ω′=p10≤2,16≤γ=p11≤1,0.01≤γ′=p12≤0.092,Vi=0.01,i=1,2,⋯,5,1×10−4,i=6,1×10−11,i=7,0.02,i=8,0.05,i=9,10,0.02,i=11,0.01,i=12.Based on the potential global minimum ofgP, the unknown parameter P∗ is obtained:
(71)λ1=0.32333,λ2=0.000346,λ0=0.27650,α1=0.98135,α2=0.99672,β=4.3321×10‐4,κ=1.0035×10‐11,μ=0.50000,ω=2.00000,ω=2.00000,γ=0.16667,γ′=0.16667,with g=3.7399. As seen in Figure 5, the three-term fractional SEIAR model (52) can provide numerical results that agree well with the real data.Figure 5
Quantity of the Norovirus infectionsIt compared with the numerical results of the SEIAR model (52) obtained by MGAM method and g=3.7399.By observing how the parameters influence the variety of infected peopleIt when the remaining parameters are unchanged, we can obtain the conclusion when the parameters λ2,α2,k,γ′ changed; the number of infected people It basically did not change while other parameters have a certain impact on It. And the influences of other orders and parameters are given in Figures 6 and 7, which show that the parameter P∗ we estimated is indeed the ideal parameter.Figure 6
The influences ofλ1,λ0,α1,β,μ,ω on the quantity of infections It when the remaining parameters remain unchanged.Figure 7
The influences ofω′,γ on the quantity of infections It when the remaining parameters remain fixed.In the following, we will study the effects of the isolation which is taken on different days. In reference [42], they study the data about a Norovirus infectious diarrhea incident reported in a middle school in a city. On March 8th, the department of the centers for disease control and prevention of a city received a telephone report from a middle school, saying that there were more than ten cases of vomiting, abdominal pain, and diarrhea in the school recently. The following case definitions were established: vomiting or diarrhea and other symptoms such as abdominal pain, fever, headache, and dizziness have occurred among the students and staff of the school since March 5th. The number of cases reached a peak on March 8th. After the intervention on March 8th, isolation measures were taken. The epidemic situation began to decline gradually. In the following, we will assess the effects of the isolation on different days. From the numerical simulation and Figure 8, the conclusion about the effects of isolation can be obtained in the following. When the intervention and isolation measures were taken on March 8th, the number of infected people reached a peak on March 8th. When the isolation measure were taken after March 8th, the number of the infected people would increase more than the real data. In particular, when the isolation measures were taken on March 12th, the changes of the number of the infected people would be similar to those with no isolation measure, which indicates that the intervention and isolation are ineffective for controlling the number of the infected people. These conclusions have practical significance for controlling the spread of the Norovirus. The earlier the intervention and isolation are taken, the less people are infected.Figure 8
Changes in the number of the infectionsIt of the multiterm fractional-order SEIAR model when the intervention and isolation are taken on different days.


## 7. Conclusion

In this research work, we devoted to studying general and multiterm fractional-order SEIAR models considering the Caputo fractional derivative. And we also research the inverse problem of parameter estimation for the fractional-order SEIAR systems. To derive the numerical solutions of the fractional-order SEIAR differential equation, the GMMP scheme and Newton method are introduced. Simultaneously, the way we estimate orders and parameters is by applying the MH-NMSS-PSO algorithm. To guarantee the correctness and effectiveness of the proposed methods, the data of a 2007 Norovirus outbreak in a middle school [42] is used as the real data to conceive the parameter estimation. With the new parameters, all figures demonstrate that the numerical solutions fit very well with the real data, which prove whether the single-term or the multiterm fractional-order SEIAR models can forecast the quantity of the infectious people accurately. And it also verifies the effectiveness of our method for parameter estimation. And the impact of every parameter on the quantity of infected humans It is studied with the remaining parameters unchanged. The result shows that the parameters we got are indeed the most ideal parameters, and the fractional-order SEIAR model we proposed can provide a better fitting among the numerical solutions and the real data than other models.The hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) algorithms are a valid method of estimating the fractional derivative orders and coefficients for fractional differential equations. In the paper [58], they propose a feasible parameter estimation method based on a hybrid Nelder-Mead simplex search and particle swarm optimization to perform the curve fitting for the multiterm time-fractional Bloch equations. From the simulated results, shown in Figure 5 of their paper, the numerical results of the two-term TFBE model (solid line) are in good agreement with the experimental data (asterisk), which shows that the parameter estimation method (MH-NMSS-PSO) is effective. In the paper [10], they apply a parameter fitting approach to the classical monoexponential model, a previously developed anomalous relaxation model, and the extended time-fractional relaxation model. They find that the extended time-fractional model is able to fit the experimental data with smaller mean-squared error than the classical monoexponential relaxation model and the anomalous relaxation model, which is shown in Figure 2. Other papers about the inverse problem also show that the MH-NMSS-PSO algorithm is an efficient and valid method of estimating parameters for fractional differential equations.In our paper, we use the modified hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) algorithm to estimate the parameters for fractional differential equations and the multiterm fractional differential equations. The modified MH-NMSS-PSO algorithm is more efficient than the classical one. With the new parameters, our multiterm fractional-order SEIAR system is capable of providing numerical results that agree very well with the real data. It also demonstrates that our dengue model can predict the number of infected human with other data, and our method can be applied to other fractional-order epidemic models.



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*Source: 5568897-2021-06-29.xml*