1 Introduction

As we all know, mathematical models play an important role in researching the dynamical behavior of infectious diseases. In the classical epidemic model, it is generally considered that individuals are completely mixed, and everyone has the same possibility of infection. However, due to the differences in age, geographical distribution, and other factors, it is more realistic to divide the total population into several different populations, that is, to establish a multigroup epidemic model. Lajmanovich et al. first proposed the SIS multigroup systems and researched the stability of the endemic equilibrium point [11]. Subsequently, there are many research efforts devoted to investigating the importance of multigroup epidemic models [6, 8, 14]. Guo et al. were the first to successfully establish the complete global dynamics of the multigroup epidemic model based on the basic reproduction number [7]. Boosted by the work of Guo et al., many researchers discussed the stability of various multigroup systems [3, 15, 20, 21, 25].

Meanwhile, individual diffusion behavior is widespread in the actual propagation of infectious diseases. With the development of global transportation, individuals in incu- bation period can easily travel from one place to another, which is thought to be one of the main reasons of the global pandemic of infectious diseases. For instance, SARS first appeared in China’s Guangdong Province in November 2002 and then quickly spread to other parts of China and even the world [26]. Also, COVID-19 was first detected at the end of December 2019 with successive cases occurring worldwide. Therefore, in order to better understand the impact of population mobility on the spread of infectious diseases, it is necessary to incorporate human movement into epidemic model to provide more theoretical guidance for epidemic control. Li et al. analyzed the stability and the uniform persistence of a SIRS epidemic model with diffusion [12]. Xu et al. studied the stability and the existence of traveling wave solutions of a SIS epidemic model with diffusion [28]. Recently, many diffusive epidemic models have been used to model within-group and inter-group interactions in spatially environments, for example, Wu et al. investigated a multigroup epidemic model with nonlocal diffusion and obtained the asymptotic behav- ior of traveling wave fronts [27].

It is worth noting in real life that the spread of infectious diseases not only depends on its current state, but also on its past state. Actually, it can be achieved that current state of fractional-order epidemic models depends on the past information since any fractional derivative contains a kernel function [30]. Furthermore, Smethurst et al. found that the patient waits for the doctor’s time to follow a power law model [24]. More importantly, Angstmann et al. proposed a infectivity SIR model with fractional- order derivative, and they showed how fractional-order derivative arise naturally by con- tinuous time random walk [2]. As generalized of classical integers ones, Hethcote firstly proposed a fractional-order SIR model with a constant population [8]. Then Almeida et al. considered the local stability of two equilibrium points of a fractional SEIR epidemic model [1].

Typically, the reaction term describes a birth-death reaction occurring in a habitat or reactor. The diffusion term simulates the movement of the individual in the environment in real-world applications. The diffusion is often described by a power law where is the diffusion coefficient, and is the elapsed time. In normal diffusion, the order But if particle undergoes superdiffusion, which mainly describes the process of active cell transport; if , this phenomenon is called subdiffusion, which can be the diffusion of proteins within cells or the diffusion of viruses between individuals [29]. And it results in a Caputo time-fractional reaction– diffusion system with fractional order . Meanwhile, it is pointed out in [19] that long waiting times model particle sticking, and the density of this process spreads slower than normal diffusion. Also, as shown in [19], Caputo time-fractional reaction– diffusion curve has a sharper peak and heavier tails, which can be used to describe the ability to control the transmission of the disease when only a small number of people are infected, such as COVID-19. The study of subdiffusion system has attracted widespread interest in recent years. Mahmoud et al. studied the Cauchy problem of the fractional- order evolution equation and obtained the expression of the solution of the time fractional- order reaction diffusion system [18]. The subdiffusive predator–prey system is discussed, and the analytical solution of the system is studied in [29]. However, few works have been devoted to studying the subdiffusion epidemic model. Motivated by this, in this work, we focus on time-fractional reaction–diffusion epidemic system, which means the spread of infectious diseases is slower than a Brown motion.

Based on the above discussion, the dynamics of the multigroup SIR epidemic model with generally incidence rates is investigated in this paper. Particularly, the susceptible individuals, infective individuals, and recovered individuals are assumed to follow Fickian diffusion.

The organization of this paper is as follows. A class of diffusive SIR epidemic model with time fractional-order derivatives is formulated and some preliminaries are introduced in Section 2. In Section 3, global dynamics of the proposed model are studied, and numerical simulations are presented to illustrate theoretical results in Section 4. Finally, a brief discussion is given in Section 5.

2 Model development

Before presenting a class of multigroup reaction–diffusion SIR epidemic model with time fractional-order derivatives, some necessary preliminaries are presented.

2.1 Preliminaries

This section begins with some notations, definitions, and results.

Notation. Let be a continuous function; be the positive cone of with the norm where and be a open set of such that where is the boundary of be the positive cone of

Definition 1. (See [22].) Caputo fractional derivative of order for a function is defined by

where and

Lemma 1. Let be nonnegative and be the solution to the following system, respectively:

p2

Proof. Let then satisfies the following system:

Based on and we have Therefore, it can be deduced that

Lemma 2. (See [29].) Consider the following system:

Suppose is mixed quasimonotonous and satisfies the local Lipschitz condition

where is constant, and where is a given constant. If the upper solution and the lower solutions satisfy system (1) has a unique solution in

Lemma 3. The system with time fractional-order derivatives

has a unique global asymptotic stability of constant equilibrium

Proof. Define the Lyapunov function

Calculating the fractional derivative of along the trajectories of system (2), one has

and .. = 0 if and only if . = ... Then according to [4], there exists a unique global asymptotic stability of constant equilibrium .. = b/µ for system (2).

Lemma 4. Consider the following system:

where satisfies the local Lipschitz condition, and Then for one has with

Proof. According to the definition of Caputo fractional-order derivative, one has

Let then

By calculation, Hence, is a solution of system (9). Further, By the uniqueness of the solution it is deduced that

2.2 System description

In [10], Korobrinikov et al. studied a multigroup SIR model as follows:

But individual movement is not be considered in system (4) that is unrealistic, then Wu et al. considered the following SIR epidemic model with diffusion [27]:

Based on the previous analysis, since fractional order has the long-term memory, which can describe the spread of infectious diseases more accurate. In addition, it is traditionally assumed that the incidence of disease transmission is bilinear with respect to the number of susceptible individuals and the number of infected individuals. But in reality, it is often difficult to obtain detailed information on the spread of infectious diseases because they may change with the surrounding environment. Therefore, the general incidence rates will be chose in this paper. Motivated by the above work, as an extension of system (5), a class of multigroup SIR epidemic model are investigated as follows:

Where implies Caputo fractional-order operator denotes the Laplace operator; denotes the outward normal derivative on the smooth bound- ary and represent the number of the susceptible, infective, and recovered individuals in group at time and spatial location respectively; imply the nature death rates of and in , respectively; denotes the disease-related death rates of in ; represents the recruitment rate of the total population; implies the recovery rate of the infected individuals in kth group; denotes the diffusion rate of and in group; represents the infection rate of infected by Furthermore, and are positive constants for and are nonnegative constants for

Before giving the main results, hypothesis in terms of generalized incidence rates and is made as follows:

and satisfy the local Lipschitz condition and

is strictly monotone increasing on and is strictly monotone increasing on for all

for all where

is nonnegative and irreducible. Furthermore, if and only if for and if and only if

for all

Remark 1. Note that under hypothesis (H), many existing models can be regarded as a special form of system (6), such as and other nonlinear incidence rate in [16].

3 Model analysis

Some dynamical behavior of system (6) are investigated in this section. Here it can be found that the susceptible class and the infected class are not effected by the recovered class of system (6). Hence, we will focus our attention on the following reduced system:

Then some basic properties of system (7) are discussed in following parts.

3.1 Nonnegative and boundedness

It is significant to demonstrate the existence, uniqueness, and boundedness of a nonnega- tive solutions for system (7) before implementing its stable process. Thus, this subsection moves to the discussion of proprieties mentioned above.

Theorem 1. Under hypothesis (H), there exists a unique nonnegative solutionof system (7), and it is also ultimately bounded for any given initial function where and

Proof. Consider these two function andAccording to condition (i) in hypothesis (H), it is obvious that and are mixed quasimonotonous. Consider the following auxiliary system:

It is obvious that is a pair of the lower solution to system (7). Then, according to Lemma 1, one has

Furthermore, the following auxiliary system is introduced:

then the above system (8) has a solution as follows:

Therefore, then there exists a constant satisfied Further, consider the following auxiliary system:

then the solution for the above system (9) is

Similarly, Since then

However, it is easy to see that

It can be deduced from Eqs. (10) and (11) that

Similar to the above analysis, it can be obtained the following equation:

Based on the above analysis and Lemma 2, system (7) has a unique nonnegative global solution. Furthermore, the expression for the solution of system (7) is

where

with represents a probability density; represent generated strong continuous operator semigroups by denoting generated strong continuous operator semigroups by can be rewritten by [23]

where is the Green function yielded

with be the eigenvalue of with the eigenfunction satisfying

Hence, by the boundedness of the eigenfunction one has

According the upper solution of system (7), one has which implies is ultimate bounded. Further, the ultimate bounded of will be analyzed. Let and Adding the first two equations of system (7) and integrating it on one has

Therefore, by [13] one has

then there exist two constants and satisfyingAccording to [17], the operator families is uniformly bounded. Hence, there exist two constants and satisfying for Finally, the uniformly boundedness of the infected group can be studied as follows:

where thus is ultimate bounded. Therefore, there exists a unique positive global solution of system (7), and it is also ultimately bounded.

3.2 Stability analysis

In this subsection, the global stability analysis of system (7) will be discussed. It is easy to find that the disease-free equilibrium point of system (7) always exists where Define the following fuction:

where and is the spectral radiuses of the matrix

Lemma 5.The basic reproduction number

Proof. Linearizing system (7) at the disease-free equilibrium point , one has

Let and Then it is easy to find Obviously, we have By the definition of the basic reproduction number [5] one has Thus, by the properties of matrix eigenvalues it can be deduced that

Therefore, is considered as a threshold parameter in place of In the following, the uniqueness and the global stability of are studied.

Theorem 2. Under hypothesis (H) and there exists the unique equilibrium point of system (7), and it is globally asympototically stable in domain where

Proof. Let and where Define

It is clear to find from that Then one has Since is irreducible, it can be obtained that and are irreducible. is also irreducible. If , the inequality holds. Further, it can be deduced that if and Thus, has a only trivial solution This shows that is the unique equilibrium of system (7) when Further, is positive, then is an eigenvalue of the matrix , and has a nonnegative eigenvector corresponding to Let be the positive left eigenvector of corresponding to the spectral radius that is,Define Lyapunov function

Calculating the time fractional derivative of along the trajectories of system (7), one has

Let ..(.)

Let which is a positive definition function in Then it is concluded from [4] that is globally asymptotically stable in domain

Theorem 3. Under hypothesis (H) and system (7) is uniform persistence, that is, for any initial value the solution satisfies

whereis a constant.

Proof. Define a set

and

Let be the solution of system (7) under the initial value For any it can be known that all nonnegative solutions generate a solution semiflow with Thus, we have and it is obvious that where is the identity matrix. It can be deduced from Lemma 4 that

Then Based on the above analysis, one has -semigroup on Obviously, is compact for and point dissipative in The following system is considered:

It can be found from Lemma 3 that is globally asymptotically stable. Thus, system (7) is globally asymptotically stable at the disease-free equilibrium pointIt can be deduced that the disease-free equilibrium in is a global attractor of which implies Let where Considering there exists a sufficiently small constant such that where If there exists a solution of system (7) with the initial value such that then there exists a constant such that and Since is irreducible, is irreducible. Let be the positive left eigenvector ofcorresponding to the spectral radius that is,

Define the following arbitrary function:

Calculating the time fractional derivative of along the trajectories of system (7), one has

which leads to a contradiction with Therefore, Thus, it can be deduced from [25] that is uniformly persistent. It is concluded that system (7) is uniformly persistent.

The ultimate boundedness and the uniform persistence imply the existence of a pos- itive equilibrium point of system (7). Therefore, the existence and global stability of the positive endemic equilibrium point of system (7) can be further discussed.

Theorem 4. Under hypothesis (H) and system (7) has at least one endemic equilibrium satisfying

Furthermore, if

system (7) is globally asymptotically stable at the endemic equilibrium point

Proof. According to Theorem 1, for any given initial condition the corresponding solution is ultimately bounded, and system (7) is uniformly persistent when Therefore, there exists a positive equilibrium point of system (7) that satisfies Eqs. (12), (13).

Next, the global stability of will be analyzed. Define the Lyapunov function

Where

and the coefficients will be determined in Eq. (20). Calculating the time fractional- order derivative of along the trajectories of system (7), it can be conclude that

For each it can be deduced from the divergence theorem that

Thus, Eq. (15) can be deduced that

Let then

Substituting Eq. (17) into Eq. (16), the following inequality holds:

Calculating the fractional-order derivative of along any solution of system (7), one has

Since is the endemic equilibrium point of system (7), one has

where denotes the cofactor of the diagonal entry of e , where

with It can be deduced from [7] that exists a unique positive solution Therefore,

Thus, substituting Eq. (19) into Eq. (21), it can be obtained that

Further, it is concluded from Eqs. (18) and (22) that

Based on [4], the endemic equilibrium point of system (7) is globally asymptotically stable.

Corollary 1.When the endemic equilibrium point is globally asymptotically stable if hypothesis (H) and satisfied.

Remark 2. It can be seen that Corollary 1 is similar with Theorem 6 of [17] when

Remark 3. Not considering infection between populations, that is, when the reproduction number of group k is Furthermore,the disease-free equilibrium point is globally asymptotically stable when , and the endemic equilibrium point is globally asymptotically stable when

4 Numerical simulations

In order to verify theoretical results numerically, numerical simulations are presented in this section. We consider system (7) with two-group case, which is suitable for infectious diseases transmitted between two cities or communities. Furthermore, system (7) with two groups (n = 2) can be calculated by the central difference method in L₁-type space and Alikhanov-type discretization in time [9]. Furthermore, we consider the following incidence rate as an example: which τ is a positive parameter measuring the psychological or inhibitory effect. Obviously, satisfy hypothesis (H). The corresponding system can be expressed as

Let assign the following values to the parameters of system (23):

It is easy to calculate that Based on by Theorem 2, the disease-free equilibrium point of system (23) is global stable which is verified by Figs. 1 and 2. Further, the following parameters is chose:

System (23) has a unique equilibrium point It can be calculated that and Eq. (14) is satisfied. Based on the above analysis, the endemic equilibrium point of system (23) is global stable, which is verified by Figs. 3 and 4.

Further, with regard to the disease-free equilibrium point of the first group, the influ- ence of different fractional order . on the stability of the infected are discussed. The error

images of the infected of are described in Fig. 5, respectively. It is easy to seen from Fig. 5 that although the infected will disappear, different order will have a sensitive effect on the change of solution. Further, when tends to 1, the numerical solutions of system (7) are also convergent to the solutions of the classical ones [17]. But the relationship between the change of the solution for system (7), and fractional order is not discussed.

5 Discussion

In this article, incorporating the population diffusion and time fractional-order derivatives, theory analysis of a class of multigroup SIR epidemic model are investigated. Firstly, the existence and uniqueness of the nonnegative solution for system (7) are established. By using Lyapunov functions the global stability of the disease-free equilibrium point E0 is obtained when the basic reproduction number Besides, when the uniform persistence and the global stability of the endemic equilibrium point E^∗ are discussed. The proposed model, a more accurate epidemic model, can help us to understand some dynamical behaviors of infectious diseases. Moreover, theoretical results may provide some useful guidance for making effective countermeasures on infectious diseases. However, the relationship between system (7) and fractional order α is still an open question, which will be our future work.

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Notes