# Dynamics of a Diffusive Multigroup SVIR Model with Nonlinear Incidence

Authors: Jinhu Xu; Yan Geng
Journal: Complexity
 (2020)

---

## Abstract

In this paper, a multigroup SVIR epidemic model with reaction-diffusion and nonlinear incidence is investigated. We first establish the well-posedness of the model. Then, the basic reproduction numberℜ0 is established and shown as a threshold: the disease-free steady state is globally asymptotically stable if ℜ0<1, while the disease will be persistent when ℜ0>1. Moreover, applying the classical method of Lyapunov and a recently developed graph-theoretic approach, we established the global stability of the endemic equilibria for a special case.


---

## Body

## 1. Introduction

As is well-known, mathematical models have played a key role in describing the dynamical evolution of infectious diseases [1–10]. For example, during the special battle for preventing the novel coronavirus (COVID-19) spreading, not only medical and biological research but also theoretical studies based on mathematical modeling may play a crucial role in analyzing and forecasting the spreading of the disease (see, for example, [4–10] and references therein; also, some useful advices for controlling the disease spreading have been given by the researchers).The spatial heterogeneity has been regarded as an important role that affects the spatial spreading of disease. Therefore, complex models are needed to investigate the spread of diseases. In recent years, reaction-diffusion models involving environmental heterogeneity have been proposed and studied to find reliable measures to control disease spreading [11–22]. In particular, the basic reproduction number is an important threshold value for investigation of the dynamics of epidemic models. Then, Wang and Zhao [14] defined the basic reproduction number and obtained its computation formula for general reaction-diffusion epidemic models. Xu et al. [21] studied an SVIR epidemic model with reaction-diffusion and the existences of travelling waves were investigated, while Zhang and Liu [22] also studied the existence of travelling waves for an SVIR epidemic model but with nonlocal dispersal and time delay.Notice that the essential assumption in classical compartmental epidemic models is that the individuals are homogeneously mixed, which means each individual has the same probability to get infected. However, infected probability may be different for each individual in terms of the impact of different factors, such as education levels, age, gender, and communities. To overcome this problem, multigroup models have been proposed by many researchers by dividing the individuals into different groups and most of these work focus on the global dynamics of the models (see, for example, [23–30]). Particularly, one of the earliest works of multigroup models was proposed by Lajmanovich and York [23] when studied the gonorrhea disease transmission in a nonhomogeneous population. In order to investigate the global dynamics of multigroup models, a subtle grouping method in estimating the derivatives of Lyapunov functionals guided by graph theory for multigroup models was developed in [28–30]. For example, Kuniya [25] considered a multigroup SVIR epidemic model with vaccination and the global stability of endemic equilibria were established by the method of Lyapunov function. However, spatially heterogeneous was not considered in the model in [25]. To the best of our knowledge, there are few results of multigroup SVIR epidemic model with reaction-diffusion. The existing results are focusing on SIS and SIR models (see [31–34]). On the contrary, incidence rate also plays an important role in modeling the epidemic models, which has been frequently used to describe the nature of certain phenomena and obtained much exact results. In addition, applying general incidence rates can obtain the unification theory by the omission of unessential detail, see, for example, [21, 35–37] and references therein. Hence, inspired by [21, 25] and the above considerations, in this paper, we consider the following multigroup SVIR epidemic model with reaction-diffusion and general incidence rate:(1)∂Si∂t=▽⋅d1ix▽Siλix−μiSx+ξixSi−∑j=1nβijxSifjIj,t≥0,x∈Ω,1≤i≤n,∂Vi∂t=▽⋅d2ix▽Vi+ξixSi−∑j=1nβ˜ijxVifjIj−μiVxVi,t≥0,x∈Ω,1≤i≤n,∂Ii∂t=▽⋅d3ix▽Ii+∑j=1nβijxSi+β˜ijxVifjIj−μiIx+δix+γixIi,t≥0,x∈∂Ω,1≤i≤n,∂Ri∂t=▽⋅d4ix▽Ri+γixIi−μiRxRi,t≥0,x∈Ω,1≤i≤n,with the homogeneous Neumann boundary conditions(2)∂Si∂ν=∂Vi∂ν=∂Ii∂ν=∂Ri∂ν=0,t≥0,x∈∂Ω,1≤i≤n,and the initial conditions(3)Si0,x,Vi0,x,Ii0,x,Ri0,x=ϕ1ix,ϕ2ix,ϕ3ix,ϕ4ix,for x∈Ω,1≤i≤n, where Ω is a domain in ℝn with smooth boundary Ω and ν is the outward normal vector to the boundary ∂Ω. Initial functions ϕkixk=1,2,3,4,1≤i≤n are nonnegative and continuously defined on Ω¯. Si=Sit,x, Vi=Vit,x, Ii=Iit,x, and Ri=Rit,x stand for the densities of the susceptible, vaccinated, infective, and recovered individuals in the i-th group at time t and spatial location x, respectively. λix is the input rate of Si in spatial location x; μiSx, μiVx, μiIx, and μiRx denote the natural death rates of Si, Vi, Ii, and Ri in spatial location x, respectively; δix is the death rate induced by the disease in spatial location x; ξix is the vaccination rate of Si in spatial location x; γix is the rate of recovery from infection in spatial location x; βijx is the infection rate of Si infected by Ij in spatial location x; β˜ijx is the infection rate of Vi infected by Ij in spatial location x; and d1ix, d2ix, d3ix, and d4ix are the diffusion rate of Si, Vi, Ii, and Ri in spatial location x, respectively. All location-dependent parameters are continuous and strictly positive defined on Ω¯, and fjIj denotes the force of infection. We assume the function fjIj satisfies the following properties:(4)fj0=0,fjIj>0forIj>0,fj′Ij>0,fj″Ij≤0for allIj>0.It is natural to assume thatfj0=0 due to the fact that disease cannot spread if there is no infection. The disease spreads heavily with the increasing number of infected individuals; thus, it reasonable to suppose fj′Ij>0. Since the infectivity of infected individuals cannot be unbounded, it should reach a certain level when the infected individuals are heavy. Therefore, the assumption fj′′Ij≤0 implies that there exists a peak level for the infectivity of the infected individuals at some certain time. Similar to [25, 34], we also assume the n-square matrix βijxn×n is nonnegative and irreducible.Because the last equation of model (1) is decoupled from other equations, we indeed need to study the following subsystem of (1):(5)∂Si∂t=▽⋅d1ix▽Si+λix−μiSx+ξixSi−∑j=1nβijxSifjIj,t≥0,x∈Ω,1≤i≤n,∂Vi∂t=▽⋅d2ix▽Vi+ξixSi−∑j=1nβ˜ijxVifjIj−μiVxVi,t≥0,x∈Ω,1≤i≤n,∂Ii∂t=▽⋅d3ix▽Ii+∑j=1nβijxSi+β˜ijxVifjIj−μiIx+δix+γixIi,t≥0,x∈Ω,1≤i≤n.The rest of this paper is organized as follows. In Section2, some preliminaries are introduced for the well-posedness of the model. In Section 3, we define the basic reproduction number ℜ0. In Section 4, the threshold dynamics are established in terms of ℜ0. An special case is performed as a supplementary to the theoretical results in Section 5. A brief conclusion ends the paper.


## 2. Well-Posedness

Throughout this paper, we denotef¯=maxx∈Ω¯fx and f¯=minx∈Ω¯fx. Let Y=CΩ¯,ℝn with the norm ⋅Y and Y+=CΩ¯,ℝ+n. It is easy to see that Y,Y+ is a strongly ordered Banach space. Let X=Y×Y×Y with norm ϕX=maxϕ1Y,ϕ2Y,ϕ3Y, where ϕ=ϕ1,ϕ2,ϕ3∈X, ϕi∈Y. Denote X+=Y+×Y+×Y+ be the positive cone of X. For convenience, set ϕk=ϕk1,ϕk1,…,ϕk1 for k=1,2,3. S=S1,S2,…,Sn, V=V1,V2,…,Vn, and I=I1,I2,…,In. Denote(6)T1i,T2i,T3i:CΩ¯,ℝ⟶CΩ¯,ℝbe the C0 semigroup associated with ▽⋅d1ix▽−μiSx+ξix, ▽⋅d2ix▽−μiVx, and ▽⋅d3ix▽−μiIx+δix+γix, respectively, subject to the Neumann boundary condition. Let Γkit,x,y be the Green function associated with ▽⋅d1ix▽−μiSx+ξix, ▽⋅d2ix▽−μiVx, and ▽⋅d3ix▽−μiIx+δix+γix, respectively, subject to the Neumann boundary condition. Then, for any ϕ∈CΩ¯,ℝ and t>0, we have(7)Tkitϕx=∫ΩΓkit,x,yϕydy,k=1,2,3,1≤i≤n.Applying Corollary 7.2.3 in [38], we know that, for each t>0,Tkit:CΩ¯,ℝ⟶CΩ¯,ℝ is compact and strongly positive. Then, there exist constants Aki>0 (k=1,2,3), satisfying Tkit≤Akieαkit for each t≥0, where αki denotes the principle eigenvalue of ▽⋅d1ix▽−μiSx+ξix, ▽⋅d2ix▽−μiVx, and ▽⋅d3ix▽−μiIx+δix+γix subject to the Neumann boundary condition. Define(8)F1iϕx=λix−∑j=1nβijxϕ1ixfjϕ3jx,1≤i≤n,F2iϕx=ξixϕ1ix−∑j=1nβ˜ijxϕ2ixfjϕ3jx,1≤i≤n,F3iϕx=∑j=1nβijxϕ1ixfjϕ3jx+∑j=1nβ˜ijxϕ2ixfjϕ3jx,1≤i≤n,for t≥0, x∈Ω¯, and ϕ=ϕ1,ϕ2,ϕ3∈X+. Set ut,⋅,ϕ=St,⋅,ϕ,Vt,⋅,ϕ,It,⋅,ϕ be the solution of model (5) with initial function ϕ=ϕ1,ϕ2,ϕ3∈X+, and then, model (5) can be rewritten as(9)Sit,⋅,ϕ=T1itϕ1+∫0tT1it−sF1ius,⋅,ϕds,Vit,⋅,ϕ=T2itϕ2+∫0tT2it−sF2ius,⋅,ϕds,Iit,⋅,ϕ=T3itϕ3+∫0tT3it−sF3ius,⋅,ϕds,for t>0 and 1≤i≤n. By virtue of Corollary 4 in [39], we have the following.Lemma 1.
Forϕ=ϕ1,ϕ2,ϕ3∈X+, model (5) has a unique mild solution ut,⋅,ϕ=St,⋅,ϕ,Vt,⋅,ϕ,It,⋅,ϕ∈X+on0,τ∞andτ∞≤∞. Moreover, this solution is a classical solution.
Then, we give the existence of solutions of model (5).Theorem 1.
The model (5) has a unique solution ut,⋅,ϕ=St,⋅,ϕ,Vt,⋅,ϕ,It,⋅,ϕ∈X+ on 0,∞ with ϕ=ϕ1,ϕ2,ϕ3∈X+. Furthermore, the solution semiflow Φt=ut,⋅:X+⟶X+ of model (5) defined by(10)Φtϕ=ut,⋅,ϕ,t≥0,admits a global compact attractor.Proof.
Suppose to the contrary thatτ∞<∞, then ut,⋅,ϕ⟶+∞ as t⟶τ∞ by Theorem 2 in [39]. It follows from the first equation of model (5) that(11)∂Si∂t≤▽⋅d1ix▽Si+λ¯i−μ¯iS+ξ¯iSi,t∈0,τ∞,x∈Ω,1≤i≤n.
By the comparison principle and Lemma 2 in [40], there exists a constant ℳ1>0 such that Sit,x≤ℳ11≤i≤n for t∈0,τ∞, x∈Ω¯. Furthermore, similar procedure can be applied to the second equation of model (5), and then, there exists a constant ℳ2>0 such that Vit,x≤ℳ21≤i≤n for t∈0,τ∞, x∈Ω¯. Hence, from the third equation of model (5), we have(12)∂Ii∂t≤▽⋅d3ix▽Ii+∑j=1nβ¯ijℳ1+β˜¯ijℳ2fj′0Ij−μ¯iI+δ¯i+γ¯iIi.
Now, we consider the following comparison system:(13)∂ωi∂t=▽⋅d3ix▽ωi+∑j=1nβ¯ijℳ1+β˜¯ijℳ2fj′0ωj−μ¯iI+δ¯i+γ¯iωi,t>0,x∈Ω,1≤i≤n,∂ωi∂t=0,t>0,x∈∂Ω.
It follows from the standard Krein–Rutman theorem (see [41]) that the eigenvalue problem of system (13) admits a principle eigenvalue λ with a strongly positive eigenfunction ϕ^2=ϕ^21,ϕ^22,…,ϕ^2n. Thus, system (13) has a solution ςeλtϕ^2x for t≥0, where ς is a positive constant, satisfying ςϕ^2≥I10,x,I20,x,…,In0,x for x∈Ω¯. By using the comparison principle, we have(14)I1t,x,I2t,x,…,Int,x≤ςeλtϕ^2x,t∈0,τ∞,x∈Ω¯,1≤i≤n.
Thus, there exists a constantℳ3 such that(15)Iit,x≤ℳ3,x∈Ω¯,1≤i≤n,which leads to a contradiction. Hence, the global existence of ut,⋅,ϕ is derived.
Now, we are in the position to show the solution is also ultimately bounded. Indeed, it follows from the comparison principle, (11), and Lemma 2 in [40] that there exist t1>0 and A1>0 such that Sit,x≤A1, t≥t1, ∀x∈Ω¯. Moreover, from the second equation of model (5) and applying similar procedures, we can find A2>0 and t2>0, satisfying Vit,x≤A2 for t≥t2, ∀x∈Ω¯.
Denote(16)Qit=∫ΩSit,x+Vit,x+Iit,xdx,1≤i≤n.
Then, we have(17)dQidt=∫Ωλix−μiSxSit,x−μiVxVt,x−μiIx+δix+γixIit,xdx≤∫Ωλixdx−minx∈Ω¯μ¯iS,μ¯iV,μ¯iI+δ¯i+γ¯iQi,t≥0.
Thus, there existt3>0 and A3>0 such that Qit≤A31≤i≤n for any t≥t3. Next, we denote τji be the eigenvalue of ▽⋅d3ix▽−μiIx+δix+γix subject to the Neumann boundary condition with eigenfunction φjix, which satisfies τ1i>τ2i≥τ3i≥⋯τji≥⋯. From Chapter 5 in [42], one obtains(18)Γ3it,x,y=∑j≥1eτjitφjixφjiy,1≤i≤n.
Sinceφjix is uniformly bounded, there exists constant κ3i>0 such that Γ3it,x,y≤κ3i∑j≥1eτjit for t>0. Moreover, assume πji are eigenvalues of ▽⋅d¯3i▽−μ¯iI+δ¯i+γ¯i subject to Neumann boundary condition, which satisfies π1i=−μ¯iI+δ¯i+γ¯i>π2i≥π3i≥⋯πji≥⋯. By Theorem 2.4.7 in Wang [43], one gets πji≥τji for all j∈ℕ+. Since πji decreases like −n2, there exists κ3>0 such that(19)Γ3i≤κ3i∑j≥1eπjit≤κ3eπ1it=κ3e−μ¯iI+δ¯i+γ¯it,∀t>0.
Lett^=maxt1,t2,t3. For all t≥t^, by (9) and (4), we have(20)Iit,x=T3itIit^,x+∫t^tT3it−s∑j=1nβijxSis,x+β˜ijxVis,xfjIjs,xds≤A3ieα3it−t^Iit^,x+∫t^t∫ΩΓ3it−s,x,y∑j=1nβijySis,y+β˜ijyVis,y×fj′0Ijs,ydyds≤A3ieα3it−t^Iit^,x+∫t^tκ3e−μ¯iI+δ¯i+γ¯⇀it−s∑j=1nβ¯ijA1+β˜¯ijA2fj′0×∫ΩIjs,ydyds≤A3ieα3it−t^Iit^,x+κ3A3∫t^te−μ¯iI+δ¯i+γ¯⇀it−s⋅∑j=1nβ¯ijA1+β˜¯ijA2fj′0ds=A3ieα3it−t^Iit^,x+κ3A3∑j=1nβ¯ijA1+β˜¯ijA2fj′01−e−μ¯iI+δ¯i+γ¯⇀it−t^μ¯iI+δ¯i+γ¯⇀i≤A3ieα3it−t^Iit^,x+κ3A3∑j=1nβ¯ijA1+β˜¯ijA2fj′0μ¯iI+δ¯i+γ¯⇀i,which yields that(21)limsupt⟶∞Iit,x≤κ3A3∑j=1nβ¯ijA1+β˜¯ijA2fj′0μ¯iI+δ¯i+γ¯i,1≤i≤n.
Thus, the above discussion implies that the system (5) is point dissipative. Furthermore, by Theorem 2.2.6 in [44], the solution semiflow Φt is compact for any t>0. Therefore, it follows from Theorem 3.4.8 in [45] that Φt has a global compact attractor in X+.


## 3. Basic Reproduction Number

For each1≤i≤n, consider the following subsystem of model (5):(22)∂Si∂t=▽⋅d1ix▽Si+λix−μiSx+ξixSi,t>0,x∈Ω,∂Vi∂t=▽⋅d2ix▽Vi+ξixSi−μiVxVi,t>0,x∈Ω,∂Si∂ν=∂Vi∂ν=0,t>0,x∈∂Ω.It follows from the Lemma 2.2 in [40] that the following system(23)∂Si∂t=∇⋅d1ix▽Si+λix−μiSx+ξixSi,t>0,x∈Ω,∂Si∂ν=0,t>0,x∈∂Ω,admits a unique positive steady state Si0x which satisfies the equation(24)▽⋅d1ix▽Si0x+λix−μiSx+ξixSi0x=0with ∂Si0x/∂ν=0 for x∈∂Ω, which is globally asymptotically stable in CΩ¯,ℝ+. Hence, the second equation of system (22) is asymptotic to(25)∂Vi∂t=▽⋅d2ix▽Vi+ξixSi0x−μiVxVi.By Lemma 2.2 in [40] and Corollary 4.3 [46], there exists a globally asymptotically stable steady state Vi0x. Therefore, concluding from the above discussion, we know that model (5) admits a unique disease-free steady state E0x=S0x,V0x,0 with S0x=S10x,S20x,…,Sn0x, V0x=V10x,V20x,…,Vn0x, and 0=0,0,…,0. Furthermore, if all the parameters of model (5) are positive constants, then we have Si0x=λi/μiS+ξi and Vi0x=λiξi/μiVμiS+ξi1≤i≤n.Linearizing model (5) at E0, we obtain the linearized system(26)∂Ii∂t=∇⋅d3ix▽Ii+∑j=1nθijxfj′0Ij−μiIx+δix+γixIi,t>0,x∈Ω,1≤i≤n,∂Ii∂ν=0,t>0,x∈∂Ω,1≤i≤n.SubstitutingIit,x=eλtϕ3ix into (26), we obtain(27)λϕ3ix=∇⋅d3ix▽ϕ3ix+∑j=1nθijxfj′0ϕ3ix−μiIx+δix+γixϕ3ix,t>0,x∈Ω,1≤i≤n,∂ϕ3i∂ν=0,t>0,x∈∂Ω,1≤i≤n.From Theorem 7.6.1 in [38], we have the following result.Lemma 2.
The eigenvalue problem (27) admits a principle eigenvalue λ0 with a strictly positive eigenfunction.DenoteT3=T31,T32,…,T3n. Assume that the distribution of initial infection is ϕ3x∈Y+. Then, ℱxT3ϕ3x is the distribution of new infective part as time evolves. Therefore, we use(28)ℒϕ3x=∫0∞ℱxT2tϕ3xdtto describe the total distribution of the new infective numbers produced during the infection period.According to [14], the basic reproduction number is defined by ℜ0=rℒ, where rℒ is the spectral radius of the operator ℒ. Furthermore, following Theorem 3 in [14], we have the following lemma.Lemma 3.
The principle eigenvalueλ0 and ℜ0−1 have the same sign, and the disease-free steady state E0x is locally asymptotically stable.


## 4. Extinction/Persistence Result

Based on the discussion above, we now investigate the extinction and persistence of the disease.Theorem 2.
Ifℜ0<1, then the disease-free steady state E0x is globally asymptotically stable.Proof.
By Lemma2, we have λ0<0 when ℜ0<1. Thus, there exists a small enough ε>0 such that λ0ε0<0. According to Theorem 1, there exists a t∗>0 such that Sit,x≤Si0x+ε0 and Vit,x≤Vi0x+ε0 (1≤i≤n) for all x∈Ω. Thus, from the third equation of model (5) and assumption (4) that(29)∂Ii∂t≤∇⋅d3ix▽Ii+∑j=1nβijxSi0x+ε0+β˜ijxVi0x+ε0fj′0Ij−μiIx+δix+γixIi,x∈Ω,t≥t∗,1≤i≤n,∂Ii∂ν=0,x∈∂Ω,t≥t∗,1≤i≤n,let ϕ¯31x,ϕ¯32x,…,ϕ¯3nx be the eigenfunction corresponding to the principal eigenvalue λ0ε0<0. Assume that(30)I1t∗,x,I2t∗,x,…,Int∗,x≤αϕ¯31x,ϕ¯32x,…,ϕ¯3nx,where α>0 is a constant. With the aid of comparison principle, we can obtain(31)It,x=I1t,x,I2t,x,…,Int,x≤αϕ¯31x,ϕ¯32x,…,ϕ¯3nxeλ0ε0t−t∗,t≥t∗.
This yieldslimt⟶∞It,x=0 uniformly for x∈Ω¯. Thus, the model (5) is asymptotic to (22). Then, by Lemma 2.2 in [40] and Corollary 4.3 in [46], we have limt⟶∞St,x=S0x and limt⟶∞Vt,x=V0x.
Before proving the main results on disease persistence, we first need to establish the following lemma.Lemma 4.
Letu⋅,t,ϕ=S⋅,t,ϕ,V⋅,t,ϕ,I⋅,t,ϕ is the solution of model (5) with u⋅,0,ϕ=ϕ=ϕ1,ϕ2,ϕ3∈X+.(i)
For anyϕ∈X+, we always have Si⋅,t,ϕ>0 and Vi⋅,t,ϕ>0 for all t>0, and there exist constants ρ1>0,ρ2>0 such that(32)liminft⟶∞Si⋅,t,ϕ≥ρ1,liminft⟶∞Vi⋅,t,ϕ≥ρ2.(ii)
If there existst∗≥0 such that Ii⋅,t∗,ϕ≡0, then we have Ii⋅,t,ϕ>0, ∀t>t∗, 1≤i≤n.Proof.
(i)
It follows from Theorem1 that there exists t0>0 and M>0 such that(33)Ii⋅,t,ϕ≤M,∀t≥t0.Then, by the first equation of model (5) and the assumption (4), we have(34)∂Si∂t≥▽⋅d1i▽Si+λix−μiSx+ξixSi−∑j=1nβijxMfj′0Si,x∈Ω,t≥t0,∂Si∂ν=0,x∈Ω¯,t≥t0.Thus, by Lemma 2.2 in [40], it follows that the comparison system(35)∂wi∂t=▽⋅d1i▽wi+λix−μiSx+ξixwi−∑j=1nβijxMfj′0wi,x∈Ω,t≥t0,∂wi∂ν=0,x∈∂Ω,t≥t0,admits a unique positive steady statewi∗x which is globally asymptotically stable in CΩ¯,ℝ. By the standard parabolic comparison theorem, there exists a constant ρ1>0 such that liminft⟶∞Si⋅,t,ϕ≥ρ1 is uniformly for x∈Ω¯. Moreover, it follows from the second equation of model (5) that(36)∂Vi∂t≥▽⋅d2i▽Vi+ξixρ1−μiVxVi−∑j=1nβ˜ijxMfj′0Vi,x∈Ω,t≥t0,∂Vi∂ν=0,x∈∂Ω,t≥t0.Similarly, there exists a constantρ2 such that liminft⟶∞Vi⋅,t,ϕ≥ρ2 is uniformly for x∈Ω¯.(ii)
AssumeIi⋅,t∗,ϕ≡0 for some t∗≥0. According to Theorem 1, it follows from model (5) that(37)∂Ii∂t≥▽⋅d3i▽Ii−μiIx+δix+γixIi,x∈Ω,t>0,∂Ii∂ν=0,x∈∂Ω,t>0.Thus, the strong maximum principle (see, e.g., [47], Theorem 4) and Hopf boundary lemma (see, e.g., [47], Theorem 3) imply that Ii⋅,t,ϕ>0, ∀t>t∗, and x∈Ω¯.Now, we are in position to state the main results of this section.Theorem 3.
Ifℜ0>1, then there exists a constant ϱ>0 such that, for any ϕ=ϕ1,ϕ2,ϕ3∈X+ with ϕ3i≡01≤i≤n, solution ut,⋅,ϕ=St,⋅,ϕ,Vt,⋅,ϕ,It,⋅,ϕ of model (5) satisfies(38)liminft⟶∞Sit,⋅,ϕ≥ϱ,liminft⟶∞Vit,⋅,ϕ≥ϱ,liminft⟶∞Iit,⋅,ϕ≥ϱ,1≤i≤n,uniformly for all x∈Ω¯. Moreover, model (5) has at least one endemic steady state E∗x.Proof.
Define(39)X0=ϕ=ϕ1,ϕ2,ϕ3∈X+:ϕ3i≡0,1≤i≤n,∂X0=X+\X0=ϕ=ϕ1,ϕ2,ϕ3∈X+:ϕ3i≡0,1≤i≤n.
According to Lemma4, for any ϕ∈X0, we have Iit,⋅,ϕ>0, ∀t>0, and x∈Ω¯. Thus, ΦtX0⊆X0, which implies that X0 is the invariant set for solution semiflow Φt of model (5). Define(40)ℳ∂=ϕ∈X+:Φtϕ∈∂X0,∀t≥0,and ωϕ be the omega limit set of the orbit O+ϕ=Φtϕ:t≥0. We first prove the following claim.Claim 1.
ωϕ=E0=S0x,V0x,0, ∀ϕ∈ℳ∂. Since ϕ∈ℳ∂, we have Φtϕ∈∂X0. Thus, Iit,⋅,ϕ≡01≤i≤n, ∀t≥0. Then, model (5) reduces to model (22); by Lemma 2.2 in [40] and Corollary 4.6 in [46], we have limt⟶∞Sit,⋅,ϕ=Si0x and limt⟶∞Vit,⋅,ϕ=Vi0x uniformly for x∈Ω¯. Hence, ωϕ=E0=S0x,V0x,0, ∀ϕ∈ℳ∂; i.e., the claim holds.
Sinceℜ0>1, we have λ0>0 by Lemma 3. Using the continuity of λ0, there exists a sufficient small enough positive constant η0>0 such that λ0η0>0.Claim 2.
E0x is uniform weak repeller for X0 in the sense that(41)limsupt⟶∞Φtϕ−E0x≥η0,∀x∈X0.
Suppose, by contradiction, there exists aϕ0∈X0 such that(42)limsupt⟶∞Φtϕ−E0x<η0.
Then, there exists an enough larget∗>0 such that(43)Si0x−η0<Sit,⋅,ϕ<Si0x+η0,Vi0x−η0<Vit,⋅,ϕ<Vi0x+η0,0<Iit,⋅,ϕ<η0,1≤i≤n,∀t≥t∗,x∈Ω¯.
Therefore, from model (5) and assumptions (4), we have(44)∂Ii∂t≥∇⋅d3ix∇Ii+∑j=1nβijxSi0x−η0+β˜ijxVi0x−η0fj′η0Ij−μiIx+δix+γixIi,x∈Ω,t≥t∗,1≤i≤n,∂Ii∂ν=0,x∈∂Ω,t≥t∗,1≤i≤n.
Setφ31x,φ32x,…,φ3nx be the strongly positive eigenfunction associated with principle eigenvalue λ0η0>0. Since Iit,⋅,ϕ0>0 for all x∈Ω¯ and t>0, there exists a constant ξ0>0 such that It∗,⋅,ϕ0≥ξ0φ31x,φ32x,…,φ3nx for x∈Ω¯. It is clear that ωt,x=ξ0eλ0η0t−t∗φ31x,φ32x,…,φ3nx is a solution of the following linear system:(45)∂ωi∂t=∇⋅d3ix▽ωi+∑j=1nβijxSi0x−η0+β˜ijxVi0x−η0fj′η0ωj−μiIx+δix+γixωi,x∈Ω,t≥t∗,1≤i≤n,∂ωi∂ν=0,x∈∂Ω,t≥t∗,1≤i≤n.
Applying the comparison principle, we have(46)It,⋅,ϕ0≥ξ0eλ0η0t−t∗φ31x,φ32x,…,φ3nx,∀x∈Ω¯,t≥t∗.
Due toλ0η0>0, we conclude that Iit,⋅,ϕ0 is unbounded. This is a contradiction.
Define a continuous functionp:X+⟶0,+∞ by(47)pϕ=minminx∈Ω¯ϕ3ix,1≤i≤n,∀ϕ∈X+.
Clearly,p−10,+∞⊆X0 and has the property that if pϕ>0 or pϕ=0 and ϕ∈X0, then pΦtϕ>0 for all t>0. Hence, for the semiflow Φt:X+⟶X+, p is a generalized distance function [48]. From Claim 1, it follows that any forward orbit of Φt in ℳ∂ converges to E0x. Moreover, Claim 2 implies that E0x is isolated in X+ and WsE0x∩X0=∅, where WsE0x is the stable set of E0x. Furthermore, there is no cycle in ℳ∂ from E0x to E0x. It then follows from Theorem 3 in [48] that there exists a constant ϱ>0 such that liminft⟶∞pΦtϕ≥〉 for all ϕ∈X0. This implies the uniform permanence of It,⋅,ϕ. By Lemma 4 and Theorem 4.7 in [49], we know that Φt has a positive steady state E∗x of model (5). The proof is complete.


## 5. Global Dynamics for a Special Case

In this section, we consider a special case with all the parameters of model (5) as constants, except for the diffusion coefficients. Then, model (5) degenerates into the following model:(48)∂Si∂t=▽⋅d1ix▽Si+λi−μiS+ξiSi−∑j=1nβijSifjIj,t≥0,x∈Ω,1≤i≤n,∂Vi∂t=▽⋅d2ix▽Vi+ξiSi−∑j=1nβ˜ijVifjIj−μiVVi,t≥0,x∈Ω,1≤i≤n,∂Ii∂t=▽⋅d3ix▽Ii+∑j=1nβijSi+β˜ijVifjIj−μiI+δi+γiIi,t≥0,x∈Ω,1≤i≤n,with the homogeneous Neumann boundary conditions (2) and the initial conditions (3). We mainly focus on the global stability of the endemic steady states of model (48). The proofs of the main results applying the Lyapunov functions and a subtle grouping technique guided by graph theory were developed in [28–30].It follows from the Lemma2 in [13] that model (48) has a disease-free steady state E0=S0,V0,0, where S0=S10,S20,…,Sn0 and V0=V10,V20,…,Vn0 with Si0=λi/μiS+ξi and Vi0=λiξi/μiVμiS+ξi. Define matrices(49)ℱ=θ110f1′0θ120f2′0…θ1n0fn′0θ210f1′0θ220f2′0…θ2n0fn′0⋮⋮⋱⋮θn10f1′0θn20f2′0…θnn0fn′0,V=μ1I+δ1+γ10…00μ2I+δ2+γ2…0⋮⋮⋱⋮00…μnI+δn+γn,where θij0=βijSi0+β˜ijVi0, then we have(50)ℱV−1=θij0fj′0μjI+δj+γj1≤i,j≤n.Then, it follows from [26] that the basic reproduction number is defined as the spectral radius of ℱV−1, i.e., ℜ0=rℱV−1. As a consequence of Theorems 2 and 3, we can obtain the following results without proofs.Theorem 4.
The disease-free steady stateE0 of model (48) is globally asymptotically stable when ℜ0<1.Theorem 5.
Ifℜ0>1, then there exists a constant ϱ>0 such that, for any ϕ=ϕ1,ϕ2,ϕ3∈X+ with ϕ3i≡01≤i≤n, solution ut,⋅,ϕ=St,⋅,ϕ,Vt,⋅,ϕ,It,⋅,ϕ of model (48) satisfies(51)liminft⟶∞Sit,⋅,ϕ≥ϱ,liminft⟶∞Vit,⋅,ϕ≥ϱ,liminft⟶∞Iit,⋅,ϕ≥ϱ,1≤i≤n,uniformly for all x∈Ω¯. Moreover, model (48) has at least one endemic steady state.Furthermore, set the endemic steady stateE∗=S∗,V∗,I∗ with S∗=S1∗,S2∗,…,Sn∗, V∗=V1∗,V2∗,…,Vn∗, and I∗=I1∗,I2∗,…,In∗ satisfying(52)λi=μiS+ξiSi∗+∑j=1nβijSi∗fjIj∗,1≤i≤nξiSi∗=∑j=1nβ˜ijVi∗fjIj∗+μiVVi∗,i≤n,∑j=1nβijSi∗+β˜ijVi∗fjIj∗=μiI+δi+γiIi∗,1≤i≤n.For the global stability of the endemic steady state E∗, we have the following conclusion.Theorem 6.
Ifℜ0>1, then the endemic steady state E∗ is globally asymptotically stable.Proof.
Define(53)aij=βijSi∗+β˜ijVi∗fjIj∗,1≤i,j≤n,Θ=∑l≠1na1l−a21…−an1−a12∑l≠2na2l…−an2⋮⋮⋱⋮−a1n−a2n…∑l≠nnanl,which is a Laplacian matrix whose column sums are zero [30]. Then, Θ is irreducible because βijn×n is irreducible. It follows from Lemma 1 in [29] that the solution space of the linear system Θζ=0 has dimension 1 with a base ζ=ζ1,ζ2,…,ζn with ζi=cii, where cii>0 is the cofactor of the i-th diagonal entry of Θ.
Define(54)Ψt=∑i=1nζiHit,where(55)Hit=∫ΩSi−Si∗−Si∗lnSiSi∗+Vi−Vi∗−Vi∗lnViVi∗+Ii−Ii∗−Ii∗lnIiIi∗dx.
DifferentiatingHit, we obtain(56)dHitdt=∫Ω1−SiSi∗▽⋅d1ix▽Si+λi−∑j=1nβijSifjIj−μiS+ξiSi+1−ViVi∗▽⋅d2ix▽Vi+ξiSi−∑j=1nβ˜ijVifjIj−μiVVi+1−IiIi∗×▽⋅d3ix▽Ii+∑j=1nβijSi+β˜ijVifjIj−μiI+δi+γiIidx.
It follows from [50] that(57)∫Ω▽⋅d1ix▽Sidx=0,∫Ω1Si▽⋅d1ix▽Sidx=∫Ωd1ix▽Si2Si2dx,∫Ω▽⋅d2ix▽Vidx=0,∫Ω1Vi▽⋅d2ix▽Vidx=∫Ωd2ix▽Vi2Vi2dx,∫Ω▽⋅d3ix▽Iidx=0,∫Ω1Ii▽⋅d3ix▽Iidx=∫Ωd3ix▽Ii2Ii2dx.
Then, using steady state equations (52), we have(58)dHitdt=∫ΩμiSSi∗2−SiSi∗−Si∗Si+μiVVi∗3−ViVi∗−Si∗Si−SiVi∗Si∗Vi+∑j=1nβijSi∗fjIj∗2−Si∗Si−IiIi∗+fjIjfjIj∗−SiIi∗fjIjSi∗IifjIj∗+∑j=1nβ˜ijVi∗fjIj∗3−Si∗Si−IiIi∗+fjIjfjIj∗−SiVi∗Si∗Vi−ViIi∗fjIjVi∗IifjIj∗−d1ixSi∗▽Si2Si2−d2ixVi∗▽Vi2Vi2−d3ixIi∗▽Ii2Ii2dx=∫ΩμiSSi∗2−SiSi∗−Si∗Si+μiVVi∗3−ViVi∗−Si∗Si−SiVi∗Si∗Vi+∑j=1nβijSi∗fjIj∗φSi∗Si+φSiIi∗fjIjSi∗IifjIj∗+φIifiIi∗Ii∗fiIi+fiIifiIi∗−IiIi∗1−fiIi∗fiIi+fjIjfjIj∗−fiIifiIi∗+lnfjIj∗fiIifjIjfiIi∗+∑j=1nβ˜ijVi∗fjIj∗φSi∗Si+φSiVi∗Si∗Vi+φViIi∗fjIjVi∗IifjIj∗+φIifiIi∗Ii∗fiIi+fiIifiIi∗−IiIi∗1−fiIi∗fiIi+fjIjfjIj∗−fiIifiIi∗+lnfjIj∗fiIifjIjfiIi∗−d1ixSi∗▽Si2Si2−d2ixVi∗▽Vi2Vi2−d3ixIi∗▽Ii2Ii2dx,where φx=1+lnx−x with global maximum value φx=0. From the assumption (4), it is easy to validate that(59)fiIifiIi∗−IiIi∗1−fiIi∗fiIi≤0.
Moreover, according to the property that arithmetic mean is not less than the associated geometric mean,Vi/Vi∗+Si∗/Si+SiVi∗/Si∗Vi≥3 and Si/Si∗+Si∗/Si≥2. Thus, we have(60)dHitdt≤∫Ω∑j=1nβijSi∗+β˜ijVi∗fjIj∗fjIjfjIj∗−fiIifiIi∗+lnfjIj∗fiIifjIjfiIi∗dx.
Consequently, we obtain(61)dΦtdt≤∫Ω∑i=1nζi∑j=1nβijSi∗+β˜ijVi∗fjIj∗fjIjfjIj∗−fiIifiIi∗+lnfjIj∗fiIifjIjfiIi∗dx.
It follows from the equalityΘζ=0 that ∑j=1najiζj=∑l=1nailζi which is equivalent to ∑j=1nβjiSj∗+β˜jiVj∗fiIi∗ζj=∑l=1nβilSi∗+β˜ilVi∗flIl∗ζi. Then,(62)∑i=1n∑j=1nβijSi∗+β˜ijVi∗fjIjζi=∑i=1n∑j=1nβjiSj∗+β˜jiVj∗fiIiζj=∑i=1nfiIifiIi∗∑j=1nβjiSj∗+β˜jiVj∗fiIi∗ζj=∑i=1nfiIifiIi∗∑l=1nβilSi∗+β˜ilVi∗flIl∗ζi=∑i=1n∑j=1nβijSi∗+β˜ijVi∗fiIifiIi∗fjIj∗ζi,and thus, ∑i=1n∑j=1nβijSi∗+β˜ijVi∗fjIj∗fjIj/fjIj∗−fiIi/fiIi∗=0 for all I1,I2,…,In>0. Following the graph-theoretic approach as proposed in [28–30], similar procedures as in [26] can be applied to verify that(63)∑i=1nζi∑j=1nβijSi∗+β˜ijVi∗fjIj∗lnfjIj∗fiIifjIjfiIi∗=0.
Thus, we havedΦt/dt≤0. Furthermore, it can be shown that the largest invariant set in S,V,I:dΦt/dt=0 is the singleton E∗. Therefore, by the LaSalle’s invariance principle, E∗ is globally asymptotically stable.


## 6. Conclusions

In this paper, a multigroup SVIR model with diffusion and nonlinear incidence rate has been investigated. We defined the basic reproduction numberℜ0 for spatially heterogeneous environment, and we further proved that ℜ0 served as a threshold index which predicts the extinction and persistence of the disease. Particularly, by using comparison principle, we proved that the disease-free steady state E0x is globally asymptotically stable when ℜ0 is less than one. If ℜ0 is great than one, then the disease will persist. Consequently, we obtained the existence of endemic steady state E∗x. Furthermore, we established the criteria on the global stability of the disease-free steady state and the endemic steady state of the model in a special case.Although the existence of the endemic steady state is established, it is necessary to point that the uniqueness and stability of the endemic steady state remains an open problem. Moreover, the impact of the latent individuals should also be considered for some diseases during its spread, such as AIDS and COVID-19 (see, for example, [4–8, 51]), in which it has been validated that the latent individuals for these two diseases may also have probability of infection. Thus, an improved model with latent compartment should be investigated. We leave these problems for future investigation.



---

*Source: 8847023-2020-12-09.xml*