# Threshold Dynamics of a Diffusive Herpes Model Incorporating Fixed Relapse Period in a Spatial Heterogeneous Environment

Authors: Yueming Lu; Wei Yang; Desheng Ji
Journal: Complexity
 (2021)

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## Abstract

In this paper, we aim to establish the threshold-type dynamics of a diffusive herpes model that assumes a fixed relapse period and nonlinear recovery rate. It turns out that when considering diseases with a fixed relapse period, the diffusion of recovered individuals will lead to nonlocal recovery term. We characterize the basic reproduction number,ℜ0, for the model through the next generation operator approach. Moreover, in a homogeneous case, we calculate the ℜ0 explicitly. By utilizing the principal eigenvalue of the associated eigenvalue problem or equivalently by ℜ0, we establish the threshold-type dynamics of the model in the sense that the herpes is either extinct or close to the epidemic value. Numerical simulations are performed to verify the theoretical results and the effects of the spatial heterogeneity on disease transmission.


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## Body

## 1. Introduction

During the past three decades, the incidence of herpes has increased in the developing countries [1]. One typical feature of herpes is that the virus will be reactivated and reactivated periodically by close physical or sexual contact, leading to a relapse period of infectiousness (see, e.g., [1–5]). Mathematical models have been used to explore the transmission of herpes. It is highlighted in [5] that an ordinary differential equations (ODEs) compartment model for herpes is also suitable for pseudorabies in pigs [6]. In [5], the population was divided into three disjoint classes, that is, susceptible individuals (who have not previously been exposed to the virus), infective individuals (who have been infected and shed the virus), and recovered (or latency) individuals (who have previously been infected with the virus but have not shed the virus), denoted by St, It, and Rt, respectively. Considering that the relapse phenomenon occurs when the virus is reactivated, they used a relapse term from recovered class to infective class to describe a disease with relapse, giving the Susceptible-Infective-Recovered-Infective (SIRI) model. With the standard incidence rate, ℜ0 is determined as the sharp threshold for determining whether the herpes is extinct or close to the epidemic value. The ODEs model in [5] was further extended to the more with general incidence function in [4], and similar threshold results were obtained. Subsequently, Blower et al. [3] formulated a model to investigate how much resistance of herpes will be produced when the rate of antiviral treatment is enlarged. For more different herpes transmission models, we refer to [2].Unlike the ODEs models for herpes that the relapse period was assumed to obey negative exponential distribution, van den Driessche and Zou et al. [7] utilized a more general relapse distribution Pt to explore the results of distinct settings on the relapse period, where Pt stands for the proportion of recovered individuals still remaining in recovered class after recovery. In particular, the authors took a step function distribution for the relapse period and obtained a delay differential equations (DDEs) model for herpes. They also found that there is no sustained oscillatory solutions. After determining ℜ0, the threshold-type results of the model were also addressed.The aforementioned models are for a spatially homogeneous environment, meaning that only ODE and DDEs models are involved. In recent years, spatial-temporal dynamics of infectious diseases governed by the reaction-diffusion models have attracted many researchers. The spatial heterogeneity (SH) and diffusion play important roles in disease transmission. Under different infection mechanisms, some new insights in disease control and new phenomenon in disease spread will be obtained; see, for instance, [8–13]. It is found in [8] that SH would increase the risk of influenza transmission so that the SH of the recovery rate and transmission rate must be increased for controlling the influenza transmission. In [9], the authors proposed a spatial nonlocal diffusive model with delay and no-flux boundary condition. Here the nonlocal delay is caused by introducing a fixed incubation period in a continuous bounded domain. By utilizing the classical theory, the threshold-type dynamics are determined by ℜ0. Here, ℜ0 was achieved by the spectral radius of the next generation operator. In a homogeneous case that all model parameters are constant, ℜ0 can be explicitly obtained. Besides, another method of calculating ℜ0 in one-dimensional space was also presented in [9]. In a recent work [14], the authors studied the dual-functionality of physical contacts driven via variations of individual spatial behavior and provided insights on mechanisms that generate spatial heterogeneity. By using an epidemic model with nonlocal delay and logistic growth, the authors in [15] studied the dynamics of model and investigated how nonlocal delay and logistic growth affect the disease transmission. It was advocated in [16] that the transitions between patterns are an emergent property in spatial epidemics that can serve as a potential trend indicator of disease spread when considering the spread of diseases in both time and space. In [17], the authors studied the existence and nonexistence of the traveling wave solutions for the model with spatial structure.The authors in [18] studied the local and global long-term dynamics of the Banana Black Sigatoka Disease with delay and seasonality. In [19], the spatial-temporal characteristics and effective control measures of brucellosis transmission are investigated. Very recently, a human-vector malaria transmission model incorporating age, time since infection, and waning immunity was studied in [20], where the well-posedness of the model, the existence of endemic equilibria, and the effect of the above structural variables on key important epidemiological traits of the human-vector association are demonstrated. Wu and Zhao [21] studied a nonlocal and delayed diffusive HIV latent infection model with spatial heterogeneity and the effects of spatial heterogeneity and delays on viral dynamics are investigated. With a simple mathematical model, Gaythorpe and Adams [22] examine how demographic and environmental heterogeneities, population behavior, and behavioural change respond to the provision of facilities, and they also studied how to reduce epidemic size and endemic prevalence by the optimal configurations of limited numbers of facilities.This paper is also inspired by nonlocal and delayed reaction-diffusion systems in bounded domains [9, 10], which discussed the common influence of incubation period and SH on the spatial spread of disease. When considering infectious diseases with a fixed incubation period, the migration of infected individuals will lead to nonlocal infection [9, 10]. We assume that the host population lives in a bounded spatial habitat Ω with smooth boundary ∂Ω. At time t and location x, we denote by St,x, It,x, and Rt,x the densities of susceptible individuals, infectious individuals, and recovered individuals, respectively. In the absence of disease, susceptible individuals would approach a steady state; that is, St,x will be governed by the following equation:(1)∂S∂t=dS△S+λx−μSxS,t≥0,x∈Ω,∂St,x∂n=0,t>0,x∈∂Ω,where dS>0 represents the dispersal rate of susceptible individuals and Δ is the Laplacian operator. With respect to space variable x, λx and μSx, respectively, represent the recruitment rate and death rate of susceptible individuals depending on spatial variable x. If infection occurs, we adopt disease transmission functions as Beddington–DeAngelis functional response and use the following equations to describe the interactions between susceptible and infectious individuals:(2)∂S∂t=dSΔS+λx−μSxS−βxSI1+axS+bxI,t>0,x∈Ω,∂I∂t=dIΔI+βxSI1+axS+bxI−μIx+kxI,t>0,x∈Ω,∂St,x∂n=∂It,x∂n=0,t>0,x∈∂Ω,where dI>0 represents the diffusion rate of infective individuals. n is the outward normal along ∂Ω. βx is the transmission rate between susceptible and infectious individuals. Note that the diffusion rate may be different from susceptible to infectious individuals depending on the disease. Here, the Beddington–DeAngelis functional response can be considered as the extensions of Holling’s type II and saturation functional response, allowing a behavioural state, namely, ’mutual interference with competitors.’ ax measures the susceptible individuals interference. bx determine how fast the transmission/infection rate will approach saturation [23]. μIx and kx, respectively, represent the death rate and removed rate of infective individuals. All these functions are positive and Hölder continuous functions, which allow the SH due to the fact that spatial habitat environment is always different.We introduce agea representing the relapse age to recovered individuals. With relapse age a, time t, and location x, we denote by Rt,a,x the density of recovered individuals. By the standard arguments as in [24], we suppose recovered individuals are dominated by(3)∂Rt,a,x∂t+∂Rt,a,x∂a=dRΔRt,a,x−μRx+ra,xRt,a,x,t>0,x∈Ω,Rt,0,x=kxIt,x1+mxIt,x,t>0,x∈Ω,∂Rt,a,x∂n=0,t>0,x∈∂Ω,where dR>0 stands for the dispersal rate of recovered individuals. μRx is the death rate. ra,x represent the relapse rate from recovered individuals to infectious individuals with relapse age a and location x. mx measures how fast the recovery rate of infectious individuals will approach saturation.Biologically, we suppose thatτ>0 is the average (fixed) relapse period, denoted by(4)Ret,x=∫0τRt,a,xda,Rrt,x=∫τ∞Rt,a,xda.Assume that the functionra,x satisfies(5)ra,x=0,fora∈0,τandx∈Ω,rx,fora∈τ,∞andx∈Ω.We calculate the derivative ofRet,x and Rrt,x by using (3) and (5) to get(6)∂Re∂t=dR△Re−μRxRe+Rt,0,x−Rt,τ,x,and(7)∂Rr∂t=dR△Rr−μRx+rxRr+Rt,τ,x−Rt,∞,x,respectively. Assuming that Rt,∞,x=0, this means that ∂Ret,x/∂t and ∂Rrt,x/∂t will be obtained if Rt,τ,x is known. To this end, we integrate (3) along the characteristic line t=a+r by introducing ρr,a,x=Ra+r,a,x,r≥0. Hence, for a∈0,τ, we directly have(8)∂ρ∂a=∂R∂t+∂R∂at=a+r=dRΔρ−μRxρ,with(9)ρr,0,x=Rr,0,x=kxI1+mxI.It then follows that(10)ρr,a,x=∫ΩGa,x,ykyIr,y1+myIr,ydy,where G stands for the Green function to the operator dRΔ−μR⋅+rx. Changing the variable a by τ (hence r=t−τ), we yield(11)Rt,τ,x=ρt−τ,τ,x=∫ΩGτ,x,ykyIt−τ,y1+myIt−τ,ydy,∀t≥τ.Putting (11) into (6) and (7), respectively, and noting that Ret,x is decoupled from S, I, and Rr equation, we arrive at the following system:(12)∂S∂t=dSΔS+λx−μSxS−βxSI1+axS+bxI,t>0,x∈Ω,∂I∂t=dIΔI+βxSI1+axS+bxI−μIx+kxI+rxRr,t>0,x∈Ω,∂Rr∂t=dR△Rr−μRx+rxRr+∫ΩGτ,x,ykyIt−τ,y1+myIt−τ,ydy,t>0,x∈Ω,∂St,x∂n=∂It,x∂n=∂Rrt,x∂n=0,t>0,x∈∂Ω.For convenience, we denote(13)u1,u2,u3=S,I,Rr,d1,d2,d3=dS,dI,dR.Then in the sequel, we will study the following system:(14)∂u1∂t=d1Δu1+λx−μSxu1−βxu1u21+axu1+bxu2,t>0,x∈Ω,∂u2∂t=d2Δu2+βxu1u21+axu1+bxu2−μIx+kxu2+rxu3,t>0,x∈Ω,∂u3∂t=d3Δu3−μRx+rxu3+∫ΩGτ,x,ykyu2t−τ,y1+myu2t−τ,ydy,t>0,x∈Ω,∂u1t,x∂n=∂u2t,x∂n=∂u3t,x∂n=0,t>0,x∈∂Ω.We arrange the rest of this paper as follows. Section2 is devoted to the well-posedness of system (14). We follow the standard procedures in [25] to define ℜ0 for (14) by the next generation operator approach in Section 3. Moreover, in a homogeneous case, we calculate ℜ0 explicitly. In Section 4, ℜ0 will be verified that it takes a role of a threshold index for herpes extinction and persistence.


## 2. Well-Posedness of System

For convenience, we introduce the spaces and notations used in this paper.(i)
DenoteX≔CΩ¯,ℝ3 and the positive cone of X is denoted by X+≔CΩ¯,ℝ+3(ii)
DenoteC=C−τ,0,X and its positive cone is denoted by C+≔C−τ,0,X+, associated with the norm ϕ=maxθ∈−τ,0ϕθX, for τ≥0(iii)
Denote byut∈C the utθ=ut+θ,θ∈−τ,0, where u:−τ,σ⟶X for σ>0(iv)
DenoteY≔CΩ¯,ℝ and its positive cone is denoted by Y+≔CΩ¯,ℝ+(v)
Denote byTiti=1,2,3:Y⟶Y, t≥0, the strongly continuous semigroups with respect to the operators d1Δ−μS⋅ , d2Δ−μI⋅+k⋅, and d3Δ−μR⋅+r⋅, associated with no-flux boundary condition, respectively(vi)
DenoteE≔C−τ,0,Y×Y and its positive cone is denoted by E+≔C−τ,0,Y+×Y+Letut,⋅=u1t,⋅,u2t,⋅,u3t,⋅ be the solution corresponding to initial data u0=u10⋅,u20⋅,u30⋅. Let F=F1,F2,F3:C+⟶Y be defined as(15)F1ϕ⋅=λx−β⋅ϕ10,⋅ϕ20,⋅1+a⋅ϕ10,⋅+bxϕ20,⋅,F2ϕ⋅=β⋅ϕ10,⋅ϕ20,⋅1+a⋅ϕ10,⋅+b⋅ϕ20,⋅+r⋅ϕ30,⋅F3ϕ⋅=∫ΩGτ,⋅,ykyϕ2−τ,y1+myϕ2−τ,ydy,∀x∈Ω¯, ϕ=ϕ1,ϕ2,ϕ3T∈C+. It allows us to rewrite (14) as(16)u=u˜tϕ+∫0tu˜t−sFutsds,t>0u0=ϕ∈C+,,where u˜t=T1t,T2t,T3tT. According to [26] [Section 7.1 and Corollary 7.2.3]), we obtain that T1(t), T2(t), and T3(t) are compact and strongly positive on Y. Further, for small enough σ>0, we have(17)ϕ0,⋅+σFϕ⋅=ϕ10,⋅+σλ⋅−β⋅ϕ10,⋅ϕ20,⋅1+a⋅ϕ10,⋅+b⋅ϕ20,⋅ϕ20,⋅+σβ⋅ϕ10,⋅ϕ20,⋅1+a⋅ϕ10,⋅+b⋅ϕ20,⋅+r⋅ϕ30,⋅ϕ30,⋅+σ∫ΩGτ,⋅,ykyϕ2−τ,y1+myϕ2−τ,ydy,≥ϕ10,⋅1−σβ¯b¯ϕ20,⋅ϕ30,⋅,where β¯=maxx∈Ω¯βx and b¯=minx∈Ω¯bx. It follows that(18)limσ⟶0+1σdistϕ0+σFϕ,X=0,∀ϕ∈C+.It then follows from [27] [Corollary 4] and [28] [Corollary 8.1.3]) that, for any u0=ϕ∈C+, there exists a unique noncontinuable mild solution ut,ϕ on 0,tmax. Moreover, ut,ϕ∈X, ∀t∈0,tmax and ut,ϕ is a classical solution of (14) for t>τ.Before going into details, we first introduce a useful lemma; see also in [10] Lemma 1.Lemma 1.
Consider the following system:(19)∂ω∂t=d1Δω+λx−μSxω,t>0,x∈Ω,∂ωt,x∂n=0,t>0,x∈∂Ω.
Then system (19) admits a unique and global asymptotic stable positive steady state (PSS) Ex in Y+. Moreover, if both λ and μS are positive constants, then U=v/μS.
Notice that∂u1/∂t≤d1Δu1+λx−μSxu1. By the comparison theorem,(20)limsupt⟶∞u1t,x≤limsupt⟶∞ωt,x=Ex,uniformly forx∈Ω¯.
Hence, there existsK>0 such that(21)u1t,x≤K,t≥0,where K depends on initial data.Theorem 1.
For eachϕ∈C+, (14) admits a unique solution ut,ϕ on 0,∞. Let Φt=ut⋅:C+⟶C+,t≥0 be the solution semiflow of (14), defined by Φtϕθ,x=ut+θ,x,ϕ,∀θ∈−τ,0,x∈Ω¯, which has a global compact attractor.Proof.
It follows from (20) that u1t,ϕ is bounded on 0,tmax. Then there exists Q>0 such that u3-equation of (14) is dominated by(22)∂ω∂t=d3Δω−μRx+rxω+Q,t>0,x∈Ω,∂ωt,x∂n=0,t>0,x∈∂Ω.
By the comparison principle and Lemma1, u3t,ϕ is bounded on 0,tmax. Similarly, u2t,ϕ is bounded on 0,tmax. Hence, ut,ϕ=u1t,ϕ,u2t,ϕ,u3t,ϕT is bounded on 0,tmax, and hence tmax=+∞ for each ϕ∈C+. Therefore, the solution semiflow of system (14) is well defined.
For anyϕ∈C+, there exists t1>0, and we have(23)∂u3∂t≤d3Δu3−μRx+rxu3+k¯m¯,t>t1,x∈Ω,∂u3t,x∂n=0,t>t1,x∈∂Ω,where k¯=maxx∈Ω¯kx, m¯=minx∈Ω¯mx. Again, from Lemma 1, there is a time t2ϕ>t1 such that u3t,ϕ≤2k¯/m¯μR¯+r¯, ∀t>t2, where μR¯=minx∈Ω¯μRx and r¯=minx∈Ω¯rx. Further,(24)∂u2t,x∂t≤d2Δu2t,x+β¯λ¯b¯μs¯−μIx+kxu2t,x+2r¯k¯m¯μR¯+r¯,t>t2,x∈Ω,∂It,x∂n=0,t>t2,x∈∂Ω,where λ¯=maxx∈Ω¯λx and μs¯=minx∈Ω¯μsx. By Lemma 1, there is t2ϕ>t1 such that u2t,ϕ≤2β¯λ¯m¯μR¯+r¯+2b¯μS¯r¯k¯/b¯μS¯m¯μR¯+r¯μI¯+k¯, ∀t>t2, where μI¯=minx∈Ω¯μIx and k¯=minx∈Ω¯kx. Hence, Φt:C+⟶C+ is point dissipative. Further from [28] [Theorem 2.1.8], Φt:C+⟶C+ is compact for any t>τ. Hence, the last assertion directly follows from [29] [Theorem 3.4.8].Lemma 2.
For eachu0=ϕ∈C+, denote by ut,x,ϕ the solution of (14).(i)
Ifu2t0,⋅,ϕ≠0 for some t0≥0, then u2t0,⋅,ϕ≠0 and u2t,x,ϕ>0, ∀t>t0, x∈Ω¯.(ii)
u1t,x,ϕ>0, ∀t>0, x∈Ω¯ and(25)liminft⟶∞u1t,x≥λ¯μ¯S+β¯/b¯,uniformly forx∈Ω¯,where λ¯=minx∈Ω¯λx, μ¯S=maxx∈Ω¯μSx.Proof.
It is easy to see thatu2-equation satisfies(26)∂u2∂t≥d2Δu2−μ¯I+k¯u2,t>0,x∈Ω,∂u2t,x∂n=0,t>0,x∈∂Ω,where μ¯I=maxx∈Ω¯μIx. Then (i) holds directly from the comparison principle.
Denote byu^1t,x,ϕ the solution of(27)∂u^1∂t≥d1Δu^1+λx−μSx+β¯b¯u^1,∂u^1t,x∂n=0,x∈∂Ω.
Then from Lemma1, and the comparison principle, u1t,x,ϕ≥u^1t,x,ϕ>0, ∀t>0,x∈Ω¯, and last assertion of (ii) directly follows. This completes the proof.


## 3. Basic Reproduction Number

By settingu2=0, it is easy to find that the density of the susceptible individuals satisfies (19). Equation (14) possesses a disease-free steady state (DFSS), denoted by Ex,0,0. Linearizing system (14) at DFSS,(28)∂u2∂t=d2Δu2+βxEx1+axEx−μIx−kxu2+rxu3,t>0,x∈Ω,∂u3∂t=d3Δu3−μRx+rxu3+∫ΩGτ,x,ykyu2t−τ,ydy,t>0,x∈Ω,∂u2∂n=∂u3∂n=0,t>0,x∈∂Ω,which is a time-delayed and nonlocal linear system. In this circumstance, we first consider the following system:(29)∂u2∂t=d2Δu2+βxEx1+axEx−μIx−kxu2+rxu3,t>0,x∈Ω,∂u3∂t=d3Δu3−μRx+rxu3+∫ΩGτ,x,ykyu2t,ydy,t>0,x∈Ω,∂u2∂n=∂u3∂n=0,t>0,∀x∈∂Ω.The following nonlocal eigenvalue problem is obtained by insertingu2=eλtψx and u3=eλtψx into (29):(30)λψ1x=d2Δψ1x+βxEx1+axEx−μIx−kxψ1x+rxψ2x,x∈Ω,λψ2x=d3Δψ2x−μRx+rxψ2x+∫ΩGτ,x,ykyψ1ydy,x∈Ω,∂ψ1x∂n=∂ψ2x∂n=0,x∈∂Ω.A direct application of the result in [26] [Theorem 7.6.1] gives that (30) admits a principal eigenvalue λE equipped with a positive eigenvector.We now focus our attention on system (28). For any ψ∈E+/0, denote by ut,ψ,t≥0, the solution of (28). The following claim is valid.(i)
u2t,ψx>0 and u3t,ψx>0 for all x∈Ω¯ and t>τ.In fact, ifψ10,⋅≡0 or ψ20,⋅≡0, we then directly have(31)u2t,ψx>0,andu3t,ψx>0,for allx∈Ω¯,t>τ,from parabolic maximum principle. If there is some θ0∈0,τ that ψ1−θ0,⋅≡0, then u3τ−θ0,ψ≡0. If u3τ−θ0,ψ≡0, we have(32)∂u3τ−θ0,x∂t=∫ΩGτ,x,ykyu2−θ0,y1+myu2−θ0,ydy>0,∀x∈Ω.Byu3t,ψ≥0, t≥0, and u3τ−θ0,ψx=0, ∀x∈Ω¯, which implies that ∂u3τ−θ0,x/∂t≤0, a contradiction. This results in u3t,ψx>0, ∀t>τ−θ0, x∈Ω¯. Similarly, u2t,ψx>0, ∀t>τ−θ0, x∈Ω¯.With the help of [30] [Theorem 2.2], we give the result on the following eigenvalue problem:(33)λψ1x=d2Δψ1x+βxEx1+axEx−μIx−kxψ1x+rxψ2x,x∈Ω,λψ2x=d3Δψ2x−μRx+rxψ2x+e−λτ∫ΩGτ,x,ykyψ1ydy,x∈Ω,∂ψ1x∂n=∂ψ2x∂n=0,x∈∂Ω.Lemma 3.
Eigenvalue problem (33) has a principal eigenvalue λ¯E,τ with a strongly positive eigenvector. Moreover, λ¯E,τ has the same sign as λE,τ≥0.
Inspired by the standard procedures in [25], we next define ℜ0 for (14) by the next generation operator approach.(i)
Suppose that both the infective and the recovered individuals are near DFSS.(ii)
Introduceψ2x,ψ3xT as the spatial initial distribution of the infective and recovered individuals.(iii)
At timet, denote by Stψ≔T2tψ2,T3tψ3T the remaining distribution of infective and recovered individuals.(iv)
∀ψ∈Y×Y, let(34)Vψx=V1ψx,V2ψx,x∈Ω¯,where(35)V1ψx=βxEx1+axExψ2x+rxψ3x,and(36)V2ψx=∫ΩGτ,x,ykyψ2ydy.
As a result,VStψ is the newly infective and recovered distribution. Hence, by the general results in [31] (see also in [31], Lemma 2.2),(37)Lψ≔∫0∞VStψdt=V∫0∞Stψdt,which is the total infective and recovered distribution, which is called the next infection operator. The spectral radius of L is defined as ℜ0 of (14), i.e.,(38)ℜ0≔rL.
The following observation comes from [31].Lemma 4.
ℜ0−1 has the same sign as λE (or λ¯U,τ).
Generally, the above definition ofℜ0 is inconvenient for an application. For a special case, we compute it when all parameters are all independent of space variable, that is,(39)λx≡λ,μSx≡μS,βx≡β,ax≡a,bx≡b,μIx≡μI,kx≡k,rx≡r,μRx≡μR,mx≡m.
Then (14) reduces to(40)∂u1∂t=d1Δu1+λ−μSu1−βu1u21+au1+bu2,t>0,x∈Ω,∂u2∂t=d2Δu2+βu1u21+au1+bu2−μI+ku2+ru3,t>0,x∈Ω,∂u3∂t=d3Δu3−μR+ru3+∫ΩGτ,x,yku2t−τ,y1+mu2t−τ,ydy,t>0,x∈Ω,∂u1t,x∂n=∂u2t,x∂n=∂u3t,x∂n=0,t>0,x∈∂Ω.
Clearly, system (40) has the disease-free equilibrium λ/μS,0,0. The next generation operator L of (40) is given by(41)Lψ≔∫0∞VStψdt=V∫0∞Stψdt,ψ∈Y×Y,where(42)V=βU1+aUrk∫ΩGτ,x,ydy0,and(43)Stψ=∫ΩG2t,⋅,ydyψ2∫ΩG3t,⋅,ydyψ3,ψ∈Y×Y.
Recall thatGit,⋅,yi=2,3 are Green functions associated with d2Δ−μI+k and d3Δ−μR+r obeying the no-flux boundary condition. Then ∫ΩG2t,⋅,ydy=e−μI+kt and ∫ΩG3t,⋅,ydy=e−μR+rt. For any s>0, we get(44)∫ΩGs,x,ydy=e−μR+rs.
Then the next infection operator defined by (41) becomes(45)Lψ2ψ3=βU1+aUrke−μR+rτ01μI+k001μR+rψ2ψ3=βUμI+k1+aUrμR+rke−μR+rτμI+k0ψ2ψ3.
It follows that(46)ℜ0=βU2μI+k1+aU+β2U24μI+k21+aU2+kre−μR+rτμI+kμR+r.


## 4. Threshold Dynamics

Theorem 2.
For anyu0=ϕ∈C+, denote by ut,x,ϕ the solution of (14). The DFSS is globally attractive provided that ℜ0<1.Proof.
From Lemma4, we know that λE<0. Due to(47)limε⟶0λE+ε=λE<0,we can choose ε0>0 where λE+ε0<0. Fixing ε0>0, by Lemma 1, and choosing t0=t0ϕ as u1≤Ex+ε0, ∀t≥t0, x∈Ω¯. It follows that(48)∂u2∂t≤d2Δu2+βxEx+ε01+axEx+ε0−μIx−kxu2+rxu3,t≥t0,x∈Ω,∂u3∂t≤d3Δu3−μRx+rxu3+∫ΩGτ,x,ykyu2t−τ,ydyt≥t0,x∈Ω.
From Lemma3, let ψ0 be the strongly positive eigenfunction to λ¯E+ε0,τ<0. Hence the linear system(49)∂v1∂t=d2Δv1+βxEx+ε01+axEx+ε0−μIx−kxv1+rxv2,t>0,x∈Ω,∂v2∂t=d3Δv2−μRx+rxv2+∫ΩGτ,x,ykyv1t−τ,ydy,t>0,x∈Ω,∂v1t,x∂n=∂v2t,x∂n=0,x∈∂Ω,has a solution vt,⋅=eλ¯E+ε0,τtψ0⋅. Choose α>0 large enough that(50)u2t,⋅,ϕ,u3t,⋅,ϕT≤αvt,⋅,∀t∈t0−τ,t0.
From the comparison principle, we directly have(51)u2t,⋅,ϕ,u3t,⋅,ϕT≤αeλ¯E+ε0,τtψ0⋅,∀t≥t0,which implies that(52)limt⟶∞u2t,x,ϕ,u3t,x,ϕ=0,0,uniformly forx∈Ω¯.
Consequently,u1-equation is asymptotic to (19). From the standard arguments for asymptotically autonomous semiflows (we refer to [32] [Corollary 4.3]), limt⟶∞u1t,x,ϕ=Ex. This proves Theorem 2.Theorem 3.
For eachu0=ϕ∈C+, denote by ut,x,ϕ the solution of (14). If ℜ0>1 and ϕi0,⋅≡0 for i=2,3, then there is sufficiently small η>0 that(53)liminft⟶∞Γt,x≥η,uniformly forx∈Ω¯,where Γ=u1,u2,u3, respectively. Further, (14) possesses at least one PSS u∗x=u1∗x,u2∗x,u3∗x.Proof.
From Lemma3, we get λ¯U,τ>0. Define(54)W0=ϕ∈C+:ϕ20,⋅≡0andϕ30,⋅≡0,and(55)∂W0=C+W0=ϕ∈C+:ϕ20,⋅≡0orϕ30,⋅≡0.
In these settings, Lemma2 tells us that u2t,x,ϕ>0 and u3t,x,ϕ>0,∀x∈Ω¯,t>0; that is, ΦtW0⊂W0. Define(56)M∂≔ϕ∈∂W0:Φtϕ∈∂W0,t≥0.
In what follows, we prove two claims.Claim 1.
ωψ=DFSS,∀ψ∈M∂, where ωϕ is the omega limit set of the orbit Φtϕ:∀t≥0.
In fact, for eachψ∈M∂, Φtψ∈∂W0, ∀t≥0. Obviously, for each t≥0, either u2t,⋅,ψ≡0 or u3t,⋅,ψ≡0. We next confirm the claim in two cases. If u2t,⋅,ψ≡0 for all t≥0, from Lemma 1, we directly have limt⟶∞u1t,x,ψ=Ex. As to u3-equation, it gives that limt⟶∞u3t,x,ψ=0. If u2t0,⋅,ψ≡0 for some t0≥0, then u2t,x,ψ>0,∀t>t0,x∈Ω¯ (from Lemma 2). Hence, u3t,⋅,ψ≡0,∀t≥t0. Then from u2-equation of (14), we get limt⟶∞u2t,x,ψ=0. Consequently, with the help of the standard arguments of asymptotically autonomous semiflows, u1 satisfies limt⟶∞u1t,x,ψ=Ex.Claim 2.
limsupt⟶∞Φtϕ−DFSS≥δ0, for all ϕ∈W0.
We will prove this claim by way of contradiction. Iflimsupt⟶∞Φtϕ0−DFSS<δ0 for some ϕ0∈W0, we can choose large enough t∗>0 as u1t,⋅,ϕ0>Ex−δ0 and u2t,⋅,ϕ0<δ0,∀t>t∗,x∈Ω¯. Hence, we have(57)∂u2∂t≥d2t,xΔu2+βxEx−δ01+axEx−δ0+bxδ0−μIx−kxu2+rxu3,∂u3∂t≥d3Δu3−μRx+rxu3+∫ΩG2τ,x,ykyu2t−τ,y1+myδ0dy,for ∀t>t∗,x∈Ω. Denote by φ0 the positive eigenvector to λ¯E−δ0,τ. Then system(58)∂v1∂t=d2Δv1+βxEx−δ01+axEx−δ0+bxδ0−μIx−kxv1+rxv2,∂v2∂t=d3Δv2−μRx+rxv2+∫ΩG2τ,x,ykyv1t−τ,y1+myδ0dy,∂v1t,x∂n=∂v2t,x∂n=0,x∈∂Ω,possesses the solution vt,⋅=eλ¯U−δ0,τtφ0⋅. Since u2t,x,ϕ0≫0 and u3t,x,ϕ0≫0, we can choose small enough ξ>0 that(59)u2t,⋅,ϕ0,u3t,⋅,ϕ0T≥ξvt,x,∀t∈t1−τ,t1,x∈Ω¯.
By (57), together with the comparison principle, we directly obtain(60)u2t,⋅,ϕ0,u3t,⋅,ϕ0T≥ξeλ¯E−δ0,τtφ0⋅,∀t>t1,x∈Ω¯.
By choosing a small enoughδ0 that λ¯U−δ0,τ>0, it implies that u2t,⋅,ϕ0 and u3t,⋅,ϕ0 are unbounded, a contradiction. This proves Claim 2.
LetD⋅:C+⟶ℝ+ be a generalized distance function defined by(61)Dϕ=minminx∈Ω¯ϕ20,x,minx∈Ω¯ϕ30,x,∀ϕ∈C+.
Clearly,D−10,∞⊂W0. By Lemma 2, D⋅ enjoy the property that if Dϕ=0 and ϕ∈W0 or Dϕ>0, then DΦtϕ>0 for all t>0 (see [33]). Notice that any forward semiflow of Φt in M∂ converges to DFSS, and DFSS is isolated in C+ with WsDFSS∩W0=0. Here WsDFSS represents the stable set of DFSS. Moreover, no cycle is formed in M∂ from DFSS to DFSS. In summary, by [19] [Theorem 2.3], we arrive at the conclusion that, for any ϕ∈W0, choosing η>0 small enough can give minDψ:ψ∈ωϕ>η. Hence,(62)liminft⟶∞u2t,⋅≥η,liminft⟶∞u3t,⋅≥η,∀x∈Ω¯.
On the other hand, from Lemma2 and letting η small enough, we obtain that liminft⟶∞u1t,x≥η,∀x∈Ω¯. This proves the first part of Theorem 3.
By [34] [Theorem 3.7 and Remark 3.10], Φt:W0⟶W0 has a global compact attractor. Further from [34] [Theorem 4.7], the semiflow Φt admits a PSS u∗x∈W0. Lemma 2 ensures that u∗x is a strictly PSS of (14).


## 5. Simulations

In this section, we mainly focus on the effects of the spatial heterogeneity and the diffusion rates of individuals on the disease dynamics. For simplicity, we assume thatΩ=0,1. Let dS=0.3, dI=0.1, dR=0.2, τ=0.4, λx=5, μSx=0.01, μIx=0.05, μRx=0.15, rx=0.2, ax=1,bx=1, mx=0.1, βx=0.15×1+0.5cos2πx, kx=0.1×1+0.2sin2πx. Then ℜ0=1.2644 and the disease is persistent (see Figure 1(a) with initial values St,x=20×1+0.2sin2πx, It,x=5×1+0.5sin2πx, Rrt,x=10×1+0.2cos2πx for t,x∈−τ,0×0,1). Assume that βx=0.09×1+0.5cos2πx, It,x=100×1+0.5sin2πx for t,x∈−τ,0×0,1), and other parameters and initial values are the same as those in Figure 1(a); then ℜ0=0.9496 and the disease is extinct (see Figure 1(b)).Figure 1
The time evolution of the densities ofIt,x for different ℜ0. (a)ℜ0>1; (b)ℜ0<1.
(a)(b)Now, we show the influence of diffusion rate onℜ0 and the number of infected individuals. Let βx=0.45×1+0.5cos2πx, kx=0.3×1+0.2sin2πx, and let other parameters be the same as those in Figure 1 except for dI; it is shown that ℜ0 decreases as dI increases (see Figure 2(a)). Furthermore, let βx=0.27×1+0.5cos2πx and other parameters be the same as those in Figure 1 except for dI; we find that the number of infected individuals It,x at steady state also decreases as dI increases (see Figure 2(b) with the same initial values as those in Figure 1).Next, we show the influence of spatial heterogeneity onℜ0. Let βx=0.45×1+0.5sin2πx, and let other parameters be the same as those in Figure 1 except for kx; then ℜ0 is nonincreasing as the heterogeneity parameter c of kx increases (see Figure 3(a) with kx=0.3×1+0.8csin2πx). However, if we take βx=0.45×1+0.5cos2πx and let other parameters be the same as those in Figure 3(a), then ℜ0 is nondecreasing as the heterogeneity parameter c of kx increases (see Figure 3(b) with kx=0.3×1+0.8csin2πx). Therefore, the effects of the spatial heterogeneity may be dependent on the actual environment.Figure 2
The influence of diffusion rate on infected individuals. (a)dI=0.1c. (b) The evolution of It,x.
(a)(b)Figure 3
The influence of spatial heterogeneity onℜ0 with kx=0.3×1+0.8csin2πx. (a) βx=0.45×1+0.5sin2πx; (b) βx=0.45×1+0.5cos2πx.
(a)(b)


## 6. Conclusion and Discussion

This paper performs a complete analysis on the threshold-type dynamics of a diffusive herpes model, like in [10] where the spatial movement of mosquitoes in EIP will result in nonlocal infection. Here we introduce a relapse age to be a continuous variable and nonlinear recovery rate. We conducted a complete analysis of model (14) by adopting a fixed relapse period, which can be regarded as a continuous work of [10]. In Theorem 1, we confirmed that (14) admits a unique solution ut,ϕ on 0,∞. Consequently, the solution semiflow generated by the solution of (14) possesses a global attractor in C+. By using the next generation operator approach, we characterize ℜ0 as its spectral radius. To build up the relation between ℜ0 and the principle eigenvalue of associated eigenvalue problem, we establish the relation between the principle eigenvalue of (30) and the principle eigenvalue of (33). To proceed further, we consider the special case where parameters are all independent of x and calculate ℜ0 explicitly. By utilizing the principal eigenvalue of the associated eigenvalue problem or equivalently by ℜ0, we establish the threshold-type dynamics of the model in the sense that the DFSS is globally attractive provided that ℜ0<1, while if ℜ0>1, system (14) is uniformly persistent and (14) possesses at least one PSS u∗x=u1∗x,u2∗x,u3∗x. This threshold-type result implies that, under the changing of the sign of ℜ0−1, the herpes will be extinct or close to the epidemic value. From the explicit expression of ℜ0 in (46), we know that it is closely related to the parameters. Compared to ℜ0 defined by the spectral radius of next generation operator, it is a challenging to analyze the impact of parameters on ℜ0, if it is not impossible. On the other hand, ℜ0 is related to the principal eigenvalue of an elliptic system with two infected components, which makes the analysis more difficult. We perform the numerical simulations to verify the theoretical results and study the effect of the spatial heterogeneity in one-dimensional space Ω=0,1. Threshold-type results are illustrated in Figure 1. In Figure 2, we can see the influence of diffusion rate on ℜ0 and the number of infected individuals. Specifically, ℜ0 decreases as dI increases (see Figure 2(a)) and the number of infected individuals It,x at steady state also decreases as dI increases (see Figure 2(b)). Figure 3 demonstrates the influence of spatial heterogeneity on ℜ0. We found that, with different βx, ℜ0 is nonincreasing or nondecreasing in terms of kx. Therefore, the effects of the spatial heterogeneity may be dependent on the actual environment.



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*Source: 6039640-2021-12-30.xml*