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

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 byR0, 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.


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][2][3][4][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 S(t), I(t), and R(t), 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, R 0 is determined as the sharp threshold for determining whether the herpes is extinct or close to the epidemic value. e 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 P(t) to explore the results of distinct settings on the relapse period, where P(t) 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. ey also found that there is no sustained oscillatory solutions. After determining R 0 , the threshold-type results of the model were also addressed. e 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 reactiondiffusion models have attracted many researchers. e 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][9][10][11][12][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 R 0 . Here, R 0 was achieved by the spectral radius of the next generation operator. In a homogeneous case that all model parameters are constant, R 0 can be explicitly obtained. Besides, another method of calculating R 0 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. e authors in [18] studied the local and global longterm 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 humanvector 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.
is 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 zΩ. At time t and location x, we denote by S(t, x), I(t, x), and R(t, 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, S(t, x) will be governed by the following equation: where d S > 0 represents the dispersal rate of susceptible individuals and Δ is the Laplacian operator. With respect to space variable x, λ(x) and μ S (x), 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: where d I > 0 represents the diffusion rate of infective individuals. n is the outward normal along zΩ. β(x) 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.' a(x) measures the susceptible individuals interference. b(x) determine how fast the transmission/infection rate will approach saturation [23]. μ I (x) and k(x), 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 age a representing the relapse age to recovered individuals. With relapse age a, time t, and location x, we denote by R(t, a, x) the density of recovered individuals. By the standard arguments as in [24], we suppose recovered individuals are dominated by 2 Complexity where d R > 0 stands for the dispersal rate of recovered individuals. μ R (x) is the death rate. r(a, x) represent the relapse rate from recovered individuals to infectious individuals with relapse age a and location x. m(x) measures how fast the recovery rate of infectious individuals will approach saturation. Biologically, we suppose that τ > 0 is the average (fixed) relapse period, denoted by We calculate the derivative of R e (t, x) and R r (t, x) by using (3) and (5) to get and respectively. Assuming that R(t, ∞, x) � 0, this means that zR e (t, x)/zt and zR r (t, x)/zt will be obtained if with It then follows that where G stands for the Green function to the operator Putting (11) into (6) and (7), respectively, and noting that R e (t, x) is decoupled from S, I, and R r equation, we arrive at the following system: For convenience, we denote en in the sequel, we will study the following system: We arrange the rest of this paper as follows. Section 2 is devoted to the well-posedness of system (14). We follow the standard procedures in [25] to define R 0 for (14) by the next generation operator approach in Section 3. Moreover, in a homogeneous case, we calculate R 0 explicitly. In Section 4, R 0 will be verified that it takes a role of a threshold index for herpes extinction and persistence.

Well-Posedness of System
For convenience, we introduce the spaces and notations used in this paper.
Before going into details, we first introduce a useful lemma; see also in [10] Lemma 1.

Lemma 1. Consider the following system:
en system (19) admits a unique and global asymptotic stable positive steady state (PSS) E(x) in Y + . Moreover, if both λ and μ S are positive constants, then Hence, there exists K > 0 such that where K depends on initial data. Proof. It follows from (20) that u 1 (t, ϕ) is bounded on [0, t max ). en there exists Q > 0 such that u 3 -equation of (14) is dominated by By the comparison principle and Lemma 1, , and hence t max � +∞ for each ϕ ∈ C + . erefore, the solution semiflow of system (14) is well defined.
where λ � min Proof. It is easy to see that u 2 -equation satisfies where μ I � max x∈Ω μ I (x). en (i) holds directly from the comparison principle. Denote by u 1 (t, x, ϕ) the solution of en from Lemma 1, and the comparison principle, u 1 (t, x, ϕ) ≥ u 1 (t, x, ϕ) > 0, ∀t > 0, x ∈ Ω, and last assertion of (ii) directly follows. is completes the proof.

Basic Reproduction Number
By setting u 2 � 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 (E(x), 0, 0). Linearizing system (14) at DFSS, which is a time-delayed and nonlocal linear system. In this circumstance, we first consider the following system: 6 Complexity e following nonlocal eigenvalue problem is obtained by inserting u 2 � e λt ψ(x) and u 3 � e λt ψ(x) into (29): A direct application of the result in [26] [ eorem 7.6.1] gives that (30) admits a principal eigenvalue λ(E) equipped with a positive eigenvector.
(i) Suppose that both the infective and the recovered individuals are near DFSS. (ii) Introduce (ψ 2 (x), ψ 3 (x)) T as the spatial initial distribution of the infective and recovered individuals.
the remaining distribution of infective and recovered individuals. where and As a result, V(S(t)ψ) is the newly infective and recovered distribution. Hence, by the general results in [31] (see also in [31], Lemma 2.2), which is the total infective and recovered distribution, which is called the next infection operator. e spectral radius of L is defined as R 0 of (14), i.e., e following observation comes from [31].
Lemma 4. R 0 − 1 has the same sign as λ(E) (or λ(U, τ)). Generally, the above definition of R 0 is inconvenient for an application. For a special case, we compute it when all parameters are all independent of space variable, that is, en (14) reduces to Clearly, system (40) has the disease-free equilibrium (λ/μ S , 0, 0). e next generation operator [L] of (40) is given by where and Recall that G i (t, ·, y)(i � 2, 3) are Green functions associated with d 2 Δ − (μ I + k) and d 3 Δ − (μ R + r) obeying the no-flux boundary condition. en Ω G 2 (t, ·, y)dy � e − (μ I +k)t and Ω G 3 (t, ·, y)dy � e − (μ R +r)t . For any s > 0, we get en the next infection operator defined by (41) becomes 8 Complexity It follows that

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 that Ω � (0, 1).
Next, we show the influence of spatial heterogeneity on R 0 . Let β(x) � 0.45 × (1 + 0.5 sin(2πx)), and let other parameters be the same as those in Figure 1 except for k(x); then R 0 is nonincreasing as the heterogeneity parameter c of k(x) increases (see Figure 3(a) with k(x) � 0.3 × (1 + 0.8c sin(2πx))). However, if we take β(x) � 0.45 × (1 + 0.5 cos(2πx)) and let other parameters be the same as those in Figure 3(a), then R 0 is nondecreasing as the heterogeneity parameter c of k(x) increases (see Figure 3(b) with k(x) � 0.3 × (1 +0.8c sin(2πx))). erefore, the effects of the spatial heterogeneity may be dependent on the actual environment.

Conclusion and Discussion
is paper performs a complete analysis on the thresholdtype 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 eorem 1, we confirmed that (14) admits a unique solution u(t, 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 R 0 as its spectral radius. To build up the relation between R 0 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 R 0 explicitly. By utilizing the principal eigenvalue of the associated eigenvalue problem or equivalently by R 0 , we establish the threshold-type dynamics of the model in the sense that the DFSS is globally attractive provided that R 0 < 1, while if R 0 > 1, system (14) is uniformly persistent and (14) possesses at least one PSS u * (x) � (u * 1 (x), u * 2 (x), u * 3 (x)). is threshold-type result implies that, under the changing of the sign of R 0 − 1, the herpes will be extinct or close to the epidemic value. From the explicit expression of [R 0 ] in (46), we know that it is closely related to the parameters. Compared to R 0 defined by the spectral radius of next generation operator, it is a challenging to analyze the impact of parameters on R 0 , if it is not impossible. On the other hand, R 0 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  e influence of spatial heterogeneity on R 0 with k(x) � 0.3 × (1 + 0.8c sin(2πx)). (a) β(x) � 0.45 × (1 + 0.5 sin(2πx)); (b) β(x) � 0.45 × (1 + 0.5 cos(2πx)). 12 Complexity results and study the effect of the spatial heterogeneity in one-dimensional space Ω � (0, 1). reshold-type results are illustrated in Figure 1. In Figure 2, we can see the influence of diffusion rate on R 0 and the number of infected individuals. Specifically, R 0 decreases as d I increases (see Figure 2(a)) and the number of infected individuals I(t, x) at steady state also decreases as d I increases (see Figure 2(b)). Figure 3 demonstrates the influence of spatial heterogeneity on R 0 . We found that, with different β(x), R 0 is nonincreasing or nondecreasing in terms of k(x). erefore, the effects of the spatial heterogeneity may be dependent on the actual environment.

Data Availability
is paper is a theoretical research and therefore there is no underlying data.

Conflicts of Interest
e authors declare that they have no conflicts of interest.