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, , for the model through the next generation operator approach. Moreover, in a homogeneous case, we calculate the explicitly. By utilizing the principal eigenvalue of the associated eigenvalue problem or equivalently by , 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.

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., [15]). 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 , , and , 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, 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 to explore the results of distinct settings on the relapse period, where 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 , 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, [813]. 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 . Here, was achieved by the spectral radius of the next generation operator. In a homogeneous case that all model parameters are constant, can be explicitly obtained. Besides, another method of calculating 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 with smooth boundary . At time and location , we denote by , , and 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, will be governed by the following equation:where represents the dispersal rate of susceptible individuals and is the Laplacian operator. With respect to space variable , and , respectively, represent the recruitment rate and death rate of susceptible individuals depending on spatial variable . 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 represents the diffusion rate of infective individuals. is the outward normal along . 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.’ measures the susceptible individuals interference. determine how fast the transmission/infection rate will approach saturation [23]. and , 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 representing the relapse age to recovered individuals. With relapse age , time , and location , we denote by the density of recovered individuals. By the standard arguments as in [24], we suppose recovered individuals are dominated bywhere stands for the dispersal rate of recovered individuals. is the death rate. represent the relapse rate from recovered individuals to infectious individuals with relapse age and location . measures how fast the recovery rate of infectious individuals will approach saturation.

Biologically, we suppose that is the average (fixed) relapse period, denoted by

Assume that the function satisfies

We calculate the derivative of and by using (3) and (5) to getandrespectively. Assuming that , this means that and will be obtained if is known. To this end, we integrate (3) along the characteristic line by introducing . Hence, for , we directly havewith

It then follows thatwhere stands for the Green function to the operator . Changing the variable by (hence ), we yield

Putting (11) into (6) and (7), respectively, and noting that is decoupled from , , and equation, we arrive at the following system:

For convenience, we denote

Then 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 for (14) by the next generation operator approach in Section 3. Moreover, in a homogeneous case, we calculate explicitly. In Section 4, 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)Denote and the positive cone of is denoted by (ii)Denote and its positive cone is denoted by , associated with the norm , for (iii)Denote by the , where for (iv)Denote and its positive cone is denoted by (v)Denote by , , the strongly continuous semigroups with respect to the operators , , and , associated with no-flux boundary condition, respectively(vi)Denote and its positive cone is denoted by

Let be the solution corresponding to initial data . Let be defined as, . It allows us to rewrite (14) aswhere . 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 . Further, for small enough , we havewhere and . It follows that

It then follows from [27] [Corollary 4] and [28] [Corollary 8.1.3]) that, for any , there exists a unique noncontinuable mild solution on . Moreover, , and is a classical solution of (14) for .

Before going into details, we first introduce a useful lemma; see also in [10] Lemma 1.

Lemma 1. Consider the following system:Then system (19) admits a unique and global asymptotic stable positive steady state (PSS) in . Moreover, if both and are positive constants, then .
Notice that . By the comparison theorem,Hence, there exists such thatwhere depends on initial data.

Theorem 1. For each , (14) admits a unique solution on . Let be the solution semiflow of (14), defined by , which has a global compact attractor.

Proof. It follows from (20) that is bounded on . Then there exists such that -equation of (14) is dominated byBy the comparison principle and Lemma 1, is bounded on . Similarly, is bounded on . Hence, is bounded on , and hence for each . Therefore, the solution semiflow of system (14) is well defined.
For any , there exists , and we havewhere , . Again, from Lemma 1, there is a time such that , , where and . Further,where and . By Lemma 1, there is such that , , where and . Hence, is point dissipative. Further from [28] [Theorem 2.1.8], is compact for any . Hence, the last assertion directly follows from [29] [Theorem 3.4.8].

Lemma 2. For each , denote by the solution of (14).(i)If for some , then and , , .(ii), , andwhere , .

Proof. It is easy to see that -equation satisfieswhere . Then (i) holds directly from the comparison principle.
Denote by the solution ofThen from Lemma 1, and the comparison principle, , , and last assertion of (ii) directly follows. This completes the proof.

3. Basic Reproduction Number

By setting , 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 . Linearizing system (14) at DFSS,which is a time-delayed and nonlocal linear system. In this circumstance, we first consider the following system:

The following nonlocal eigenvalue problem is obtained by inserting and into (29):

A direct application of the result in [26] [Theorem 7.6.1] gives that (30) admits a principal eigenvalue equipped with a positive eigenvector.

We now focus our attention on system (28). For any , denote by , the solution of (28). The following claim is valid.(i) and for all and .

In fact, if or , we then directly havefrom parabolic maximum principle. If there is some that , then . If , we have

By , , and , , which implies that , a contradiction. This results in , , . Similarly, , , .

With the help of [30] [Theorem 2.2], we give the result on the following eigenvalue problem:

Lemma 3. Eigenvalue problem (33) has a principal eigenvalue with a strongly positive eigenvector. Moreover, has the same sign as ,.
Inspired by the standard procedures in [25], we next define for (14) by the next generation operator approach.(i)Suppose that both the infective and the recovered individuals are near DFSS.(ii)Introduce as the spatial initial distribution of the infective and recovered individuals.(iii)At time , denote by the remaining distribution of infective and recovered individuals.(iv), letwhereandAs a result, 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. The spectral radius of is defined as of (14), i.e.,The following observation comes from [31].

Lemma 4. has the same sign as (or ).
Generally, the above definition of is inconvenient for an application. For a special case, we compute it when all parameters are all independent of space variable, that is,Then (14) reduces toClearly, system (40) has the disease-free equilibrium . The next generation operator of (40) is given bywhereandRecall that are Green functions associated with and obeying the no-flux boundary condition. Then and . For any , we getThen the next infection operator defined by (41) becomesIt follows that

4. Threshold Dynamics

Theorem 2. For any , denote by the solution of (14). The DFSS is globally attractive provided that .

Proof. From Lemma 4, we know that . Due towe can choose where . Fixing , by Lemma 1, and choosing as , , . It follows thatFrom Lemma 3, let be the strongly positive eigenfunction to . Hence the linear systemhas a solution . Choose large enough thatFrom the comparison principle, we directly havewhich implies thatConsequently, -equation is asymptotic to (19). From the standard arguments for asymptotically autonomous semiflows (we refer to [32] [Corollary 4.3]), . This proves Theorem 2.

Theorem 3. For each , denote by the solution of (14). If and for , then there is sufficiently small thatwhere , respectively. Further, (14) possesses at least one PSS .

Proof. From Lemma 3, we get . DefineandIn these settings, Lemma 2 tells us that and ; that is, . DefineIn what follows, we prove two claims.

Claim 1. , where is the omega limit set of the orbit .
In fact, for each , , . Obviously, for each , either or . We next confirm the claim in two cases. If for all , from Lemma 1, we directly have . As to -equation, it gives that . If for some , then (from Lemma 2). Hence, . Then from -equation of (14), we get . Consequently, with the help of the standard arguments of asymptotically autonomous semiflows, satisfies .

Claim 2. , for all .
We will prove this claim by way of contradiction. If for some , we can choose large enough as and . Hence, we havefor . Denote by the positive eigenvector to . Then systempossesses the solution . Since and , we can choose small enough thatBy (57), together with the comparison principle, we directly obtainBy choosing a small enough that , it implies that and are unbounded, a contradiction. This proves Claim 2.