Minimal Wave Speed for a Nonlocal Viral Infection Dynamical Model

: To provide insights into the spreading speed and propagation dynamics of viruses within a host, in this paper, we investigate the traveling wave solutions and minimal wave speed for a degenerate viral infection dynamical model with a nonlocal dispersal operator and saturated incidence rate. It is found that the minimal wave speed c ∗ is the threshold that determines the existence of traveling wave solutions. The existence of traveling fronts connecting a virus-free steady state and a positive steady state with wave speed c ≥ c ∗ is established by using Schauder’s fixed-point theorem, limiting arguments


Introduction
Although huge advances have been made in preventing and treating HIV and viral hepatitis, such as antiretroviral treatment for HIV and vaccination programs for the hepatitis B virus (HBV), HIV and HBV pandemics remain a major global public health problem.It is reported that there were 36.9 million people living with HIV worldwide in 2017 and 257 million people and 71 million people in 2015 were living with HBV and hepatitis C virus (HCV), respectively.Meanwhile, 0.94 million people in 2017 and 1.3 million people in 2015 died from AIDS-related and hepatitis-related illnesses, respectively [1,2].Therefore, we have a long way to go to control and extinguish these viral infectious diseases.
To understand the pathogenesis of viruses within the host and then propose more effective control measures, many different methods have been developed.In particular, mathematical models have been verified as an effective method [3,4].In 2000, Nowak and May [5] proposed the following basic viral infection model: du(t) dt = βw(t)v(t) − µu(t), dv(t) dt = pu(t) − γv(t), (1) where w(t), u(t), and v(t) denote the concentrations of healthy target cells, infected cells, and free virions at time t, respectively.s is the recruitment rate of healthy target cells.b, µ, and γ represent the death rates of healthy target cells, infected cells, and free virions, respectively.The infectious incidence rate is βw(t)v(t) and p is the virus production rate.All the parameters in model (1) are positive.System (1) has a virus-free equilibrium point E 0 = (w 0 , 0, 0), which is globally asymptotically stable when R 0 ≤ 1, where w 0 = s/b and R 0 = pβs/(bµγ) = pβw 0 /(µγ).If R 0 > 1, system (1) admits at a unique positive equilibrium point that is globally asymptotically stable [6].Since then, many works concerning the impacts of various factors on within-host viral dynamics have been conducted using mathematical models [7,8].Although the incidence rate in most of these viral models adopts a bilinear function response, this may be not so appropriate when the concentration of virions is high.In this case, the saturation effect may cause a viral response rate that is less than linear.Hence, it is more reasonable to adopt a saturation nonlinear incidence rate βw(t)v m (t)/(1 + αv n (t)), where m, n, α > 0. The case where m = n = 1 has been studied in viral infection models by several researchers, including [9,10].
Note that many studies on viral infection models assume that the within-host environments are homogeneous, and ignore the impact of heterogeneous environments and the mobility of virions or cells.However, virions or cells may move within and between tissues and may face different environments in different locations within the host, which would consequently impact the dynamics of the virus [11,12].Thus, it is more reasonable to incorporate spatial factors into the models, which have been studied by some researchers [13][14][15].Strain et al. [13] introduced a lattice cellular automaton model to investigate the contribution of three-dimensional spatial correlations in viral propagation.Wang and Wang [14] proposed a degenerate HBV infection model with a local dispersal operator and investigated the existence of traveling wave solutions.Lai and Zou [15] established a reaction-diffusion viral infection model with a repulsion effect and investigated its spreading speed and the existence of traveling wave solutions.Most of these studies assume that the virions or cells diffuse in the form of local dispersal and follow Fickian diffusion, which can only be used to study situations where the density of the species is relatively low and the species diffuses in a small range [16].However, the concentrations of virions and cells are relatively high within tissues, which suggests that nonlocal dispersal may be more reasonable in viral infection models.Moreover, the nonlocal dispersal operator can be viewed as an approximation of the local dispersal operator when the kernel function takes a special form [17]. Recently, Zhao and Ruan [18] assumed that the virions diffuse in the form of the nonlocal mode in domain Ω ∈ R n (n ≥ 1), and subsequently proposed and analyzed the following nonlocal viral infection model: where w(x, t), u(x, t), and v(x, t) are the concentrations of the target cells, infected cells, and free virions at time t and location x, respectively.d v represents the diffusion rate of the virions.Here, J(x − y) can be viewed as the probability that virions jump from location y to location x and J(x − y) = J(y − x).Thus, the nonlocal dispersal operator Ω J(x − y)[v(y, t) − v(x, t)]dy includes not only the rate that virions arrive at location x from other locations ( Ω J(x − y)v(y, t)dy), but also the leaving rate of virions at location x ( Ω J(y − x)v(x, t)dy).Other parameters have the same meanings as those in model (1).The authors in [18] investigated the threshold dynamics of model ( 2) and the impact of the dispersal rate on solutions of (2).
In the process of viral transmission, there is evidence exhibiting that virions can spread in a way like a traveling wave front [13].Thus, if the virus diffusion takes the form of nonlocal dispersal in an unbounded domain, two interesting questions arise: (1) Can the model exhibit traveling wave solutions or not?(2) What is the spreading speed of the virus?Additionally, accurate estimates of the spreading speed, especially at the early stage of viral infection, can provide insights into how the virus propagates.From a mathematical point of view, estimates of the spreading speed can usually be obtained by studying the asymptotic spreading speed, which is relative to minimal wave speed.In this paper, inspired by the above-mentioned arguments, we intend to study the traveling wave solutions and minimal wave speed problems of the following viral infection model with the saturation incidence rate: where Here, the domain is R, and the incidence rate is in saturated mass action βw(x, t)v(x, t)/(1 + αv(x, t)).Other parameters are the same as those in (2).Throughout this paper, we assume that the dispersal kernel J satisfies J is compactly supported and R J(x)e λx dx < +∞ for all λ > 0. Clearly, system (3) always admits in a virus-free steady state E 0 = (w 0 , 0, 0), where w 0 = s/b.Moreover, the ODE system associated with system (3) admits a unique positive steady state is the basic reproduction number of the corresponding ODE system.In the rest of this paper, we always assume that R 0 > 1 holds.The traveling wave solution of ( 3) is a positive solution (w(x, t), u(x, t), v(x, t)) of (3) which has the form where c > 0 is the wave speed.A positive traveling wave solution (W(ξ), U(ξ), V(ξ)) is called a traveling semifront of (3) if it satisfies lim It is clear that the traveling wave solution (W(ξ), U(ξ), V(ξ)) satisfies (5) where The viral infection model is neither a cooperative system nor a competitive system, which together with the existence of the recruitment term of healthy target cells infers that the classic methods, such as the monotone semiflow method, the shooting method, and connection index theory, are all not valid.Meanwhile, as far as we know, few mathematical works have been performed to study the existence of traveling wave solutions and the minimal wave speed in viral infection models [14,[19][20][21], especially for nonlocal systems.Furthermore, the nonlocal dispersal operator causes the solutions of system (3) to lack regularity and compactness, which may lead to new difficulties in analysis.Recently, Wang and Ma [22] investigated the traveling wave solution problem for a nonlocal HIV infection model with a Beddington-DeAngelis functional response, where they assumed that all cells and virions can nonlocally diffuse but have the same diffusion ability.They proved the existence of traveling wave solutions for c ≥ c * , but there are some additional conditions for c = c * .The existence of traveling wave solutions for c = c * and the nonexistence for c < c * were further studied in [23].It is worth noting that using the Lyapunov function is an effective method to show the traveling wave solutions connect to the positive steady state.However, not only upper-bound estimations of the solution are required, but also lower-bound estimations, which is also a challenge for nonlocal systems.In particular, only free virions can diffuse in our model, which may also lead to some challenges.In this paper, we will overcome the aforementioned difficulties to obtain traveling wave solutions and the minimal wave speed of system (3) by utilizing Schauder's fixed-point theorem, the rescaling method, the comparison principle, and so on.
The paper is organized as follows.In the next section, we establish the existence or nonexistence of traveling wave solutions with wave speed c > 0, and give the minimal wave speed of system (3).In Section 3, we discuss the results.Some conclusions are presented in the final section.

Traveling Wave Solutions
In this section, we mainly focus on the traveling wave solutions and the minimal wave speed of system (3).Firstly, we establish the existence of traveling fronts for c > c * .Secondly, we show the existence of traveling fronts for c = c * .Finally, the nonexistence of traveling fronts is investigated for 0 < c < c * .

The Existence of Traveling Fronts for c > c *
In this subsection, we first give the definition of c * and then study the existence of traveling semifronts for c > c * .
In the following, we always assume that R 0 > 1 and c > c * .Let where σ,ϵ 1 , ϵ and M can be defined later.We always assume that Other restrictions on M can be found later.For convenience, denote Proof.By the definitions of W(ξ) and V(ξ), we have In the case of ξ < x 1 , we have U(ξ) = φ 0 e λ c ξ .It follows from V(ξ) ≤ ϕ 0 e λ c ξ that one has Therefore, the above two cases yield that Note that by the fact that U(ξ) ≤ φ 0 e λ c ξ .This completes the proof.□ Lemma 4. For ϵ 1 ∈ (0, λ c ) and σ > max{w 0 , βw 0 ϕ 0 /b}, the function W(ξ) satisfies Proof.If ξ > x 3 , then W(ξ) = 0, and the conclusion clearly holds.Thus, it needs only to be shown that the conclusion is valid for ξ < This completes the proof.□ Lemma 5. Let ϵ be small enough to satisfy 0 < ϵ < min{λ c , ϵ 1 } and M be large enough.Then, the functions U(ξ) and V(ξ) satisfy Proof.We only consider the case in which x 4 ≤ x 5 ; the others can be considered similarly.Let Then, Obviously, the first inequality holds for ξ > x 4 , and the second one is valid for ξ > x 5 . If Directly, one has which together with ξ < 0 yields that where Then, it follows from the standard theory of ordinary differential equations that system (11) admits a unique solution (W Lemma 6.The operator F satisfies F(Γ X ) ⊂ Γ X .Moreover, operator F is completely continuous.
Proof.By using Lemmas 3-5 and similar arguments to those in ([24], Theorem 2.5), it is easy to show that F(Γ X ) ⊂ Γ X .Now, we show that F is completely continuous.Let (W X (ξ), U X (ξ), V X (ξ)) be the unique solution of system (11) with (ψ(ξ), φ(ξ), ϕ(ξ)) ∈ Γ X .Then, we can obtain that For any (ψ Then, by the definition of (W X are uniformly bounded on [−X, X] according to Equation (11).Therefore, we can get that operator F is compact on Γ X .This completes the proof.□ It is obvious that Γ X is a closed and convex set.Then, it follows from Lemma 6 and Schauder's fixed-point theorem that operator F admits a fixed point For simplicity, we drop the superscript * and denote the fixed point as (W X (ξ),U X (ξ),V X (ξ)) in the following. Define Theorem 1.There exists a positive constant C * independent of X such that and where Obviously, VX (ξ) ≤ pβw 0 /(αµγ) for any ξ ∈ R.
By simple calculations, we can obtain that Similarly, we can obtain It follows from assumption (H) that the kernel function J is Lipschitz continuous.Let Q be its Lipschitz constant.Then, by similar arguments to the proofs in ([25], Theorem 2.8), it is easy to show Combining the above arguments, the conclusion follows.This completes the proof.
Proof.Suppose by contradiction that there exists a nondecreasing sequence
With the aid of Lemmas 7 and 8, the existence of traveling fronts of system (3) for c > c * can be obtained as follows.
Theorem 2. Suppose that R 0 > 1.For any c > c * , system (3) admits a traveling front with wave speed c.
Inspired by [22,[26][27][28], we use the Lyapunov method to show that this conclusion holds.Let f (z) = z − 1 − ln z, χ 1 (z) = +∞ z J(y)dy, χ 2 (z) = z −∞ J(y)dy.By the assumption (H), without loss of generality, we assume that the compact support of J is [−r, r].Then, it is clear that and (W(ξ), U(ξ), V(ξ)) is the solution of system (5).It is clear that L(W, U, V)(ξ) is bounded from below by Lemmas 7 and 8. Following similar calculations to those in [22,27], one has Thus, Proof.The proof is divided into the following three steps.
For a bounded function φ(ξ) ≥ 0, define its two-sided Laplace transform as Denote the two-sided Laplace transform of U(ξ) and V(ξ) by L U (λ) and L V (λ), respectively.Obviously, λ + U ≥ ρ and λ + V ≥ ρ.Taking the two-sided Laplace transform on both sides of the second and third equations of system (5), we get that the first equation of (27) implies that λ + U ≥ λ + V .Then, for λ ∈ (0, min {λ + (c), λ + V }), where λ + (c) is defined in Lemma 1, the first equation of ( 27) yields that and the second equation of ( 27) implies that Hence, Since 0 < c < c * , Lemmas 1 and 2 infer that a contradiction.This completes the proof.□ Remark 2. Theorems 2, 3 and 4 imply that c * defined in Lemma 2 is the minimal wave speed of system (3).

Discussion of Results
It was found that when the kernel function takes a special form, the model with a nonlocal dispersal operator exhibits similar wave propagation properties to the model with a fractional Laplacian operator [32].In fact, fractional Laplacian and fractional derivatives are special cases of nonlocal dispersal operators [33,34].As far as we know, there are few results on the propagation dynamics of the degenerate viral dynamical model with fractional diffusion or a nonlocal dispersal operator.Thus, the results obtained in this paper can not only provide some insights into the spreading speed and the propagation dynamics of a virus but also provide a basis for the propagation properties of a viral dynamical model with fractional diffusion.
Recall that system (3) is neither a cooperative system nor a competitive system.At present, there are still some difficulties in giving an exact expression for the asymptotic spreading speed of system (3) and in elucidating the relationship between the minimal wave speed and the asymptotic spreading speed.In the following, we show some numerical arguments by using MATLAB R2016a.We divide the simulation into two steps.

•
Choose an appropriate spatial domain and then discretize it.We take the domain to be [−500, 500].The discretization step size is 0.2, which results in 5001 ordinary differential equations.Under our specified parameters and initial values, the viruses are always away from the boundaries of the domain during our simulation.

•
Let 0.05 be the time step.We use the ode45 function in Matlab to solve the ordinary differential equations for numerical simulation.
In addition, inspired by [35], we use the slope of the boundaries of the virus's spreading domain to estimate the asymptotic spreading speed of the virus.
• Let 0.05 be the time step.We use the ode45 function in Matlab to solve the ordinary differential equations for numerical simulation.
In addition, inspired by [35], we use the slope of the boundaries of the virus's spreading domain to estimate the asymptotic spreading speed of the virus.
We now give the estimation of the asymptotic spreading speed and show the relationship between the minimal wave speed and the asymptotic spreading speed of system (3) by simulations.Let the parameters values be s = 2.6 × 10 4 cells mL −1 day −1 , µ = 0.26 day −1 , p = 2.9 virions day −1 cells −1 , β = 2.25 × 10 −7 mL day −1 virions −1 , γ = 6.0 day −1 , b = 0.0026 day −1 , which were used for HCV infectious transmission [36].Then, the basic reproduction number R 0 = 4.1827 > 1.Additionally, we assume that α = 1 × 10 −7 mL virions −1 , d v = 0.1 mm 2 day −1 , J(x) with compact support [−r, r] satisfies and the initial data w(x, 0) = 1 × 10 7 for x ∈ R, u(0, 0) = 200, v(0, 0) = 1500, u(x, 0) = v(x, 0) = 0 for all x ∈ R \ {0}.Setting the radius of compact support as r = 2, we can get that the minimal wave speed c * = 0.3177 by Lemma 2 and find that system (3) admits a non-monotonic traveling front which has a hump in the profile (see Figure 1a).Let v * = 0.0001 be the threshold value above which the virus can be detected.It is found that the asymptotic spreading speed is approximately equal to 0.32 > c * (see Figure 1b), which implies that the asymptotic spreading speed may be larger than the minimal wave speed.Next, we studied the influences of the diffusion ability d v and the radius r of compact support on the minimal wave speed c * .Figure 2 shows that c * increases as d v or r increases (the parameter values are fixed to those in Figure 1 except for d v or r).Hence, decreasing the diffusion ability or diffusion radius may postpone the spread of the virus.Next, we studied the influences of the diffusion ability d v and the radius r of compact support on the minimal wave speed c * .Figure 2 shows that c * increases as d v or r increases (the parameter values are fixed to those in Figure 1 except for d v or r).Hence, decreasing the diffusion ability or diffusion radius may postpone the spread of the virus.
Finally, we investigated the influences of the diffusion mode on the spreading speed.Assume that the virions can move either in the form of nonlocal dispersal or in the form of local dispersal (Laplace diffusion).Let the parameter values and initial data be the same as those in Figure 1 except for the radius r of compact support.Figure 3 shows that the solutions have a large hump for both local and nonlocal dispersals.It also shows that the virus with nonlocal dispersal spreads faster than the virus with local dispersal when the radius r is larger, while the inverse is true when the radius r is smaller.Thus, there may exist a threshold value r * such that a virus with nonlocal dispersal and a virus with local dispersal have the same asymptotic spreading speed when r = r * and a virus with nonlocal dispersal spreads faster (slower) than a virus with local dispersal when r > r * (r < r * ).
Hence, nonlocal dispersal can postpone the spread of a virus when the diffusion radius is smaller and accelerate the spread of a virus when the diffusion radius is larger.In fact, it is found that the minimal wave speed for nonlocal dispersal is smaller than the minimal wave speed for local dispersal when the diffusion radius is small enough, and it can surpass the minimal wave speed for local dispersal when the radius increases (see Figure 2b), where the minimal wave speed for local dispersal can be defined by similar arguments to those in Section 2. The radius r of compact support Finally, we investigated the influences of the diffusion mode on the spreading speed.Assume that the virions can move either in the form of nonlocal dispersal or in the form of local dispersal (Laplace diffusion).Let the parameter values and initial data be the same as those in Figure 1 except for the radius r of compact support.Figure 3 shows that the solutions have a large hump for both local and nonlocal dispersals.It also shows that the virus with nonlocal dispersal spreads faster than the virus with local dispersal when the radius r is larger, while the inverse is true when the radius r is smaller.Thus, there may exist threshold value r * such that a virus with nonlocal dispersal and a virus with local dispersal have the same asymptotic spreading speed when r = r * and a virus with nonlocal dispersal spreads faster (slower) than a virus with local dispersal when r > r * (r < r * ).Hence, nonlocal dispersal can postpone the spread of a virus when the diffusion radius is smaller and accelerate the spread of a virus when the diffusion radius is larger.In fact, it is found that the minimal wave speed for nonlocal dispersal is smaller than the minimal wave speed for local dispersal when the diffusion radius is small enough, and it can surpass the minimal wave speed for local dispersal when the radius increases (see Figure 2b), where the minimal wave speed for local dispersal can be defined by similar arguments to those in Section 2.

Conclusions
Inspired by the phenomenon of viruses spreading like traveling waves [13], and considering the actual situation of virus transmission, we established a degenerate viral infection dynamical model with a nonlocal dispersal operator and analyzed the existence of traveling wave solutions of the model.We proved the existence of traveling wave solutions connecting the virus-free steady state and the positive steady state with wave speed c ≥ c * ,  The radius r of compact support Finally, we investigated the influences of the diffusion mode on the spreading speed.Assume that the virions can move either in the form of nonlocal dispersal or in the form of local dispersal (Laplace diffusion).Let the parameter values and initial data be the same as those in Figure 1 except for the radius r of compact support.Figure 3 shows that the solutions have a large hump for both local and nonlocal dispersals.It also shows that the virus with nonlocal dispersal spreads faster than the virus with local dispersal when the radius r is larger, while the inverse is true when the radius r is smaller.Thus, there may exist a threshold value r * such that a virus with nonlocal dispersal and a virus with local dispersal have the same asymptotic spreading speed when r = r * and a virus with nonlocal dispersal spreads faster (slower) than a virus with local dispersal when r > r * (r < r * ).Hence, nonlocal dispersal can postpone the spread of a virus when the diffusion radius is smaller and accelerate the spread of a virus when the diffusion radius is larger.In fact, it is found that the minimal wave speed for nonlocal dispersal is smaller than the minimal wave speed for local dispersal when the diffusion radius is small enough, and it can surpass the minimal wave speed for local dispersal when the radius increases (see Figure 2b), where the minimal wave speed for local dispersal can be defined by similar arguments to those in Section 2.

Conclusions
Inspired by the phenomenon of viruses spreading like traveling waves [13], and considering the actual situation of virus transmission, we established a degenerate viral infection dynamical model with a nonlocal dispersal operator and analyzed the existence of traveling wave solutions of the model.We proved the existence of traveling wave solutions connecting the virus-free steady state and the positive steady state with wave speed c ≥ c * , as well as the nonexistence of traveling wave solutions with 0 < c < c * .Thus, we can conclude that c * defined in Lemma 2 is the minimal wave speed of system (3).It is worth

Conclusions
Inspired by the phenomenon of viruses spreading like traveling waves [13], and considering the actual situation of virus transmission, we established a degenerate viral infection dynamical model with a nonlocal dispersal operator and analyzed the existence of traveling wave solutions of the model.We proved the existence of traveling wave solutions connecting the virus-free steady state and the positive steady state with wave speed c ≥ c * , as well as the nonexistence of traveling wave solutions with 0 < c < c * .Thus, we can conclude that c * defined in Lemma 2 is the minimal wave speed of system (3).It is worth mentioning that the lower-bound estimation of the traveling wave solutions was achieved by adopting rescaling methods and the comparison principle, which is a challenge for some nonlocal models.While other methods may exist, our method is much simpler and can be easily adapted for application to other models with nonlocal dispersal.
Furthermore, the relationship between the minimal wave speed and the asymptotic spreading speed and the influences of the diffusion mode and diffusion ability on the minimal wave speed or the asymptotic spreading speed were investigated via simulations.Both the theoretical and numerical simulation results indicate the existence of traveling wave solutions of system (3), which is consistent with the evidence presented in [13].Based on the simulations, we conclude that the asymptotic spreading speed may be larger than the minimal wave speed, and decreasing the diffusion ability or diffusion radius may postpone the spread of the virus.Nonlocal dispersal can postpone the spread of the virus when the diffusion radius is smaller and accelerate the spread when the diffusion radius is larger.
For the proposed model in this study, there is a typical characteristic, i.e., the target cell cannot move freely within the host, which is suitable for HBV or HCV infections.However, due to the diversity of viruses, there also exists some viruses, such as HIV or HTLV, for which their susceptible target cells and infected cells can move freely within the host and may have different mobilities.Therefore, if we consider nonlocal dispersal and different mobilities in both the target cells and virions, two interesting questions naturally arise that are worth further study: can the virions propagate as a traveling wave front, and what is its minimal wave speed?Moreover, during our analysis, we assumed that the kernel function K(x) is symmetric.However, the actual environment is very complex, and the virions may diffuse asymmetrically within the host.The traveling wave solution and minimal wave speed of a model with an asymmetric dispersal kernel function should be further studied.

Theorem 3 .
Then, by similar arguments to those in ([22], Theorem 2.1) or ([29], Theorem 2.3), the conclusion is valid.□ 2.2.The Existence of a Traveling Front with Wave Speed c = c * Assume that R > 1 and c = c * .Then, system (3) admits a traveling front with wave speed c * .

Figure 1 .
Figure 1.Solutions of system (3).(a) Evolution of virus population.(b) Evolution of the virus spreading domain.

Figure 1 .
Figure 1.Solutions of system (3).(a) Evolution of virus population.(b) Evolution of the virus spreading domain.

Figure 2 .
Figure 2. The influence of parameters on minimal wave speed.(a) The influence of diffusion ability on c * (nonlocal dispersal).(b) The influence of radius r of compact support.

Figure 3 .
Figure 3.The concentration of v(x, 1300) for local or nonlocal operator.

Figure 2 .
Figure 2. The influence of parameters on minimal wave speed.(a) The influence of diffusion ability on c * (nonlocal dispersal).(b) The influence of radius r of compact support.

Figure 2 .
Figure 2. The influence of parameters on minimal wave speed.(a) The influence of diffusion ability on c * (nonlocal dispersal).(b) The influence of radius r of compact support.

Figure 3 .
Figure 3.The concentration of v(x, 1300) for local or nonlocal operator.

Figure 3 .
Figure 3.The of v(x, 1300) for local or nonlocal operator.