GLOBAL DYNAMICS AND TRAVELLING WAVE SOLUTIONS FOR A CLASS OF NON-COOPERATIVE REACTION-DIFFUSION SYSTEMS WITH NONLOCAL INFECTIONS

. We consider a class of non-cooperative reaction-diﬀusion system, which includes diﬀerent types of incidence rates for virus dynamical models with nonlocal infections. Threshold dynamics are expressed by basic reproduction number R 0 in the following sense, if R 0 < 1, the infection-free steady state is globally attractive, implying infection becomes extinct; while if R 0 > 1, virus will persist. To study the invasion speed of virus, the existence of trav- elling wave solutions is studied by employing Schauder’s ﬁxed point theorem. The method of constructing super-solutions and sub-solutions is very techni- cal. The mathematical diﬃculty is the problem constructing a bounded cone to apply the Schauder’s ﬁxed point theorem. As compared to previous mathe- matical studies for diﬀusive virus dynamical models, the novelty here is that we successfully establish the general existence result of travelling wave solutions for a class of virus dynamical models with complex nonlinear transmissions and nonlocal infections.


1.
Introduction. Mathematical models have been shown to be an effective and valuable approach to understand virus infection dynamics in the within-host environment. The dynamical properties of human immunodeficiency virus-1 (HIV-1), hepatitis B virus (HBV) and human T-cell leukemia type-1 (HTLV-1) infections have been discussed with the help of mathematical models. Most of these works are based on the assumption that cells and viruses are well mixed.
Spatial structure is important to understand the dynamical behaviour of virus infection. In [38], Wang and Wang proposed a model to simulate the HBV infection with spatial dependence. They assume that target cells and infected cells cannot move, while virus can move according to Fickian diffusion. For this model, the non-existence of travelling wave solutions is studied by employing the geometric singular perturbation method. The existence of travelling wave solutions is observed numerically. In [53], Xu and Ma made a further investigation for the HBV model with saturation response. Global dynamics for steady states were discussed by constructing the coupled lower-upper solution. In [61], a diffusive HBV model 3214 WEI WANG AND WANBIAO MA with delayed Beddington-DeAngelis response was proposed. Global dynamics and travelling wave solutions were investigated, only considering the diffusion of virus. Some further developments have been performed on diffusive virus dynamical models (see, [44,45]).
Some recent studies reveal that high concentration of infected cells can promote the diffusion of virus [9]. [26] established a diffusive virus model to study the repulsion effect of super-infecting virions by infected cells (see, also [46]). Numerical computations of the spreading speed have shown that the repulsion of super-infecting virions can promote the spread of virus (see, [26,46]).
As is known to all, the process of productive infection of CD4 + T cell with HIV-1 is very complicated, which can be decomposed into several steps: entry of virus into the cell, reverse transcription of virus RNA to DNA, and integration of viral DNA into the host-cell genome [1]. In the past few years, it has been realized that the decreasing of CD4 + T cells induced by two factors: one is the natural death, and the other is apoptosis which has been considered as the main pathway to cause the decreasing of CD4 + T cells (see, for example, [2,29]). However, recent studies in [1] and [4] reveal that only 5% of CD4 + T cells die due to caspase-3-mediated apoptosis. Most of CD4 + T cells die due to the caspase-1-mediated pyroptosis in non-activated CD4 + T cells that have undergone abortive infection [1,8]. Dying infected CD4 + T cells can release inflammatory signals which attract more uninfected CD4 + T cells to die [4]. [51] firstly proposed the following virus infection dynamical model with the caspase-1-mediated pyroptosis ∂V (x, t) ∂t = D 0 ∆V + Ω Γ(τ, x, y) βU (y, t − τ )ω(y, t − τ ) 1 + aω(y, t − τ ) dy (1) In model (1), U (x, t), V (x, t), M (x, t) and ω(x, t) describe the concentrations of uninfected cells, infected cells, inflammatory cytokines IL-1 and virus at time t and location x, respectively. ξ is the target cell production rate. β is the infection rate. k is the virus production rate. α 1 is the death rate of infected CD4 + T cells which are caused by pyroptosis. α 2 is the production rate of inflammatory cytokines IL-1 released from infected cells. Inflammatory cytokines IL-1 is assumed to induce the death of uninfected CD4 + T cells qU M/(1 + bM ). The natural death rates of uninfected cells, infected cells, inflammatory cytokines and virus are d U , d V , d M and d ω . The diffusion rates of uninfected cells, infected cells, inflammatory cytokines and virus are D 0 , D 0 , D 1 and D 2 . Time delay τ is the average incubation period. Here Γ is the Green function associated with ∆ and zero-flux boundary conditions, which satisfies Ω Γ(τ, x, y)dy = 1, ∀x ∈ Ω, τ > 0. Γ(τ, x, y) = Γ(τ, y, x) and Γ(τ, x, y) > 0 ∀x ∈ Ω, x = y, τ > 0 (see, [10]). All the parameters in model (1) are assumed to be positive constants.
Usually, an infectious case is firstly found at one location and then the virus spreads to other areas. Thus, an important question for virus infection is: what is the spreading speed? In general, it is challenging to compute the asymptotic spreading speed. For most of cooperative systems, the asymptotic speed equals the minimal wave speed [19]. However, for some non-cooperative systems, it is challenging to obtain the asymptotic spreading speed, especially for diffusive SIR models and virus models. Travelling wave solutions are an important tool which can be used to describe the spreading speed of population [6,15,7,14,22,56,59,16,54,43,25,17,18,57,3,41,49,62,30]. [20] and [36] have developed the general theory on the existence of travelling wave solutions for monotonic (or cooperative) systems. Obviously, this method can not be applied to the non-monotonic model (1) . In the recent years, many methods for the existence of travelling wave solutions for non-cooperative systems have been established. Shooting method is firstly proposed by Dunbar in [6] to investigate the existence of travelling wave solutions for the Lotka-Volterra predatorprey system (non-monotonic). Recently, [17] developed a geometric method to study the existence of travelling wave solutions for a class of non-monotonic systems consisting of two equations with general functions. In [18], Huang made a further investigation for the model proposed in [17] to abandon the restriction condition on the diffusion coefficients. The Schauder's fixed point theorem is also widely applied to show the existence of travelling wave solutions connecting two steady states [15,7,22,56,59,54,43,25,17,18,57,3]. In [57], Zhang et al. developed another method to show the existence of weak travelling wave solutions for a class of non-cooperative systems, which allows us to avoid the difficulties in studying the detailed final state (i.e., steady states, periodic solutions, etc.) The non-cooperative models in above literatures mostly consist of two equations. To the best of our knowledge, there are few literatures about the existence result of minimal wave speed for non-cooperative systems consisting of more than three equations with complicated nonlinear interaction functions. In [58], Zhang investigated the existence of weak travelling wave solutions for a class of non-cooperative reaction-diffusion systems consisting of three equations. However, the results obtained in [58] cannot be directly applied to study the existence of travelling wave solutions for a class of non-cooperative reaction-diffusion systems with a discrete delay and spatial non-locality. Thus, it is important and necessary to establish the general result for travelling wave solutions for the non-cooperative system with complicated nonlinear interaction functions and a discrete delay and spatial non-locality, which is constituted by more than three equations.
In this paper, we use zero-flux boundary conditions where ν is the outward normal to ∂Ω, and the initial conditions The purpose of the paper is to discuss global dynamics of model (2) in the case of a bounded domain and the existence of travelling wave solutions of model (2) in the case of an unbounded domain.
The remainder of the paper is organised as follows. In Section 2, we establish global dynamics for virus dynamical models with nonlinear transmissions and nonlocal infections in the case of a bounded spatial domain. In Section 3, the existence of travelling wave solutions is studied in the case of an unbounded domain. Finally, some discussions and conclusions are given in Section 4.
We are now in the position to study the well-posedness of model (2) in the sense of the following theorem.
and has a compact global attractor.
Proof. We find that model (2) defines a semiflow: From [21], we have that ξ r1 is the globally attractive steady state for the parabolic equations The comparison principle implies there exists t 1 (φ) > 0 such that H(x, t) ≤ 2ξ r1 =: Hence, the existence of solutions u(t, ·, φ) of model (2) claimed in Lemma 2.1 is indeed global (i.e., t φ = ∞). The solution semiflow is point dissipative. According to Theorem 2.2.6 in [52], we get that Φ(t) is compact for any t ≥ 0. Thus, from Theorem 3.4.8 in [13], we know that Φ(t) has a compact global attractor in C + .
In the following, we establish the threshold-type result on the extinction and uniform persistence of the virus in terms of the basic reproduction number for model (2) in a bounded spatial domain.
Obviously, it is easy to see that model (2) always exists a unique infection-free satisfying the following boundary conditions We consider the nonlocal eigenvalue problem of model (5) associated with We firstly consider the following model Substituting u 2 (x, t) = e λt φ 1 (x) and u 4 (x, t) = e λt φ 2 (x) into equations of u 2 and u 4 , we have the following eigenvalue problem of model (7) λφ By the similar argument to Theorem 7.6.1 in [32], eigenvalue problem (8) has a principal eigenvalue λ 0 with a positive eigenfunction. From Theorem 2.2 in [34], we get the following result.
Lemma 2.3. The eigenvalue problem (6) has a principal eigenvalue λ 0 with a strictly positive eigenfunction, and for any τ ≥ 0, λ 0 has the same sigh as λ 0 .
From Theorem 2.3 in [47] (see, also [5,35,37,11]), we can show that the basic reproduction number R 0 of model (2) equals the spectral radius of the following and hence By Theorem 3.1 (i) in [37], we then obtain the following Lemma.
Therefore, there exists a unique λ c ∈ (0, c 2D ) such that Based on the analysis above, we have the following Lemma.
Hence, we have that In the first equality, the mean value theorem is used and P 11 = (U 0 − θ 2 φ, θ 2 γ), We need to show If t > − 1 ε ln Q, then ψ(t) = 0. It is easy to obtain the result. If t ≤ − 1 ε ln Q, theṅ and If t > − 1 ε ln Q, then γ(t) = 0. It is easy to obtain the result. If t ≤ − 1 ε ln Q, theṅ . Hence, we get that This completes the proof.
Note that Ξ ∈ Π is a fixed point of the operator of F . By employing L'Höspital rule to the maps F i , (i = 1, 2, 3, 4), it then follows that lim The proof of the main result stated in Theorem 3.2 is completed.
Remark 1. Generally speaking, it is a challenging problem to study the existence of travelling wave solutions by constructing suitable super-solutions and sub-solutions connecting two steady states for the applications of Schauder's fixed point theorem. In Theorem 3.2, the profile of travelling wave solutions has the infection-free steady state as its limit as −∞. However, we cannot show the asymptotic property of profile in the other end +∞ in mathematics. However, in [24], Li and Zou first offer an algorithm for numerically computing travelling wave solutions for spatially non-local equations which is very challenging. By employing the method proposed in [24], numerically, we find that the profile of travelling wave solutions has the infection steady state as its limit as +∞ for model (1) (see, [51] Section 6).

Discussions and conclusions.
In this paper, we study a class of non-cooperative reaction-diffusion system, which includes different types of incidence rates for virus dynamical models with nonlinear transmissions and nonlocal infections. In Theorem 2.5, we have obtained the threshold-type result in a bounded domain. Our threshold-type result (Theorem 2.5) shows that basic reproduction number R 0 may be used to design the control strategies of the disease transmission and to estimate the infection level. In the case where R 0 > 1, we may obtain an approximate value of the infection level from the persistence of model (2), and then change some parameters to drive R 0 < 1 so that the disease can be eradicated ultimately.
To study the invasion speed of virus, in Theorem 3.2, we investigate the existence of travelling wave solutions by employing Schauder's fixed point theorem. To the best of our knowledge, there are few literatures about the existence result of minimal wave speed for non-cooperative systems consisting of more than three equations with complicated nonlinear interaction functions and nonlocal infection.
The mathematical difficulty in the proof of Theorem 3.2 is the problem constructing a bounded cone to apply the Schauder's fixed point theorem. Thanks for Lemma 3.1, we construct two continuous functions Φ(t) and Ψ(t), which are useful for constructing the profile set. Then the bounded cone is achieved through a pair of super-solutions and sub-solutions. This approach is very technical, and has been proven successfully in establishing the general existence result of travelling wave solutions for virus dynamical models with more complex nonlinear transmissions and nonlocal infection.
As a final remark, we should point out that, in the present paper, we did not derive whether c * is the minimal wave speed, that is, there do not exist travelling wave solutions for 0 < c < c * . A detailed analysis of this problem will be challenging and we leave this as a further project.