Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment

Nonlinear Analysis: Real World Applications 46 (2019) 178–194

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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa

Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment Yongli Cai a , Xinze Lian b , Zhihang Peng c , Weiming Wang a ,∗ a b c

School of Mathematical Science, Huaiyin Normal University, Huaian, 223300, PR China Oujiang College, Wenzhou University, Wenzhou, 325035, PR China School of Public Health, Nanjing Medical University, Nanjing, 211166, PR China

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Article history: Received 29 January 2018 Received in revised form 28 August 2018 Accepted 28 September 2018 Available online xxxx Keywords: Heterogenous environment Basic reproduction number Disease-free equilibrium Uniformly persistent Influenza disease

abstract In this paper, we propose an SIRS influenza model with general incidence rate to describe disease transmission in a heterogenous environment. The basic reproduction number R0 is defined for the model, which can be used to govern the threshold dynamics of influenza disease: if R0 < 1, the unique disease-free equilibrium is globally asymptotic stable and there is no endemic equilibrium, while R0 > 1, there is at least one endemic equilibrium and the disease is uniformly persistent. Epidemiologically, we find that the spatial heterogeneity can enhance the infectious risk of the influenza and thus, in order to control the spread of the influenza, we must increase the recovery rate and the spatial heterogeneity in the transmission rate, or people should change their travelling plan and stay at home to reduce the value of the diffusion coefficient. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Influenza is an acute viral infection caused by an influenza virus, which is transmitted from one person to another through respiratory droplets produced as a consequence of coughing or sneezing by an infected person [1,2]. Influenza is a serious public health problem that causes severe illness and death in high risk populations. World Health Organization (WHO) reported that influenza occurs globally with an annual attack rate estimated at 5%–10% in adults and 20%–30% in children, and worldwide, these annual epidemics are estimated to result in about 3,000,000 to 5,000,000 cases of severe illness, and about 290,000 to 650,000 deaths [3]. Research estimates that 99% of deaths in children under 5 years of age with influenza related lower respiratory tract infections are found in developing countries [4]. There are 4 types of seasonal influenza viruses, types A, B, C and D, which circulate worldwide and can affect anybody in any age group [3]. According to the combinations of various virus surface proteins, type ∗

Corresponding author. E-mail addresses: [email protected] (Y. Cai), [email protected] (X. Lian), [email protected] (Z. Peng), [email protected] (W. Wang). https://doi.org/10.1016/j.nonrwa.2018.09.006 1468-1218/© 2018 Elsevier Ltd. All rights reserved.

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A influenza viruses are further classified into subtypes, such as influenza A(H1N1) and A(H3N2) which are currently circulating among humans. Influenza A virus infection has been one of the most serious public health challenges globally and attained an unprecedented degree of attention in recent years [5–8]. Also, only influenza type A viruses are known to have caused pandemics [3]. It is now widely believed that the mathematical models have been revealed as a powerful tool to understand the mechanism that underlies the spread of influenza. The simplest scheme that can be considered to model influenza spread is a deterministic homogeneous susceptible-infected-recovered-(re)susceptible (SIRS) [9–13] or SIRS-type [14–18] compartmental models which are usually reasonable to provide valuable insights into evolutionary dynamics of influenza A in humans by antigenic drift. These influenza epidemic models, it is worthy to note, are based on the assumption of “homogeneous mixing”, with susceptible, infectious and recovered individuals mixing uniformly, without regard to location or other such factors [19]. A classical SIRS model proposed by Anderson and May [20] is as follows: ⎧ dS(t) ⎪ ⎪ = Λ − µS(t) − βS(t)I(t) + γR(t), ⎪ ⎪ ⎪ ⎨ dt dI(t) = βS(t)I(t) − (µ + α + δ)I(t), ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dR(t) = δI(t) − (µ + γ)R(t), dt

(1.1)

where S, I, R denote the number of the population that are susceptible, infectious and recovered with temporary immunity, respectively. All parameters are assumed to be positive constants, and Λ is a constant recruitment of susceptible individuals, µ the natural mortality of populations, α the mortality caused by the disease, γ the rate constant for loss of immunity, δ the recovery rate. And βSI is called the transmission function or the incidence rate, β the transmission rate. In mathematical epidemic models, e.g., model (1.1), the transmission function plays a key role in determining disease dynamics [21,22]. There are two extreme forms of disease transmission functions frequently used in well-known epidemic models. One is the density-dependent transmission or the bilinear incidence rate, i.e., βSI in model (1.1); the other is the frequency-dependent transmission or the standard βSI . Besides, there are several different nonlinear transmission functions [23–28]. incidence rate, S+I In general, we can write the transmission function as g(S, I) = βf (I)S, here, the function f (I) is assumed to satisfy (H1′ ) f : R+ → R+ is continuously differentiable with f (0) = 0, f ′ (0) > 0 and f (I) > 0 for I ∈ (0, +∞); (H2′ ) I/f (I) is monotone increasing on (0, +∞). The general transmission function g(S, I) = βf (I)S can be used in some specific forms for the incidence rate that have been commonly used, for example: (i) bilinear type [20]: f (I) = I. Il (ii) saturated incidence rate: [23]: f (I) = with l = 1, where parameter h is positive constant, α a 1 + αI h nonnegative constant measuring the psychological or inhibitory effect; the term I measures the infection force 1 measures the inhibition effect from the behavioural change of the susceptible of the disease, and 1 + αI h individuals when their number increases or from the crowding effect of the infective individuals. (ii 1) [24]: f (I) = I/(1 + αI); (ii 2) [12,25]: f (I) = I/(1 + αI 2 ). (iii) incidence rates with “media coverage” effect as shown below: (iii 1) [27]: f (I) = I exp(−mI), where m is a positive constant;

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(iii 2) [28–30]: βf (I) = (β1 − β2 h(I))I, where β1 > β2 and the function h(I) satisfies h(0) = 0, h′ (I) ≥ 0,

lim h(I) = 1.

I→+∞

In this sense, omitting the mortality induced by the influenza, i.e., α = 0 (it is reported α = 4.1 × 10−8 by Chinese CDC in China in 2016 [31]), we can improve model (1.1) as the following general form: ⎧ dS(t) ⎪ ⎪ = Λ − µS(t) − βf (I(t))S(t) + γR(t), ⎪ ⎪ ⎪ ⎨ dt dI(t) (1.2) = βf (I(t))S(t) − (µ + δ)I(t), ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dR(t) = δI(t) − (µ + γ)R(t). dt The similar model to (1.2) can be found in [32] with a special case Λ = µN . For such a homogeneously mixed population, e.g., model (1.2), we can define the basic reproduction number βΛfI (0) , (1.3) R0 = µ(µ + α + δ) which can be used to determine whether there is an endemic outbreak or not: the disease dies out if R0 < 1 and persists if R0 > 1. On the other hand, the subpopulations S, I, R move randomly in the space [33–36], and the actual spread of many infectious diseases (such as influenza disease) occurs in a diverse or dispersed population [19,37]. Thus it is appropriate to consider a population divided into subpopulations, such as S, I, R in model(1.2), which differ from each other. Subpopulations can be determined not only on the basis of disease-related factors such as mode of transmission, latent period, infectious period, and genetic susceptibility or resistance, but also on the basis of social, cultural, economic, demographic, and geographic factors [37]. Hence, in epidemiological models, the factor of spatially heterogeneity must be considered [38–49]. Based on the discussions above, in this paper, we will focus on the spatiotemporal dynamics of the following SIRS model for spreading of the influenza disease through a spatially heterogenous region: ⎧ ∂t S = ∇ · (d(x)∇S) + Λ(x) − β(x)f (x, I)S − µ(x)S + γ(x)R, x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎨∂ I = ∇ · (d(x)∇I) + β(x)f (x, I)S − (µ(x) + δ(x))I, x ∈ Ω , t > 0, t (1.4) ⎪ ∂ R = ∇ · (d(x)∇R) + δ(x)I − (µ(x) + γ(x))R, x ∈ Ω , t > 0, t ⎪ ⎪ ⎩ [d(x)∇S(x, t)] · n = [d(x)∇I(x, t)] · n = [d(x)∇R(x, t)] · n = 0, x ∈ ∂Ω , t > 0, subject to the initial conditions S(0, x) = S0 (x) ≥ 0, I(0, x) = I0 (x) ≥ 0, R(0, x) = R0 (x) ≥ 0, x ∈ Ω ,

(1.5)

where S(x, t), I(x, t) and R(x, t) represent respectively the population density of susceptible, infective and recovered individuals with the diffusion coefficient d(x), and the habitat Ω ⊂ Rm is a bounded domain with smooth boundary ∂Ω . The Neumann (or zero–flux) boundary condition means that no individual crosses the boundary of the habitat and n is the outward unit normal vector on ∂Ω . The symbol ∇ is the gradient ¯ . Similar operator. The incidence rate is β(x)Sf (x, I), and β(x) is positive H¨older continuous function on Ω ′ ′ to the assumptions (H1 ) and (H2 ), the function f (x, I) is assumed to satisfy: (H1 ) f (·, ·) ∈ C 1 (Ω × R+ ) with f (x, 0) = 0, fI (x, 0) > 0 and f (x, I) > 0 for (x, I) ∈ Ω × (0, +∞); (H2 ) I/f (·, I) is monotone increasing for (x, I) ∈ Ω × (0, +∞). Specifically, we require that ¯ ). S0 (x), I0 (x), R0 (x), d(·), Λ(·), µ(·), β(·), δ(·), γ(·) ∈ C 1 (Ω

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In the remainder of this paper, we use the following notations: ¯ , R3 ) be the Banach space with the supremum norm ∥ · ∥. Define X+ := C(Ω ¯ , R3+ ), then Let X := C(Ω + (X, X ) is a strongly ordered space. The rest of this article is organized as follows: In Section 2, we give the dynamical analysis of the extinction and persistence of the influenza disease analytically. In Section 3, we give an example and some numerical results to show the influenza disease dynamics with the diffusion and spatial heterogeneity. And in Section 4, we provide a brief discussion and the summary of our main results. 2. Dynamical analysis of the extinction and persistence of the influenza disease 2.1. Global existence and uniqueness of the solutions In this subsection, we will focus on the existence and uniqueness of the global solutions of model (1.4). The main results are as follows: Theorem 2.1. For every initial value function ϕ := (ϕ1 , ϕ2 , ϕ3 ) ∈ X+ , model (1.4) has a unique solution U (·, t; ϕ) = (S(·, t; ϕ), I(·, t; ϕ), R(·, t; ϕ)) on [0, ∞) with U (·, 0; ϕ) = ϕ and the semiflow Ψt : X+ → X+ generated by (1.4) is defined by ¯ , t ≥ 0. Ψt (ϕ) = (S(·, t; ϕ), I(·, t; ϕ), R(·, t; ϕ)), ∀x ∈ Ω

(2.1)

Furthermore, the semiflow Ψt : X+ → X+ is point dissipative and the positive orbits of bounded subsets of X+ for Ψt are bounded. ¯ , R) → C(Ω ¯ , R) be the C0 semigroups associated with ∇ · (d∇) − µ, Proof . Suppose T1 (t), T2 (t), T3 (t) : C(Ω ∇ · (d∇) − (µ + δ) and ∇ · (d∇) − (µ + γ) subject to the Neumann boundary condition, respectively. It then follows that T (t) := (T1 (t), T2 (t), T3 (t)) is strongly positive and compact for each t > 0 (see, [50]). For every ¯ , R3 ), we define F = (F1 , F2 , F3 ) : X+ → X by initial value functions ϕ = (ϕ1 (x), ϕ2 (x), ϕ3 (x)) ∈ C(Ω F1 (ϕ)(x) = Λ(x) − β(x)f (x, ϕ2 (x))ϕ1 (x) + γ(x)ϕ2 (x), F2 (ϕ)(x) = β(x)f (x, ϕ2 (x))ϕ1 (x), F3 (ϕ)(x) = δ(x)ϕ2 (x). Then model (1.4) can be rewritten as the integral equation ∫ U (t) = T (t)ϕ +

t

T (t − s)F (U (s))ds, 0

where U (t) = (S(t), I(t), R(t))T and T is the transpose of the row vector (S(t), I(t), R(t)). It is easy to show that lim dist (ϕ + hF (ϕ), X+ ) = 0, ∀ϕ ∈ X+ .

h→0+

( ) By Corollary 4 in [51], model (1.4) has a unique positive solution S(·, t; ϕ), I(·, t; ϕ), R(·, t; ϕ) on [0, τe ), where 0 < τe ≤ ∞. In what follows, we prove that the local solution can be extended to a global one, that is τe = ∞. For this purpose, by a standard argument, we only need to prove that the solution is bounded in Ω × [0, τe ). To this end, we let N (x, t) = S(x, t) + I(x, t) + R(x, t).

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Then N (x, t) satisfies the following system ⎧ x ∈ Ω , t > 0, ⎪ ⎨∂t N = ∇ · (d(x)∇N ) + Λ(x) − µ(x)N, [d(x)∇N (x)] · n = 0, x ∈ ∂Ω , t > 0, ⎪ ⎩ N (x, 0) = N0 (x) = S0 (x) + I0 (x) + R0 (x) ≥ 0, x ∈ Ω .

(2.2)

Thanks to [52], model (2.2) admits a unique positive steady state S∗ which is globally asymptotically stable ( ) in C(Ω , R), where S∗ satisfies (2.8). It follows that S(·, t; ϕ), I(·, t; ϕ), R(·, t; ϕ) is bounded on [0, τe ), which implies the Theorem. □ 2.2. Extinction of the influenza disease In this subsection, we will focus on the extinction of influenza disease through studying the disease-free dynamics of model (1.4). We first determine the basic reproduction number, R0 , which is defined as the average number of secondary infections generated by a single infected individual introduced into a completely susceptible population, is one of the important quantities in epidemiology [53,54]. Setting I(x, t) = R(x, t) = 0 in (1.4), we obtain the following equation for the density S(x, t) of susceptible population: { ∂t S = ∇ · (d(x)∇S) + Λ(x) − µ(x)S, x ∈ Ω , t > 0, (2.3) [d(x)∇S(x, t)] · n = 0, x ∈ ∂Ω t > 0. It is easy to see that (2.3) admits a positive equilibrium S∗ (x) which is unique, globally asymptotically stable. Generally, the function (S∗ , 0, 0) is called the disease-free equilibrium (DFE) of model (1.4). Linearizing model (1.4) around DFE, we obtain the following model for the infection related variable { ∂t I = ∇ · (d(x)∇I) + (β(x)S∗ fI (x, 0) − (µ(x) + δ(x)))I, x ∈ Ω , t > 0, (2.4) [d(x)∇I(x, t)] · n = 0, x ∈ ∂Ω , t > 0. Substituting I(x, t) = eλt φ(x) into (2.4), we obtain the following eigenvalue problem { λφ(x) = ∇ · (d(x)∇φ(x)) + (β(x)S∗ fI (x, 0) − (µ(x) + δ(x)))φ(x), x ∈ Ω , [d(x)∇φ(x)] · n = 0, x ∈ ∂Ω .

(2.5)

By a similar argument as Theorem 7.6.1 in [50], it follows that (2.5) has a principal eigenvalue λ∗ (S∗ ) = s(∇ · (d∇) + βS∗ fI (0) − (µ + δ)) with a positive eigenfunction φ∗ (x), where s(A) denotes the spectral bound of a closed linear operator A. Observe that (λ∗ , φ∗ ) satisfies { ∗ ∗ λ φ (x) = ∇ · (d(x)∇φ∗ (x)) + (β(x)S∗ fI (x, 0) − (µ(x) + δ(x)))φ∗ (x), x ∈ Ω , [d(x)∇φ∗ (x)] · n = 0, x ∈ ∂Ω . It is well-known that λ∗ (S∗ ) is given by the following variational characterization: {∫ } ∫ 2 λ∗ (S∗ ) = − inf d(x)|∇φ| + (µ(x) + δ(x) − β(x)S∗ fI (x, 0))φ2 : φ ∈ H 1 (Ω ), φ2 = 1 . Ω

(2.6)

Ω

¯ , R) associated with ∇ · (d∇) − (µ + δ) and ϕ2 (x) be the spatial Let T2 (t) be the semigroup on C(Ω distribution of infective I(x, t). Then the distribution of infective individuals as time evolves becomes T2 (t)ϕ2 (x). Thus, the distribution of new infection at time t is β(x)S∗ fI (x, 0)T2 (t)ϕ2 (x). Consequently, the distribution of total new infections is ∫ ∞ β(x)S∗ fI (x, 0)T2 (t)ϕ2 (x)dt. 0

Define

∫ L(ϕ2 )(x) :=

∞

∫

0

∞

T2 (t)ϕ2 dt.

β(x)S∗ fI (x, 0)T2 (t)ϕ2 dt = β(x)S∗ fI (x, 0) 0

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Then L is the next infection operator, which maps the initial distribution ϕ2 to the distribution of the total infectious produced during the infection period. Following [53–56], we define the spectral radius of L as the basic reproduction number for model (1.4), that is, ∫ ⎫ ⎧ ⎪ ⎪ 2 ⎪ ⎪ β(x)S f (x, 0)φ ∗ I ⎬ ⎨ Ω ∫ R0 := ρ(L) = sup . (2.7) 2 2⎪ φ∈H 1 (Ω) ⎪ ⎪ ⎪ ⎭ ⎩ d(x)|∇φ| + (µ(x) + δ(x))φ φ̸=0 Ω

We now explore the properties of the basic reproduction number. By the general results in [57] and the same arguments as in [55,56,58,59], we have the following observations. Lemma 2.2.

R0 − 1 has the same sign as λ∗ (S∗ ).

Before giving the main results of this section, we give the following useful lemma. Lemma 2.3. Let (S(x, t; ϕ), I(x, t; ϕ), R(x, t; ϕ)) be the solution of model (1.4) with initial value ϕ = (ϕ1 , ϕ2 , ϕ3 ) ∈ X+ . If there exist some t0 ≥ 0 such that I(·, t0 ; ϕ), R(·, t0 ; ϕ) ̸≡ 0, then I(·, t; ϕ), R(·, t; ϕ) > ¯ . Moreover, for any ϕ ∈ X+ , we have S(·, t; ϕ) > 0, ∀t ≥ t0 , x ∈ Ω ¯ and 0, ∀t ≥ t0 , x ∈ Ω lim inf S(x, t; ϕ) ≥ t→∞

¯ , where Λ ˇ := min Λ(x) and θ := uniformly for x ∈ Ω ¯ x∈Ω

ˇ Λ ∥µ∥ + θ∥β∥

max

I∈[0,∥S∗ ∥]

f (I).

Proof . It is easy to see that I(x, t; ϕ) and R(x, t; ϕ) satisfy ⎧ x ∈ Ω , t > 0, ⎨∂t I ≥ ∇ · (d(x)∇I) − (µ(x) + δ(x))I, ∂t R ≥ ∇ · (d(x)∇R) − (µ(x) + γ(x))R, x ∈ Ω , t > 0, ⎩ [d(x)∇I(x, t)] · n = [d(x)∇R(x, t)] · n = 0, x ∈ ∂Ω , t > 0. If I(·, t0 ; ϕ), R(·, t0 ; ϕ) ̸≡ 0 for some t0 ≥ 0, it then follows from the strong maximum principle (see, [60], ¯ . From the first equation of model (1.4), we Proposition 13.1) that I(·, t; ϕ), R(·, t; ϕ) > 0, ∀ t ≥ t0 , x ∈ Ω get { ˇ − (∥µ∥ + θ∥β∥)S, x ∈ Ω , t > 0, ∂t S ≥ ∇ · (d(x)∇S) + Λ [d(x)∇S(x, t)] · n = 0, x ∈ ∂Ω , t > 0. By the comparison principle, we have lim inf S(x, t; ϕ) ≥ t→∞

ˇ Λ ¯ , which completes uniformly for x ∈ Ω ∥µ∥ + θ∥β∥

the proof. □ Next, we consider the properties of the DFE, including its existence, uniqueness, and stability. Theorem 2.4.

For model (1.4), there exists a unique DFE (S∗ , 0, 0), where S∗ is a positive solution of { −∇ · (d(x)∇S) = Λ(x) − µ(x)S, x ∈ Ω , (2.8) [d(x)∇S(x, t)] · n = 0, x ∈ ∂Ω .

(i) If R0 < 1, then any positive solutions of (1.4) converge to the DFE (S∗ , 0, 0) as t → ∞. That is, the DFE (S∗ , 0, 0) is globally asymptotically stable. (ii) If R0 > 1, then there exists ϵ0 > 0 such that any positive solution of model (1.4) satisfies lim sup ∥(S(·, t), I(·, t), R(·, t)) − (S∗ , 0, 0)∥ > ϵ0 . t→∞

(2.9)

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Proof (i). Assume that R0 < 1. By Lemma 2.2, it implies that λ∗ (S∗ ) < 0. By the continuity, there is a ε > 0 such that λ∗ (S∗ + ε) < 0. According to Theorem 2.1, there is a τ1 > 0 such that S(x, t) ≤ N (x, t) ≤ S∗ + ε, ∀t ≥ τ1 . Using the fact that (H2 ), we have f (·, I) f (·, I) ≤ lim = fI (·, 0). I↑0 I I Observe from the second equality of model (1.4) that ⎧ ( ) ⎪ ⎨∂t I ≤ ∇ · (d(x)∇I) + β(x)fI (x, 0)(S∗ + ε) − (µ(x) + δ(x)) I, [d(x)∇I(x, t)] · n = 0, ⎪ ⎩ I(x, 0) = I0 (x),

(2.10)

x ∈ Ω , t > 0, x ∈ ∂Ω , t > 0, x ∈ Ω.

Let Z(x, t) with Z(x, 0) = I0 (x) be the solution of the following linear system { ( ) ∂t Z = ∇ · (d(x)∇Z) + β(x)fI (x, 0)(S∗ + ε) − (µ(x) + δ(x)) Z, x ∈ Ω , t > 0, [d(x)∇Z(x, t)] · n = 0, x ∈ ∂Ω , t > 0. By the comparison principle, 0 ≤ I(x, t) ≤ Z(x, t) for all t > τ1 and x ∈ Ω . Let Q(t) be the solution semigroup by the operator ∇ · (d∇) + βfI (·, 0)(S∗ + ε) − (µ + δ), and let ϖ(Q) be the exponential growth bound of Q(t). Then ϖ(Q) = λ∗ (S∗ + ε) < 0. Choose 0 < a < −ϖ(Q) and fixed it, it follows from [61] that there is a constant C > 0 such that ∥I(·, t)∥ ≤ ∥Z(·, t)∥ = ∥Q(t)Z(·, 0)∥ ≤ Ce−at ∥I0 (·)∥ → 0 as t → ∞.

(2.11)

We now show that R(·, t) tends to 0 as t → ∞. Let T3 (t) be the semigroup generated by the operator ∇ · (d∇) − (µ + γ). Choosing a smaller when necessary, and assume that it also satisfies the condition 0 < a < −ϖ(T3 ). It follows that there is a constant C > 0 such that ∥T3 (t)∥ ≤ Ce−at , t > 0.

(2.12)

Applying the formula of variation of constants [61] and (2.11), (2.12), we have ∫ t ∥R(·, t)∥ ≤ ∥T3 (t)R0 (·)∥ + ∥T3 (t − s)δ(·)I(·, s)∥ds 0

≤ Ce−at ∥R0 (·)∥2 + Cte−at → 0 as t → ∞.

(2.13)

Now set Sˆ = S − S∗ , then Sˆ satisfies ⎧ ˆ ˆ ˆ ⎪ ⎨∂t S ≤ ∇ · (d(x)∇S) − µ(x)S + γ(x)R, x ∈ Ω , t > 0, ˆ t)] · n = 0, [d(x)∇S(x, x ∈ ∂Ω , t > 0, ⎪ ⎩ˆ S(x, 0) = S0 (x) − S∗ (x), x ∈ Ω. ˆ 0) be the solution of the following system Let Z(x, t) with Z(x, 0) = S(x, { ∂t Z = ∇ · (d(x)∇Z) − µ(x)Z + γ(x)R, x ∈ Ω , t > 0, [d(x)∇Z(x, t)] · n = 0, x ∈ ∂Ω , t > 0. ˆ t) ≤ Z(x, t) for all t > 0 and x ∈ Ω . Choose 0 < a < −ϖ(T1 ) and fix By the comparison principle, 0 ≤ S(x, it, then there is a constant C > 0 such that ∥T1 (t)∥ ≤ Ce−at , t > 0.

(2.14)

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Taking advantage of (2.13) and (2.14), we arrive at ˆ t)∥ ≤ ∥Z(·, t)∥ ∥S(·,

t

∫ ≤ ∥T1 (t)Z(·, 0)∥ +

∥T1 (t − s)γ(·)R(·, t)∥ds 0

≤ Ce−at ∥S(·, 0)∥2 + Ct(1 + 1/2t)e−at →0

as t → ∞.

It follows that S(x, t) → S∗ (x) as t → ∞, which completes the proof. (ii) Assume, for the sake of contradiction, that there exists a positive solution of model (1.4) such that lim sup ∥(S(·, t), I(·, t), R(·, t)) − (S∗ , 0, 0)∥ < ϵ0 . t→∞

Then there exists t1 > 0 such that S∗ − ϵ0 < S(x, t) < S∗ + ϵ0 , 0 < I(x, t) < ϵ0 for t ≥ t1 . It follows from (H2 ) that f (·, I)/I ≥ f (·, ϵ0 )/ϵ0 ≥ fI (·, ϵ0 ). It follows that { ( ) ∂t I ≥ ∇ · (d(x)∇I) + β(x)fI (·, ϵ0 )(S∗ − ϵ0 ) − (µ(x) + δ(x)) I, x ∈ Ω , t > t1 , (2.15) [d(x)∇I(x, t)] · n = 0, x ∈ ∂Ω , t > t1 . ( ) For any ϵ ∈ 0, minx∈Ω¯ S∗ , we consider the following eigenvalues problem { ∇ · (d(x)∇I) + (β(x)fI (x, ϵ)(S∗ − ϵ) − (µ(x) + δ(x)))I = λI, x ∈ Ω , [d(x)∇I(x, t)] · n = 0, x ∈ ∂Ω . Define Rϵ := ρ(Lϵ ) as the spectral radius of the operator ∫

∞

Lϵ : φ(x) → β(x)fI (x, ϵ)(S∗ − ϵ)

T (t)φdt. 0

Since limϵ→0 Rϵ = R0 > 1, we restrict ϵ small enough such that Rϵ > 1. Hence λ∗ϵ = s(∇·(d∇)+βfI (·, ϵ)(S∗ − ( ) ϵ) − (µ + δ)) > 0. As a consequence, we can fix a small ϵ0 ∈ ϵ, minx∈Ω¯ S∗ such that λ∗ϵ0 > 0. Since I(·, t0 ) > 0, by Lemma 2.3, we can choose a sufficiently small number η > 0 such that I(·, t0 ) ≥ ∗ ηϕ∗ϵ0 (·), where ϕ∗ϵ0 (·) is a strongly positive eigenfunction corresponding to λ∗ϵ0 . Note that ηeλϵ0 (t−t0 ) ϕ∗ϵ0 (x) is a solution o the following linear system { ( ) ∂t I = ∇ · (d(x)∇I) + β(x)fI (x, ϵ0 )(S∗ − ϵ0 ) − (µ(x) + δ(x)) I, x ∈ Ω , t > t0 , [d(x)∇I(x, t)] · n = 0, x ∈ ∂Ω , t > t0 . It then follows from (2.15) and the comparison principle that ∗

I(x, t) ≥ ηeλϵ0 (t−t0 ) ϕ∗ϵ0 (x), ∀t ≥ t0 , and, hence, I(x, t) is unbounded as t → ∞ which contradicts (2.3). □ 2.3. Persistence of the influenza disease In this subsection, we study the existence and the persistence of the endemic equilibrium (EE) of model (1.4). Theorem 2.5. If R0 > 1, model (1.4) admits at least one EE (S ∗ , I ∗ , R∗ ), and there exists ε > 0 such that for any ϕ ∈ X+ with ϕi ̸≡ 0, i = 1, 2, 3, we have lim inf S(x, t; ϕ), lim inf I(x, t; ϕ), lim inf R(x, t; ϕ) ≥ ε t→∞

¯. uniformly for all x ∈ Ω

t→∞

t→∞

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Proof . Let W0 = {ϕ ∈ X+ : ϕ2 (·) ̸≡ 0 and ϕ3 (·) ̸= 0}, and ∂W0 = X+ \W0 = {ϕ ∈ X+ : ϕ2 (·) ≡ 0 or ϕ3 (·) ̸= 0}. ¯ , t > 0. In By Lemma 2.3, it follows that for any ϕ ∈ W0 , we have I(x, t; ϕ) > 0, R(x, t; ϕ) > 0, ∀ x ∈ Ω other words, Ψt W0 ⊆ W0 , ∀ t ≥ 0. Define M∂ := {ϕ ∈ ∂W0 : Ψt ϕ ∈ ∂W0 , ∀ t ≥ 0} and ω(ϕ) the omega limit set of the orbit O+ (ϕ) := {Ψ (t)ϕ : ∀ t ≥ 0}. For any given ψ ∈ M∂ , we have Ψt ψ ∈ ∂W0 , ∀ t ≥ 0. It then follows that for each t ≥ 0, either I(·, t; ψ) ≡ 0 or R(·, t; ψ) ≡ 0. In the case where I(·, t; ψ) ≡ 0 for all t ≥ 0, then R(x, t; ψ) satisfies { ∂t R = ∇ · (d(x)∇R) − (µ(x) + γ(x))R, x ∈ Ω , t > 0, [d(x)∇R(x, t)] · n = 0, x ∈ ∂Ω , t > 0. ¯ . It then follows that for any sufficiently small ε > 0, there Hence, lim R(·, t; ψ) = 0 uniformly for x ∈ Ω t→∞ exists τ2 > 0 such that R(·, t; ψ) < ε for t ≥ τ2 . Thus, from the first equation of (1.4), we can get { ∂t S = ∇ · (d(x)∇S) + Λ(x) − µ(x)S + εγ(x), x ∈ Ω , t > τ2 , [d(x)∇S(x, t)] · n = 0, x ∈ ∂Ω , t > τ2 . It follows from Theorem 2.1 and ε arbitrary that lim S(·, t; ψ) = S∗ . In the case where I(·, t˜0 ; ψ) ̸≡ 0, for t→∞ some t˜0 ≥ 0. Then Lemma 2.3 implies that I(·, t; ψ) > 0, ∀ x ∈ Ω , ∀ t > t˜0 . Hence, R(·, t; ψ) ≡ 0, ∀ t˜0 ≥ 0. From the last equation of (1.4), it follows that I(·, t; ψ) ≡ 0, ∀ t > t˜0 , which is a contradiction. Hence, we have ω(ϕ) = {(S∗ , 0, 0)}, ∀ ψ ∈ ∂ M. Moreover, if R0 > 1, it follows from Theorem 2.4 (ii) that (S∗ , 0, 0) is a uniform weak repeller for W0 in the sense that lim sup ∥(S(·, t), I(·, t), R(·, t)) − (S∗ , 0, 0)∥ > ϵ, for all ψ ∈ W0 . t→∞

Define a continuous function p : X+ → [0, ∞) by { } p(ψ) := min min ψ2 (x), min ψ2 (x) , ∀ ψ ∈ X+ . ¯ x∈Ω

¯ x∈Ω

It follows from Lemma 2.3 that p−1 (0, ∞) ⊆ W0 and p has the property that if p(ψ) > 0 or ψ ∈ W0 with p(ψ) = 0, then p(Ψt ψ) > 0, ∀ t > 0. Thus, p is a generalized distance function for the semiflow Ψt : X+ → X+ (see, [62]). Note that any forward orbit of Ψt in M∂ converges to (S∗ , 0, 0). Moreover, the claim above implies that (S∗ , 0, 0) is isolated in X+ and W s (S∗ , 0, 0) ∩ W0 = ∅, where W s (S∗ , 0, 0) is the stable set of (S∗ , 0, 0). Further, there is no cycle in M∂ from (S∗ , 0, 0) to (S∗ , 0, 0). It then follows from Theorem 3 in [62] that there exists an η > 0 such that min{p(ψ) : ψ ∈ ω(ψ)} > η for any ψ ∈ W0 . Hence lim inf I(·, t; ψ), lim inf R(·, t; ψ) ≥ η, ∀ ψ ∈ W0 . t→∞

t→∞

Further, it follows from Lemma 2.3 that we can choose η small enough such that lim inf S(·, t; ψ) ≥ η, ∀ ψ ∈ W0 . t→∞

Thus, the uniform persistence stated in the conclusion holds. By Theorem 3.7 and Remark 3.10 in [63], it follows that ψt : W0 → W0 has a global attractor. It then follows from Theorem 4.7 in [63] that Ψt has an equilibrium (S ∗ , I ∗ , R∗ ) ∈ W0 . Clearly, Lemma 2.3 implies that (S ∗ , I ∗ , R∗ ) is the EE of model (1.4). This ends the proof. □

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3. Applications and influenza disease dynamics via numerical simulations In this section, we apply the obtained abstract results to specific forms of the function f (x, I). Motivated by Xiao and Ruan [25], we consider the function f (x, I) as f (x, I) =

I . 1 + α(x)I 2

To be more specific, we consider the following system: ⎧ ⎪ ⎪∂t S = ∇ · (d(x)∇S) + Λ(x) − β(x)SI − µ(x)S + γ(x)R, x ∈ Ω , t > 0, ⎪ ⎪ ⎪ 1 + α(x)I 2 ⎪ ⎪ ⎨ β(x)SI ∂t I = ∇ · (d(x)∇I) + − (µ(x) + δ(x))I, x ∈ Ω , t > 0, ⎪ 1 + α(x)I 2 ⎪ ⎪ ⎪ ⎪ ∂t R = ∇ · (d(x)∇R) + δ(x)I − (µ(x) + γ(x))R, x ∈ Ω , t > 0, ⎪ ⎪ ⎩ [d(x)∇S(x, t)] · n = [d(x)∇I(x, t)] · n = [d(x)∇R(x, t)] · n = 0, x ∈ ∂Ω , t > 0,

(3.1)

and the initial conditions are the(same as)in (1.5). It is easy to verify that the conditions (H1 ) and (H2 ) ∂ I are satisfied with fI (·, 0) = 1, = 2α(·)I > 0. ∂I f (·, I) Following the procedure in Section 2, for model (3.1), we define the reproduction number R0 as ∫ ⎧ ⎫ ⎪ ⎪ 2 ⎪ ⎪ β(x)S∗ φ ⎨ ⎬ Ω ∫ , (3.2) R0 = sup 2 2⎪ φ∈H 1 (Ω) ⎪ ⎪ ⎪ ⎩ ⎭ d(x)|∇φ| + (µ(x) + δ(x))φ φ̸=0 Ω

I where S∗ is the solution of (2.3). Specially, if Λ, µ, β, δ, γ, d are all positive constants and f (x, I) = , 1 + αI 2 Λβ . then R0 = µ(µ + δ) Following the procedure in Section 2, we can obtain the following results. For model (3.1),

Theorem 3.1.

(a) if R0 < 1 the disease-free equilibrium (S∗ , 0, 0) is globally asymptotically stable, here S∗ =

Λ(x) ; µ(x)

(b) if R0 > 1, (b1) there exists ϵ0 > 0 such that any positive solution of model (3.1) satisfies lim sup ∥(S(·, t), I(·, t), R(·, t)) − (S∗ , 0, 0)∥ > ϵ0 .

(3.3)

t→∞

(b2) there exists ε > 0 such that for any ϕ ∈ X+ with ϕi ̸≡ 0, i = 1, 2, 3, we have lim inf S(x, t; ϕ), lim inf I(x, t; ϕ), lim inf R(x, t; ϕ) ≥ ε t→∞

t→∞

t→∞

¯ . Moreover, there is at least one EE (S ∗ , I ∗ , R∗ ) and uniformly for all x ∈ Ω S∗ = ∗

I = R∗ =

Λ(1 + αI ∗2 ) , R0 µ −R0 µ (µ + γ + δ) +

√

2

(R0 µ (µ + γ + δ))2 + 4 Λ2 α (µ + γ) (R0 − 1) 2Λα(µ + γ)

,

(3.4)

δ I ∗. δ+γ

(b3) if Λ, µ, β, δ, γ, d are all positive constants and f (x, I) = asymptotically stable in the interior of X+ .

I , then EE (S ∗ , I ∗ , R∗ ) is globally 1 + αI 2

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Fig. 1. The relation between R0 and c in β(x) = 0.2 · 10−4 (1 + c cos πx).

In the following, we implement numerical simulations in order to show how to derive some epidemiological insights from our analytic results. For the sake of convenience, from Section 3.1 to 3.4, we concentrate on Ω = [0, 1] ⊂ R, and in Section 3.5, Ω = [−2, 2] ⊂ R. Based on the insightful work about dynamics of 10000 −1 influenza A [5,6,11] , the parameters in model (3.1) are taken as: Λ = d which means the total 70 · 52 · 7 1 population size is assumed as N = 10,000, and µ = d−1 which suggests the average life expectancy 70 · 52 · 7 1 d−1 which means the new antigenic variants can rise with a frequency of one per is 70 years, γ = 1.5 · 52 · 7 1.5 years. 3.1. The influence of spatially heterogenous transmission rate β(x) on R0 In this subsection, we will focus on the role of the spatial heterogeneity of environment in R0 . As an example, we only consider the disease transmission rate β(x) as the key factor of spatial heterogeneity, and choose δ(x) = 0.25, β(x) = 0.2 · 10−4 (1 + c cos πx), where 0 ≤ c ≤ 1 is the magnitude of spatially heterogenous transmission rate. The spatially homogeneous case occurs at c = 0, and the bigger c means the higher heterogeneity of spatial transmission rate. In Fig. 1, we show the numerical results of the relations between R0 and the parameter c in β(x). From the definition of R0 in (3.2) and Fig. 1, we can know that R0 is an increasing function of c. Note that R0 takes the minimum at the spatially homogeneous case c = 0, and becomes greater than one when c passes through the critical value c∗ = 0.6465 (see Fig. 1). That is, if c < c∗ = 0.6465, then R0 < 1, the influenza disease dies out; while if c > c∗ = 0.6465, then R0 > 1, the influenza disease will break out. 3.2. The effect of the spatial heterogenous β(x) on the influenza disease dynamics Let α(x) = 0.005, δ = 0.25, when β = 0.2 · 10−4 (1.0 + 0.5 cos 2πx), we can calculate R0 = 0.9387 < 1, from Theorem 2.4, we know that the solutions of model (3.1) tend to the DFE, i.e., the influenza disease extinct (See Fig. 2(a)). While we choose β = 0.2 · 10−4 (1.0 + 1.0 cos 2πx), we can obtain R0 = 1.166 > 1, from Theorem 2.5, we know that model (3.1) admits at least one EE (S ∗ , I ∗ , R∗ ), i.e., the influenza disease will break out (see Fig. 2(b)). It should be noted that in the spatially homogeneous case, i.e., c = 0, then β = 0.2·10−4 and R0 = 0.8 < 1, from Theorem 2.4, we know that the influenza disease will extinct (similar to that in Fig. 2(a)). Comparing with Fig. 2(b), we can conclude that the spatial heterogeneity can enhance the infectious risk of influenza.

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10000 1 1 Fig. 2. The long time behaviour of the solution I(x, t) of model (3.1) with Λ = 70·52·7 , µ = 70·52·7 , γ = 1.5·52·7 , α(x) = 0.005, δ = 0.25, the initial value is (S0 , I0 , R0 ) = (9980, 20, 0). (a) β = 0.2 · 10−4 (1.0 + 0.5 cos 2πx); (b) β = 0.2 · 10−4 (1.0 + 1 cos 2πx).

Fig. 3. Distribution of infected hosts I(x, t) prevalence at different time t = 1, 7, 14, 21, 28, 100d for various diffusion coefficient 1 1 d(x) = 1.25 · 10−2 , 1.25 · 10−3 , 1.25 · 10−4 , and fixed Λ = 10000 , µ = 70·52 , γ = 1.5·52 , β(x) = 0.8 · 10−4 , α(x) = 0.05 (1.1 + cos 2πx), 70·52 11 the initial value is (S0 , I0 , R0 ) = (9980, 20, 0).

3.3. The effects of the spatial heterogeneity in β(x) with different diffusion coefficient d(x) on the influenza disease dynamics

We now take the disease transmission rate β(x) = 0.8 · 10−4 , the psychological effect α(x) =

0.05 11 (1.1

+

cos 2πx) and observe how the spatial distribution of infected hosts prevalence changes with the diffusion coefficient d(x) = 1.25 × 10−2 , 1.25 × 10−3 , 1.25 × 10−4 . The numerical results shown in Fig. 3 reveal that the effect of diffusion on the distribution of infected hosts I(x, t) prevalence is highly sensitive due to the spatial environmental heterogeneity. It is found that a higher diffusion coefficient has a tendency to increase infected hosts I(x, t) prevalence towards the boundary while decreasing the prevalence in the middle. On the other hand, we can realize that as time goes by, the influence of diffusion on the distribution of infected hosts I(x, t) is getting stronger and stronger. More precisely, in the first day 3(a), the diffusion nearly has no effect on the distribution of I(x, t), while from t = 7d 3(b) to t = 100d 3(f), it is revealed that the distribution of infected hosts I(x, t) prevalence is highly sensitive due to the value of diffusion coefficient.

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Fig. 4. Distribution of infected hosts I(x, t) prevalence at time t = 21d for different recovery rate δ(x) = 0.05, 0.25, 0.5 and disease 10000 1 1 transmission rate β(x) = 0.8 · 10−4 (1.1 + c · cos(2πx)) (c = 0, 0.5, 1), with fixed Λ = 70·52·7 , µ = 70·52·7 , γ = 1.5·52·7 , α(x) = 0.05 (1.1 + cos 2πx), d(x) = 0.0125, the initial value is (S , I , R ) = (9980, 20, 0). 0 0 0 11

3.4. The effects of the spatial heterogenous β(x) and different recovery rate δ(x) on the influenza disease dynamics

In this subsection, we will focus on the effects of the spatial heterogenous β(x) and different recovery rate δ(x) on the distribution of infected hosts I(x, t). As an example, we adopt β(x) = 0.8 · 10−4 (1.1 + c · cos(2πx)) (c = 0, 0.5, 1) and consider three different cases with recovery rate δ(x) = 0.05, 0.25 and 0.5, respectively. From Fig. 4, we can know that a lower δ(x) has a tendency to increase infected hosts I(x, t) prevalence (Fig. 4(a)), while a higher δ(x) has a tendency to decrease infected hosts I(x, t) prevalence (Fig. 4(d)). In addition, we can realize spatial heterogeneity in β(x) (measured by c) has big influence on the distribution of I(x, t). It is found that, in the case without spatial heterogeneity in β(x) (i.e., c = 0), the solution I(x, t) of model (3.1) has a tendency to increase prevalence in the middle while decreasing the prevalence towards the boundary. In the case with spatial heterogeneity in β(x) (i.e., c > 0), for small δ (e.g., δ=0.05, 0.25), a lower spatial heterogeneity in β(x) (e.g., c=0.05) has a tendency to increase infected hosts I(x, t) prevalence in the middle while decreasing the prevalence towards the boundary; in turn, for big δ (e.g., δ=0.5, 0.75), a lower spatial heterogeneity in β(x) (e.g., c=0.5) has a tendency to decrease I(x, t) prevalence in the middle while increasing the prevalence towards the boundary (see, Figs. 4(a) and 4(b)); and a higher spatial heterogeneity in β(x) (e.g., c=1) has a tendency to decrease infected hosts I(x, t) prevalence in the middle while increasing the prevalence towards the boundary.

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Fig. 5. The solutions of S(x, ·) and I(x, ·) of model (3.1) with initial condition (3.1) and with/without diffusion. The parameters are taken as: Λ = 7.14 · 10−2 , µ(x) = 5.5 · 10−5 , δ(x) = 0.2, γ(x) = 2.74 · 10−3 , α(x) = 0.005, β(x) = 0.514, d(x) = 0.0125.

3.5. The short-term influenza disease dynamics with/without diffusion Thanks to the insightful work in [33,34], in this subsection, we will focus on the short-term influenza disease dynamics with/without diffusion for mode (3.1), the parameters are all taken as positive constants: 1 1 −4 Λ = 10000 , δ(x) = 0.25, d(x) = 0.0125, Ω = [−2, 2] ⊂ 70·52 , µ = 70·52 , γ = 1.5·52 , α(x) = 0.005, β(x) = 0.88 · 10 R, and the initial conditions are taken as: ⎧ ⎨S0 = 9980 · 0.96 exp(−10x2 ), −2 ≤ x ≤ 2, t > 0, I0 = 20 · 0.04 exp(−100x2 ), −2 ≤ x ≤ 2, t > 0, (3.5) ⎩ R0 = 0, −2 ≤ x ≤ 2, t > 0. The initial condition (3.5) has both populations S(x, t) and I(x, t) spread all over the domain but more populations concentrated at the origin. More precisely, initially, the susceptible population is concentrated in the domain [-1.25, 1.25] (see the curve t = 0 in Fig. 5(c), S(−1.25, 0) = S(1.25, 0) ≈ 1.56 · 10−3 ), and the infectious population I(x, t) is concentrated in the domain [-0.4, 0.4] (see the curve t = 0 in Fig. 5(d), I(−0.4, 0) = S(0.4, 0) ≈ 9 · 10−8 ). Fig. 5 shows the output of the solutions S(x, t) and I(x, t) of model (3.1) with the initial condition (3.5). In the case without diffusion, i.e., d(x) = 0, with the increase of time, there is an increasing in susceptible population S(x, t) outside the domain of initial concentration (see Fig. 5(a)), after t = 7, there is rapid spread of susceptible in the domain Ω = [−2, 2], and more populations are concentrated in the domain at the origin; and for the infectious population I(x, t), there is an increasing with a pulse of high amplitude as compared to initial condition (3.5) after t = 7 and more populations are concentrated in the domain of initial concentration [-0.4, 0.4], but it does not, however, happen outside the domain of initial concentration (see Fig. 5(b)). In the case with diffusion, as an example, we adopt d(x) = 0.0125, the susceptible population S(x, t) spreads to domain Ω = [−2, 2] with the increase of time. In Fig. 5(c), S(x, t) is showing spread in

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the domain [-1.75, 1.75] at t = 7 with a pulse of high amplitude as compared to initial condition (3.5), and at t = 14, S(x, t) spreads further to domain Ω = [−2, 2]. The spread of infection I(x, t) spreads further to domain Ω = [−2, 2] (see Fig. 5(d)), which has the same pattern as susceptible population (see Fig. 5(c)). 4. Conclusion In this paper, we investigate the threshold influenza disease dynamics of an SIRS model with a general nonlinear incidence. The novelty of the model is that it includes both diffusion and spatial heterogeneity of the environment. We introduce the basic reproduction number R0 defined in (2.7), which exhibits the effect of spatiotemporal factors on the extinction and persistence of the disease. In a nutshell, we summarize our main findings as well as their related biological implications. • The threshold dynamics governed by R0 : Mathematical analysis of the model (1.4) allowed us to achieve a formula for the basic reproduction number R0 and a threshold condition for the disease to die out or not: if R0 < 1, the unique DFE is globally asymptotic stable and there is no EE (c.f., Theorem 2.4 and Fig. 2(a)), while if R0 > 1, there is at least one EE and the disease is uniformly persistent (c.f., Theorem 2.5 and Fig. 2(b)). The most interesting finding is that the diffusion comes into play to define the basic reproduction number due to spatial heterogeneity in the environmental condition (c.f., definition (2.7) and Fig. 1), and the basic reproduction number is independent of the diffusion coefficient, however, in the absence of environmental heterogeneity (c.f., definition (1.3) ). • The spatial heterogeneity can enhance the infectious risk of the influenza: From the numerical results in Figs. 1 and 2, we can know more about the effect of the spatial heterogenous β(x) on the dynamics of model (3.1). It is worthy to note that, in the spatially homogeneous case, i.e., c = 0, R0 takes the minimum and R0 < 1, the influenza disease will die out forever. On the other hand, R0 is an increasing function with respect to c (see Fig. 1), hence there is a threshold value c∗ which can be used to determine relation between R0 with one: if c < c∗ , then R0 < 1, the influenza disease dies out (c.f., Fig. 2(a)); while if c > c∗ , then R0 > 1, the influenza disease will break out (c.f., Fig. 2(b)). As a consequence, our results suggest that the combination of spatial heterogeneity tends to enhance the persistence of the influenza disease for the model (3.1). In other words, the infectious risk of influenza would attract great importance if spatial heterogeneity is taken into account. • The spatial distribution of the infectious and the control strategy: It is found in Fig. 3 that a higher diffusion coefficient d(x) has a tendency to increase infected hosts I(x, t) prevalence towards the boundary while decreasing the prevalence in the middle and reduce the spread of the influenza disease. In addition, with the increase of time, the influence of diffusion on the distribution of infected hosts I(x, t) is getting stronger and stronger. Thus, when an influenza disease appears and spreads in a region, people should change travelling plan and stay at home to reduce the value of the diffusion coefficient to decrease infection risk. In addition, in Fig. 4, it is also found that we can know that a lower δ(x) has a tendency to increase infected hosts I(x, t) prevalence (Fig. 4(a)), while a higher δ(x) has a tendency to decrease infected hosts I(x, t) prevalence (Fig. 4(d)). And for big δ(x), a lower spatial heterogeneity in β(x) has a tendency to decrease I(x, t) prevalence in the middle while increasing the prevalence towards the boundary (see, Figs. 4(a) and 4(b)); and a higher spatial heterogeneity in β(x) has a tendency to decrease infected hosts I(x, t) prevalence in the middle while increasing the prevalence towards the boundary. Thus in order to control the spread of the influenza, we must increase the recovery rate and the spatial heterogeneity in the transmission rate. Acknowledgements The authors would like to thank the anonymous referees for very helpful suggestions and comments which led to improvement of our original manuscript. This research was supported by the National Natural

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