## Abstract

We obtain full information about the existence and non-existence of travelling wave solutions for a general class of diffusive Kermack–McKendrick SIR models with non-local and delayed disease transmission. We show that this information is determined by the basic reproduction number of the corresponding ordinary differential model, and the minimal wave speed is explicitly determined by the delay (such as the latent period) and non-locality in disease transmission, and the spatial movement pattern of the infected individuals. The difficulty is the lack of order-preserving property of the general system, and we obtain the threshold dynamics for spatial spread of the disease by constructing an invariant cone and applying Schauder’s fixed point theorem.

## 1. Introduction

The basic compartmental models to describe the transmission of communicable diseases are contained in a sequence of three papers by W. O. Kermack and A. G. McKendrick (1927, 1932, 1933) (Anderson 1991; Brauer 2008), where the SIR model
1.1
was formulated and analysed, where *S*(*t*), *I*(*t*) and *R*(*t*) denote the sizes of the susceptible, infected and removed individuals, respectively. The constant *β* is the transmission coefficient, and *γ* is the recovery rate. Let *S*_{0} = *S*(0) be the density of the population at the beginning of the epidemic with everyone susceptible, then it is well known that the so-called basic reproduction number *R*_{0} = *βS*_{0}/*γ* completely determines the transmission dynamics and epidemic potential: if *R*_{0}>1, the *I*(*t*) first increases to its maximum and then decreases to zero and hence an epidemic occurs; if *R*_{0} < 1, then *I*(*t*) decreases to zero and epidemic does not happen.

Such a model and the aforementioned threshold result have been playing a pivotal role in subsequent developments in the mathematical modelling-assisted study of infectious disease transmission dynamics. This model is based on the assumption of a high degree of homogeneity in the population, including the mobility. In reality, however, individuals can be exposed to the infection from contact with infectives in different spatial locations. This consideration led to the Kendall non-local model in 1957 that involves space-dependent integro-differential equations
1.2
where the kernel *K*(*x* − *y*) ≥ 0 weights the contributions of the infected individuals at location *y* to the infection of susceptible individuals at location *x*, with . Disease propagation in space is relevant to the so-called travelling waves, solutions of the form (*S*(*x* + *c t*), *I*(*x* + *c t*), *R*(*x* + *c t*)) for which *c* is called the wave speed. In the case where *R*_{0}>1, Kendall (1965) proved that there exists *c** > 0 such that equation (1.2) admits non-trivial travelling wave solutions for all speeds *c* ≥ *c** and no non-trivial travelling wave solution with speeds less than *c**. Aronson (1977) later showed that equation (1.2) can be reduced to a scalar integro-differential equation, and this reduction enabled him to formally link the wave speed to the asymptotic speed of propagation. Similar reductions can be done for more general situations, for example, the nonlinear (double) integral equation model
1.3
was used to include the incubation period in Diekmann (1978, 1979) and Thieme (1977*a*,*b*, 1979). When , the existence of travelling wave solutions and the asymptotic speed of propagation were considered in Diekmann (1979) and Thieme (1979) for equation (1.3), see also Thieme and Zhao (2003), Ruan (2007) and references therein for other subsequent works in the subject area. An important feature of the reduced model is a certain order-preserving property that permits the applications of the powerful monotone dynamical systems and comparison arguments.

Random movement of individuals in space was further incorporated into the Kendall model by De Mottoni *et al*. (1979) by adding some diffusion terms as follows:
1.4
subject to the Neumannn boundary condition. When the space is unbounded, *μ* = *σ* = 0 and *K*(·) is a constant (*β*) multiple of the delta function, system (1.4) reduces to the following reaction–diffusion model:
1.5
This equation was considered by Hosono and Ilyas (1994), where it is proved that if *S*(0, *x*) = *S*_{0} is a constant and if *γ*/*β S*_{0}<1, then for each there exists a positive constant *ϵ* < *S*_{0} such that system (1.5) has a travelling wave solution (*S*(*x* + *c t*), *I*(*x* + *c t*)) satisfying .

Extension of the above result for system (1.4) with a general kernel *K*(·), even if *μ* = *σ* = 0, is difficult due to the fundamental issue that the system of equations governing the wave solutions is no longer an ordinary differential system: it is a system of functional differential equations with both advanced and delayed arguments and it is a system without any obvious order-preserving property. In addition, if we wish to consider, as pointed out in Ruan (2007), the effect of spatial heterogeneity (geographical movement), non-local interaction and time delay such as latent period on the spread of the disease, we need to examine an even more general model of the following form:
1.6
where *d*_{1}, *d*_{2} and *d*_{3} are the diffusion rates for the susceptible, infective and removed individuals, respectively. The kernel *K*(*x* − *y*, *t* − *s*) ≥ 0 describes the interaction between the infective and the susceptible individuals at location *x* and the present *t* which occurred at location *y* and at earlier instance *s*, which throughout this paper is assumed to satisfy the following conditions:

(K1)

*K*is non-negative and integrable, and satisfies(K2) For every

*c*≥ 0, there exists such that for any*λ ∈*[0,*λ*_{c}), and as*λ*→*λ*_{c}− 0.

It is not difficult to verify that *λ*_{c} is non-decreasing on *c* ≥ 0.

We focus here on the existence and non-existence of travelling wave solutions of system (1.6). We shall prove that if *R*_{0} = *βS*_{0}/*γ* > 1, then there exists *c*_{*} > 0 such that for every *c* > *c*_{*}, system (1.6) admits a non-trivial travelling wave solution with wave speed *c*, and if *R*_{0} = *βS*_{0}/*γ* < 1, then for any *c* ≥ 0, system (1.6) admits no non-trivial travelling wave solutions with wave speed *c*. Therefore, the existence and non-existence of travelling wave solutions is determined completely by the basic reproduction number and the condition *R*_{0} = *βS*_{0}/*γ* = 1 coincides with the threshold for the existence of wavefronts. Furthermore, when *βS*_{0}/*γ* > 1 we show that equation (1.6) admits no non-trivial travelling wave solutions with wave speed *c ∈* [0, *c*_{*}). Therefore, we confirm that *c*_{*}>0 is indeed the minimal wave speed. We do anticipate that *c*_{*} is the asymptotic speed of propagation for equation (1.6), following the work described in Murray (1989), though verification of this requires some additional work. Our approach for the existence of travelling wave solutions is to construct a suitable invariant set and then apply Schauder’s fixed point theorem, see also Li *et al*. (2006) and . Our construction of the invariant set is motivated by the work of Ducrot and Magal (2009). For *c ∈* [0, *c*_{*}), we conclude the non-existence of non-trivial travelling wave solutions by an argument applying the Laplace transform to the *I*(*x* + *c t*) component, this argument was first introduced by Carr and Chmaj (2004) and further used by Wang *et al*. (2008, 2009).

We should point out that when , the existence and non-existence of non-trivial travelling wave solutions of equation (1.6) were proved by Ducrot and Magal (2009) (see also Ducrot *et al*. 2009). These studies considered the following infection-age structured model with diffusion:
1.7
with *a* being the time since the infection and the maximum attainable age of infection. When the incubation is exactly equal to *τ* > 0, then the function *β*(*a*) takes the form . So, if we further assume that and *γ*(*a*) ≡ *γ*, then equation (1.7) reduces to the two former equations of (1.6) with and . However, the aforementioned papers did not prove the existence of the minimal wave speed. We also refer to Faria *et al*. (2006), Gourley *et al*. (2004), Li & Zou (in press), Ou & Wu (2007), Wang *et al*. (2006, 2008) and references therein for some relevant progress on the existence of travelling wave solutions of reaction–diffusion equations with non-local interaction and time delay.

Our analytic study about the minimal speed permits discussion of the effect on the minimal wave speed *c** of (i) the diffusion rate *d*_{2} of infective individuals, (ii) non-local interaction between the infective and the susceptible individuals and (iii) the latent period of disease. We confirm that the latent period of disease can slow down the speed of the disease, the non-local interaction between the infective and the susceptible individuals and the spatial movement of infective individuals can increase the speed of the spread of the disease.

## 2. Main results

In this section, we will state precisely and prove the main results of this paper. In the sequel, we always assume that the initial disease free equilibrium is (*S*_{0},0,0).

Because the first equation and the second equation of system (1.6) form a closed system, we consider only the following system:
2.1
We look for the non-trivial travelling wave solutions (*S*_{c}(*x* + *c t*), *I*_{c}(*x* + *c t*)) of (2.1) satisfying the following conditions:
2.2
and
Let ξ = *x* + *c t*. Then the system describing travelling wave solutions is as follows:
2.3

Linearizing the second equation of (2.3) at the initial disease free point (*S*_{0},0), we have
2.4
Let *J*(ξ) = e^{λξ}, then we get a characteristic equation
2.5
For the sake of convenience, we set
It is easy to prove the following lemma, see also Li *et al*. (2007, lemma 3.27) and Wang *et al*. (2006, lemma 2.2).

## Lemma 2.1.

*Assume that S*_{0} > *γ*/*β. Then there exists c*_{*} > 0 *and λ** > 0 *such that* ∂/∂ *λ Θ*(*λ*, *c*)|_{(λ*, c*)} = 0 and *Θ*(*λ**, *c*_{*}) = 0. *Furthermore*,

*if*0<*c*<*c*_{*},*then Θ*(*λ*,*c*) > 0*for all λ ∈*[0,*λ*_{c});*if c*>*c*_{*},*then the equation Θ*(*λ*,*c*) = 0*has two positive real roots λ*_{1}(*c*)*and λ*_{2}(*c*)*with*0 <*λ*_{1}(*c*) <*λ**<*λ*_{2}(*c*) <*λ*_{c}*such that*,*and*

In the following, we always assume that *S*_{0} > *γ*/*β*. In addition, we fix *c* > *c*_{*} and always denote *λ*_{i}(*c*) by *λ*_{i}, *i* = 1,2.

## Lemma 2.2.

*The function I*^{+}*(ξ) = e*^{λ1ξ}*satisfies the following linear equation:*
2.6

## Lemma 2.3.

*For α > 0 sufficiently small and σ > S*

_{0}

*large enough, the function*

*satisfies*2.7

*for any*

*.*

## Proof.

When , we have *S*^{−}(ξ) = 0, which implies that equation (2.7) holds.

When , namely *S*^{−}(ξ) = *S*_{0}− *σ* e^{αξ} > 0, we need to prove that
2.8
Obviously, it is sufficient to ensure
That is,
2.9
Keeping *α σ* = 1 and letting , for some *σ* large enough we have that equation (2.9) holds. This completes the proof. ▪

## Lemma 2.4.

*Let ϵ > 0 satisfy ϵ < α/2 and ϵ < λ*

_{2}

*−*

*λ*_{1}

*. Then for M > 0 sufficiently large, the function I*

^{−}

*(ξ): = e*

^{λ1ξ}

*(1−Me*

^{ϵξ}

*) satisfies*

## Proof.

If *S*_{0}− *σ*e^{αξ}≤0, that is, , it is needed to prove that
2.10
Namely
Consequently, we need to verify that
Since , it is sufficient to prove
Then, for sufficiently large *M* > 0 with
we have that equation (2.10) holds.

If *S*_{0}− *σ* e^{αξ} > 0, that is, , it is sufficient to prove that
that is,
Consequently, we need only to show that
Since (see lemma 2.3, *σ* > *S*_{0}), we have . Therefore, we have
if . Thus, the proof is completed. ▪

Define
and
where *α*_{1}≥ *S*_{0} and *α*_{2}≥ *γ* satisfy −Λ_{11} > 2 *λ*_{1} and −Λ_{21} > 2 *λ*_{1}. Furthermore, define an operator by
where
In the following, for *u ∈ Γ* we denote
2.11

## Lemma 2.5.

*The set Γ is closed and convex in*

*.*

The proof is very easy and we omit it.

## Lemma 2.6.

*The operator F maps Γ into Γ.*

## Proof.

Give (*S*(·), *I*(·)) *∈ Γ*. It is obvious that we only need to prove that
and
for all .

First, we consider *F*_{1}[ *S*(·), *I*(·)](*x*). Since *α*_{1}*S*(*x*)− *S*(*x*)(*K** *I*)(*x*)≤ *α*_{1}*S*_{0}, for any we have
Since satisfies
for , we have
It follows that
When , we have
Similarly, when , we have

Secondly, we consider *F*_{2}[ *S*(·), *I*(·)](*x*). Since *S*(*x*)(*K** *I*)(*x*) + ((*α*_{2}− *γ*)/*β*) *I*(*x*)≥0 for all , we have *F*_{2}[(*S*(·), *I*(·))](*x*)≥0 for all . Because and , we have that *I*^{−}(ξ): = e^{λ1ξ}(1− *M*e^{ϵξ}) satisfies
Hence, we have
Because *I*(ξ) ≤ *I*^{+}(ξ) and *S*(ξ) ≤ *S*_{0}, by equation (2.6) we have
Consequently, we have
This completes the proof. ▪

Define
with norm
where *μ* > 0 is a constant and satisfies .

## Lemma 2.7.

*The map**is continuous with respect to the norm |·|*_{μ}*in**.*

## Proof.

For any (*S*_{1}(·), *I*_{1}(·)) *∈ Γ* and (*S*_{2}(·), *I*_{2}(·)) *∈ Γ*, we have
Note that (*K** *I*)(*x*)≤ *G*(*λ*_{1}, *c*) e^{λ1x}. When ξ≥0, we have
Similarly, for ξ<0 we have
Then, it is sufficient to prove that | *S*_{1}(·)− *S*_{2}(·)|_{μ}→0 and | *I*_{1}(·)− *I*_{2}(·)|_{μ}→0 imply | *S*_{1}(·)− *S*_{2}(·)|_{μ/2}→0 and |(*K** *I*_{1})(·)−(*K** *I*_{2})(·)|_{μ}→0, respectively. Given ε > 0 sufficiently small. Note that | *S*_{1}(*x*)− *S*_{2}(*x*)|≤ *S*_{0} for any . Then there exists *N* > 0 such that | *S*_{1}(*x*)− *S*_{2}(*x*)|e^{− μ| x|/2}≤ *S*_{0}e^{− μ N/2}≤ε for any | *x*|≥ *N*. Furthermore, when | *S*_{1}(·)− *S*_{2}(·)|_{μ}<εe^{− μ N/2}, for | *x*|< *N* we have
Thus, we conclude that | *S*_{1}(·)− *S*_{2}(·)|_{μ/2}→0 as | *S*_{1}(·)− *S*_{2}(·)|_{μ}→0. Consider
Since , then there exists *N** > 0 such that

Furthermore, when | *I*_{1}(·)− *I*_{2}(·)|_{μ}≤ε e^{− μ(1 + c) N*}, we have
Combining the above arguments and the fact | *I*_{1}(*x*)− *I*_{2}(*x*)|≤e^{λ1x} for any , we have that
Thus, we conclude that is continuous with respect to the norm |·|_{μ} in . Similarly, we can prove that is continuous with respect to the norm |·|_{μ} in . This completes the proof. ▪

## Lemma 2.8.

*The map**is compact with respect to the norm |·|*_{μ}*in**.*

## Proof.

For any (*S*, *I*) *∈ Γ*, we have
Therefore, for any we have
Similarly, for any we have
For each integer , define an operator *F*^{n} by
By Ascoli–Arzela lemma, we have that is compact with respect to supremum norm in because *F*^{n}[ *S*(·), *I*(·)](·) is also uniformly bounded and equicontinuous for (*S*, *I*) *∈ Γ*. Consequently, we have that is compact with respect to the norm |·|_{μ} in . Furthermore, since is a compact series and
by proposition 2.1 in Zeilder (1986) we have that converges to *F* in *Γ* with respect to the norm |·|_{μ} and hence, is compact with respect to the norm |·|_{μ} in . The proof is completed. ▪

## Theorem 2.9.

*Assume that S*_{0}*> γ/β. For every c > c*

_{*}

*, system (2.1) admits a travelling wave solution (S*

_{c}

*(x + ct),I*

_{c}

*(x + ct)) such that (2.2),*

*for any*

*and S*

_{c}

*(·) is non-increasing in*

*. In addition, we have*

## Proof.

When *c* > *c*_{*}, Schauder’s fixed point theorem implies that there exists a pair of (*S*_{c}(·), *I*_{c}(·)) *∈ Γ*, which is a fixed point of the operator *F*. Consequently, the solution (*S*_{c}(*x* + *c t*), *I*_{c}(*x* + *c t*)) is a non-negative travelling wave solution of equation (2.1). It is obvious that , , 0≤ *S*_{c}(ξ)≤ *S*_{0} and 0≤ *I*_{c}(ξ)≤e^{λ1ξ} for any . In the following, we first prove that *S*_{c}(ξ) is non-increasing and equation (2.2) holds.

Note that (*S*_{c}(·), *I*_{c}(·)) *∈ Γ* is a fixed point of the operator *F*. Applying the L’Hospital theorem to the maps *F*_{1} and *F*_{2}, it is easy to show that
Consequently, it follows from equation (2.3) that and . Integrating the two sides of
2.12
from to ξ, we have
Since 0≤ *S*_{c}(ξ)≤ *S*_{0} for any , we conclude that and hence, is bounded on . Otherwise, if , then there exists a constant δ_{0} > 0 such that for large ξ > 0, which contradicts the fact 0≤ *S*_{c}(*x*)≤ *S*_{0}. Therefore, and is bounded on . Multiplying (1/*d*_{1}) e^{−(c/d1)ξ} for the two sides of the equality (2.12), we have
Integrating the last equality from ξ to , we have
which implies that *S*_{c}(ξ) is non-increasing in . Because (*S*_{c}, *I*_{c}) *∈ Γ*, for ξ<0 with |ξ| sufficiently large, we have
Hence, there exists ξ*<0 such that *d*/dξ(*S*_{c}(ξ))<0 for ξ<ξ*. Therefore, we have .

Furthermore, since 2.13 we have for any , where In view of , we have that Since we have Consequently, it follows that for any . Thus, we have because is bounded.

Now by lemma 2.3 in Wu & Zou (2001), we have
Consequently, integrating equation (2.12) on yields
Furthermore, by integrating equation (2.13) on , we obtain
To prove that , we define a function for any which satisfies the following equation:
Obviously, , and . Furthermore, we can show that is non-decreasing in . In fact, *N*_{c}(ξ) satisfies
Following this, we have that
In view of , we have that for any . This completes the proof. ▪

## Theorem 2.10.

*Assume that S*_{0}*> γ/β. For*

*, there exist no non-trivial travelling wave solutions*

*of equation (2.1) such that equation (2.2) and*

*and*

*.*

## Proof.

Now we consider the case *c ∈* (0, *c*_{*}). Fix *c ∈* (0, *c*_{*}). We prove the theorem by way of contradiction. Assume that there exists a non-trivial travelling wave solutions (*S*_{c}(*x* + *c t*), *I*_{c}(*x* + *c t*)) of (2.1) such that equation (2.2). Since , there exists such that *S*(ξ) > (*β S*_{0} + *γ*)/2 *β* for any . Therefore, we have
2.14
for any ξ≤ξ′. Let for any . It is not difficult to verify , see also Wang & Li (2009, theorem 3.5), where the convolution is defined by equation (2.11). Then, integrating two sides of inequality (2.14) from to ξ with ξ≤ξ′, we have
2.15
In view of
we have that (*K** *J*)(ξ)− *J*(ξ) is integrable on for any . Consequently, from equation (2.15) we have that *J*(ξ) is integrable on for any . Now integrating the two sides of inequality (2.15) from to ξ with ξ≤ξ′, we have
Since (*y* + *c s*) *J*(ξ−θ(*y* + *c s*)) is non-increasing on θ *∈* [0,1], we have
Let . Since the kernel *K*(*y*, *s*) is an even function of *y*, we have . Then, we have
2.16
Therefore, for any ξ≤ξ′ we have
Since *J*(·) is increasing, then for any ξ≤ξ′ and any η > 0 we have
Thus, there exists η_{0} > 0 sufficiently large and some θ_{0}*∈* (0,1) such that
Let *p*(*x*) = *J*(*x*) e^{− μ0x} with . Then,
By virtue of as , we have that there exists *p*_{0} > 0 such that
which implies that
Consequently, there exists *q*_{0} > 0 such that for any . Furthermore, by equations (2.14)–(2.16), we have
Now consider *S*_{0}− *S*(ξ). By *c S*′(ξ) = *d*_{1}*S*′′(ξ)− *β S*(ξ)(*K** *I*)(ξ), integrating from to ξ yields
Let . It is obvious that *f*(*x*)≤ *C*_{0} e^{μ0x} for any and some constant *C*_{0} > 0. Let *R*(ξ) = *S*_{0}− *S*(ξ)≥0 for any . Then, we have
Solving the last ODE yields
where . Since *f*(ξ) = *O*(e^{μ0ξ}) as , it is easy to see that as , where . In view of 0≤ *R*(ξ)≤ *S*_{0}, we have

For with 0<Re *λ*< *μ*_{0}, we can define a two-sided Laplace transform of *I* by
Applying the property of Laplace transforms (see Widder 1941), we know that either there exists a real number *α* > 0 such that is analytic for with 0<Re *λ*< *α* and *λ* = *α* is singular point of , or for with Re *λ* > 0, is well defined. Now we use this property to conclude that for *c ∈* (0, *c*_{*}), equation (2.1) admits no travelling wave solutions (*S*(*x* + *c t*), *I*(*x* + *c t*)) satisfying equation (2.2).

By Fubini’s theorem, we have
where . In view of
we have
2.17
for with 0<Re *λ*< *μ*_{0}, where *Θ*(*λ*, *c*) is defined by equation (2.5). Note that the right-hand side of equation (2.17) is defined for with 0<Re *λ*< *μ*_{0} + *μ*′_{0}. For *c ∈* (0, *c*_{*}), since *Θ*(*λ*, *c*) > 0 for *λ ∈* (0, *λ*_{c}), we have that is defined for all with *λ*_{c} > Re *λ* > 0 and there are no singularity of in *λ ∈* [0, *λ*_{c}). Because equation (2.17) can be re-written as
However, for *c ∈* (0, *c*_{*}), we have that as , which implies that the last equality is false. This is a contradiction. The proof is complete. ▪

## Theorem 2.11.

*Assume that S*_{0}*< γ/β. Then for any c≥0, there exists no travelling wave solutions (S(x + ct),I(x + ct)) satisfying*
2.18

## Proof.

Assume that there exists non-trivial travelling wave solution (*S*(*x* + *c t*), *I*(*x* + *c t*)) such that equation (2.18). Then, we have
Integrating the two sides of the last equality, we have
This is a contradiction. This completes the proof. ▪

## Remark 2.12.

When *d*_{1} · *d*_{2} = 0, the conclusions in theorems 2.9, 2.10 and 2.11 remain valid. In fact, if *d*_{1} = 0, then it is sufficient to define *F*_{1}by
where *α*_{1}*≥S*_{0} satisfies *β α*_{1}*/c > 2λ*_{1}. Similarly, if *d*_{2} = 0, then we need only to redefine *F*_{2}by
where *α*_{2} ≥ *γ* satisfies *α*_{2}*/c > 2λ*_{1}.

## 3. Discussion

In this paper, we study the existence and non-existence of non-trivial travelling wave solutions for model (1.6). As the travelling wave solutions obtained or excluded in this work describe the transition from a disease-free equilibrium to an endemic equilibrium, the existence and non-existence of non-trivial travelling wave solutions indicates whether or not the disease can spread.

Theorems 2.9 and 2.11 combined provide a threshold condition for the existence of travelling wave solutions in terms of the basic reproduction number *βS*_{0}/*γ* of the corresponding ODE system in the absence of non-local interaction, time delays and spatial diffusion. Therefore, whether disease spreads or not is independent of the non-local delayed interaction and spatial movement patterns of the population.

The speed at which the disease spreads (if it does), however, depends on the aforementioned factors. We have shown that if the basic reproduction number is larger than one, then system (1.6) admits a non-trivial travelling wave solution with wave speed *c* > *c*_{*}, where *c*_{*} is the minimal wave speed. As discussed in §1, this minimal wave speed *c*_{*} should be the asymptotic speed of propagation of the disease. This minimal wave speed *c*_{*} is defined by lemma 2.1, from which it is easy to see that *c*_{*} is dependent on the diffusion rate *d*_{2} of the infected individuals, the pattern of non-local interaction between the infected and the susceptible individuals, and the latent period of disease.

More specifically, for *c* > 0 and *λ ∈* (0, *λ*_{c}), direct calculations yield that
3.1
In the case where *K*(*x*, *t*) = δ(*t*− *τ*)δ(*x*) with *τ* > 0, we have
Therefore, from equation (3.1), we can conclude that *c*_{*} = *c*_{*}(*τ*) is a decreasing function of *τ* > 0.

In the case where , we have
Consequently, *c*_{*} = *c*_{*}(ρ) is an increasing function of ρ > 0. Indeed, direct calculations also give
3.2
and
3.3
Hence, we observed that the latent period can reduce the speed of the spread of the disease, and the non-locality of interaction can increase the speed of disease spread, an observation in coincidence with those reported by Li *et al*. (2007) and Wang *et al*. (2008).

Similar conclusions can be made for the case where with *τ* > 0. This case reduces to equation (1.7), and we have
This implies that *c*_{*} is an increasing function of *d*_{2} > 0 and hence, we know that the geographical movement of infected individuals can increase the speed of the spread of the disease. Now fix *d*_{2} > 0, then for any *τ*_{0} > 0, there exists a unique pair of *λ*_{*}(*τ*_{0}) > 0 and *c*_{*}(*τ*_{0}) > 0 such that *Θ*(*λ*_{*}(*τ*_{0}), *c*_{*}(*τ*_{0}), *d*_{2}, *τ*_{0}) = 0 and *Θ*(*λ*, *c*_{*}(*τ*_{0}), *d*_{2}, *τ*_{0})≥0 for any *λ* ≥ 0. It is easy to see that . Then we have
which implies that *c*_{*} is a decreasing function of *τ* > 0.

## Acknowledgements

Z.-C.W. was supported by NSF of Gansu Province of China (0710RJZA020) and The Fundamental Research Fund for Physics and Mathematic of Lanzhou University (LZULL200807). This work was partially supported by the Canada Research Chairs Program, by Natural Sciences and Engineering Research Council of Canada (NSERC), by Canadian Institute of Health Research (CIHR), by Mathematics for Information Technology and Complex Systems (MITACS) and by Geomatics for Informed Decisions (GEOIDE).

## Footnotes

- Received July 17, 2009.
- Accepted September 7, 2009.

- © 2009 The Royal Society