Traveling Wave Solutions in a Reaction-Diffusion Epidemic Model

Lemma 1. If [H1] holds, then E₂ and $$ are endemic points of model (3) and model (7), respectively.

Lemma 2. For model (7), if [H1] holds, then $$ is unstable, and $$ is stable.

Lemma 3. Model (14) is a quasi-monotone decreasing system in $$.

Proof. Let

From [17], we can know that the functions F₁(S, I) and F₂(S, I) are said to possess a quasi-monotone nonincreasing system, if the sign of ∂F₁(S, I)/∂I and ∂F₂(S, I)/∂S are both nonpositive.

Since

Then, Let then obviously, $$ is the unique real root of G^′(z).

Since $$, consider α₀ = (R_d − 1)/R_d, then we can get

And hence, G(z) has two positive roots.

Since ν ≥ R₀/(2 − R_d), thus G(α₀) = R₀ + νR_d − 2ν ≤ 0.

According to conditions [H1] and [H2], we can get

Then, $$. Hence, ∂F₁(S, I)/∂I ≤ 0.

That is to say, model (14) is a quasi-monotone system in $$.

Remark 4. Model (24) is also a quasi-monotone system in $$.

Definition 5. A smooth function $$  (ξ ∈ ℝ) is an upper solution of model (24) if its derivatives $$ and $$ are continuous on ℝ, and $$ satisfies

with the following boundary value conditions

Definition 6. A smooth function $$  (ξ ∈ ℝ) is a lower solution of model (24) if its derivatives $$ and $$ are continuous on ℝ, and $$ satisfies

with the following boundary value conditions

Lemma 7 (see [1], [15].)Corresponding to every $$, model (29) has a unique (up to a translation of the origin) monotonically increasing traveling wave solution w(ξ) for ξ ∈ R. The traveling wave solution w has the following asymptotic behaviors.

• (i)

  For the wave solution with noncritical speed $$, one has

  $$ ()

where a_ω and b_ω are positive constants.

• (ii)

  For the wave with critical speed $$, one has

  $$ ()

where the constant d_c is negative, b_c is positive, and a_c ∈ R.

Lemma 8. For each $$, (33) is a smooth upper solution of model (24).

Proof. On the boundary,

As for the I component, we have As for the S component, since ν > 1, then $$. And

• (i)

  if $$, then $$;

• (ii)

  if $$, then $$.

Thus we can get:

Hence, $$ forms a smooth upper solution for model (24).

Lemma 9. For each fixed $$, (38) is a lower solution of model (24).

Proof. On the boundary,

As for the I component, we have As for the S component, we have Thus $$ forms a smooth lower solution for model (24).

Lemma 10. Let $$, $$ and $$ be the upper solution and the lower solution defined in (33) and (38), then there exists a positive number r, such that $$ for all ξ ∈ R.

Proof. Our proof is only for $$, and the proof for the case of $$ is similar to it.

First, we derive the asymptotic behaviors of the upper solution and the lower solution at infinities.

According to Lemma 7, when ξ → −∞, we can obtain:

And let $$, $$, when ξ → +∞, we can get

where, A₁, A₂, B₁, B₂ are all positive constants.

Since for any $$, $$ is also a solution of model (32). Thus, $$ is an upper solution of model (24). So, according to Lemma 7, when ξ → −∞, we can get:

Since $$, we can choose a large enough number $$, such that

hence, there exists a large number N₁ ≫ 1, such that

By using a similar argument as above, there exists a large enough number N₂ ≫ 1, such that

Second, we show that

We deal with such two possible cases:

Case  1. If

then, the proof is completed.

Case  2. If there exists a point ξ₀ ∈ (−N₁, N₂), such that

satisfying  $$ or $$.

In this case, we use the Sliding Domain method [15].

Step  1. we shift $$ to the left by increasing the number $$ until finding a new number $$ such that $$ on the smaller interval $$.

Step  2. we shift $$ back to the right by decreasing r₁ to a smaller number $$ such that one of the branches of the upper solution touches its counterpart of the lower solution at some point ξ₁ in the interval $$. On the endpoints of the interval $$, we still have $$.

Let $$ and $$, where

For $$, we get that

where ζ_i ∈ [0,1], i = 1,2, 3,4. Since the above model is monotone and the cube $$ is convex, thus we can deduce by Maximum Principle that $$ for $$. So ξ₁ does not exist and we can decrease r₂ further to $$. It is calculated that the point ξ₀ does not exist either. The proof of this lemma is completed.

Theorem 11. For $$, model (24) has a unique (up to a translation of the origin) traveling wave solution. The traveling wave solution is strictly increasing and has the following asymptotic properties:

• (i)

  if $$, when ξ → −∞,

  $$ ()

when ξ → +∞, and if μ₁ ≠ μ₂, then while μ₁ = μ₂: where, μ = min {μ₁, μ₂} > 0, $$, A₁, A₂, $$, $$, $$ and $$ are all positive constants.

• (ii)

  if $$, when ξ → −∞,

  $$ ()

when ξ → +∞, and if μ₁ ≠ μ₂, then while μ₁ = μ₂, where μ = min {μ₁, μ₂} > 0, B₁₂, B₂₂ < 0, $$, and $$, $$, $$, $$ are all positive constants.

Proof. From Lemma 3 and Remark 4, we know that model (24) is a quasi-monotone nonincreasing system in $$, and by using the monotone iteration scheme given in [3, 13], we can obtain the existence of the solution (S, I) ^T to the first two equations in model (24) for every $$, which satisfies

According to the above inequality, we can get that, on the boundary, the solution tends to (0,0) ^T as ξ → −∞ and $$ as ξ → +∞.

To derive the asymptotic decay rate of the traveling wave solutions as ξ → ±∞, we just let $$ and

be the traveling wave solution of model (24) generated form the monotone iteration, since the case of (ii) $$ is similar to it.

We differentiate model (24) with respect to ξ, and note that U^′(ξ) = (χ₁, χ₂) ^T(ξ) satisfies

where

Now, we study the exponential decay rate of the traveling wave solution as ξ → −∞. The asymptotic model of model (62) as ξ → −∞ is

where

The second equation of model (64) has two independent solutions with the following form:

Relating the second equation of model (62) with the second equation of model (64), we can deduce that χ₂(ξ) has the following property as ξ → −∞:

for some constants α and β. Thus, we can obtain that where So we obtain that Thus, $$.

Now, we consider the first equation of model (64). We rewrite it as

One can verify that $$ is not a characteristic of

The above equation has two independent solutions of the following form:

Thus, when ξ → −∞, χ₁(ξ) has the following property: for some constants $$; γ ≠ 0. Since χ₁(−∞) = 0, thus $$. So, when ξ → −∞, we have the following formula:

Then, we study the exponential decay rate of the traveling wave solution as ξ → +∞. The asymptotic model of model (62) as ξ → +∞ is

By setting $$, i = 1,2, we rewrite model (76) as a first order model of ordinary differential equation in the four components $$:

In the case of (i) μ₁ ≠ μ₂, we can obtain that the solution of model (77) has the following form:

where and h_i  (i = 1,2, 3,4) are the eigenvectors of the constant matrix with λ_i  (i = 1,2, 3,4) as the corresponding eigenvalues, c_i  (i = 1,2, 3,4) are arbitrary constants. Since thus $$, so when ξ → +∞, we can get that

Furthermore, we can obtain that

where κ₁, κ₂, Λ₁, Λ₂, Γ₁, and Γ₂ are all constants.

Let μ = min {μ₁, μ₂}, then

where thus, when ξ → +∞, we can get that

In the case of (ii) μ₁ = μ₂, we can obtain that the solution of model (77) has the following form:

where H_1,2 is the eigenvector of the constant matrix with $$ as the corresponding eigenvalues, H_3,4 is the eigenvector of the constant matrix with $$ as the corresponding eigenvalues, G_i  (i = 1,2, 3,4) are arbitrary constants.

Since $$, thus

So, when ξ → +∞, we can get that

By comparing the upper solution and roughness of the exponential dichotomy [24], we obtain the asymptotic decay rate of the traveling wave solutions at +∞ given in Theorem 11.

According to the monotone iteration process [3], the traveling wave solution U(ξ) is increasing; thus U^′(ξ) ≥ 0 and $$ hold

satisfying

The strong Maximum Principle implies that $$. So the strict monotonicity of the traveling wave solutions is concluded.

Now, we use the Sliding domain method to prove the uniqueness of the traveling wave solution. Let $$ and $$ be the traveling wave solution of model (24), with $$. Thus, there are some positive numbers A_i, B_i  (i = 1,2, 3,4), such that for a big enough number N ≫ 1, when ξ < −N, we have

when ξ > N, Since the traveling wave solutions of model (24) are translation-invariant, then for any θ ∈ R, $$ is also a traveling wave solution of model (24). Thus, by using the same method as above, when ξ < −N, we can get when ξ > N, If θ is large enough, then we can obtain the following inequalities: Thus, if θ is large enough, then $$, for all ξ ∈ R∖[−N, N].

Now, we consider model (24) on the interval [−N, N].

First, suppose that

then where, ζ_i ∈ (0,1)  (i = 1,2, 3,4), ξ ∈ (−N, N). Since the above model is monotone, by the Maximum Principle, we can deduce that W(ξ) > 0,  ξ ∈ [−N, N]. Consequently, we get that $$.

Second, we suppose that there exists a point ξ_* ∈ (−N, N) such that

or

In this case, we increase θ, that is shifting $$ to the left, so that $$ and $$. According to the monotonicity of $$ and U₂, we can find a number $$ such that $$, ξ ∈ (−N, N). Shifting $$ back until one component of $$ touches its counterpart of U₂(ξ) at some point $$. Since $$ and U₂(ξ) are strictly increasing, $$, thus, we get that $$, ξ = ±N. However, by the Maximum Principle for that component again, we find that components of $$ and U₂ are identically equal for all ξ ∈ [−N, N] for a larger number θ. This is a contradiction, thus $$, ξ ∈ R. Here, θ is a new number which is chosen by the above mean.

Now, decrease the θ until one of the following happens.

Case (a). There is a $$, such that $$, ξ ∈ R. In this case, we have finished the proof.

Case (b). There are a $$ and a point ξ₁ ∈ R, such that one of the components of $$ and U₂ are equal. And $$, ξ ∈ R. On R for that component, according to the Maximum Principle, we find that $$ and U₂ must be identical on that component. We can return to Case (a).

Consequently, in either situation, their exists a number $$ such that

This ends of the proof.

Theorem 12. For each $$, model (3) has a unique (up to a translation of the origin) traveling wave solution. The traveling wave solution is strictly increasing and has the following asymptotic properties:

• (i)

  $$: when ξ → −∞,

  $$ ()

when ξ → +∞, and if μ₁ ≠ μ₂, then if μ₁ = μ₂, where μ = min {μ₁, μ₂} > 0, $$  A₁, A₂, $$, $$, $$ and $$ are all positive constants.

• (ii)

  $$: when ξ → −∞,

  $$ ()

when ξ → +∞, and if μ₁ ≠ μ₂, then if μ₁ = μ₂, then where μ = min {μ₁, μ₂} > 0, B₁₂, B₂₂ < 0, $$, $$, $$, $$, $$ are all positive constants.

Theorem 13. There is no monotone traveling wave solution of model (24) for any $$. In other words, there is no monotone traveling wave solution of model (3) for any $$.

Proof. Suppose there is a monotone traveling wave solution $$ of model (24) with the wave speed c₀, where $$.

The asymptotic model of $$ as ξ → −∞ is

The second function of (108) has two characteristics as the following ones: $$, $$. Thus it has two independent solutions of the following form:

Similar to the proof of Theorem 11, we can get that, when ξ → −∞, χ₂(ξ) can be described as the following equation:

where tan(τ(ξ)) = K₁/K₂, and h.o.t is the short notation for the higher order terms.

That is to say, l₂(ξ) is oscillating. Thus, any solution of model (24) with $$ is not strictly monotone.