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Abstract
Scientists have been seeking ways to use Wolbachia to eliminate the mosquitoes that spread human diseases. Could Wolbachia be the determining factor in controlling the mosquito-borne infectious diseases? To answer this question mathematically, we develop a reaction-diffusion model with free boundary in a one-dimensional environment. We divide the female mosquito population into two groups: one is the uninfected mosquito population that grows in the whole region while the other is the mosquito population infected with Wolbachia that occupies a finite small region. The mosquito population infected with Wolbachia invades the environment with a spreading front governed by a free boundary satisfying the well-known one-phase Stefan condition. For the resulting free boundary problem, we establish criteria under which spreading and vanishing occur. Our results provide useful insights on designing a feasible mosquito releasing strategy that infects the whole mosquito population with Wolbachia and eradicates the mosquito-borne diseases eventually.
1. Introduction
Recently, several public health projects were launched in China [Citation27], USA [Citation1] and France [Citation22], with an aim to fight mosquito populations that transmit Zika virus, Dengue fever and Chikungunya. All of these projects involve the release of male Aedes aegypti mosquitoes infected with the Wolbachia bacteria to the wild. For instance, 20000 male Aedes aegypti mosquitoes carrying Wolbachia bacteria were released on Stock Island of the Florida Keys in the week of April 20, 2017. Google's Verily is about to release 20 million machine-reared Wolbachia-infected mosquitoes in Fresno (see [Citation1]). A factory in Southern China is manufacturing millions of ‘mosquito warriors’ (male Aedes aegypti mosquitoes carrying Wolbachia bacteria) to combat epidemics transmitted by mosquitoes [Citation27].
The science behind these projects is based on the following two facts: (i) Wolbachia often induces cytoplasmic incompatibility (CI) which leads to early embryonic death when Wolbachia-infected males mate with uninfected females and (ii) Wolbachia-infected females produce viable embryos after mating with either infected or uninfected males, resulting in a reproductive advantage over uninfected females. In practice, Wolbachia has been successfully transferred into Aedes aegypti or Aedes albopictus by embryonic microinjections, and the injected infection has been stably maintained with complete CI and nearly perfect maternal transmission [Citation2,Citation16,Citation17,Citation23,Citation31,Citation32,Citation34]. Thus, the bacterium is expected to invade host population easily driving the host population to decline. Successful Wolbachia invasion in Aedes aegypti has been observed by Xi et al. in the laboratory caged population within seven generations [Citation33].
By releasing Aedes albopictus mosquitoes infected with Wolbachia bacteria into the wild, it is expected that over a long time period, the wild Aedes aegypti mosquito population would decline drastically and hopefully be completely replaced by infected mosquitoes so that the mosquito-borne infectious diseases such as Zika, Dengue fever and Chikungunya would be eradicated. To qualitatively examine if Wolbachia can effectively invade the wild uninfected mosquito population, Zheng, Tang and Yu [Citation38] considered the following model:
(1)
(1) where u denotes the number of reproductive infected insects and v denotes uninfected ones,
and
denote half of the constant birth rates for the infected and uninfected insects respectively. The parameter
(resp.
) denotes the density-dependent death rate for the infected (resp. uninfected) population. The birth rate of uninfected mosquitoes is diminished by the factor
due to the sterility caused by cytoplasmic incompatibility (CI) for mating between infected males and uninfected females.
Let us now recall the origin of system (Equation1(1)
(1) ) with some details. Let
and
denote the number of released female mosquitoes and the number of released males, respectively, and suppose the released mosquitoes were infected with Wolbachia. Also, assume that
and
satisfy
(2)
(2) where
denotes the total population size, with
,
and
standing for the numbers of uninfected reproductive females, uninfected reproductive males, and infected reproductive females and males other than those from releasing, respectively. Let
(resp.
) be the natural birth rate of the infected (resp. uninfected) mosquitos and
be the proportion of mosquitos born female. Then the proportion of mosquitos born male is
. With complete CI (see Table ) and perfect maternal transmission, we have
(3)
(3)
Table 1. Strong CI, × means ‘no offspring’.
We also note that female Aedes aegypti mosquitoes infected with the Wolbachia bacteria were initially released at a specific site. Hence, the infected female mosquitoes initially occupy only a small region, while the wild uninfected females are distributed over the whole area.
To model the spatial spreading of Wolbachia in the wild mosquito population and explore the possibility that the infection can indeed occupy the whole region, it is natural to consider system (Equation5(5)
(5) ) under the setting of a free boundary problem.
In this work, we consider the following free boundary problem in one-dimensional space:
(6)
(6) The equation governing the movement of the spreading front
is deduced in a manner similar to that in Section 1.3 of [Citation3]. It is known as the one-phase Stefan condition in the literature. This type of free boundary condition has been widely used in previous work such as [Citation5–7,Citation10,Citation18–20,Citation24,Citation28–30].
We will first analyse system (Equation6(6)
(6) ) with constant birth rates
and
in Section 3. Environmental variables such as available water surfaces and humidity have huge impacts on birth rates [Citation8]. This is why we also extend our study to the case with space-dependent birth rates
and
in Section 4, while we assume that the natural death rate is spatially independent for simplicity.
Throughout this paper, we assume that and
satisfy the following conditions, unless otherwise stated:
is the Hölder space with Hölder exponent θ. The initial conditions
and
are assumed to be bounded and satisfy
(7)
(7) For the free boundary problem (Equation6
(6)
(6) )–(Equation7
(7)
(7) ), the main question we are concerned about is whether the infected population can eventually occupy the whole space or not.
Definition 1.1
The notion of vanishing and spreading
If the infected population eventually occupies the whole space, i.e.
we say spreading occurs; otherwise, we say vanishing occurs.
The main goal of this work is to derive conditions under which the spreading occurs. If spreading occurs, then the whole mosquito population will become infected with Wolbachia bacteria and this leads to the extinction of the mosquito population and eventually the eradication of mosquito-borne diseases.
Organization of the paper. The paper is organized as follows. We first establish the global existence and uniqueness of solutions to the free boundary problem (Equation6(6)
(6) ) in Section 2. In Section 3, we present a detailed analysis of a specific case of model (Equation6
(6)
(6) ). In Section 4, we study the population dynamics of infected mosquitoes in a heterogeneous environment with a free boundary condition. In order to better understand the effects of dispersal and spatial variations on the outcome of the competition, we study system (Equation6
(6)
(6) ) over a bounded domain with Neumann boundary conditions. We summarize our results in the last section.
2. Global existence of smooth solutions
Using arguments that are similar to those in [Citation11], we can establish the following result concerning the existence and uniqueness of solutions to system (Equation6(6)
(6) )–(Equation7
(7)
(7) ).
Theorem 2.1
Local existence
Consider system (Equation6(6)
(6) ) with initial conditions (Equation7
(7)
(7) ). Assume that
and
satisfy (
). Then, there exists T>0 such that (Equation6
(6)
(6) ) admits a unique solution
satisfying
where is the Hölder exponent in (
),
K and T are constants that depend only on
θ,
and
.
The next result provides some bounds on the solutions to system (Equation6(6)
(6) ) with initial conditions (Equation7
(7)
(7) ).
Lemma 2.1
Let be a solution of (Equation6
(6)
(6) ) for
for some T>0. Then,
for all
and
where
for all
and
where
for all
where
depends on μ,
and
.
Proof.
The strong maximum principle yields that for all
and
and
for all
and
. Note that
yields that
Thus,
for
. Next, we consider the initial value problem
(8)
(8) From the comparison principle, we know that
Similarly, we can show that
To prove (iii), we first consider the auxiliary function
(9)
(9) for
and
, where
We have
(10)
(10) We also note that
and
Thus,
. Applying the comparison principle, we get
Since
, we then have
Consequently,
with
.
Bearing the above result in mind, we can show that the local solution obtained in Theorem 2.1 can indeed be extended to all t>0.
Theorem 2.2
Global existence and uniqueness
System (Equation6(6)
(6) )–(Equation7
(7)
(7) ) admits a unique solution for
.
Proof.
Let be the maximal time interval in which the unique solution exists. We will show that
. Suppose to the contrary that
. In view of Lemma 2.1, there exists positive constants
,
and Λ, independent of
, such that for
,
Fix
and
Using the standard
estimates together with the Sobolev embedding theorem and the Hölder estimates for parabolic equations (see Lunardi [Citation21] for eg.), we can find
depending only on δ, K,
and
such that
where we used the convention that
for
. By virtue of the proof of Theorem 2.1 in [Citation11], there exists a
depending only on
,
and
such that the solution of (Equation6
(6)
(6) ) with the initial time
can be extended uniquely to the time
, which contradicts the definition of
Thus,
and the proof is complete.
3. The special case of constant birth rates
System (Equation5(5)
(5) ) was investigated in [Citation14,Citation15] for two disjoint cases. Namely, the fitness benefit case and the fitness cost case. Define
and
as
and
. Wolbachia is said to have the fitness benefit if
, which means that the local area is more (or at least equally) favourable for infected mosquitoes. The fitness cost case is represented by
, see [Citation38].
In this section, we assume that for i = 1, 2, where
are positive constants. In other words, we have the constant-coefficient free boundary problem given by
(11)
(11) System (Equation11
(11)
(11) ) is essentially a competition model. For the fitness benefit case,
, u is the so-called superior competitor and v the inferior competitor (see [Citation11]). For the fitness cost case,
, (Equation11
(11)
(11) ) represents a strong competition [Citation25]. Throughout this section, we always assume u is a superior competitor. That is, the Wolbachia infection has a fitness benefit. The strong competition case is usually more complicated to be studied mathematically. To the best of our knowledge, results for competition models with a free boundary are very limited in strong competition case. Further details can be seen in [Citation41,Citation42].
We organize this section as follows. In Subsection 3.1 we present some preliminary results, which play a role in proving our main results. Subsection 3.2 is devoted to the vanishing case. The invasion dynamics is studied in detail in Subsection 3.3. A rough estimation of asymptotic spreading speed of Wolbachia invasion is given in Subsection 3.4.
3.1. Preliminary results
Consider the system
(12)
(12) The following result holds.
Lemma 3.1
Let and
. Then,
if
all positive solutions of (Equation12
(12)
(12) ) tend to zero in
as
If
there exists a unique positive stationary solution φ of (Equation12
(12)
(12) ) such that all positive solutions of (Equation12
(12)
(12) ) approach φ in
as
.
Proof.
(i) and (ii) follow from Propositions 3.1, 3.2 and 3.3 of [Citation4].
We recall the following comparison principle.
Lemma 3.2
Comparison principle [Citation11]
Assume that and
. Denote by
and
Let
and
Suppose that
(13a)
(13a)
(13b)
(13b) and
(13c)
(13c) Let
be the unique solution of (Equation11
(11)
(11) ). Then,
and
for all
in
and
for all
in
The following follows from Lemmas A.2 and A.3 in [Citation37].
Lemma 3.3
Let a, b and q be fixed positive constants. For any given
and
there exists
such that, if the continuous and non-negative function
satisfies
(14)
(14) with
for all
then
Let a, b and q be fixed positive constants. For any given
and
there exists
such that
where
is a continuous and non-negative function satisfying
(15)
(15) and
for all
.
We are now in the position to present part of our main results.
3.2. The vanishing case
We consider the vanishing case in this subsection.
Theorem 3.1
Let be the solution of system (Equation11
(11)
(11) ) with initial data (Equation7
(7)
(7) ). If
then
uniformly in any bounded subset of
.
Proof.
Theorem 2.1 yields that for , there is a constant
depending on θ,
,
and
such that
(16)
(16) where
Suppose that
Then, there exists a sequence
in
, where
as
, such that
Note that
By passing to a subsequence if necessary, it follows that
as
. Define
for
and
. It follows from (Equation16
(16)
(16) ) and standard parabolic regularity that
has a subsequence
satisfying
as
, where
is the solution to the following system
(17)
(17) with
for all
. Since
the maximum principle implies that
in
. Hence, we can apply Hopf Lemma at the point
to obtain
Therefore, we have
for large i. This, together with the Stefan condition, implies that
.
On the other hand, implies
as
(see Lemma 3.3 in [Citation12]). This is a contradiction. Thus,
Next, we prove that
. Having
implies that, for any
, there exists T>0 such that
for all
and
. Thus,
(18)
(18) By Lemma 3.3 and the arbitrariness of ϵ, we have
uniformly in any bounded subset of
. This, together with the fact
shows that
.
3.3. The invasion dynamics
Theorem 3.2
Suppose is the solution of system (Equation11
(11)
(11) ) under conditions (Equation7
(7)
(7) ). If
then
and
uniformly in any compact subset of
.
Proof.
Consider the system
(19)
(19) Then,
and
. Consequently, we have
In a similar manner, we can obtain that
Since
then, for
, there exists
such that
for all
and
. If
, then for any given L, there exists
such that u satisfies
(20)
(20) By Lemma 3.3, we know that for sufficiently small
uniformly in any compact subset of
. Since
, there exists
such that
and
for all
and
. Then,
satisfies
(21)
(21) Let
be the solution to the following problem:
(22)
(22) It follows from the comparison principle that
By Corollary 3.6 of [Citation26], we have
Here,
satisfies
(23)
(23) Letting
, it follows from standard elliptic regularity and a diagonal procedure that
converges to
uniformly on any compact subset of
, where
satisfies
(24)
(24) We consider now the following system:
(25)
(25) Since
then
as
(see Lemma 2.2 of [Citation38], for e.g.). Then, the solution
of the problem
(26)
(26) satisfies
as
uniformly in
. By the comparison principle, we have
and
for
, which immediately yields that
The criteria for spreading and vanishing are given in the following theorem.
Theorem 3.3
If then
.
Proof.
Note that is nondecreasing. We only need to show that
implies
. It follows from Theorem 3.1 that
implies
uniformly in any bounded subset of
. Assume that
. Then for sufficiently small
, there exists T>0 such that
Let
be the solution of the following problem
(27)
(27) By the comparison principle, we have
for all
and
. Since
for t>T then, by Lemma 3.1, we know that
uniformly in any compact subset of
, where
is the unique positive solution of
(28)
(28) Thus,
which is a contradiction. Therefore,
and this completes the proof.
Theorem 3.4
If , then there exists
such that
as
.
Proof.
Since uniformly for
, then there exists
such that
when
. So,
satisfies
(29)
(29) Note that,
depends on μ. So, we consider the following problem.
(30)
(30) It follows from the comparison principle that
Clearly,
is independent of μ. Now, we consider the following system.
(31)
(31) By Lemma 3.2, we know that
for
. It follows from [Citation10, Lemma 3.7] that
if
where
This implies that
.
By Theorems 3.3 and 3.4, we can also derive spreading criteria in terms of the diffusion coefficient , for any fixed
.
Theorem 3.5
Spreading criteria
Let where
is any prefixed positive constant. Then, spreading occurs provided that either
or
and
.
Our next result is a criterion on ‘vanishing’.
Theorem 3.6
Assume that
Then, there exists
such that
whenever
.
Proof.
Consider the following problem
(32)
(32) Lemma 3.2 applies and yields that
Furthermore, by Lemma 3.8 of [Citation10], there exists
such that
in the case
, where
and
,
are such that
and
Therefore,
.
3.4. The spreading speed
If spreading occurs, it is important to estimate the spreading speed of . Following an idea in [Citation12], one can obtain a rough estimate of the spreading speed as stated in the following theorem.
Theorem 3.7
[Citation12]
Suppose that and let
be the solution of (Equation11
(11)
(11) ). If
in
in
and
then
where
is the minimal speed of the travelling waves to the problem related with (Equation11
(11)
(11) ) in the entire space. This estimation of the spreading speed is independent of μ.
However, in the fitness benefit case, we can derive an estimate better than the one in Theorem 3.4. We first recall Proposition 5.1 of [Citation11].
Proposition 3.1
[Citation11]
For any given constants
and
the problem
(33)
(33) admits a unique positive solution
which depends on
and satisfies
as
. Moreover,
for
and for each
, there exists a unique
such that
.
Our result reads:
Theorem 3.8
Assume . If
then
where
is determined by Proposition 3.1.
Proof.
Note that
(34)
(34) Thus, the pair
is a subsolution to the problem
(35)
(35) By the comparison principle,
for t>0. Theorem 4.2 of [Citation10] yields that
Hence
Note that
uniformly for
and
Then, there exists
such that
and
Next, we consider the following problem
(36)
(36) By the comparison principle, we obtain
for
. From Theorem 3.3, we know that
. Using a similar argument as above, we have
. Therefore,
4. The free boundary problem with a heterogeneous birth rate
In this section, we consider the free boundary problem (Equation6(6)
(6) )–(Equation7
(7)
(7) ) with the heterogeneous birth rates
and
.
4.1. Some useful lemmas
In this subsection, we first study a related eigenvalue problem:
(37)
(37) Problem (Equation37
(37)
(37) ) admits a positive principal eigenvalue
determined by
(38)
(38) We state two hypotheses that we refer to when needed. We use a generic symbol
in the statement of the hypotheses. The function
will be replaced accordingly (by b,
or
) in the rest of this Section.
Remark 4.0.1
In order to compare the principal eigenvalues associated with different parameters, we denote the principal eigenvalue
by
. When we fix
and study the property of
as d varies, we write
. Similarly, we write
when d is fixed while
varies.
We gather the following known results about the dependance of on d and h.
Lemma 4.1
[Citation40]
Suppose that satisfies (
), where
is replaced by
. Then,
has the following properties:
is increasing with respect to d.
as
and
as
.
For any fixed
there exists
such that
for
for
and
for
.
Lemma 4.2
[Citation40]
Assume that () holds, where
is replaced by
. Then,
has the following properties:
is monotone decreasing with respect to
.
as
and
.
For any fixed
there exists
such that
for
for
for
.
For the reader's convenience, we also recall some facts related to the following problem
(39)
(39) The proof of the next lemma follows from Lemma 5.2 and Lemma 6.2 of [Citation40].
Lemma 4.3
Assume that satisfies (
), where
is replaced by
. Let
be the unique solution of (Equation39
(39)
(39) ) with an initial condition
Then,
where
is the unique positive solution of the following elliptic problem
(40)
(40)
4.2. Sharp criteria for spreading and vanishing
Let us first consider the vanishing case.
Theorem 4.1
Let be the solution of system (Equation6
(6)
(6) ) subject to initial conditions (Equation7
(7)
(7) ). If
and
satisfies (
), where we replace
by
then
uniformly in any bounded subset of
.
The proof is similar to that of Theorem 3.1, above.
In order to obtain sharp criteria for spreading, we require stronger conditions on and
. Namely, we assume that
(41)
(41) Our assumption (Equation41
(41)
(41) ) is not excessive in the sense that, when
and
are constant, we have
. Consequently,
is a positive constant over the interval
.
Theorem 4.2
Assume that satisfies (
) and
satisfies (
) (where B is replaced accordingly). If
then spreading occurs.
Proof.
First, we consider the following equation:
(42)
(42) Since
satisfies the hypotheses of Lemma 4.3, all solutions of (Equation42
(42)
(42) ) with non-trivial non-negative initial values converge to
as
.
It follows, from the comparison principle, that for all t>0 and x>0. Since
uniformly in any compact subset of
then, for any
, there exists T>0 such that
for
.
Consider the following eigenvalue problem:
(43)
(43) It is well known that the principal eigenvalue
can be characterized by
Using (iii) of Lemma 4.1, for any fixed
, there exists
such that
In this theorem, we have
Let us set
, for
and
(here
is the corresponding eigenfunction of
). Choose
, small enough, so that
A straightforward calculation leads to
(44)
(44) By the comparison principle, we have
, for
and
. Thus,
By Theorem 4.1, we have
. Therefore, spreading occurs.
Theorem 4.3
Suppose that satisfies (
) and
satisfies the hypotheses of Lemma 4.3. If
then
(i.e. the species u spreads eventually).
Proof.
Similarly, we consider the following equation
(45)
(45) Since
satisfies the hypotheses of Lemma 4.3, all solutions of (Equation45
(45)
(45) ) with nontrivial and nonnegative initial conditions converge to
as
.
It follows from the comparison principle that for t>0, x>0. Since
uniformly in any compact subset of
. So for any
, there exists T>0 such that
for
.
Consider the following eigenvalue problem:
(46)
(46) The principal eigenvalue
is characterized by
Since
satisfies the hypotheses of (B3). Then by Lemma 4.2, for any fixed
, there exists
such that
for all
,
for
, and
for
.
If , then we set
, for
,
(here
is the corresponding eigenfunction of
). Choose
small enough so that
for
. After a straightforward calculation, we obtain
(47)
(47) By the comparison principle, we have
for
,
. Hence,
Similarly, we have
; hence, spreading occurs.
Theorem 4.4
If and
is small enough, then ‘vanishing’ occurs.
Proof.
We consider the following problem as an auxiliary to the first equation of (Equation6(6)
(6) ):
(48)
(48) Denote the principal eigenvalue
and the corresponding positive eigenfunction
satisfy
(49)
(49) One can verify that there exists
such that
, when
. Furthermore, it follows, from Theorem 4.2 in [Citation40], that there exists a constant
such that
for all
. Now, we can use the following auxiliary functions, which were constructed in [Citation40]. Let
The conditions on α and β will be determined later. If we let
, direct calculations show that
Since
as
, we can find sufficiently small
, such that
Moreover, there exists
, small enough, such that
Let
. Direct calculation leads to
Furthermore, we choose
. Then,
In order to apply the comparison principle, we choose
small enough such that
Thus, we have
(50)
(50) Form the comparison principle, we have
for t>0 and
So,
This implies that vanishing occurs.
Moreover, we can derive vanishing criteria in terms of the coefficient μ when .
Theorem 4.5
Suppose that . For any given
there exists
depending on
and
such that vanishing occurs whenever
.
Proof.
As in the proof of the Theorem 4.4, let and
satisfy Equation (Equation49
(49)
(49) ). We still define
,
as follows
Here, we also let
and choose
large enough such that
For this fixed
, we choose
such that
Then, we have
(51)
(51) Form the comparison principle, we have
, for
and
Thus,
This implies that vanishing occurs.
Next, we will prove the following conclusions.
Theorem 4.6
Assume that satisfies (
), where
is replaced by
. If
, then the species u vanishes eventually.
Proof.
Choose . Consider the following equation:
(52)
(52) It follows from the comparison principle that
for t>0 and
. Since
, Proposition 3.1 of [Citation4] yields that
Consequently,
.
Under some assumptions, stated below, we can obtain the asymptotic spreading speed from Theorem 3.6 of [Citation9].
Theorem 4.7
Assume that satisfies (
), where
is replaced by
. If
, then
Furthermore, if
satisfies (
), then
5. Summary and conclusions
We studied a reaction-diffusion model with a free boundary in one-dimensional environment. The model is developed to better understand the dynamics of Wolbachia infection under the assumptions supported by recent experiments such as perfect maternal transmission and complete CI.
In the special case of constant birth rates, we only considered the fitness benefit case. For the fitness benefit case, where the environment is more favourable for infected mosquitoes, our results show that the spreading of Wolbachia infection occurs if either the size of the initial habitat of infected population is large enough, say
(Theorem 3.3), or the boundary moving coefficient μ is sufficiently large (
) in case of
(Theorem 3.4). A rough estimate on the spreading speed of
is also provided. Moreover, if
and
, then the infection cannot spread and
.
The case of inhomogeneous (spatially dependent) birth rates is treated in Section 4. Detailed criteria for spreading and vanishing are derived in Subsection 4.2 with the aid of spectral properties of relevant eigenvalue problems.
Acknowledgments
The authors wish to thank the anonymous reviewers for their valuable comments and suggestions, which greatly helped us improve the presentation of this work. Y. Liu and Z. Guo were supported by National Science Foundation of China (No. 11371107, 11771104), Program for Chang Jiang Scholars and Innovative Research Team in University (IRT-16R16). Y. Liu was supported by the National Natural Science Foundation of China under Grant No.11271093 and the Innovation Research for the Postgraduates of Guangzhou University under Grant No.2017GDJC-D05. M. El Smaily and L. Wang acknowledge partial support received through NSERC-Discovery grants from the Natural Sciences and Engineering Research Council of Canada (NSERC).
Disclosure statement
No potential conflict of interest was reported by the author(s).
Additional information
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References
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