# Dynamics of Mosquito Population Models with Spatial Diffusion

Authors: Urbain Traoré
Journal: International Journal of Differential Equations
 (2021)

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## Abstract

In this paper, we study some reaction-diffusion models of interactive dynamics of the wild and sterile mosquitoes. The well-posedness of the concerned model is proved. The stability of the steady states is discussed. Numerical simulations are presented to illustrate our theoretical results.


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## Body

## 1. Introduction

Mosquito-borne diseases (e.g., malaria, dengue fever, and West Nile virus) remain a public health problem in developing countries despite many efforts of controls. The mortality and morbidity associated to this disease are still important for these countries. Many control strategies have been explored to control this disease among which are the development of effective insecticides and drug treatments, vaccine, and insecticide-treated bed nets. Nowadays, biological control of wild mosquito population by using genetically modified or transgenic mosquitoes is also used (see [1, 2]). This strategy control releases male mosquitoes (sterilized by radiological or chemical means) into the environment to mate with wild mosquitoes that are present in the environment. Female mosquitoes successfully mating with a sterile male will not breed or produce eggs that will not hatch. Repetitive releases of sterile mosquitoes or the release of a significantly large number of sterile mosquitoes can possibly wipe out or suppress a population of wild mosquitoes (see [3, 4]). Recently, several mathematical models have been proposed to measure the effectiveness of this control strategy (see [5–11]).The model that we are studying by integrating the diffusion is obtained by assuming that the dynamics of the two mosquito populations follow a logistic growth, the birth rate of sterile mosquitoes being their release rate (see [5]):(1)dwtdt=aCNw2twt+gt−μ1+ξ1wt+gtwt,dgtdt=B·−μ2+ξ2wt+gtgt.The termswt and gt are the numbers of wild and sterile mosquitoes consecutively, μi and ξi, i=1,2 are the density independent and dependent death rates of the wild and sterile mosquitoes, respectively. The parameter CN is the number of matings per individual, per unit of time, with N=w+g; a is the number of wild offspring produced per mate. B· designs the releasing rate of the sterile mosquitoes. Furthermore, it is well known that, among a number of factors that contribute to the upsurgence of malaria, we have vector immigration (see [12]). Mobility is well known to play an important role in the study of population dynamics (see [13–16] and the references therein). Thus, it becomes important to take into account this aspect on the propagation of the disease. However, in the works cited above, we noticed that their authors do not take into account population mobility in space. Thus, motivated by the works in [5], we present a model that includes spatial diffusion of wild and sterile mosquitoes. This model is given by(2)∂wdt=dwΔw+aCNw2x,twx,t+gx,t−μ1+ξ1wx,t+gx,twx,t,∂gdt=dgΔg+B·−μ2+ξ2wx,t+gx,tgx,t.Here, the parametersdw and dg represent the diffusion coefficients of wild and sterile mosquitoes, respectively. Symbol Δ is the Laplacian operator, and Ω is a connected and bounded spatial domain in Rn with a smooth boundary ∂Ω. For model 2, we consider homogeneous Neumann boundary conditions(3)∂w∂ν=0,∂g∂ν=0,on∂Ω×0,+∞,and initial conditions(4)wx,0=ϕ1x>0,gx,0=ϕ2x>0,∀x,0∈Ω¯.The functionsϕ1 and ϕ2 are nonnegative continuous functions in Ω¯ and denotes the outward normal derivative on the boundary ∂Ω.In this paper, we proposed a complete mathematical analysis of model (2). The analysis of model (2) in the case of constant releasing rate of sterile mosquitoesB· is studied Section 2. The well-posedness is established, and the stability of the equilibria is derived. Section 3 is devoted to the analysis of model (2) in the case of proportional releasing rate of sterile mosquitoes. In Section 4, numerical simulations are carried out to illustrate the theoretical results.


## 2. Existence and Positivity

In this paper, we study model (2) with a constant release rate of sterile mosquitoes, i.e.,B·=b>0. We assume that the interaction rate CN is constant, and in order to simplify the writing, we always write a instead of CNa with a>μ1; then, system (2) becomes(5)∂wdt=dwΔw+aw2x,twx,t+gx,t−μ1+ξ1wx,t+gx,twx,t∂gdt=dgΔg+b−μ2+ξ2wx,t+gx,tgx,t,with initial condition (4) and Neumann boundary condition (3).We now derive the result about the existence and the positivity of the solution of system (5).In this section, we establish the global existence, positivity, and boundedness of solutions of problem (5) under the conditions (3) and (4).We state the following result.Theorem 1.
For any given initial data satisfying condition (4), there exists a unique solution of system (5) defined on 0,+∞, and this solution remains nonnegative and bounded for all t≥0.Proof. Let ϕ=ϕ1,ϕ2T∈CΩ¯×CΩ¯. System (5) can be rewritten abstractly in the Banach space X as follows:(6)u′t=Aut+Fut,t>0u0=ϕ∈X,where u=w,gT, u0=ϕ1,ϕ2T, and Aut=dwΔw,dgΔgT and(7)Fut=aw2w+g−μ1+ξ1w+gwb−μ2+ξ2w+gg.
SinceF is locally Lipschitzian in X, thus there exists a unique local solution of system (5) on the interval 0,Tmax, where Tmax denotes the maximal existence time for solution of system (6) (see [7, 14, 15, 17]).
To achieve the global existence in0,+∞, we need to prove the positiveness and boundedness of this solution.
Sincewx,0=ϕ1x>0 and g0=ϕ2x>0, for t∈0,Tmax,(8)wx,t−dw∫0tΔwx,sds=e−∫0tμ1+ξ1wx,s+gx,sdswx,0++∫0tαw2x,se−∫0tμ1+ξ1wx,η+gx,ηdηdswx,s+gx,sand(9)gx,t−dg∫0tΔgx,sds=e−∫0tμ2+ξ2wx,s+gx,sdsgx,0+b∫0te−∫0tμ2+ξ2wx,η+gx,ηdηds.Thus,(10)wt,x−dw∫0tΔwx,sds>0andgx,t−dg∫0tΔgx,sds>0,∀x∈Ω.Then, integrating (10) over Ω and using the fact that ∂w/∂ν=∂g/∂ν=0, we obtain(11)∫Ωwx,tdx>0and∫Ωgx,tdx>0,∀t∈0,Tmax.Hence, it follows that(12)wx,t>0a.e.Ωandgx,t>0a.e.Ω,∀t∈0,Tmax,where a.e.Ω defines almost everywhere in Ω.Furthermore, from (5), we deduce that(13)∂gdt−dgΔg≤b−μ2gx,t,∂g∂ν=0,gx,0=ϕ2x,0≤ϕ2∞=maxx∈Ω¯ϕ2x.By the principle comparison [18], it follows that gx,t≤g1t, where g1t=ϕ2x,0e−μ2t+b/μ21−e−μ2t is the solution of(14)dg1dt=b−μ2g1,g10=ϕ2∞.Sinceg1t≤maxb/μ2,ϕ2∞ for all t∈0,+∞, thus(15)gx,t≤maxb/μ2,ϕ2∞,∀x,t∈Ω¯×0,Tmax.In order to proveL∞ uniformly boundedness of wx,t, it is sufficient to show that wx,t is L1 uniformly bounded (see [1], Theorem 2).Let(16)ux,t=ξ2γwx,t+agx,t,withγ=mint∈0,Tmax×Ω¯gx,t.Thus,(17)∂udtx,t−Δξ2γdwwx,t+adggx,t,=ξ2γ∂wdt+a∂gdt−ξ2γdwΔw−adgΔgx,t,=ξ2γaw2x,twx,t+gx,t−ξ2γμ1+ξ1wx,t+gx,twx,t+ab−aμ2+ξ2wx,t+gx,tgx,t,≤ξ2γawx,t−ξ2γμ1wx,t+αb−aμ2gx,t−aξ2wx,tgx,t,≤ab+ξ2awx,tγ−gx,t−ξ2γμ1wt,x−μ2agx,t,≤ab−minμ1,μ2ξ2γwx,t+αgx,t,≤ab−minμ1,μ2ux,t.Since∂w/∂ν=∂g/∂ν=0, it follows that(18)ddt∫Ωux,tdx≤measΩab−minμ1,μ2∫Ωux,tdx.Hence,(19)∫Ωux,tdx≤measΩmaxabminμ1,μ2,ξ2γϕ1+αϕ2∞,which implies that(20)supt≥0∫Ωwx,tdx≤K,where(21)K=1ξ2γmeansΩmaxabminμ1,μ2,ξ2γϕ1+αϕ2∞.Therefore, from [13], we deduce that there exists a positive constant K∗ that depends on K such that(22)supt≥0w.,t∞≤K∗,that is, wx,t is L∞.Sincewx,t and gx,t are L∞ bounded on 0,Tmax, it follows from the standard theory for semilinear parabolic systems [19] that Tmax=+∞. Thus, the global existence follows. This completes the proof.


## 3. Analysis of the Equilibrium Points

Here, we are interested in the equilibrium points of model (5). Clearly, the presence of the reaction-diffusion term does not affect the number of the equilibrium. As in [5], for model (5), there is only one equilibrium point whose number of wild mosquitoes is zero, E00,g0, where(23)g0=μ22+4bξ2−μ22ξ2>0.Concerning the positive equilibrium of system (5), its components verify(24)aww+g=μ1+ξ1w+g,bg=μ2+ξ2w+g,which implies(25)aw=Nμ1+ξ1N,ag=abμ2+ξ2N.From (23), we deduce that(26)aN=Nμ1+ξ1N+abμ2+ξ2N.Hence, we introduce the following function:(27)PN≔Nμ1+ξ1Nμ2+ξ2N−aNμ2+ξ2N+ab,=ξ1ξ2N3−ξ2a−μ1−ξ1μ2N2−a−μ1μ2N+ab.Consequently, there is an equivalence between the fact thatw,g is an equilibrium point for system (5) and the fact that N=w+g is the solution of the problem PN=0.We have(28)P′N=3ξ1ξ2N2−2ξ2a−μ1−ξ1μ2N−a−μ1μ2,that if N¯>0 is an extrema of PN with P′N¯=0, then(29)N¯=N¯τ:=13ξ1ξ2ξ2α−μ1−ξ1μ22+3ξ1ξ2α−μ1μ2+ξ2α−μ1−ξ1μ2.Let us observe that(30)2PN¯=2PN¯−P′N¯N¯=−ξ1ξ2N¯3−a−μ1μ2N¯+2ab.Thus, by defining the release threshold value of sterile mosquitoes as follows:(31)b0τ≔12αξ1ξ2N¯2+α−μ1μ2N¯,then PN¯<0 is equivalent to b<b0. Moreover, since P0=ab>0 and limN⟶+∞PN=+∞, then problem (26) admits one and only one positive solution N∗=N¯ if and only if b=b0, and two positive solutions, N1∗ and N2∗ where N1∗<N¯<N2∗ if and only if b<b0.From the above discussion as in [5], we have the following result.Theorem 2.
Boundary equilibriumE00,g0, see (22), is the unique equilibrium of (5) if and only if b>b0.
Model (5) admits a unique positive equilibrium E∗w∗,g∗ with(32)w∗=N¯μ1+ξ1N¯a,g∗=bμ2+ξ2N,where N¯ given in (27), if and only if b=b0. For b<b0, we have two positive equilibria E1∗w1∗,g1∗ and E2∗w2∗,g2∗ for the model with(33)w1∗=N1∗μ1+ξ1N1∗a<w2∗=N2∗μ1+ξ1N2∗a,g2∗=bμ2+ξ2N2∗<g1∗=bμ2+ξ2N1∗.Using (27), we verify that N¯ does not depend on b; therefore, b0 is also independent of b. As soon as the mosquito release rate is higher than this rate, wild mosquitoes end up disappearing.We now discuss the stability of the corresponding steady states of the model. Let us assume that0=λ1<λ2<⋯<λn<⋯ be the eigenvalues of the operator −Δ on Ω with the homogeneous Neumann boundary conditions and Eλi be the eigenfunction space corresponding to λi in C1Ω. Let Y=C1Ω2, ψi,j:j=1,2,…,dimEλi be an orthonormal basis of Eλi and Yij=dψi,j:d∈R2. Then,(34)Y=⊕i=1∞YiandYi=⊕j=1dimEλiYij.LetE∗w∗,g∗ be an arbitrary steady state of system (5). We define the following perturbation about E∗w∗,g∗:(35)U1x,t=wx,t−w∗andU2x,t=gx,t−g∗.By linearizing system (5) at E∗w∗,g∗, we obtain(36)∂U∂t=DΔU+AUx,t,where(37)A=agw∗+g∗2−ξ1w∗+aw∗w∗+g∗−μ1+ξ1w∗+g∗−aw∗w∗+g∗2+ξ1w∗−ξ2g∗−μ2−ξ2w∗+2g∗and(38)D=diagdw,dgandU=U1,U2T.We define the operatorL by(39)LU=DΔU+A∗Ux,t.Yi is invariant under the operator L for all i≥1. Furthermore, η is an eigenvalue of L if and only if it is a root of the following equation:(40)det−λiD+A−ηℐ=0for some i≥1, and in this case, there exists an eigenvector in Yi.At the boundaryE0, matrix A becomes(41)−μ1+ξ1g00−ξ2g0−μ2+2ξ2g0,and in this case, characteristic (33) has only two roots η1=−μ1+ξ1g0+λidw<0, η2=−μ2+2ξ2g0+λidg<0. So, E0 is locally asymptotically stable.It is easy to verify that at the equilibriaE1∗ and E2∗ (when 0<b<b0), characteristic (33) takes the form(42)Lη=η2+B1η+B2=0,where with N=N1∗ or N2∗(43)B1=μ2+ξ2N+g+λidw+dg−awgN2+ξ1w,B2=λidwλidg+μ2+ξ2N+g−λiagwN2−ξ1w+−agwN2+ξ1wμ2+ξ2N+g−awN2+ξ1ξ2wg.So,(44)B2=wNP′N+λidwμ2+ξ2g+N+λidg−λiagwN2−ξ1w.SinceP′N1∗<0 (Cai et al.,2014), then for λi=0, we get L0=w/NP′N−λidg<0, since Lη>0 for sufficiently large η>0. Therefore, there is at least one positive root for Lη=0. This yields that E1∗ is unstable.At the boundaryE2, we have(45)B2=wN2∗P′N2∗+λidwμ2+ξ2g+N2∗+λidwμ2−dg.Thanks to Cai et al.,P′N2∗>0 and choosing dw and dg such as dg<dwμ2; it follows that B2>0. Moreover, using the fact that aw=Nμ1+ξ1N, we have(46)awgN2=μ1+ξ1NgN<μ1+ξ1N≤μ2+ξ2Nifμ1≤μ2,ξ1≤ξ2.Therefore this choice of the parametersμ1,μ2 and ξ1,ξ2 allowed us to deduce that B1>0. Since B1>0 and B2>0, then (40) has no positives roots; therefore, E2∗ is locally asymptotically stable. We summarize the previous discussion in the following results.Theorem 3.
(i)
Forb>b0 such that the boundary equilibrium E0 is the only equilibrium for system (5), it is locally asymptotically stable(ii)
Forb<b0, the boundary equilibrium E0 is a locally asymptotically stable node, E1∗ is a unstable and moreover if μ1≤μ2, ξ1≤ξ2, then E2∗ is stable(iii)
Forb=b0, the boundary equilibrium E∗, see (26) is an unstable saddle node


## 4. Numerical Results

In this section, we give some numerical simulations to illustrate the theoretical results obtained in Section3, we take one-dimensional spatial domain. Ω=0,50 and the grid sizes are dx=0.1 and dt=0.01. Figure 1 presents the evolution of the population size of sterile and wild mosquito in system (5).Figure 1
Parameter values are chosen to bea=10, μ0=0.1, μ1=0.5, μ2=0.4, b=21, ξ1=0.5, ξ2=0.4, dg=10, and dw=10.The threshold release value isb0=21.935161. For b=21<b0, E0=0,6.7629 is a locally asymptotically stable node. We have two positive equilibria E1=7.84,4.07 which is a saddle point an equilibrium E2=8.70,3.87 which is a locally asymptotically stable. Solutions with different initial values approach either E0 or E2∗ as shown in the lower left and right figures in Figure 1, respectively. From these numerical simulations, we can, therefore, say that wild and sterile mosquitoes can coexist or although wild mosquitoes can disappear, depending on the initial quantity of the two populations.Next, we take valuesb=30>b0 and the other parameters remain unchanged. For this set of parameter values, the boundary equilibrium E0 is the only equilibrium for system (5), and it is globally asymptotically stable and this is evident from Figure 2.Figure 2
Parameter values are chosen to bea=10, μ0=0.1, μ1=0.5, μ2=0.4, b=30, ξ1=0.5, ξ2=0.4, dg=10, and dw=10.As shown by the results of the numerical simulations, the introduction of the diffusion in the model does not suppress the dynamics of wild mosquitoes or genetically modify, and one obtains globally the configurations that in the absence of the diffusion. However, the mathematical analysis becomes more complicated.


## 5. Discussion

In this work, we carried out a theoretical and numerical study of reaction-diffusion model describing the dynamic of the interaction between wild and sterile mosquitoes. We have established the theoretical and numerical effects of the introduction of the population diffusion on the stability of equilibrium states. On the other hand, when the number of sterile mosquitoes rejections is proportional to the population of wild mosquitoes, the results show that the solutions of the model exhibit a complex dynamic behavior. However, in order to have more realistic and more economical models for the release of sterile mosquitoes, it would be interesting to have a variable release rate that is proportional to the size of the wild mosquitoes while taking into account the mobility of the mosquitoes. This approach could be the subject of future studies [20].



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*Source: 9034274-2021-12-29.xml*