Citation: Victoria Gartman, Kathrin Wichmann, Lea Bulling, María Elena Huesca-Pérez, Johann Köppel. Wind of Change or Wind of Challenges: Implementation factors regarding wind energy development, an international perspective[J]. AIMS Energy, 2014, 2(4): 485-504. doi: 10.3934/energy.2014.4.485
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Mosquitoes are small biting insects comprising of the family Culicidae. There are about 3,500 mosquito species in the world (grouped into 41 genera) [4,30,43]. Mosquito species, such as Anopheles, Aedes aegypti, Aedes albopictus and Culex, play significant roles as vectors of some major infectious diseases of humans, such as malaria, yellow fever, Chikungunya, west Nile virus, dengue fever, Zika virus and other arboviruses [4,48]. These diseases are transmitted from human-to-human via an effective bite from an infected female adult mosquito [4,51]. While adult male mosquitoes feed on plant liquids such as nectar, honeydew, fruit juices and other sources of sugar for energy, female mosquitoes, in addition to feeding on sugar sources (for energy), feed on the blood of human and other animals solely to acquire the proteins needed for eggs development [4,30,49,51].
Mosquitoes are the best known disease vector for vector-borne diseases (VBDs) of humans (VBDs account for 17% of the estimated global burden of all infectious diseases) [76,78]. Mosquito is the world's deadliest animal (accounting for more human deaths annually than any other animal), spreading human diseases such as malaria, dengue and yellow fever, which together are responsible for several million deaths and hundreds of millions of cases annually [13,49]. For example, malaria, transmitted by female Anopheles mosquitoes, is endemic in 91 countries, with about 40% of the world's population at risk, causing up to 500 million cases and over 1 million deaths annually [48,49,77]. Similarly, dengue, transmitted by female Aedes mosquitoes, causes over 20 million cases a year in more than 100 countries [48,77].
The life-cycle of the mosquito is completed via four distinct stages, namely: eggs, larva, pupa and adult stages, with the first three largely aquatic [48]. A female mosquito can lay about 100-300 eggs per oviposition [4,48], and this process is temperature dependent [4]. The eggs are laid at a convenient breeding site, usually a swamp or humid area in the aquatic environment (the Anopheles species typically lays their eggs on the surface of the water) and after about 2-3 days, they hatch into larva. Larvae develop through four instar stages [48,4]. At the end of each larval stage, the larvae molt, shedding their skins to allow for further growth (the larvae feed on microorganisms and organic matter in the water) [4]. During the fourth molt, the larvae mature into pupae (the whole process of maturation from larvae to pupae takes 4-10 days [51]). The pupae then develop into adult mosquitoes in about 2-4 days [4,51].
The duration of the entire life-cycle of the mosquito, from egg laying to the emergence of an adult mosquito, varies between 7 and 20 days [51], depending on the ambient temperature of the breeding site (typically a swamp or humid area) and the mosquito species involved [28] (for instance, Culex tarsalis, a common mosquito in California (USA), might go through its life cycle in 14 days at
The introduction, abundance and distribution of mosquitoes worldwide have been affected by various environmental (climatic) factors such as temperature, humidity, rainfall and wind [2,15,47,54,55,57,58,60,67,79]. These factors have direct effect on different ecological aspects of the mosquito species which includes the oviposition process, development during aquatic stages and the biting rate of mosquitoes [2,47,60]. Furthermore, the oviposition process, development at aquatic stages, emergence of the adult and other developmental processes in the larval habitats of mosquitoes, play a key role in the determination of abundance and distribution of mosquitoes [3,56].
Understanding mosquito population dynamics is crucial for gaining insight into the abundance and dispersal of mosquitoes, and for the design of effective vector control strategies (that is, understanding mosquito population dynamics has important implications for the prediction and assessment of the effects of many vector control strategies [51,52]). The purpose of the current study is to qualitatively assess the impact of temperature and rainfall on the population dynamics of female mosquitoes in a certain region, and taking into consideration the dynamics of the human-vector interaction. This study extends earlier mosquito population biology in literature such as the model in [1], by designing a new temperature-and rainfall-dependent mechanistic mosquito population model that incorporates some more notable features of mosquitoes population ecology such as four stages for larval development and three different dispersal (questing for blood meal, fertilized and resting at breeding site) states of female adult mosquitoes. The non-autonomous model is formulated in Section 2 and its autonomous version is analyzed in Section 3. The full non-autonomous model is analyzed and simulated in Section 4. Since malaria is the world's most important parasitic infectious disease [34], numerical simulations of the model will be carried out using parameters relevant to the population biology of adult female Anopheles mosquitoes in Section 5 (it is worth stating that there are approximately 430 species of the Anopheles mosquitoes, of which 30-40 transmit malaria in humans (i.e., they are vectors) [4,43]).
This study is based on the formulation and rigorous analysis of a mechanistic model for the dynamics of female Anopheles mosquitoes in a population. The model splits the total immature mosquito population at time
$ dEdt = ψU(T)(1−UKU)U−[σE(R,ˆT)+μE(ˆT)]E,dL1dt = σE(R,ˆT)E−[ξ1(N,R,ˆT)+μL(ˆT)+δLL]L1,dLidt = ξ(i−1)(N,R,ˆT)L(i−1)−[ξi(N,R,ˆT)+μL(ˆT)+δLL]Li;i=2,3,4,dPdt = ξ4(N,R,ˆT)L4−[σP(R,ˆT)+μP(ˆT)]P,dVdt = σP(R,ˆT)P+γUU−ηVHH+FV−μA(T)V,dWdt = ηVHH+FV−[τWH+μA(T)]W,dUdt = ατWHW−[γU+μA(T)]U,L = 4∑i=1Li. $ | (1) |
In (1),
$T(t)=T0[1+T1 cos(2π365(ωt+θ))],$ |
(where
Pupae mature into the female adult mosquitoes of type
Variables | Description |
|
Population of female eggs |
|
Population of female larvae at Stage |
|
Population of female pupae |
|
Population of fertilized female mosquitoes that have laid eggs at the breeding site |
(including unfertilized female mosquitoes not questing for blood meal) | |
|
Population of fertilized, but non-reproducing, female mosquitoes questing for blood meal |
|
Population of fertilized, well-nourished with blood, and reproducing female mosquitoes |
Parameters | Description |
|
Deposition rate of female eggs |
|
Maturation rate of female eggs |
|
Maturation rate of female larvae from larval stage |
|
Maturation rate of female pupae |
|
Natural mortality rate of female eggs |
|
Natural mortality rate of female larvae |
|
Natural mortality rate of female pupae |
|
Natural mortality rate of female adult mosquitoes |
|
Density-dependent mortality rate of female larvae |
|
Constant mass action contact rate between female adult mosquitoes of type |
|
Probability of successfully taking a blood meal |
|
Rate of return of female adult mosquitoes of type |
|
Rate at which female adult mosquitoes of type |
|
Constant population density of humans at human habitat sites |
|
Constant alternative source of blood meal for female adult mosquitoes |
|
Environmental carrying capacity of female adult mosquitoes |
|
The daily survival probability of Stage |
|
The average duration spent in Stage |
|
Rate of nutrients intake for female larvae in Stage |
|
Total available nutrient for female larvae |
|
Cumulative daily rainfall |
|
Daily mean ambient temperature |
|
Daily mean water temperature in the breeding site |
|
Maximum daily survival probability of aquatic Stage |
|
Rainfall threshold |
The functional forms of the nutrient-, rainfall- and temperature-dependent parameters of the model (1) are formulated as follows. This functional forms derived from [2,47,59,60], characterizes the female Anopheles mosquitoes (which transmits malaria in humans). The per-capita rate of deposition of female eggs (
$ψU(T)=−0.153T2+8.61T−97.7.$ |
Similarly, following [47], the per-capita death rate of the female adult mosquitoes (
$μA(T)=−ln(−0.000828T2+0.0367T+0.522).$ |
Furthermore, following [60], the per-capita death rate of female eggs (
$μE(ˆT)=11.011+20.212[1+(ˆT12.096)4.839]−1,μL(ˆT)=18.130+13.794[1+(ˆT12.096)4.839]−1,μP(ˆT)=18.560+20.654[1+(ˆT19.759)6.827]−1.$ |
Similarly, following [60], the per-capita maturation rate of eggs (into larvae) and pupae (into female adult mosquitoes) are given by
$σi(R,ˆT)=(1−pi)pdii1−pdii; i={E,P},$ |
where
$di(ˆT)=1μi(ˆT),$ |
and
$ p_i(R, \hat{T}) = p_i(R)p_i(\hat{T}), $ | (2) |
with
$ p_{i}(R) = R(R_{I_M}-R)(4p_{Mi}/R^2_{I_M}), \, \, i = \{E, L_1, L_2, L_3, L_4, P\}, $ | (3) |
where
$ξj(N,R,ˆT)=ξj(N)ξj(R,ˆT); j=1,2,3,4,$ |
where
$ξj(R,ˆT)=(1−pj)pdjj1−pdjj; j=1,2,3,4,$ |
with
Furthermore, since almost all communities within tropical and sub-tropical regions of the world record temperatures in the range
Temperature ( |
|
|
|
|
|
16-40 | 0.892-23.431 | 0.194-0.932 | 0.091-0.122 | 0.040-0.115 | 0.074-0.408 |
The non-autonomous model (1) is an extension of the autonomous model for the population biology of the mosquito developed in [51,52], by including:
(ⅰ) aquatic stages of the mosquito (i.e., adding the
(ⅱ) the effect of climate variables (i.e., adding the dependency on temperature and rainfall).
It also extends the model by Lutambi et. al. [41] by, inter alia:
(ⅰ) incorporating the effect of climate variable (temperature and rainfall);
(ⅱ) using logistic growth rate for egg oviposition rate (a constant rate was used in [41]);
(ⅲ) incorporating four larval stages (only one larval class was used in [41]).
Furthermore, the model (1) extends the non-autonomous climate-driven mosquito population biology model developed by Abdelrazec and Gumel [1] by
(ⅰ) including the dynamics of adult female mosquitoes (i.e., including compartments
(ⅱ) including four larval classes (a single larval class is considered in [1]);
(ⅲ) including dependence on (constant and uniform availability of) nutrients for the four larval stages.
Since, as stated in Section 2,
$ ˙M ≤ ψU(t)(1−UKU)U−δLL2−μM(t)M−(1−α)τWHW,≤ ψU(t)(1−UKU)U−μM(t)M, t>0. $ | (4) |
In order to study the asymptotic dynamics of the mosquito population, subject to fluctuations in temperature and rainfall, we assume that the mosquito population stabilizes at a periodic steady-state. Furthermore, following [40,55], it is assumed that for the time
$ \psi_U(t)\biggl(1 -\frac{U}{K_U}\biggr)U -\mu_M(t)A < 0 \ {\rm \ for\ all}\ A \geq h_0. $ |
Lemma 2.1. For any
Proof. Let
$ \frac{d\phi}{dt} = f(t, \phi(t)), \, t\geq 0 \, \, \, \phi(0) = \phi_0, $ |
where,
$f(t,ϕ(t))=[ψU(t)[1−ϕ9(0)KU]ϕ9(0)−[σE(t)+μE(t)]ϕ1(0)σE(t)ϕ1(0)−[ξ1(t)+μL(t)+δLϕL(0)]ϕ2(0)ξ(i−2)(t)ϕ(i−1)(0)−[ξ(i−1)(t)+μL(t)+δLϕL(0)]ϕi(0);i=3,4,5ξ4(t)ϕ5(0)−[σP(t)+μP(t)]ϕ6(0)σP(t)ϕ6(0)+γUϕ9(0)−ηVHH+Fϕ7(0)−μA(t)ϕ7(0)ηVHH+Fϕ7(0)−[τWH+μA(t)]ϕ8(0)ατWHϕ8(0)−[γU+μA(t)]ϕ9(0)],$ |
For the total mosquito population
$lim supt→∞(E(t)+L(t)+P(t)+V(t)+W(t)+W(t))≤M∗(t),$ |
where
$ \dot{M}^* = \psi_U(t)\left(1 -\frac{U}{K_U}\right)U -\mu_M(t)M^*, \ \ t>0, $ | (5) |
given by,
$M∗(t) = e−∫t0μM(s)ds×{∫t0[ψU(s)(1−U(s)KU)U(s)e∫s0μM(τ)dτ]ds+ ∫ω0ψU(s)(1−U(s)KU)U(s)exp[∫s0μM(ζ)dζ]exp[∫ω0μM(s)ds]−1}.$ |
Thus, all solutions of the model (1) are ultimately-bounded [40]. Moreover, it follows from (5) that
It is instructive to, first of all, analyze the dynamics of the autonomous equivalent of the non-autonomous model (1) to determine whether or not it has some qualitative features that do not exist in the model (1). Consider, now, the non-autonomous model (1) with all rainfall-and temperature-dependent parameters set to a constant (i.e.,
$dEdt = ψU(1−UKU)U−(σE+μE)E,dL1dt = σEE−[ξ1+μL+δLL]L1,dLidt = ξ(i−1)L(i−1)−[ξi+μL+δLL]Li;i=2,3,4,dPdt = ξ4L4−(σP+μP)P,dVdt = σPP+γUU−ηVHH+FV−μAV,dWdt = ηVHH+FV−(τWH+μA)W,dUdt = ατWHW−(γU+μA)U,L = 4∑i=1Li, $ | (6) |
where, now,
In this section, some results for the existence and linear asymptotic stability of the trivial equilibrium point of the autonomous model (6) will be provided. It is convenient to introduce the following parameter groupings (noting that
$ {τ∗W=τWH, η∗V=ηVHH+F, CE=σE+μE, CP=σP+μP,Ci=ξi+μL (for i=1,2,3,4), C5=η∗V+μA, C6=τ∗W+μA, C7=γU+μA,B=σEσPξ1ξ2ξ3ξ4, C=C1C2C3C4CECP, D=C5C6C7−ατ∗Wη∗VγU>0. $ | (7) |
The autonomous model (6) has a trivial equilibrium solution, denoted by
$T0=(E∗,L∗1,L∗2,L∗3,L∗4,P∗,V∗,W∗,U∗)=(0,0,0,0,0,0,0,0,0).$ |
The linear stability of
$F=[00F1000000]andV=[V100V2V300V4V5],$ |
where
$F1=[00ψU000000],V1=[CE00−σEC100−ξ1C2],V2=[00−ξ2000000],V3=[C300−ξ3C400−ξ4CP],V4=[00−σP000000],V5=[C50−γU−η∗VC600−ατ∗WC7].$ |
It follows from [73] that the associated vectorial reproduction number of the autonomous model (6) [63], denoted by
$ \mathcal{R}_0 = \dfrac{\alpha\tau^*_W\eta^*_V\psi_UB}{CD}, $ | (8) |
where
Theorem 3.1. The trivial equilibrium (
Theorem 3.2. The trivial equilibrium point (
Proof. Consider the Lyapunov function
$K1= ατ∗Wη∗Vξ4σP[σEξ1ξ2ξ3E+CEξ1ξ2ξ3L1+C1CEξ2ξ3L2+C1C2CEξ3L3+C1C2C3CEL4]+C1C2C3C4CE[σPη∗Vατ∗WP+CPη∗Vατ∗WV+CPC5ατ∗WW+CPC5C6U].$ |
It is convenient to define
$S=ατ∗Wη∗Vξ4σP[CEξ1ξ2ξ3L1+C1CEξ2ξ3L2+C1C2CEξ3L3+C1C2C3CEL4].$ |
Thus, the Lyapunov derivative is given by
$˙K1= ατ∗Wη∗Vξ4σP[σEξ1ξ2ξ3˙E+CEξ1ξ2ξ3˙L1+C1CEξ2ξ3˙L2+C1C2CEξ3˙L3+C1C2C3CE˙L4]+C1C2C3C4CE[σPη∗Vατ∗W˙P+CPη∗Vατ∗W˙V+CPC5ατ∗W˙W+CPC5C6˙U],= ατ∗Wη∗VB[ψU(1−UKU)U]+C1C2C3C4CE(CPη∗Vατ∗WγUU−CPC5C6C7U)−δLLS,= ατ∗Wη∗VBψUU−CDU−ατ∗Wη∗VBψUUKUU−δLLS,= [CD(R0−1)−ατ∗Wη∗VBψUUKU]U−δLLS,$ |
where
Theorem 3.2 shows that the mosquito population (both immature and mature) will be effectively controlled (or eliminated) if the associated vectorial reproduction threshold,
The existence of a non-trivial equilibrium of the model (6) will now be explored. Let
$ E∗∗=ψUCE(1−U∗∗KU)U∗∗,E∗∗=1σE(C1+δLL∗∗)L∗∗1,L∗∗1=1ξ1(C2+δLL∗∗)L∗∗2,L∗∗2=1ξ2(C3+δLL∗∗)L∗∗3,L∗∗3=1ξ3(C4+δLL∗∗)L∗∗4, L∗∗4=CPDU∗∗ατ∗Wη∗VσPξ4,P∗∗=DU∗∗ατ∗Wη∗VσP,V∗∗(U∗∗)=C6C7U∗∗ατ∗Wη∗V, W∗∗=C7U∗∗ατ∗W, U∗∗=ατ∗Wη∗VσPξ4L∗∗4CPD. $ | (9) |
It follows from (9) that
$ L^{**}_i \, =\, \dfrac{\bigl(C_{i+1} + \delta_L L^{**}\bigr)L^{**}_{i+1}}{\xi_i}; \, i=1, 2, 3. $ | (10) |
Multiplying the second, third, fourth and fifth equations of (9), and substituting the expressions for
$ B\alpha\tau^{*}_W\eta^{*}_V\psi_U\biggl( 1 -\dfrac{U^{**}}{K_U}\biggr) \, =\, C_EC_PD\prod\limits_{i = 1}^{4}\bigl(C_i + \bar{L}^{**}\bigr). $ | (11) |
Substituting the expression for
$ L_4^{**} =\dfrac{K_UC_PD}{\alpha\tau^*_W\eta^*_V\sigma_P\xi_4} \left[1-\dfrac{C_EC_PD \prod\limits_{i = 1}^{4}\bigl(C_i + \delta_L L^{**}\bigr)} {B\alpha\tau^*_W\eta^*_V\psi_U}\right]. $ | (12) |
Furthermore, substituting the expressions for
$ L^{**} = \dfrac{1}{\xi_1\xi_2\xi_3}\biggl[\xi_1\xi_2\xi_3 + \xi_1\xi_2\bigl( C_4 + \delta_L L^{**}\bigr) + \xi_1 \prod\limits_{i=3}^{4}\bigl( C_i + \delta_L L^{**}\bigr) + \prod\limits_{i=2}^{4}\bigl( C_i + \delta_L L^{**}\bigr) \biggr] L^{**}_4. $ | (13) |
Finally, substituting (12) into (13), and simplifying, shows that the non-trivial equilibria of the model (6) satisfy the following polynomial:
$ b7(L∗∗)7+b6(L∗∗)6+b5(L∗∗)5+b4(L∗∗)4+b3(L∗∗)3+b2(L∗∗)2+b1(L∗∗)+b0=0, $ | (14) |
where the coefficients
(ⅰ) the coefficients
(ⅱ) the polynomial (14) has at least one positive root whenever
These results are summarized below.
Theorem 3.3. The model (6) has at least one non-trivial equilibrium whenever
Furthermore, it is worth stating that, for the special case of the autonomous model (6) with no density-dependent larval mortality (i.e.,
$ L^{**} = \biggl(1 -\dfrac{1}{\mathcal{R}_0} \biggr)Q_1X_6. $ | (15) |
Thus, in the absence of density-dependent larval mortality (i.e.,
Theorem 3.4. The model (6) with
Consider the special case of the autonomous model (6) in the absence of density-dependent mortality rate for larvae (i.e.,
$J(T1)=[−CE0000000ψU(2R0−1)σE−C100000000ξ1−C200000000ξ2−C300000000ξ3−C400000000ξ4−CP00000000σP−C50γU000000η∗V−C600000000ατ∗W−C7].$ |
The eigenvalues of the matrix
$ P9(λ)= λ9+A8λ8+A7λ7+A6λ6+A5λ5+A4λ4+A3λ3+A2λ2+A1λ+CD(R0−1), $ | (16) |
where
$ P_9(\lambda) = F(\lambda)G(\lambda) + CD\bigl(\mathcal{R}_0 -2\bigr), $ | (17) |
where,
$ F(\lambda) = (\lambda + C_E)(\lambda + C_P)(\lambda + C_1)(\lambda + C_2)(\lambda + C_3)(\lambda + C_4), $ | (18) |
and,
$ G(\lambda) = \lambda^3 + (C_5 + C_6 + C_7)\lambda^2 + (C_5C_6 + C_5C_7 + C_6C_7)\lambda + D, $ | (19) |
so that,
$ F(λ)G(λ)= λ9+A8λ8+A7λ7+A6λ6+A5λ5+A4λ4+A3λ3+A2λ2+A1λ+CD. $ | (20) |
The asymptotic stability of
Theorem 3.5.(Routh-Hurwitz)[31]. Let
Definition 3.6.(Bézout Matrix)[31]. Let
$a(x)b(y)−a(y)b(x)x−y=n−1∑i,k=0bikxiyk.$ |
The symmetric matrix
$bi,j=min(i,n−1−j)∑k=max(0,i−j)(bi−kaj+1+k−ai−kbj+1+k) for all i,j≤n.$ |
Theorem 3.7.(Liénard-Chipart)[31] Let
$h(u)=−a1−a3u−⋯,g(u)=−a2−a4u−⋯.$ |
The polynomial
Theorem 3.8. (Sylvester's Criterion)[27] A real, symmetric matrix is positive definite if and only if all its principal minors are positive.
We claim the following result.
Lemma 3.9. The polynomial
Proof. It follows from the equation for
$G(λ)=λ3+(C5+C6+C7)λ2+(C5C6+C5C7+C6C7)λ+D=0.$ |
Using the Routh-Hurwitz Criterion (Theorem 3.5), the principal minors,
$Δ1= C5+C6+C7>0,Δ2= (C5+C6+C7)(C5C6+C5C7)+C6C7(C6+C7)+ατ∗Wη∗VγU>0,Δ3= DΔ2>0.$ |
Thus, all the roots of
Remark 1. It follows from Lemma 3.9 and Theorem 3.7 that the corresponding Bézout matrix of
Remark 2. Consider
Furthermore, consider the characteristic polynomial
$P9(λ)=λ9+A8λ8+A7λ7+A6λ6+A5λ5+A4λ4+A3λ3+A2λ2+A1λ+A0.$ |
To apply Theorem 3.7, let
$h(u)=A0+A2u+A4u2+A6u3+A8u4,$ |
and,
$g(u)=A1+A3u+A5u2+A7u3+u4.$ |
Thus, it follows from Definition 3.6 that the corresponding Bézout matrix of
$Bh,g(P9)=[A1A2−A0A3A1A4−A0A5A1A6−A0A7A1A8−A0A1A4−A0A5A3A4−A2A5+A1A6−A0A7A3A6−A2A7+A1A8−A0A3A8−A2A1A6−A0A7A3A6−A2A7+A1A8−A0A5A6−A4A7+A1A8−A2A5A8−A4A1A8−A0A3A8−A2A5A8−A4A7A8−A6].$ |
Sylvester's Criterion (Theorem 3.8) can be used to obtain the necessary and sufficient conditions for
(ⅰ)
(ⅱ)
Therefore,
$Bh,g(P9)=[b(FG)0,0−CDKA3b(FG)0,1−CDKA5b(FG)0,2−CDKA7b(FG)0,3−CDKb(FG)1,0−CDKA5b(FG)1,1−CDKA7b(FG)1,2−CDKb(FG)1,3b(FG)2,0−CDKA7b(FG)2,1−CDKb(FG)2,2b(FG)2,3b(FG)3,0−CDKb(FG)3,1b(FG)3,2b(FG)3,3].$ |
where
$ B_{h, g}(P_9) = \left[\begin {array}{cccc} \Delta^{(P_9)}_1&b^{(P_9)}_{0, 1}&b^{(P_9)}_{0, 2}&b^{(P_9)}_{0, 3} \\ 0&\dfrac{\Delta^{(P_9)}_2}{\Delta^{(P_9)}_1}&b^{(P_9)}_{1, 2}-\dfrac{b^{(P_9)}_{0, 1}b^{(P_9)}_{0, 3}}{b^{(P_9)}_{0, 0}}&b^{(P_9)}_{1, 3}-\dfrac{b^{(P_9)}_{0, 1}b^{(P_9)}_{0, 4}}{b^{(P_9)}_{0, 0}} \\ 0&0&\dfrac{\Delta^{(P_9)}_{3}}{\Delta^{(P_9)}_{2}}&B_1 \\ 0&0&0& \dfrac{\Delta^{(P_9)}_{4}}{\Delta^{(P_9)}_{3}} \end {array} \right], $ | (21) |
where
$Δ(P9)1= A1A2−CD(R0−1)A3=A1A2−CDA3−CD(R0−2)A3= b(FG)0,0−CD(R0−2)A3,$ |
is positive whenever the following inequality holds:
$R0<2+b(FG)0,0CDA3=2+Z1,$ |
where,
Theorem 3.10 Consider the model (6) with
$1<R0<RC0=2+min{Zk:Δ(P9)k>0 for all k=1,2,3,4},$ |
and unstable whenever
The results above (Theorem 3.4 and Theorem 3.10) show that the condition
Theorem 3.11. Consider a special case of the model (6) with
Proof. The proof of Theorem 3.11, based on using a non-linear Lyapunov function of Goh-Voltera type, is given in Appendix B.
The ecological implication of Theorem 3.11 is that mosquitoes will persist in the community whenever the associated conditions for the global asymptotic stability of the non-trivial equilibrium (
Consider the model (6) with
The rank and signature of the Bézout matrix,
Theorem 3.12. Consider the autonomous model (6) with
$ψU=ψ∗U=CD(2+Z4)ατ∗Wη∗VB=CDRC0ατ∗Wη∗VB,$ |
where
Proof. To prove Theorem 3.12, it is sufficient to establish the transversality condition [20]. Let
$Δ(P9)4=|b(FG)0,0−(ατ∗Wη∗VψUB−2CD)A3b(FG)0,1−(ατ∗Wη∗VψUB−2CD)A5b(P9)0,2b(P9)0,3b(FG)1,0−(ατ∗Wη∗VψUB−2CD)A5b(FG)1,1−(ατ∗Wη∗VψUB−2CD)A7b(P9)1,2b(P9)1,3b(FG)2,0−(ατ∗Wη∗VψUB−2CD)A7b(FG)2,1−(ατ∗Wη∗VψUB−2CD)b(P9)2,2b(P9)2,3b(FG)3,0−(ατ∗Wη∗VψUB−2CD)b(P9)3,1b(P9)3,2b(P9)3,3|.$ |
Hence,
$dΔ(P9)4(ψU)dψU|ψU=ψ∗U= Tr(Adj(Bh,g(P9)(ψU))|ψU=ψ∗UdBh,g(P9)(ψU)dψ|ψU=ψ∗U)≠0,$ |
where 'Tr' and 'Adj' denote, respectively, the trace and adjoint of a matrix. Similarly, let
$dΔ(P9)4(μA)dψ|μA=μ∗A= Tr(Adj(Bh,g(P9)(μA))|μA=μ∗AdBh,g(P9)(μA)dμA|μA=μ∗A),$ |
for all
Theorem 3.12 shows that sustained oscillations are possible, with respect to the autonomous model (6) with
In this section, a bifurcation diagram for the autonomous model (6) with
(ⅰ) Solving for
$l: ψU=ψlU=CD(μA)ατ∗Wη∗VB.$ |
(ⅱ) Solving for
$H: ψU=ψ∗U=CD(μA)[2+Z4(μA)]ατ∗Wη∗VB,$ |
where
$D1 ={(μA,ψU):0<ψU≤ψlU;μA>0},D2 ={(μA,ψU):ψlU<ψU<ψ∗U;μA>0},D3 ={(μA,ψU):ψU>ψ∗U;μA>0}.$ |
The regions can be described as follows (see also Table 3):
Threshold Condition |
|
|
Existence of Stable Limit Cycle |
|
GAS | No | No |
|
Unstable | LAS | No |
|
Unstable | Unstable | Yes |
(ⅰ) Region
(ⅱ) Region
(ⅲ) Region
Sensitivity analysis determines the effects of parameters on the model outcomes [16]. The effect of these uncertainties, as well as the determination of the parameters that have the greatest influence on the mosquitoes dispersal dynamics (with respect to a given response function), are carried out using an uncertainty and sensitivity analysis [2,14,44,45,46,55]. In particular, following [14], the Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC) will be used for the autonomous model (6). The range and baseline values of the parameters, tabulated in Table 4, will be used. Appropriate response functions are chosen for these analyses.
Parameters | Baseline Value | Range | Reference |
|
50/day | (10 -100)/day | [2, 22, 38, 40, 65] |
|
40000 | [2, 38, 65] | |
|
0.84/day | (0.7 -0.99)/day | [22] |
|
0.05/day |
|
[22] |
|
0.095/day |
|
|
|
0.11/day |
|
|
|
0.13/day |
|
|
|
0.16/day |
|
|
|
0.34/day |
|
[22] |
|
0.04/ml |
|
[29] |
|
0.8/day |
|
[22] |
|
0.17/day |
|
|
|
0.89/day |
|
[51, 52] |
|
|
|
[51, 52] |
|
16 |
|
[51] |
|
0.86 |
|
[51] |
|
0.05/day |
|
[2, 19, 38, 53, 65] |
|
|
[60] | |
|
|
||
|
|
||
|
|
||
|
|
||
|
|
[60] |
Using the population of female adult mosquitoes of type
Parameters |
|
|
|
|
|
|
+0.6863 | +0.8509 | +0.9083 | +0.8958 | +0.88 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
+0.6525 | +0.6896 | +0.7019 | +0.63 |
|
|
+0.6337 | +0.6817 | +0.6543 | +0.60 |
|
|
|
|
|
|
|
|
+0.6473 |
|
|
|
|
-0.7842 | -0.9103 | -0.9193 | -0.9427 | -0.96 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
-0.6390 |
|
|
|
|
|
+0.9284 |
|
+0.6106 | +0.6224 |
|
|
-0.8597 |
|
|
|
-0.69 |
In summary, this study identifies five parameters that dominate the population dynamics and dispersal of the mosquito, namely the probability of female adult mosquito of type
Control measure by model (1) | Effect on population dynamics of mosquitoes | Effect on vectorial reproduction number |
Environmental interpretation |
Significant reduction in the value of |
Significant decrease in the population size of adult mosquitoes of type |
Significant decrease in the value of |
Personal protection against mosquito bite plays an important role in minimizing the size of mosquito population in the community. |
Significant reduction in the value of |
Significant decrease in the population size of all three adult mosquito compartments | Significant decrease in the value |
The removal of mosquito breeding (egg laying) sites, such as removal of stagnant waters, is an effective control measure against the mosquito population. |
Significant reduction in the value of |
Significant decrease in the population size of all three adult mosquito compartments | Significant decrease in the value |
The removal of mosquito breeding sites and use of larvicides are effective control measures against the mosquito population. |
Significant increase in the value of |
Significant decrease in the population size of adult mosquitoes of type |
Significant decrease in the value of |
The use of insecticides and insecticides treated bednets (ITNs) are important control measures against the mosquito population. |
In this section, dynamical properties of the non-autonomous model (1) will be explored. The non-autonomous model (1) has a unique trivial equilibrium point denoted by
$dE∗(t)dt = ψU(t)[1−U∗(t)KU]U∗(t)−[σE(t)+μE(t)]E∗(t),dL∗1(t)dt = σE(t)E∗(t)−[ξ1(t)+μL(t)+δLL∗(t)]L∗1(t),dL∗i(t)dt = ξ(i−1)(t)L∗(i−1)(t)−[ξi(t)+μL(t)+δLL∗(t)]L∗i(t); i=2,3,4,dP∗(t)dt = ξ4(t)L∗4(t)−[σP(t)+μP(t)]P∗(t),dV∗(t)dt = σP(t)P∗(t)+γUU∗(t)−ηVHH+FV∗(t)−μA(t)V∗(t),$ |
$ dW∗(t)dt = ηVHH+FV∗(t)−[τWH+μA(t)]W∗(t),dU∗(t)dt = ατWHW∗(t)−[γU+μA(t)]U∗(t),L∗(t) = 4∑i=1L∗i(t). $ | (22) |
The vectorial reproduction ratio, associated with the non-autonomous model (6), will be computed using the approach in [5,6,7,8,9,10,75]. The next generation matrix
$F(t)=[00F1(t)000000]andV(t)=[V1(t)00V2(t)V3(t)00V4(t)V5(t)],$ |
where
$F1(t)=[00ψU(t)000000],V1(t)=[CE(t)00−σE(t)C1(t)00−ξ1(t)C2(t)],V2(t)=[00−ξ2(t)000000],V3(t)=[C3(t)00−ξ3(t)C4(t)00−ξ4(t)CP(t)],V4(t)=[00−σP(t)000000],V5=[C5(t)0−γU−η∗VC6(t)00−ατ∗WC7(t)],$ |
where
$dx(t)dt=[F(t)−V(t)]x(t)$ |
where
$dY(t,s)dt=−V(t)Y(t,s)∀t≥s,Y(s,s)=I.$ |
where
Suppose that
$Ψ(t)=∫t−∞Y(t,s)F(s)ϕ(s)ds=∫∞0Y(t,t−a)F(t−a)ϕ(t−a)da.$ |
Let
$(Lϕ)(t)=∫∞0Y(t,t−a)F(t−a)ϕ(t−a)da∀t∈R,ϕ∈Cω.$ |
The vectorial reproduction ratio of the model (22)
Theorem 4.1. The trivial equilibrium
The global asymptotic stability of the trivial equilibrium
Theorem 4.2. The trivial equilibrium
The proof of Theorem 4.2, based on using comparison theorem [69], is given in Appendix B. The epidemiological implication of Theorem 4.2 is that the mosquito population (both immature and mature) can be effectively controlled (or eliminated) if the associated vectorial reproduction threshold,
In this section, the possibility of the existence of a non-trivial positive periodic solution for the non-autonomous system (1) will be explored using uniform persistence theory [40,72,81,82] (see also [55]). Following and using notations in, Lou and Zhao [40], it is convenient to define the following sets (
$X= Ω,X0= {ϕ=(ϕ1,ϕ2,ϕ3,ϕ4,ϕ5,ϕ6,ϕ7,ϕ8,ϕ9)∈X:ϕi(0)>0 for all i∈[1,9]},∂X0= X∖X0={ϕ∈X:ϕi(0)=0forsomei∈[1,9]}.$ |
Theorem 4.3. Consider the non-autonomous model (2.1). Let
$lim inft→∞(E,L1,L2,L3,L4,P,V,W,U)(t)≥(φ,φ,φ,φ,φ,φ,φ,φ,φ).$ |
Proof. The proof is based on using the approach in [40,55]. Let
Thus, it suffices to show that model (1) is uniformly-persistent with respect to
$ K∂= {ϕ∈∂X0:Pn(ϕ)∈∂X0forn≥0},D1= {ϕ∈X:ϕi(0)=0 for all i∈[1,9]},∂X0∖D1= {ϕ∈X:ϕi(0)≥0 for some i∈[1,9]}. $ | (23) |
We claim that
Thus, from (23), it can be verified that
$T∗0=(0,0,0,0,0,0,0,0,0).$ |
Hence, the set
$lim supt→∞|Φ(nω)ϕ−T0|≥ϵ for all ϕ∈X0.$ |
Thus, it follows that
It follows, from Theorem 4.5 in [42] (see also Theorem 2.1 in [84]), that
$lim inft→∞min(E(t,ϕ),L1(t,ϕ),L2(t,ϕ),L3(t,ϕ),L4(t,ϕ),P(t,ϕ),V(t,ϕ),W(t,ϕ),U(t,ϕ))=lim inft→∞d(ϕ,∂X0)≥φ, for all ϕ∈X0.$ |
In particular,
The non-autonomous model (6) is simulated to illustrate the effect of the two climate variables (temperature and rainfall) on the population dynamics of adult mosquitoes in a community. Suitable functional forms for the temperature-and rainfall-dependent functions, relevant to Anopheles mosquitoes (mostly given in [2,47,55,60]) as defined in Section 2.1, will be used. For these simulations, water temperature
The combined effect of mean monthly temperature and rainfall is assessed by simulating the non-autonomous model using various mean monthly temperature and rainfall values in the range
Month | Jul | Aug | Sept | Oct | Nov | Dec | Jan | Feb | Mar | Apr | May | Jun |
Temperature ( |
17.5 | 18.5 | 20 | 21.0 | 22.5 | 22.0 | 25 | 25 | 25.5 | 22.5 | 20 | 17.5 |
Rainfall ( |
48.2 | 32.3 | 65.2 | 107.1 | 121 | 118.3 | 124 | 142.2 | 113 | 98.1 | 35.4 | 34.7 |
Month | Jul | Aug | Sept | Oct | Nov | Dec | Jan | Feb | Mar | Apr | May | Jun |
Temperature ( |
25.5 | 25 | 24 | 25.5 | 26 | 26.5 | 25.5 | 26 | 27 | 27.5 | 27 | 26.5 |
Rainfall ( |
255 | 115 | 162 | 113 | 57 | 15 | 20 | 55 | 80 | 150 | 210 | 320 |
Month | Jul | Aug | Sept | Oct | Nov | Dec | Jan | Feb | Mar | Apr | May | Jun |
Temperature ( |
17.5 | 18 | 19 | 20.5 | 20 | 19.5 | 20.5 | 20.5 | 21.5 | 20.5 | 19.5 | 18.5 |
Rainfall ( |
14.5 | 29.8 | 21.3 | 36.7 | 151 | 79.1 | 73.9 | 48.8 | 89.2 | 119.9 | 129.4 | 15.8 |
This study presents a new mathematical model for the population biology of the mosquito (the world's deadliest animal, which accounts for 80% of vector-borne diseases of humans). Some of the notable features of the new model are:
(ⅰ) incorporating four developmental stages of the mosquito larvae (
(ⅱ) including density dependence for the eggs oviposition process and larval mortality rates;
(ⅲ) including the dispersal states of female adult mosquitoes(
The model, which takes the form of a non-autonomous deterministic system of non-linear differential equations, is used to assess the impact of temperature and rainfall on the population dynamics of the mosquito. The main theoretical and epidemiological findings of this study are summarized below:
(ⅰ) The trivial equilibrium of the autonomous model (6) is globally-asymptotically stable whenever the associated vectorial reproduction number
(ⅱ) Uncertainty and sensitivity analysis of the autonomous version of the model shows that the top five parameters that have the most influence on the dynamics of the model (with respect to various response functions) are the probability of female adult mosquito of type
(ⅲ) The trivial periodic solution of the non-autonomous model (1) is shown to be globally-asymptotically stable, whenever the spectral radius of a certain linear operator (denoted by
Numerical simulations of the non-autonomous model, using relevant functional forms (given in Section 2.1) and parameter values associated with the Anopheles species (which causes malaria in humans), show the following:
(ⅰ) For mean monthly temperature and rainfall values in the range
(ⅱ) For mean monthly temperature and rainfall data for three cities in Africa, namely, KwaZulu-Natal, South-Africa; Lagos, Nigeria and Nairobi, Kenya (Tables 7, 8 and 9). The peak mosquito abundance for KwaZulu-Natal (Figure 6a) and Lagos (Figure 6b) occur when temperature and rainfall values lie in the range
One of the authors (ABG) is grateful to National Institute for Mathematical and Biological Synthesis (NIMBioS) for funding the Working Group on Climate Change and Vector-borne Diseases (VBDs). NIMBioS is an Institute sponsored by the National Science Foundation, the U.S. Department of Homeland Security, and the U.S. Department of Agriculture through NSF Award #EF-0832858, with additional support from The University of Tennessee, Knoxville.
$b0 = (1R0−1)Q1X6,$ |
$b1 = [(1R0−1)Q1X3+Q2X5X6]δL+1,b2 = [(1R0−1)Q1X2+Q2(X5X3+X4X6)](δL)2,b3 = [(1R0−1)Q1+Q2(X1X6+X2X4+X3X4)](δL)3,b4 = Q2(X1X3+X2X4+X5+X6)(δL)4,b5 = Q2(X1X2+X3+X4)(δL)5,b6 = Q2(X1+X2)(δL)6,b7 = Q2(δL)7,$ |
where,
$Q1= σECPDKUατ∗Wη∗VB, Q2=KU(CPD)2σECE(ατ∗Wη∗VB)2ψU, X1=C1+C2+C3+C4,X2= C2+C3+C4+ξ1, X3=C2C3+C2C4+C3C4+C3ξ1+C4ξ1+ξ1ξ2,X4= C1C2+C1C3+C1C4+C2C3+C2C4+C3C4, X5= C1C2C3+C1C2C4+C1C3C4+C2C3C4, X6=C2C3C4+C3C4ξ1+C4ξ1ξ2+ξ1ξ2ξ3,$ |
with
$A1=D(C1C2C3C4CE+C1C2C3C4CP+C1C2C3CECP+C1C2C4CECP+C1C3C4CECP+C2C3C4CECP)+C(C5C6+C5C7+C6C7)>0,$ |
$ A2=2∑i=1Ci3∑j=i+1Cj4∑k=j+1Cj5∑l=k+1Cl6∑m=l+1Cm7∑n=m+1Cn(CE+CP+7∑q=n+1Cq)+CECP(3∑i=1Ci4∑j=i+1Cj5∑k=j+1Ck6∑l=k+1Cl7∑m=l+1Cm−ατ∗Wη∗VγU3∑iCi4∑j=i+1Cj)−ατ∗Wη∗VγU2∑i=1Ci3∑j=i+1Cj4∑k=j+1Ck(4∑l=k+1Cl+CE+CP)>0,A3=3∑i=1Ci4∑j=i+1Cj5∑k=j+1Cj6∑l=k+1Cl7∑m=l+1Cm(CE+CP+7∑n=m+1Cn)+CECP(4∑i=1Ci5∑j=i+1Cj6∑k=j+1Ck7∑l=k+1Cl−ατ∗Wη∗VγU4∑iCi)−ατ∗Wη∗VγU3∑i=1Ci4∑j=i+1Cj(4∑k=j+1Ck+CE+CP)>0,A4=4∑i=1Ci5∑j=i+1Cj6∑k=j+1Cj7∑l=k+1Cl(CE+CP+7∑m=l+1Cm)+CECP(5∑i=1Ci6∑j=i+1Cj7∑k=j+1Ck−ατ∗Wη∗VγU)−ατ∗Wη∗VγU3∑i=1Ci(4∑j=i+1Cj+CE+CP)>0,A5=5∑i=1Ci6∑j=i+1Cj7∑k=j+1Cj(CE+CP+7∑l=k+1Cl)+CECP6∑i=1Ci7∑j=i+1Cj−ατ∗Wη∗VγUC>0,A6=6∑i=1Ci7∑j=i+1Cj(CE+CP+7∑k=j+1Ck)+CECP7∑i=1Ci−ατ∗Wη∗VγU>0,A7=7∑i=1Ci(CE+CP+7∑j=i+1Cj)>0,A8=CE+CP+7∑i=1Ci>0, $ |
where
Proof. Consider the model (6) with
$K2= E−E∗∗lnE+d1(L1−L∗∗1lnL1)+d2(L2−L∗∗2lnL2)+d3(L3−L∗∗3lnL3)+d4(L4−L∗∗4lnL4)+d5(P−P∗∗lnP)+d6(V−V∗∗lnV)+ d7(W−W∗∗lnW)+d8(U−U∗∗lnU),$ |
where,
$ d1=CEσE, d2=C1CEξ1σE, d3=C2C1CEξ2ξ1σE, d4=C3C2C1CEξ3ξ2ξ1σE,d5=C4C3C2C1CEξ4ξ3ξ2ξ1σE, d6=CB, d7=C5Cη∗VB, d8=C6C5Cατ∗Wη∗VB, $ | (24) |
with
$ CEE∗∗=ψU(1−U∗∗KU)U∗∗,C1L∗∗1=σEE∗∗, CiL∗∗i=ξ(i−1)L∗∗(i−1); i=2,3,4,CPP∗∗=ξ4L∗∗4,C5V∗∗=σPP∗∗+γUU∗∗, C6W∗∗=η∗VV∗∗,C7U∗∗=ατ∗WW∗∗. $ | (25) |
The Lyapunov derivative of
$ ˙K2= (1−E∗∗E)[ψU(1−UKU)U−CEE]+d1(1−L∗∗1L1)[σEE−C1L1]+d2(1−L∗∗2L2)[ξ1L1−C2L2]+d3(1−L∗∗3L3)[ξ2L2−C3L3]+d4(1−L∗∗4L4)[ξ3L3−C4L4]+d5(1−P∗∗P)[ξ4L4−CPP]+d6(1−V∗∗V)[σPP−C5V+γUU]+d7(1−W∗∗W)[η∗VV−C6W]+d8(1−U∗∗U)[ατ∗WW−C7U], $ | (26) |
Substituting (24) and (25) into (26), and simplifying, gives
$ ˙K2= −ψUUEKU(E∗∗−E)(U∗∗−U)+γUd6U∗∗(3−UV∗∗U∗∗V−VW∗∗V∗∗W−U∗∗WUW∗∗)+CEE∗∗(9−L∗∗1EL1E∗∗−L∗∗2L1L2L∗∗1−L∗∗3L2L3L∗∗2−L∗∗4L3L4L∗∗3−P∗∗L4PL∗∗4−V∗∗PVP∗∗−W∗∗VWV∗∗−U∗∗WUW∗∗−E∗∗UEU∗∗). $ | (27) |
The first term of (27) is automatically negative in
The assumption
Proof. Consider the non-autonomous model (1) with
$ψU(1−U(t)KU)U(t)≤ψUU(t) (since KU>U(t) for all t≥0),$ |
and,
$Ci(t)+δLL(t)≥Ci(t), for all t≥0,$ |
it follows that the non-autonomous model (1), subject to the aforementioned assumptions, can be re-written as
$ dEdt ≤ ψUU−CE(t)E,dL1dt ≤ σE(t)E−C1(t)L1,dLidt ≤ ξ(i−1)(t)L(i−1)−Ci(t)Li ; i=2,3,4,dPdt = ξ4(t)L4−CP(t)P,dVdt = σP(t)P+γUU(t)−C5(t)V,dWdt = η∗VV−C6(t)W,dUdt = ατ∗WW−C7(t)U. $ | (28) |
Following [75], the equations in (28), with equalities used in place of the inequalities, can be re-written in terms of the next generation matrices
$ \frac{d X(t)}{d t} = [F(t)-V(t)]X(t). $ | (29) |
It follows, from Lemma 2.1 in [80], that there exists a positive and bounded
$X(t)=eθtx(t), with θ=1ωlnρ[ϕF−V(ω)],$ |
is a solution of the linearized system (28). Furthermore, it follows from Theorem 2.2 in [75] that
$((E,L1,L2,L3,L4,P,V,W,U)(0))T≤Q∗((ˉE,ˉL1,ˉL2,ˉL3,ˉL4,ˉP,ˉV,ˉW,ˉU)(0))T.$ |
Thus, it follows, by comparison theorem [37,69], that
$(E(t),L1(t),L2(t),L3(t),L4(t),P(t),V(t),W(t),U(t))≤Q∗X(t)for all t>0,$ |
where
Hence,
$(E(t),L1(t),L2(t),L3(t),L4(t),P(t),V(t),W(t),U(t))→(0,0,0,0,0,0,0,0,0),$ |
as
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1. | Kamaldeen Okuneye, Steffen E. Eikenberry, Abba B. Gumel, Weather-driven malaria transmission model with gonotrophic and sporogonic cycles, 2019, 13, 1751-3758, 288, 10.1080/17513758.2019.1570363 | |
2. | Gbenga J. Abiodun, Babatope. O. Adebiyi, Rita O. Abiodun, Olanrewaju Oladimeji, Kelechi E. Oladimeji, Abiodun M. Adeola, Olusola S. Makinde, Kazeem O. Okosun, Ramsès Djidjou-Demasse, Yves J. Semegni, Kevin Y. Njabo, Peter J. Witbooi, Alejandro Aceves, Investigating the Resurgence of Malaria Prevalence in South Africa Between 2015 and 2018: A Scoping Review, 2020, 13, 1874-9445, 119, 10.2174/1874944502013010119 | |
3. | Abdulaziz Y.A. Mukhtar, Justin B. Munyakazi, Rachid Ouifki, Assessing the role of climate factors on malaria transmission dynamics in South Sudan, 2019, 310, 00255564, 13, 10.1016/j.mbs.2019.01.002 | |
4. | Babagana Modu, Nereida Polovina, Savas Konur, Agent-Based Modeling of Malaria Transmission, 2023, 11, 2169-3536, 19794, 10.1109/ACCESS.2023.3248292 | |
5. | Joseph Baafi, Amy Hurford, Modeling the Impact of Seasonality on Mosquito Population Dynamics: Insights for Vector Control Strategies, 2025, 87, 0092-8240, 10.1007/s11538-024-01409-7 | |
6. | Ademe Kebede Gizaw, Temesgen Erena, Eba Alemayehu Simma, Dawit Kechine Menbiko, Dinka Tilahun Etefa, Delenasaw Yewhalaw, Chernet Tuge Deressa, Modeling the variability of temperature on the population dynamics of Anopheles arabiensis, 2025, 18, 1756-0500, 10.1186/s13104-025-07153-y |
Variables | Description |
|
Population of female eggs |
|
Population of female larvae at Stage |
|
Population of female pupae |
|
Population of fertilized female mosquitoes that have laid eggs at the breeding site |
(including unfertilized female mosquitoes not questing for blood meal) | |
|
Population of fertilized, but non-reproducing, female mosquitoes questing for blood meal |
|
Population of fertilized, well-nourished with blood, and reproducing female mosquitoes |
Parameters | Description |
|
Deposition rate of female eggs |
|
Maturation rate of female eggs |
|
Maturation rate of female larvae from larval stage |
|
Maturation rate of female pupae |
|
Natural mortality rate of female eggs |
|
Natural mortality rate of female larvae |
|
Natural mortality rate of female pupae |
|
Natural mortality rate of female adult mosquitoes |
|
Density-dependent mortality rate of female larvae |
|
Constant mass action contact rate between female adult mosquitoes of type |
|
Probability of successfully taking a blood meal |
|
Rate of return of female adult mosquitoes of type |
|
Rate at which female adult mosquitoes of type |
|
Constant population density of humans at human habitat sites |
|
Constant alternative source of blood meal for female adult mosquitoes |
|
Environmental carrying capacity of female adult mosquitoes |
|
The daily survival probability of Stage |
|
The average duration spent in Stage |
|
Rate of nutrients intake for female larvae in Stage |
|
Total available nutrient for female larvae |
|
Cumulative daily rainfall |
|
Daily mean ambient temperature |
|
Daily mean water temperature in the breeding site |
|
Maximum daily survival probability of aquatic Stage |
|
Rainfall threshold |
Temperature ( |
|
|
|
|
|
16-40 | 0.892-23.431 | 0.194-0.932 | 0.091-0.122 | 0.040-0.115 | 0.074-0.408 |
Threshold Condition |
|
|
Existence of Stable Limit Cycle |
|
GAS | No | No |
|
Unstable | LAS | No |
|
Unstable | Unstable | Yes |
Parameters | Baseline Value | Range | Reference |
|
50/day | (10 -100)/day | [2, 22, 38, 40, 65] |
|
40000 | [2, 38, 65] | |
|
0.84/day | (0.7 -0.99)/day | [22] |
|
0.05/day |
|
[22] |
|
0.095/day |
|
|
|
0.11/day |
|
|
|
0.13/day |
|
|
|
0.16/day |
|
|
|
0.34/day |
|
[22] |
|
0.04/ml |
|
[29] |
|
0.8/day |
|
[22] |
|
0.17/day |
|
|
|
0.89/day |
|
[51, 52] |
|
|
|
[51, 52] |
|
16 |
|
[51] |
|
0.86 |
|
[51] |
|
0.05/day |
|
[2, 19, 38, 53, 65] |
|
|
[60] | |
|
|
||
|
|
||
|
|
||
|
|
||
|
|
[60] |
Parameters |
|
|
|
|
|
|
+0.6863 | +0.8509 | +0.9083 | +0.8958 | +0.88 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
+0.6525 | +0.6896 | +0.7019 | +0.63 |
|
|
+0.6337 | +0.6817 | +0.6543 | +0.60 |
|
|
|
|
|
|
|
|
+0.6473 |
|
|
|
|
-0.7842 | -0.9103 | -0.9193 | -0.9427 | -0.96 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
-0.6390 |
|
|
|
|
|
+0.9284 |
|
+0.6106 | +0.6224 |
|
|
-0.8597 |
|
|
|
-0.69 |
Control measure by model (1) | Effect on population dynamics of mosquitoes | Effect on vectorial reproduction number |
Environmental interpretation |
Significant reduction in the value of |
Significant decrease in the population size of adult mosquitoes of type |
Significant decrease in the value of |
Personal protection against mosquito bite plays an important role in minimizing the size of mosquito population in the community. |
Significant reduction in the value of |
Significant decrease in the population size of all three adult mosquito compartments | Significant decrease in the value |
The removal of mosquito breeding (egg laying) sites, such as removal of stagnant waters, is an effective control measure against the mosquito population. |
Significant reduction in the value of |
Significant decrease in the population size of all three adult mosquito compartments | Significant decrease in the value |
The removal of mosquito breeding sites and use of larvicides are effective control measures against the mosquito population. |
Significant increase in the value of |
Significant decrease in the population size of adult mosquitoes of type |
Significant decrease in the value of |
The use of insecticides and insecticides treated bednets (ITNs) are important control measures against the mosquito population. |
Month | Jul | Aug | Sept | Oct | Nov | Dec | Jan | Feb | Mar | Apr | May | Jun |
Temperature ( |
17.5 | 18.5 | 20 | 21.0 | 22.5 | 22.0 | 25 | 25 | 25.5 | 22.5 | 20 | 17.5 |
Rainfall ( |
48.2 | 32.3 | 65.2 | 107.1 | 121 | 118.3 | 124 | 142.2 | 113 | 98.1 | 35.4 | 34.7 |
Month | Jul | Aug | Sept | Oct | Nov | Dec | Jan | Feb | Mar | Apr | May | Jun |
Temperature ( |
25.5 | 25 | 24 | 25.5 | 26 | 26.5 | 25.5 | 26 | 27 | 27.5 | 27 | 26.5 |
Rainfall ( |
255 | 115 | 162 | 113 | 57 | 15 | 20 | 55 | 80 | 150 | 210 | 320 |
Month | Jul | Aug | Sept | Oct | Nov | Dec | Jan | Feb | Mar | Apr | May | Jun |
Temperature ( |
17.5 | 18 | 19 | 20.5 | 20 | 19.5 | 20.5 | 20.5 | 21.5 | 20.5 | 19.5 | 18.5 |
Rainfall ( |
14.5 | 29.8 | 21.3 | 36.7 | 151 | 79.1 | 73.9 | 48.8 | 89.2 | 119.9 | 129.4 | 15.8 |
Variables | Description |
|
Population of female eggs |
|
Population of female larvae at Stage |
|
Population of female pupae |
|
Population of fertilized female mosquitoes that have laid eggs at the breeding site |
(including unfertilized female mosquitoes not questing for blood meal) | |
|
Population of fertilized, but non-reproducing, female mosquitoes questing for blood meal |
|
Population of fertilized, well-nourished with blood, and reproducing female mosquitoes |
Parameters | Description |
|
Deposition rate of female eggs |
|
Maturation rate of female eggs |
|
Maturation rate of female larvae from larval stage |
|
Maturation rate of female pupae |
|
Natural mortality rate of female eggs |
|
Natural mortality rate of female larvae |
|
Natural mortality rate of female pupae |
|
Natural mortality rate of female adult mosquitoes |
|
Density-dependent mortality rate of female larvae |
|
Constant mass action contact rate between female adult mosquitoes of type |
|
Probability of successfully taking a blood meal |
|
Rate of return of female adult mosquitoes of type |
|
Rate at which female adult mosquitoes of type |
|
Constant population density of humans at human habitat sites |
|
Constant alternative source of blood meal for female adult mosquitoes |
|
Environmental carrying capacity of female adult mosquitoes |
|
The daily survival probability of Stage |
|
The average duration spent in Stage |
|
Rate of nutrients intake for female larvae in Stage |
|
Total available nutrient for female larvae |
|
Cumulative daily rainfall |
|
Daily mean ambient temperature |
|
Daily mean water temperature in the breeding site |
|
Maximum daily survival probability of aquatic Stage |
|
Rainfall threshold |
Temperature ( |
|
|
|
|
|
16-40 | 0.892-23.431 | 0.194-0.932 | 0.091-0.122 | 0.040-0.115 | 0.074-0.408 |
Threshold Condition |
|
|
Existence of Stable Limit Cycle |
|
GAS | No | No |
|
Unstable | LAS | No |
|
Unstable | Unstable | Yes |
Parameters | Baseline Value | Range | Reference |
|
50/day | (10 -100)/day | [2, 22, 38, 40, 65] |
|
40000 | [2, 38, 65] | |
|
0.84/day | (0.7 -0.99)/day | [22] |
|
0.05/day |
|
[22] |
|
0.095/day |
|
|
|
0.11/day |
|
|
|
0.13/day |
|
|
|
0.16/day |
|
|
|
0.34/day |
|
[22] |
|
0.04/ml |
|
[29] |
|
0.8/day |
|
[22] |
|
0.17/day |
|
|
|
0.89/day |
|
[51, 52] |
|
|
|
[51, 52] |
|
16 |
|
[51] |
|
0.86 |
|
[51] |
|
0.05/day |
|
[2, 19, 38, 53, 65] |
|
|
[60] | |
|
|
||
|
|
||
|
|
||
|
|
||
|
|
[60] |
Parameters |
|
|
|
|
|
|
+0.6863 | +0.8509 | +0.9083 | +0.8958 | +0.88 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
+0.6525 | +0.6896 | +0.7019 | +0.63 |
|
|
+0.6337 | +0.6817 | +0.6543 | +0.60 |
|
|
|
|
|
|
|
|
+0.6473 |
|
|
|
|
-0.7842 | -0.9103 | -0.9193 | -0.9427 | -0.96 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
-0.6390 |
|
|
|
|
|
+0.9284 |
|
+0.6106 | +0.6224 |
|
|
-0.8597 |
|
|
|
-0.69 |
Control measure by model (1) | Effect on population dynamics of mosquitoes | Effect on vectorial reproduction number |
Environmental interpretation |
Significant reduction in the value of |
Significant decrease in the population size of adult mosquitoes of type |
Significant decrease in the value of |
Personal protection against mosquito bite plays an important role in minimizing the size of mosquito population in the community. |
Significant reduction in the value of |
Significant decrease in the population size of all three adult mosquito compartments | Significant decrease in the value |
The removal of mosquito breeding (egg laying) sites, such as removal of stagnant waters, is an effective control measure against the mosquito population. |
Significant reduction in the value of |
Significant decrease in the population size of all three adult mosquito compartments | Significant decrease in the value |
The removal of mosquito breeding sites and use of larvicides are effective control measures against the mosquito population. |
Significant increase in the value of |
Significant decrease in the population size of adult mosquitoes of type |
Significant decrease in the value of |
The use of insecticides and insecticides treated bednets (ITNs) are important control measures against the mosquito population. |
Month | Jul | Aug | Sept | Oct | Nov | Dec | Jan | Feb | Mar | Apr | May | Jun |
Temperature ( |
17.5 | 18.5 | 20 | 21.0 | 22.5 | 22.0 | 25 | 25 | 25.5 | 22.5 | 20 | 17.5 |
Rainfall ( |
48.2 | 32.3 | 65.2 | 107.1 | 121 | 118.3 | 124 | 142.2 | 113 | 98.1 | 35.4 | 34.7 |
Month | Jul | Aug | Sept | Oct | Nov | Dec | Jan | Feb | Mar | Apr | May | Jun |
Temperature ( |
25.5 | 25 | 24 | 25.5 | 26 | 26.5 | 25.5 | 26 | 27 | 27.5 | 27 | 26.5 |
Rainfall ( |
255 | 115 | 162 | 113 | 57 | 15 | 20 | 55 | 80 | 150 | 210 | 320 |
Month | Jul | Aug | Sept | Oct | Nov | Dec | Jan | Feb | Mar | Apr | May | Jun |
Temperature ( |
17.5 | 18 | 19 | 20.5 | 20 | 19.5 | 20.5 | 20.5 | 21.5 | 20.5 | 19.5 | 18.5 |
Rainfall ( |
14.5 | 29.8 | 21.3 | 36.7 | 151 | 79.1 | 73.9 | 48.8 | 89.2 | 119.9 | 129.4 | 15.8 |