Although different strategies for mosquito-borne disease prevention can vary significantly in their efficacy and scale of implementation, they all require that individuals comply with their use. Despite this, human behavior is rarely considered in mathematical models of mosquito-borne diseases. Here, we sought to address that gap by establishing general expectations for how different behavioral stimuli and forms of mosquito prevention shape the equilibrium prevalence of disease. To accomplish this, we developed a coupled contagion model tailored to the epidemiology of dengue and preventive behaviors relevant to it. Under our model's parameterization, we found that mosquito biting was the most important driver of behavior uptake. In contrast, encounters with individuals experiencing disease or engaging in preventive behaviors themselves had a smaller influence on behavior uptake. The relative influence of these three stimuli reflected the relative frequency with which individuals encountered them. We also found that two distinct forms of mosquito prevention—namely, personal protection and mosquito density reduction—mediated different influences of behavior on equilibrium disease prevalence. Our results highlight that unique features of coupled contagion models can arise in disease systems with distinct biological features.
Citation: Marya L. Poterek, Mauricio Santos-Vega, T. Alex Perkins. Equilibrium properties of a coupled contagion model of mosquito-borne disease and mosquito preventive behaviors[J]. Mathematical Biosciences and Engineering, 2025, 22(8): 1875-1897. doi: 10.3934/mbe.2025068
[1] | Lukáš Pichl, Taisei Kaizoji . Volatility Analysis of Bitcoin Price Time Series. Quantitative Finance and Economics, 2017, 1(4): 474-485. doi: 10.3934/QFE.2017.4.474 |
[2] | Andres Fernandez, Norman R. Swanson . Further Evidence on the Usefulness of Real-Time Datasets for Economic Forecasting. Quantitative Finance and Economics, 2017, 1(1): 2-25. doi: 10.3934/QFE.2017.1.2 |
[3] | Samuel Asante Gyamerah . Modelling the volatility of Bitcoin returns using GARCH models. Quantitative Finance and Economics, 2019, 3(4): 739-753. doi: 10.3934/QFE.2019.4.739 |
[4] | Guillermo Peña . Interest rates affect public expenditure growth. Quantitative Finance and Economics, 2023, 7(4): 622-645. doi: 10.3934/QFE.2023030 |
[5] | Abdul Haque, Huma Fatima, Ammar Abid, Muhammad Ali Jibran Qamar . Impact of firm-level uncertainty on earnings management and role of accounting conservatism. Quantitative Finance and Economics, 2019, 3(4): 772-794. doi: 10.3934/QFE.2019.4.772 |
[6] | Arifenur Güngör, Hüseyin Taştan . On macroeconomic determinants of co-movements among international stock markets: evidence from DCC-MIDAS approach. Quantitative Finance and Economics, 2021, 5(1): 19-39. doi: 10.3934/QFE.2021002 |
[7] | Cemile Özgür, Vedat Sarıkovanlık . An application of Regular Vine copula in portfolio risk forecasting: evidence from Istanbul stock exchange. Quantitative Finance and Economics, 2021, 5(3): 452-470. doi: 10.3934/QFE.2021020 |
[8] | Md Qamruzzaman, Jianguo Wei . Do financial inclusion, stock market development attract foreign capital flows in developing economy: a panel data investigation. Quantitative Finance and Economics, 2019, 3(1): 88-108. doi: 10.3934/QFE.2019.1.88 |
[9] | David Melkuev, Danqiao Guo, Tony S. Wirjanto . Applications of random-matrix theory and nonparametric change-point analysis to three notable systemic crises. Quantitative Finance and Economics, 2018, 2(2): 413-467. doi: 10.3934/QFE.2018.2.413 |
[10] | Fredrik Hobbelhagen, Ioannis Diamantis . A comparative study of symbolic aggregate approximation and topological data analysis. Quantitative Finance and Economics, 2024, 8(4): 705-732. doi: 10.3934/QFE.2024027 |
Although different strategies for mosquito-borne disease prevention can vary significantly in their efficacy and scale of implementation, they all require that individuals comply with their use. Despite this, human behavior is rarely considered in mathematical models of mosquito-borne diseases. Here, we sought to address that gap by establishing general expectations for how different behavioral stimuli and forms of mosquito prevention shape the equilibrium prevalence of disease. To accomplish this, we developed a coupled contagion model tailored to the epidemiology of dengue and preventive behaviors relevant to it. Under our model's parameterization, we found that mosquito biting was the most important driver of behavior uptake. In contrast, encounters with individuals experiencing disease or engaging in preventive behaviors themselves had a smaller influence on behavior uptake. The relative influence of these three stimuli reflected the relative frequency with which individuals encountered them. We also found that two distinct forms of mosquito prevention—namely, personal protection and mosquito density reduction—mediated different influences of behavior on equilibrium disease prevalence. Our results highlight that unique features of coupled contagion models can arise in disease systems with distinct biological features.
The Caginalp phase-field system
∂u∂t−Δu+f(u)=θ, | (1.1) |
∂θ∂t−Δθ=−∂u∂t, | (1.2) |
has been introduced in [1] in order to describe the phase transition phenomena in certain class of material. In this context,
ψ=∫Ω(12|∇u|2+F(u)−uθ−12θ2)dx, | (1.3) |
where
H=u+θ. | (1.4) |
Then, the evolution equation for the order parameter
∂u∂t=−δuψ, | (1.5) |
where
∂H∂t=−divq, | (1.6) |
where
q=−∇θ, | (1.7) |
we obtain (1.2). Now, a well-known side effect of the Fourier heat law is the infinite speed of propagation of thermal disturbances, deemed physically unreasonable and thus called paradox of heat conduction (see, for example, [9]). In order to account for more realistic features, several variations of (1.7), based, for example, on the Maxwell-Cattaneo law or recent laws from thermomechanics, have been proposed in the context of the Caginalp phase-field system (see, for example, [19], [20], [21], [23], [24], [25], [26], [27], [28], [30], [31], [35], [36], [37], [38], [44], [45] and [46]).
A different approach to heat conduction was proposed in the Sixties (see, [47], [48] and [49]), where it was observed that two temperatures are involved in the definition of the entropy: the conductive temperature
θ=φ−Δφ. | (1.8) |
Our aim in this paper is to study a generalization of the Caginalp phase-field system based on this two temperatures theory and the usual Fourier law with a nonlinear coupling.
The purpose of our study is the following initial and boundary value problem
∂u∂t−Δu+f(u)=g(u)(φ−Δφ), | (1.9) |
∂φ∂t−Δ∂φ∂t−Δφ=−g(u)∂u∂t, | (1.10) |
u=φ=0on∂Ω, | (1.11) |
u|t=0=u0, φ|t=0=φ0. | (1.12) |
The paper is organized as follows. In Section 2, we give the derivation of the model. The Section 3 states existence, regularity and uniqueness results. In Section 4, we address the question of dissipativity properties of the system. The last section, analyzes the spatial behavior of solutions in a semi-infinite cylinder, assuming their existence.
Thoughout this paper, the same letters
In our case, to obtain equations (1.9) and (1.10), the total free energy reads in terms of the conductive temperature
ψ(u,θ)=∫Ω(12|∇u|2+F(u)−G(u)θ−12θ2)dx, | (2.1) |
where
H=G(u)+θ=G(u)+φ−Δφ, | (2.2) |
which yields thanks to (1.6), the energy equation,
∂φ∂t−Δ∂φ∂t+divq=−g(u)∂u∂t. | (2.3) |
Considering the usual Fourier law (
Remark 2.1. We can note that we still have an infinite speed of propagation here.
Before stating the existence result, we make some assumptions on nonlinearities
|G(s)|2≤c1F(s)+c2,c0,c1,c2≥0, | (3.1) |
|g(s)s|≤c3(|G(s)|2+1),c3≥0, | (3.2) |
c4sk+2−c5≤F(s)≤f(s)s+c0≤c6sk+2−c7,c4,c6>0,c5,c7≥0, | (3.3) |
|g(s)|≤c8(|s|+1),|g′(s)|≤c9c8,c9≥0, | (3.4) |
|f′(s)|≤c10(|s|k+1),c10≥0, | (3.5) |
where
Theorem 3.1. We assume that (3.1)-(3.4) hold true. For all initial data
Proof. The proof is based on the Galerkin scheme. Here, we just make formally computations to get a priori estimates, having in mind that these estimates can be rigourously justified using the Galerkin scheme see, for example, [10], [11] and [40] for details.
Multiplying (1.9) by
12ddt(‖∇u‖2+2∫ΩF(u)dx)+‖∂u∂t‖2=∫Ωg(u)∂u∂t(φ−Δφ)dx. | (3.6) |
Multiplying (1.10) by
12ddt(‖φ‖2+2‖∇φ‖2+‖Δφ‖2)+‖∇φ‖2+‖Δφ‖2=−∫Ωg(u)∂u∂t(φ−Δφ)dx. | (3.7) |
Now, summing (3.6) and (3.7), we are led to,
ddt(‖∇u‖2+2∫ΩF(u)dx+‖φ‖2+2‖∇φ‖2+‖Δφ‖2)+2(‖∂u∂t‖2+‖∇φ‖2+‖Δφ‖2)=0. | (3.8) |
Multiplying (1.9) by
12ddt‖u‖2+‖∇u‖2+∫Ωf(u)udx=∫Ωg(u)u(φ−Δφ)dx. | (3.9) |
Using (3.2)-(3.3), (3.9) becomes
12ddt‖u‖2+‖∇u‖2+c∫ΩF(u)dx≤c′∫Ω|G(u)|2dx+12(‖φ‖2+‖Δφ‖2)+c″. | (3.10) |
Adding (3.8) and (3.10), one has
dE1dt+2(‖∇u‖2+c∫ΩF(u)dx+‖∂u∂t‖2+‖∇φ‖2)+‖Δφ‖2≤c′∫Ω|G(u)|2dx+‖φ‖2+c″, | (3.11) |
where
E1=‖u‖2+‖∇u‖2+2∫ΩF(u)dx+‖φ‖2+2‖∇φ‖2+‖Δφ‖2 | (3.12) |
enjoys
E1≤c(‖u‖2H1(Ω)+‖u‖k+2k+2+‖φ‖2H2(Ω))−c′ | (3.13) |
and
E1≤c″(‖u‖2H1(Ω)+‖u‖k+2k+2+‖φ‖2H2(Ω))−c‴. | (3.14) |
Multiplying now (1.10) by
12ddt‖∇φ‖2+‖∂φ∂t‖2+‖∇∂φ∂t‖2=−∫Ωg(u)∂u∂t∂φ∂tdx. | (3.15) |
Taking into account (3.4) and using Hölder's inequality, we get
12ddt‖∇φ‖2+12‖∂φ∂t‖2+‖∇∂φ∂t‖2≤c(‖∇u‖2+1)‖∂u∂t‖2 | (3.16) |
and then, summing (3.11) and (3.16), we have
dE2dt+2(‖∇u‖2+c∫ΩF(u)dx+‖∂u∂t‖2+‖∇φ‖2+12‖Δφ‖2+12‖∂φ∂t‖2+‖∇∂φ∂t‖2)≤c∫Ω|G(u)|2dx+‖φ‖2+c″(‖∇u‖2+1)‖∂u∂t‖2+c‴, | (3.17) |
where
E2=E1+‖∇φ‖2 | (3.18) |
satisfies similar estimates as
We deduce from (3.1) and (3.17)
dE2dt+c(‖∂φ∂t‖2+‖∇∂φ∂t‖2)≤c′E2+c″, | (3.19) |
which achieve the proof.
For more regularity on solutions, we make following additional assumptions:
f(0)=0andf′(s)≥−c,c≥0. | (3.20) |
We have:
Theorem 3.2. Under assumptions of Theorem 3.1 and assuming that (3.20) is satisfied. For every initial data
Proof. As above proof, we focus on a priori estimates.
We multiply (1.10) by
12ddt‖∇φ‖2+‖∇∂φ∂t‖2+‖Δ∂φ∂t‖2=∫Ωg(u)∂u∂tΔ∂φ∂tdx. | (3.21) |
Thanks to (3.4) and Hölder's inequality:
∫Ωg(u)∂u∂tΔ∂φ∂tdx≤c∫Ω(|u|+1)|∂u∂t||Δ∂φ∂t|dx≤c(‖∇u‖2+1)‖∂u∂t‖2+12‖Δ∂φ∂t‖2 | (3.22) |
and then,
12ddt‖∇φ‖2+‖∇∂φ∂t‖2+12‖Δ∂φ∂t‖2≤c(‖∇u‖2+1)‖∂u∂t‖2. | (3.23) |
Differentiating (1.9) with respect to time, we get
∂2u∂t2−Δ∂u∂t+f′(u)∂u∂t=g′(u)∂u∂t(φ−Δφ)+g(u)(∂φ∂t−Δ∂φ∂t). | (3.24) |
Multiplying (3.24) by
12ddt‖∂u∂t‖2+‖∇∂u∂t‖2+∫Ωf′(u)|∂u∂t|2dx=∫Ωg′(u)|∂u∂t|2(φ−Δφ)dx+∫Ωg(u)∂u∂t(∂φ∂t−Δ∂φ∂t)dx. | (3.25) |
Using (1.10), we write,
∫Ωg(u)∂u∂t(∂φ∂t−Δ∂φ∂t)dx=∫Ωg(u)∂u∂t(−g(u)∂u∂t+Δφ)dx=−∫Ω|g(u)∂u∂t|2dx+∫Ωg(u)∂u∂tΔφdx. | (3.26) |
Owing to (3.26), (3.25) reads
12ddt‖∂u∂t‖2+‖∇∂u∂t‖2+∫Ωf′(u)|∂u∂t|2dx=∫Ωg′(u)|∂u∂t|2(φ−Δφ)dx+∫Ωg(u)∂u∂tΔφdx−∫Ω|g(u)∂u∂t|2dx, | (3.27) |
since
∫Ωg′(u)|∂u∂t|2(φ−Δφ)dx≤c∫Ω|∂u∂t|2(|φ|+|Δφ|)dx≤12‖∇∂u∂t‖2+c(‖φ‖2+‖Δφ‖2), | (3.28) |
∫Ωg(u)∂u∂tΔφdx=−∫Ωg′(u)∇u∂u∂t∇φdx−∫Ωg(u)∇∂u∂t∇φdx | (3.29) |
and then,
|∫Ωg′(u)∇u∂u∂t∇φdx|≤c∫Ω|∇u||∂u∂t||∇φ|dx≤16‖∇∂u∂t‖2+c‖∇u‖2‖Δφ‖2 | (3.30) |
and
|∫Ωg(u)∇∂u∂t∇φdx|≤c∫Ω(|u|+1)|∇∂u∂t||∇φ|dx≤16‖∇∂u∂t‖2+c(‖∇u‖2+1)‖∇φ‖2. | (3.31) |
Furthemore,
∫Ω|g(u)∂u∂t|2dx≤c∫Ω(|u|+1)2|∂u∂t|2dx≤c(‖∇u‖2+‖u‖2+1)‖∂u∂t‖2. | (3.32) |
Now, collecting (3.27)–(3.32) and owing to (3.20), we are led to
ddt‖∂u∂t‖2+c‖∇∂u∂t‖2≤c′(‖u‖2H1(Ω)+1)(‖∂u∂t‖2+‖φ‖2H2(Ω)). | (3.33) |
Adding (3.19),
dE3dt+c(‖∂u∂t‖2H1(Ω)+‖∂φ∂t‖2H2(Ω))≤c′E3+c″, | (3.34) |
where
E3=E2+ε1‖∇φ‖2+ε2‖∂u∂t‖2 | (3.35) |
enjoys
E3≥c(‖u‖2H(Ω)+‖u‖k+2k+2+‖φ‖2H2(Ω))−c′ | (3.36) |
and
E3≤c″(‖u‖2H(Ω)+‖u‖k+2k+2+‖φ‖2H2(Ω))−c‴. | (3.37) |
We complete the proof applying Gronwall's lemma.
We now give a uniqueness result
Theorem 3.3. Under assumptions of Theorem 3.2 and assuming that (3.5) holds true. The problem (1.9)-(1.12) has a unique solution
Proof. We suppose the existence of two solutions
∂u∂t−Δu+f(u1)−f(u2)=g(u1)(φ−Δφ)+(g(u1)−g(u2))(φ2−Δφ2), | (3.38) |
∂φ∂t−Δ∂φ∂t−Δφ=−g(u1)∂u∂t−(g(u1)−g(u2))∂u2∂t, | (3.39) |
u|∂Ω=φ|∂Ω=0, | (3.40) |
u|t=0=u01−u02,φ|t=0=φ01−φ02, | (3.41) |
with
Multiplying (3.38) by
12ddt‖∇u‖2+‖∂u∂t‖2+∫Ω(f(u1−f(u2)))∂u∂tdx=∫Ωg(u1)(φ−Δφ)∂u∂tdx+∫Ω(g(u1)−g(u2))(φ2−Δφ2)∂u∂tdx. | (3.42) |
Multiplying (3.39) by
12ddt(‖φ‖2+‖∇φ‖2)+‖∇φ‖2=−∫Ωg(u1)∂u∂tφdx−∫Ω(g(u1)−g(u2))∂u2∂tφdx. | (3.43) |
Multiplying (3.39) by
12ddt(‖∇φ‖2+‖Δφ‖2)+‖Δφ‖2=∫Ωg(u1)∂u∂tΔφdx+∫Ω(g(u1)−g(u2))∂u2∂tΔφdx. | (3.44) |
Finally, adding (3.42), (3.43) and (3.44), we get
dE4dt+‖∂u∂t‖2+‖∇φ‖2+‖Δφ‖2+∫Ω(f(u1)−f(u2))∂u∂tdx=∫Ω(g(u1)−g(u2))(φ2−Δφ2)∂u∂tdx−∫Ω(g(u1)−g(u2))(φ−Δφ)∂u2∂tdx, | (3.45) |
where
E4=‖∇u‖2+‖φ‖2+2‖∇φ‖2+‖Δφ‖2. | (3.46) |
Now, owing to (3.5), and applying Hölder's inequality for
∫Ω(f(u1)−f(u2))∂u∂tdx≤c∫Ω(|u2|k+1)|u||∂u∂t|dx≤c(‖∇u2‖2k+1)‖∇u‖2+‖∂u∂t‖2, | (3.47) |
we also get, thanks to (3.4), and applying Hölder's inequality,
∫Ω(g(u1)−g(u2))(φ2−Δφ2)∂u∂tdx≤c∫Ω|u||φ2−Δφ2||∂u∂t|dx≤c‖∇u‖2(‖φ2‖2+‖Δφ2‖2)+‖∂u∂t‖2 | (3.48) |
and
∫Ω(g(u1)−g(u2))(φ−Δφ)∂u2∂tdx≤c∫Ω|u||∂u∂t||φ−Δφ|dx≤c‖∂u2∂t‖2(‖φ‖2+‖Δφ‖2)+‖∇u‖2. | (3.49) |
From (3.45)-(3.49), we deduce a differential inequality of the type
dE4dt+c‖∂u∂t‖2≤c(‖∇u2‖2k+‖∂u2∂t‖2+‖φ2‖2+‖Δφ2‖2+1)E4. | (3.50) |
In particular,
dE4dt≤cE4 | (3.51) |
and then applying the Gronwall's lemma to (3.51), we end the proof.
This section is devoted to the existence of bounded absorbing sets for the semigroup
∀ϵ>0,|G(u)|2≤ϵF(s)+cϵ,s∈R. | (4.1) |
We then have
Theorem 4.1. Under the assumptions of the Theorem 3.3 and assuming that (4.1) holds true. Then,
Proof. Going from (3.8) and (3.10), we get, summing (3.8) and
dE5dt+2(c‖∇u‖2+δ∫ΩF(u)dx+‖∂u∂t‖2+‖∇φ‖2+‖Δφ‖2)≤2c′δ∫Ω|G(u)|2dx+δ(‖φ‖2+‖Δφ‖2)+c″≤2c′δ∫Ω|G(u)|2dx+δ(c‖∇φ‖2+‖Δφ‖2)+c″, | (4.2) |
where
E5=δ‖u‖2+‖∇u‖2+2∫ΩF(u)dx+‖φ‖2+2‖∇φ‖2+‖Δφ‖2 | (4.3) |
satisfies
E5≥c(‖u‖2H1(Ω)+‖u‖k+2k+2+‖φ‖2H2(Ω))−c′ | (4.4) |
and
E5≤c″(‖u‖2H1(Ω)+‖u‖k+2k+2+‖φ‖2H2(Ω))−c‴. | (4.5) |
From (4.2) and owing to (4.1), we obtain
dE5dt+2(c‖∇u‖2+δ∫ΩF(u)dx+‖∂u∂t‖2+‖∇φ‖2+‖Δφ‖2)≤Cϵ∫ΩF(u)dx+δ(c‖∇φ‖2+‖Δφ‖2)+C′ϵ, | (4.6) |
where
2δ≥Cϵand2>cδ, | (4.7) |
we then deduce from (4.6),
dE5dt+c(E5+‖∂u∂t‖2)≤c′, | (4.8) |
we complete the proof applying the Gronwall's lemma.
Remark 4.2. It follows from theorems 3.1, 3.2 and 4.1 that we can define the family solving operators:
S(t):Φ⟶Φ,(u0,φ0)↦(u(t),φ(t)),∀t≥0, | (4.9) |
where
The aim of this section is to study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist. This study is motivated by the possibility of extending results obtained above to the case of unbounded domains like semi-infinite cylinders. To do so, we will study the behavior of solutions in a semi-infinite cylinder denoted
u=φ=0on(0,+∞)×∂D×(0,T) | (5.1) |
and
u(0,x2,x3;t)=h(x2,x3;t),φ(0,x2,x3;t)=l(x2,x3;t)on{0}×D×(0,T), | (5.2) |
where
We also consider following initial data
u|t=0=φ|t=0=0onR. | (5.3) |
Let us suppose that such solutions exist. We consider the function
Fw(z,t)=∫t0∫D(z)e−ws(usu,1+φ(φ,1+φ,1s)+φsφ,1)dads, | (5.4) |
where
Fw(z+h,t)−Fw(z,t)=e−wt2∫R(z,z+h)(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dx+∫t0∫R(z,z+h)e−ws(|us|2+|∇φ|2+|Δφ|2)dxds+w2∫t0∫R(z,z+h)e−ws(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dxds, | (5.5) |
where
Hence,
∂Fw∂t(z,t)=e−wt2∫D(z)(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)da+∫t0∫D(z)e−ws(|us|2+|∇φ|2+|Δφ|2)dads+w2∫t0∫D(z)e−ws(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dads. | (5.6) |
We consider a second function, namely,
Gw(z,t)=∫t0∫D(z)e−ws(usu,1+φ(θ,1+φ,1s))dads, | (5.7) |
where
Similarly, we have
Gw(z+h,t)−Gw(z,t)=e−wt2∫R(z,z+h)(|u|2+|∇θ|2)dx+∫t0∫R(z,z+h)e−ws(|∇u|2+f(u)u+uΔφ+|φ|2+|∇φ|2)dxds+w2∫t0∫R(z,z+h)e−ws(|u|2+|∇θ|2)dxds+∫t0∫R(z,z+h)e−ws(G(u)−g(u)u)φdxds | (5.8) |
and then
∂Gw∂t(z,t)=e−wt2∫D(z)(|u|2+|∇θ|2)da+∫t0∫D(z)e−ws(|∇u|2+f(u)u+uΔφ+|φ|2+|∇φ|2)dads+w2∫t0∫D(z)e−ws(|u|2+|∇θ|2)dads+∫t0∫D(z)e−ws(G(u)−g(u)u)φdads. | (5.9) |
We choose
2F(u)+τu2≥C1u2,C1>0. | (5.10) |
Now, we focus on the nonliear part i.e.,
w(F(u)+τ2|u|2)+τf(u)u+τ(G(u)−g(u)u)φ+w2|φ|2. | (5.11) |
We assume that
For
w(F(u)+τ2|u|2)+τf(u)u+τ(G(u)−g(u)u)φ+w2|φ|2≥C3(|u|2+|φ|2+|Δφ|2). | (5.12) |
Taking into account previous choices, it clearly appears that the following function
Hw=Fw+τGw | (5.13) |
satisfies
∂Hw∂t(z,t)≥C4∫t0∫D(z)e−ws(|u|2+|∇u|2+|us|2+|φ|2+|∇φ|2+|Δφ|2+|∇θ|2)dads. | (5.14) |
We give now an estimate of
|Fw|≤(∫t0∫D(z)e−wsu2sdads)1/2(e−wsu2,1)1/2+(∫t0∫D(z)e−wsφ2dads)1/2(e−wsφ2,1)1/2+(∫t0∫D(z)e−wsφ2dads)1/2(e−wsφ2,1s)1/2+(∫t0∫D(z)e−wsφ2sdads)1/2(e−wsφ2,1)1/2≤C5∫t0∫D(z)e−ws(|∇u|2+|us|2+|φ|2+|∇φ|2+|φs|2+|∇φs|2)dads,C5>0. | (5.15) |
Similarly,
|Gw|≤(∫t0∫D(z)e−wsu2dads)1/2(∫t0∫D(z)e−wsu2,1dads)1/2+(∫t0∫D(z)e−wsφ2dads)1/2(∫t0∫D(z)e−wsθ2,1dads)1/2+(∫t0∫D(z)e−wsφ2sdads)1/2(∫t0∫D(z)e−wsφ2,1dads)1/2≤C6∫t0∫D(z)e−ws(|u|2+|∇u|2+|φ|2+|∇φ|2+|∇θ|2)dads,C6>0. | (5.16) |
We then deduce the existence of a positive constant
|Hw|≤C7∂Hw∂z. | (5.17) |
Remark 5.1. The inequality (5.17) is well known in the study of spatial estimates and leads to the Phragmén-Lindelöf alternative (see, e.g., [9], [39]).
In particular, if there exist
Hw(z,t)≥Hw(z0,t)eC−17(z−z0),z≥z0. | (5.18) |
The estimate (5.18) gives information in terms of measure defined in the cylinder. Actually, from (5.18), we deduce that
e−wt2∫R(0,z)(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dx+τe−wt2∫R(0,z)(|u|2+|∇θ|2)dx+∫t0∫R(0,z)e−ws(|us|2+|∇φ|2+|Δφ|2)dxds+τ∫t0∫R(0,z)e−ws(|∇u|2+f(u)u+g(u)uΔφ+|φ|2+2|∇φ|2)dxds+w2∫t0∫R(0,z)e−ws(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dxds+τw2∫t0∫R(0,z)e−ws(|u|2+|∇θ|2)dx+τ∫t0∫R(0,z)e−ws(G(u)−g(u)u)φdxds | (5.19) |
tends to infinity exponentially fast. On the other hand, if
−Hw(z,t)≤−Hw(0,t)eC−17z,z≥0, | (5.20) |
where
Ew(z,t)=e−wt2∫R(z)(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dx+τe−wt2∫R(z)(|u|2+|∇θ|2)dx+∫t0∫R(z)e−ws(|us|2+|∇φ|2+|Δφ|2)dxds+τ∫t0∫R(z)e−ws(|∇u|2+f(u)u+g(u)uΔφ+|φ|2+2|∇φ|2)dxds+w2∫t0∫R(z)e−ws(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dxds+τw2∫t0∫R(z)e−ws(|u|2+|∇θ|2)dx+τ∫t0∫R(z)e−ws(G(u)−g(u)u)φdxds | (5.21) |
and
Finally, setting
Ew(z,t)=12∫R(z)(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dx+τ12∫R(z)(|u|2+|∇θ|2)dx+∫t0∫R(z)(|us|2+|∇φ|2+|Δφ|2)dxds+τ∫t0∫R(z)(|∇u|2+f(u)u+g(u)uΔφ+|φ|2+2|∇φ|2)dxds+w2∫t0∫R(z)(|∇u|2+2F(u)+|φ|2+2|∇φ|2+|Δφ|2)dxds+τw2∫t0∫R(z)(|u|2+|∇θ|2)dx+τ∫t0∫R(z)(G(u)−g(u)u)φdxds. | (5.22) |
We have the following result
Theorem 5.2. Let
Ew(z,t)≤Ew(0,t)ewt−C−17z,z≥0, | (5.23) |
where the energy
The author would like to thank Alain Miranville for his advices and for his careful reading of this paper.
The author declares no conflicts of interest in this paper.
[1] |
S. Q. Deng, X. Yang, Y. Wei, J. T. Chen, X. J. Wang, H. J. Peng, A review on dengue vaccine development, Vaccines, 8 (2020), 63. https://doi.org/10.3390/vaccines8010063 doi: 10.3390/vaccines8010063
![]() |
[2] | S. Rajapakse, C. Rodrigo, A. Rajapakse, Treatment of dengue fever, Infect. Drug Resist., 5 (2012), 103–112. https://doi.org/10.2147/IDR.S22613 |
[3] | S. J. Thomas, D. Strickman, D. W. Vaughn, Dengue Epidemiology: Virus Epidemiology, Ecology, and Emergence, Adv. Virus Res., 61 (2003), 235–289. https://doi.org/10.1016/S0065-3527(03)61006-7 |
[4] |
N. L. Achee, F. Gould, T. A. Perkins, R. C. Reiner Jr, A. C. Morrison, S. A. Ritchie, et al., A critical assessment of vector control for dengue prevention, PLoS Negl.Trop. Dis., 9 (2015), e0003655. https://doi.org/10.1371/journal.pntd.0003655 doi: 10.1371/journal.pntd.0003655
![]() |
[5] | D. Pilger, M. De Maesschalck, O. Horstick, J. L. San Martin, Dengue outbreak response: documented effective interventions and evidence gaps, TropIKA.net, 1 (2010). |
[6] |
P. A. Reyes-Castro, L. Castro-Luque, R. Díaz-Caravantes, K. R. Walker, M. H. Hayden, K. C. Ernst, Outdoor spatial spraying against dengue: A false sense of security among inhabitants of Hermosillo, Mexico, PLoS Negl.Trop. Dis., 11 (2017), e0005611. https://doi.org/10.1371/journal.pntd.0005611 doi: 10.1371/journal.pntd.0005611
![]() |
[7] |
F. Espinoza-Gómez, C. M. Hernández-Suárez, R. Coll-Cárdenas, Educational campaign versus malathion spraying for the control of Aedes aegypti in Colima, Mexico, J. Epidemiol. Community Health, 56 (2002), 148–152. https://doi.org/10.1136/jech.56.2.148 doi: 10.1136/jech.56.2.148
![]() |
[8] |
N. Arunachalam, B. K. Tyagi, M. Samuel, R. Krishnamoorthi, R. Manavalan, S. C. Tewari, et al., Community-based control of Aedes aegypti by adoption of eco-health methods in Chennai City, India, Pathog. Global Health, 106 (2012), 488–496. https://doi.org/10.1179/2047773212Y.0000000056 doi: 10.1179/2047773212Y.0000000056
![]() |
[9] |
C. Aerts, M. Revilla, L. Duval, K. Paaijmans, J. Chandrabose, H. Cox, et al., Understanding the role of disease knowledge and risk perception in shaping preventive behavior for selected vector-borne diseases in Guyana, PLoS Negl. Trop. Dis., 14 (2020), e0008149. https://doi.org/10.1371/journal.pntd.0008149 doi: 10.1371/journal.pntd.0008149
![]() |
[10] |
N. Andersson, E. Nava-Aguilera, J. Arosteguí, A. Morales-Perez, H. Suazo-Laguna, J. Legorreta-Soberanis, et al., Evidence based community mobilization for dengue prevention in Nicaragua and Mexico (Camino Verde, the Green Way): cluster randomized controlled trial, BMJ, 351 (2015), h3267. https://doi.org/10.1136/bmj.h3267 doi: 10.1136/bmj.h3267
![]() |
[11] |
J. Arosteguí, R. J. Ledogar, J. Coloma, C. Hernández-Alvarez, H. Suazo-Laguna, A. Cárcamo, et al., The Camino Verde intervention in Nicaragua, 2004–2012, BMC Public Health, 17 (2017), 406. https://doi.org/10.1186/s12889-017-4299-3 doi: 10.1186/s12889-017-4299-3
![]() |
[12] |
S. Funk, M. Salathé, V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: a review, J. R. Soc. Interface, 7 (2010), 1247–1256. https://doi.org/10.1098/rsif.2010.0142 doi: 10.1098/rsif.2010.0142
![]() |
[13] | T. Berry, M. Ferrari, T. Sauer, S. J. Greybush, D. Ebeigbe, A. J. Whalen, et al., Stabilizing the return to normal behavior in an epidemic, medRxiv, 2023.03.13.23287222. |
[14] |
S. Del Valle, H. Hethcote, J. M. Hyman, C. Castillo-Chavez, Effects of behavioral changes in a smallpox attack model, Math. Biosci., 195 (2005), 228–251. https://doi.org/10.1016/j.mbs.2005.03.006 doi: 10.1016/j.mbs.2005.03.006
![]() |
[15] |
M. J. Ferrari, S. Bansal, L. A. Meyers, O. N. Bjørnstad, Network frailty and the geometry of herd immunity, Proc. R. Soc. B, 273 (2006), 2743–2748. https://doi.org/10.1098/rspb.2006.3636 doi: 10.1098/rspb.2006.3636
![]() |
[16] |
E. P. Fenichel, C. Castillo-Chavez, M. G. Ceddia, G. Chowell, P. A. Gonzalez Parra, G. J. Hickling, et al., Adaptive human behavior in epidemiological models, Proc. Natl. Acad. Sci. U.S.A., 108 (2011), 6306–6311. https://doi.org/10.1073/pnas.101125010 doi: 10.1073/pnas.101125010
![]() |
[17] |
L. LeJeune, N. Ghaffarzadegan, L. M. Childs, O. Saucedo, Mathematical analysis of simple behavioral epidemic models, Math. Biosci., 375 (2024), 109250. https://doi.org/10.1016/j.mbs.2024.109250 doi: 10.1016/j.mbs.2024.109250
![]() |
[18] |
C. Eksin, K. Paarporn, J. S. Weitz, Systematic biases in disease forecasting – The role of behavior change, Epidemics, 27 (2019), 96–105. https://doi.org/10.1016/j.epidem.2019.02.004 doi: 10.1016/j.epidem.2019.02.004
![]() |
[19] |
T. Boccia, M. N. Burattini, F. A. B. Coutinho, E. Massad, Will people change their vector-control practices in the presence of an imperfect dengue vaccine?, Epidemiol. Infect., 142 (2014), 625–633. https://doi.org/10.1017/S0950268813001350 doi: 10.1017/S0950268813001350
![]() |
[20] |
V. M. Alvarado-Castro, C. Vargas-De-León, S. Paredes-Solis, A. Li-Martin, E. Nava-Aguilera, A. Morales-Pérez, et al., The influence of gender and temephos exposure on community participation in dengue prevention: a compartmental mathematical model, BMC Infect. Dis., 24 (2024), 463. https://doi.org/10.1186/s12879-024-09341-w doi: 10.1186/s12879-024-09341-w
![]() |
[21] |
J. Jiao, G. P. Suarez, N. H. Fefferman, How public reaction to disease information across scales and the impacts of vector control methods influence disease prevalence and control efficacy, PLoS Comput. Biol., 17 (2021), e1008762. https://doi.org/10.1371/journal.pcbi.1008762 doi: 10.1371/journal.pcbi.1008762
![]() |
[22] |
K. Roosa, N. H. Fefferman, A general modeling framework for exploring the impact of individual concern and personal protection on vector-borne disease dynamics, Parasites Vectors, 15 (2022), 361. https://doi.org/10.1186/s13071-022-05481-7 doi: 10.1186/s13071-022-05481-7
![]() |
[23] |
J. M. Epstein, J. Parker, D. Cummings, R. A. Hammond, Coupled contagion dynamics of fear and disease: mathematical and computational explorations, PLoS One, 3 (2008), e3955. https://doi.org/10.1371/journal.pone.0003955 doi: 10.1371/journal.pone.0003955
![]() |
[24] |
K. Jain, V. Bhatnagar, S. Prasad, S. Kaur, Coupling fear and contagion for modeling epidemic dynamics, IEEE Trans. Network Sci. Eng., 10 (2023), 20–34. https://doi.org/10.1109/TNSE.2022.3187775 doi: 10.1109/TNSE.2022.3187775
![]() |
[25] |
J. M. Epstein, E. Hatna, J. Crodelle, Triple contagion: a two-fears epidemic model, J. R. Soc. Interface, 18 (2021), 20210186. https://doi.org/10.1098/rsif.2021.0186 doi: 10.1098/rsif.2021.0186
![]() |
[26] |
N. Perra, D. Balcan, B. Gonçalves, A. Vespignani, Towards a characterization of behavior-disease models, PLoS One, 6 (2011), e23084. https://doi.org/10.1371/journal.pone.0023084 doi: 10.1371/journal.pone.0023084
![]() |
[27] |
S. A. Pedro, F. T. Ndjomatchoua, P. Jentsch, J. M. Tchuenche, M. Anand, C. T. Bauch, Conditions for a second wave of COVID-19 due to interactions between disease dynamics and social processes, Front. Phys., 8 (2020), 574514. https://doi.org/10.3389/fphy.2020.574514 doi: 10.3389/fphy.2020.574514
![]() |
[28] |
A. Bernardin, A. J. Martínez, T. Perez-Acle, On the effectiveness of communication strategies as non-pharmaceutical interventions to tackle epidemics, PLoS One, 16 (2021), e0257995. https://doi.org/10.1371/journal.pone.0257995 doi: 10.1371/journal.pone.0257995
![]() |
[29] | Mathematica 14.0, 2024. Available from: http://www.wolfram.com. |
[30] | M. M. Andersen, S. Højsgaard, caracas: Computer Algebra, 2023. Available from: https://github.com/r-cas/caracas. |
[31] |
P. Driessche, J. Watmough, Further notes on the basic reproduction number, Math. Epidemiol., 2008 (2008), 159–178. https://doi.org/10.1007/978-3-540-78911-6_6 doi: 10.1007/978-3-540-78911-6_6
![]() |
[32] |
A. B. Sabin, Research on dengue during World War II, Am. J. Trop. Med. Hyg., 1 (1952), 30–50. https://doi.org/10.4269/ajtmh.1952.1.30 doi: 10.4269/ajtmh.1952.1.30
![]() |
[33] |
C. Probst, T. M. Vu, J. M. Epstein, A. E. Nielsen, C. Buckley, A. Brennan, et al., The normative underpinnings of population-level alcohol use: An individual-level simulation model, Health Educ. Behav., 47 (2020), 224–234. https://doi.org/10.1177/1090198119880545 doi: 10.1177/1090198119880545
![]() |
[34] | J. M. Epstein, Agent_Zero, Princeton University Press, 2014. |
[35] | T. M. Vu, C. Probst, J. M. Epstein, A. Brennan, M. Strong, R. C. Purshouse, Toward inverse generative social science using multi-objective genetic programming, in Proceedings of the Genetic and Evolutionary Computation Conference, ACM, Prague Czech Republic, (2019), 1356–1363. |
[36] |
L. Cattarino, I. Rodriguez-Barraquer, N. Imai, D. A. T. Cummings, N. M. Ferguson, Mapping global variation in dengue transmission intensity, Sci. Transl. Med., 12 (2020), eaax4144. https://doi.org/10.1126/scitranslmed.aax4144 doi: 10.1126/scitranslmed.aax4144
![]() |
[37] |
Y. Liu, K. Lillepold, J. C. Semenza, Y. Tozan, M. B. M. Quam, J. Rocklöv, Reviewing estimates of the basic reproduction number for dengue, Zika and chikungunya across global climate zones, Environ. Res., 182 (2020), 109114. https://doi.org/10.1016/j.envres.2020.109114 doi: 10.1016/j.envres.2020.109114
![]() |
[38] |
A. C. Morrison, R. C. Reiner, W. H. Elson, H. Astete, C. Guevara, C. del Aguila, et al., Efficacy of a spatial repellent for control of Aedes-borne virus transmission: A cluster-randomized trial in Iquitos, Peru, Proc. Natl. Acad. Sci., 119 (2022), e2118283119. https://doi.org/10.1073/pnas.2118283119 doi: 10.1073/pnas.2118283119
![]() |
[39] |
H. J. Wearing, P. Rohani, Ecological and immunological determinants of dengue epidemics, Proc. Natl. Acad. Sci., 103 (2006), 11802–11807. https://doi.org/10.1073/pnas.0602960103 doi: 10.1073/pnas.0602960103
![]() |
[40] |
N. G. Reich, S. Shrestha, A. A. King, P. Rohani, J. Lessler, S. Kalayanarooj, et al., Interactions between serotypes of dengue highlight epidemiological impact of cross-immunity, J. R. Soc. Interface, 10 (2013), 20130414. https://doi.org/10.1098/rsif.2013.0414 doi: 10.1098/rsif.2013.0414
![]() |
[41] |
C. B. F. Vogels, C. Rückert, S. M. Cavany, T. A. Perkins, G. D. Ebel, N. D. Grubaugh, Arbovirus coinfection and co-transmission: A neglected public health concern?, PLoS Biol., 17 (2019), e3000130. https://doi.org/10.1371/journal.pbio.3000130 doi: 10.1371/journal.pbio.3000130
![]() |
[42] |
M. Chan, M. A. Johansson, The incubation periods of dengue viruses, PLoS One, 7 (2012), e50972. https://doi.org/10.1371/journal.pone.0050972 doi: 10.1371/journal.pone.0050972
![]() |
[43] |
Q. A. Ten Bosch, J. M. Wagman, F. Castro-Llanos, N. L. Achee, J. P. Grieco, T. A. Perkins, Community-level impacts of spatial repellents for control of diseases vectored by Aedes aegypti mosquitoes, PLoS Comput. Biol., 16 (2020), e1008190. https://doi.org/10.1371/journal.pcbi.1008190 doi: 10.1371/journal.pcbi.1008190
![]() |
[44] |
L. C. Harrington, J. P. Buonaccorsi, J. D. Edman, A. Costero, P. Kittayapong, G. G. Clark, et al., Analysis of survival of young and old Aedes aegypti (Diptera: Culicidae) from Puerto Rico and Thailand, J. Med. Entomol., 38 (2001), 537–547. https://doi.org/10.1603/0022-2585-38.4.537 doi: 10.1603/0022-2585-38.4.537
![]() |
[45] |
T. W. Scott, P. H. Amerasinghe, A. C. Morrison, L. H. Lorenz, G. G. Clark, D. Strickman, et al., Longitudinal studies of Aedes aegypti (Diptera: Culicidae) in Thailand and Puerto Rico: blood feeding frequency, J. Med. Entomol., 37 (2000), 89–101. https://doi.org/10.1603/0022-2585-37.1.89 doi: 10.1603/0022-2585-37.1.89
![]() |
[46] |
D. L. Smith, K. E. Battle, S. I. Hay, C. M. Barker, T. W. Scott, F. E. McKenzie, Ross, Macdonald, and a theory for the dynamics and control of mosquito-transmitted pathogens, PLoS Pathogens, 8 (2012), e1002588. https://doi.org/10.1371/journal.ppat.1002588 doi: 10.1371/journal.ppat.1002588
![]() |
[47] | R Core Team, R: A Language and Environment for Statistical Computing, 2021. Available from: https://www.R-project.org/. |
[48] |
K. Soetaert, T. Petzoldt, R. W. Setzer, Solving differential equations in R: package deSolve, J. Stat. Software, 33 (2010), 1–25. https://doi.org/10.18637/jss.v033.i09 doi: 10.18637/jss.v033.i09
![]() |
[49] |
A. Puy, S. Lo Piano, A. Saltelli, S. A. Levin, Sensobol: An R package to compute variance-based sensitivity indices, J. Stat. Software, 102 (2022), 1–37. https://doi.org/10.18637/jss.v102.i05 doi: 10.18637/jss.v102.i05
![]() |
[50] | R. M. Anderson, R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, New York, 1992. |
[51] | M. Martcheva, Introduction to epidemic modeling, in An Introduction to Mathematical Epidemiology (ed. M. Martcheva), Springer US, Boston, MA, (2015), 9–31. https://doi.org/10.1007/978-1-4899-7612-3_2 |
[52] | F. Brauer, C. Castillo-Chavez, Z. Feng, Endemic disease models, Math. Models Epidemiol., 69 (2019), 63–116. https://doi.org/10.2478/acph-2019-0006 |
[53] |
R. C. Reiner, S. T. Stoddard, B. M. Forshey, A. A. King, A. M. Ellis, A. L. Lloyd, et al., Time-varying, serotype-specific force of infection of dengue virus, Proc. Natl. Acad. Sci., 111 (2014), E2694–E2702. https://doi.org/10.1073/pnas.13149331 doi: 10.1073/pnas.13149331
![]() |
[54] | M. Ryan, E. Brindal, M. Roberts, R. I. Hickson, A behaviour and disease transmission model: incorporating the Health Belief Model for human behaviour into a simple transmission model, J. R. Soc. Interface, 21 (215), 20240038. https://doi.org/10.1098/rsif.2024.0038 |
[55] |
J. Cascante-Vega, S. Torres-Florez, J. Cordovez, M. Santos-Vega, How disease risk awareness modulates transmission: coupling infectious disease models with behavioural dynamics, R. Soc. Open Sci., 9 (2022), 210803. https://doi.org/10.1098/rsos.210803 doi: 10.1098/rsos.210803
![]() |
[56] |
V. Vanlerberghe, M. E. Toledo, M. Rodriguez, D. Gomez, A. Baly, J. R. Benitez, et al., Community involvement in dengue vector control: cluster randomised trial, BMJ, 338 (2009), b1959–b1959. https://doi.org/10.1136/bmj.b1959 doi: 10.1136/bmj.b1959
![]() |
[57] |
J. Quintero, N. R. Pulido, J. Logan, T. Ant, J. Bruce, G. Carrasquilla, Effectiveness of an intervention for Aedes aegypti control scaled-up under an inter-sectoral approach in a Colombian city hyper-endemic for dengue virus, PLoS One, 15 (2020), e0230486. https://doi.org/10.1371/journal.pone.0230486 doi: 10.1371/journal.pone.0230486
![]() |
[58] |
J. Raude, K. MCColl, C. Flamand, T. Apostolidis, Understanding health behaviour changes in response to outbreaks: Findings from a longitudinal study of a large epidemic of mosquito-borne disease, Soc. Sci. Med., 230 (2019), 184–193. https://doi.org/10.1016/j.socscimed.2019.04.009 doi: 10.1016/j.socscimed.2019.04.009
![]() |
[59] |
L. S. Lloyd, P. Winch, J. Ortega-Canto, C. Kendall, The design of a community-based health education intervention for the control of Aedes aegypti, Am. J. Trop. Med. Hyg., 50 (1994), 401–411. https://doi.org/10.4269/ajtmh.1994.50.401 doi: 10.4269/ajtmh.1994.50.401
![]() |
[60] |
A. Caprara, J. W. D. O. Lima, A. C. R. Peixoto, C. M. V. Motta, J. M. S. Nobre, J. Sommerfeld, et al., Entomological impact and social participation in dengue control: a cluster randomized trial in Fortaleza, Brazil, Trans. R. Soc. Trop. Med. Hyg., 109 (2015), 99–105. https://doi.org/10.1093/trstmh/tru187 doi: 10.1093/trstmh/tru187
![]() |
[61] |
A. M. Buttenheim, V. Paz-Soldan, C. Barbu, C. Skovira, J. Q. Calderón, L. M. M. Riveros, et al., Is participation contagious? Evidence from a household vector control campaign in urban Peru, J Epidemiol. Community Health, 68 (2014), 103–109. https://doi.org/10.1136/jech-2013-202661 doi: 10.1136/jech-2013-202661
![]() |
[62] |
J. Bedson, L. A. Skrip, D. Pedi, S. Abramowitz, S. Carter, M. F. Jalloh, et al., A review and agenda for integrated disease models including social and behavioural factors, Nat. Hum. Behav., 5 (2021), 834–846. https://doi.org/10.1038/s41562-021-01136-2 doi: 10.1038/s41562-021-01136-2
![]() |
[63] |
K. Magori, M. Legros, M. E. Puente, D. A. Focks, T. W. Scott, A. L. Lloyd, et al., Skeeter Buster: a stochastic, spatially explicit modeling tool for studying Aedes aegypti population replacement and population suppression strategies, PLoS Negl. Trop. Dis., 3 (2009), e508. https://doi.org/10.1371/journal.pntd.0000508 doi: 10.1371/journal.pntd.0000508
![]() |
[64] |
E. L. Davis, T. D. Hollingsworth, M. J. Keeling, An analytically tractable, age-structured model of the impact of vector control on mosquito-transmitted infections, PLoS Comput. Biol., 20 (2024), e1011440. https://doi.org/10.1371/journal.pcbi.1011440 doi: 10.1371/journal.pcbi.1011440
![]() |
[65] |
M. Predescu, G. Sirbu, R. Levins, T. Awerbuch-Friedlander, On the dynamics of a deterministic and stochastic model for mosquito control, Appl. Math. Lett., 20 (2007), 919–925. https://doi.org/10.1016/j.aml.2006.12.001 doi: 10.1016/j.aml.2006.12.001
![]() |
[66] |
J. Elsinga, H. T. Van Der Veen, I. Gerstenbluth, J. G. M. Burgerhof, A. Dijkstra, M. P. Grobusch, et al., Community participation in mosquito breeding site control: an interdisciplinary mixed methods study in Curaçao, Parasites Vectors, 10 (2017), 434. https://doi.org/10.1186/s13071-017-2371-6 doi: 10.1186/s13071-017-2371-6
![]() |
[67] |
A. N. Rakhmani, Y. Limpanont, J. Kaewkungwal, K. Okanurak, Factors associated with dengue prevention behaviour in Lowokwaru, Malang, Indonesia: a cross-sectional study, BMC Public Health, 18 (2018), 619. https://doi.org/10.1186/s12889-018-5553-z doi: 10.1186/s12889-018-5553-z
![]() |
[68] |
A. J. Mackay, M. Amador, A. Diaz, J. Smith, R. Barrera, Dynamics of Aedes aegypti and Culex quinquefasciatus in septic tanks, J. Am. Mosq. Control Assoc., 25 (2009), 409–416. https://doi.org/10.2987/09-5888.1 doi: 10.2987/09-5888.1
![]() |