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The principle aim of this work is to simulate the invasion of two invasive mosquito species Aedes aegypti and Aedes albopictus in central Europe at a landscape scale. The spatial-temporal dynamics of invasion is investigated in dependence of predation pressure, seasonal variation of ambient temperature as well as human population density. The introduction of temperature dependent entomological parameters enables the simulation of seasonal pattern of population dynamics. The influence of temperature, predation pressure and human population density on invasion is studied in one-dimensional cases. In two dimensions, georeferenced parameters such as annual mean temperature and human population density are prepared by a geographical information system and introduced into the finite element tool COMSOL Multiphysics. The results show that under the current temperature, central Europe cannot become a permanent breeding region for Aedes aegypti . However, southwest Germany especially the regions along the Upper Rhine Valley may provide suitable habitats for the permanent establishment of Aedes albopictus . An annual temperature rise of two degrees would lead to dramatic increase of invasion speed and extension range of Aedes albopictus .

Vector-borne disease dengue fever is mainly transmitted by two invasive mosquito species: Aedes (Stegomyia) aegypti (L.) and Aedes (Stegomyia) albopictus (Skuse) [

The primary vector of dengue fever Aedes aegypti is also known as the yellow fever mosquito, since it is well-known for the transmission of yellow fever virus. It came originally from Africa, but now it distributes widely in many tropical and subtropical regions around the world. Its domestic form feeds almost solely on human blood [

Another important vector of dengue fever is Aedes albopictus. It came originally from Asia, and is also known as Asian tiger mosquito. It belongs to the top 100 invasive species of the world [

The widely geographical distribution of these two mosquito species poses big threats to the public health worldwide. Even more concerns have been aroused nowadays, since these two species may extend their expansion range to less favorable regions as a result of climate change [

Since there is no dengue vaccine, and no commercial chikungunya vaccine is available [

Several studies have already been carried out. Takahashi et al. [

Most of the above-mentioned mechanistic models were built in one-dimensional case. Variables like temperature were considered as spatially homogeneous. Moreover, many entomological parameters were assumed as temperature independent. Additionally, many models are not suitable for application in temperate regions, since the seasonal temperature variation and overwintering strategy of mosquitoes were not taken into account, which may play decisive roles for the permanent establishment of mosquito in these regions.

The aim of the present work is to simulate the invasion of Aedes agypti and Aedes albopictus in central Europe. The model system employed here is based on the compartment model of Richter et al. [

The invasion of both Aedes aegypti and Aedes albopictus was simulated in one-dimensional cases in Wolfram Mathematica 8.0 at first. The system of reaction-diffusion equations was solved by the numerical methods of lines. Based on the work of Richter et al. [^{2}. These temperature data were then interpolated into COMSOL environment using the two-dimensional linear interpolation option in the interpolation menu. Geo-referenced human population density data were obtained from DLM250 (digital landscape models at a scale of 1:250,000) of ATKIS (Amtliches Topographisch-Kartographisches Information System) of year 2007. Landscape structures were used to control the sizes of finite element grids. They were converted from shape files to CAD files by using the “Export to CAD” tool from ArcToolbox in ArcGIS. The CAD files were then imported into the COMSOL Multiphysics.

The system of reaction diffusion equations is applied in the present work. The general form of reaction diffusion equations is as follows

∂ u i ∂ t = L [ u ] + f i ( u 1 , ⋯ , u n ) (1)

In the present study, the spatial operator has the simple form

L [ u ] = ∇ ( D ∇ u ) (2)

The system of equations is presented as follows:

∂ A ∂ t = f ( T ) ϕ ( 1 − A C ( P ) ) M − ( γ ( T ) + μ A ( T ) ) A − β A A + K s f ( T ) ∂ M ∂ t = ∇ ⋅ ( D ∇ M − v M ) + r γ ( T ) A − μ M ( T ) M (3)

where A is the population density of aquatic phase, M is the population density of winged phase (adult females); Φ is the intrinsic oviposition rate; C(P) is the environmental capacity of aquatic phase, which is defined as a function of human population density P as both species are highly domestic and the available breeding sites through the use of artificial containers are closely related to P; f(T), γ(T), μA(T), μM(T) are temperature response function of oviposition rate, emergence rate, mortality rate of aquatic phase and mortality rate of winged phase, respectively; β is the predation rate, whereas K_{s} is the saturation constant of predators; D and v are the diffusion coefficient and advection coefficient of winged phase respectively; r is the proportion of females.

In the model system the life stages of mosquito are divided into an aquatic phase (egg, larva and pupa) and a winged (adult) phase. It is assumed that all the eggs laid by females can survive and hatch to larvae. The sex ratio of mosquitoes is assumed to be 1:1 throughout their whole life stage i.e. r = 0.5, which is plausible as the study of Delatte et al. [

In the work of Maidana et al. [

C ( P ) = C 0 + k P (4)

where coefficient C_{0} can be interpreted as the environmental capacity of aquatic phase in an environment without human population. In a peridomestic environment, the increase of breeding sites through the use of artificial containers by adult females for oviposition is considered as a number k of larvae per human. k is closely related to the Breteau Index, i.e. the number of breeding sites per one hundred houses. In the following simulations, k will be set as 3, which corresponds to a high Breteau Index [

The entomological data applied in the present work come from the temperature-controlled experiment of Yang et al. [

To describe the temperature response of oviposition rate f(T), the O’Neil equation with biological parameters T_{opt} (optimum temperature), T_{max} (lethal temperature) and Q_{10} is applied.

f ( T ) = ( T max − T T max − T opt ) p exp ( p ( T − T opt ) T max − T opt ) (5)

with

p = 1 400 W 2 [ 1 + 1 + 40 W ] 2 (6)

and

W = ( Q 10 − 1 ) ( T max − T opt ) (7)

According to the experimental data, T_{max} and T_{opt} are fixed as 39˚C and 31˚C, respectively. The data and temperature response curve of oviposition rate is shown in

The data and fitting curve of transition rate from aquatic phase to winged phase are shown in

γ ( T ) = n 0 [ 1 − exp ( − ( T T 1 ) r 1 ) ] exp [ − ( T T 2 ) r 2 ] (8)

Equation (9) is applied as the temperature response function of mortality rate of both aquatic phase and winged phase. The data and fitting curves are shown in

μ ( T ) = 1 − [ 1 − exp ( − ( T T 1 ) r 1 ) ] exp [ − ( T T 2 ) r 2 ] (9)

Based on the work of Richter et al. [

M = r γ A μ M (10)

By inserting Equation (10) into the right hand side of the first equation of Equation (3), the stationary solutions of aquatic phase density A can be obtained.

G ( A ) = f ( T ) ϕ ( 1 − A C ( P ) ) r γ ( T ) A μ M ( T ) − ( γ ( T ) + μ A ( T ) ) A − β A A + K s f ( T ) = 0 (11)

where G(A) is the growth rate of aquatic phase density.

Equation (11) has three stationary solutions:

A S 1 = 0 (12)

the minimal viable aquatic phase density:

A s 2 = − 1 2 r γ ( T ) ϕ f ( T ) ⋅ [ C ( P ) ) γ ( T ) μ M ( T ) + C ( P ) μ A ( T ) μ M ( T ) − C ( P ) r γ ( T ) ϕ f ( T ) + K s r γ ( T ) ϕ f ( T ) − W ] (13)

and the maximum aquatic phase density:

A s 3 = − 1 2 r γ ( T ) ϕ f ( T ) ⋅ [ C ( P ) γ ( T ) μ M ( T ) + C ( P ) μ A ( T ) μ M ( T ) − C ( P ) r γ ( T ) ϕ f ( T ) + K s r γ ( T ) ϕ f ( T ) + W ] (14)

with

The predation term causes a strong Allee effect, i.e. the growth rate is negative when the population density lies below a critical value [

The criterion of stability λ = ∂ G ( A ) ∂ A | A = A stationary [_{s}_{3} are stable, whereas state A = A_{s}_{2} is unstable.

The stationary solutions are governed by temperature T, predation pressure β and human population density P. All these three parameters determine the regions of viability. In the case of

higher than β1. If predation pressure is larger than β2, the Allee effect is so strong that it leads to only one stable state i.e. zero state. If the predation pressure lies between the two bifurcation points, there are two stable states which are separated by an unstable state (the dashed line). In _{1} and T_{2}. A region of viability exists between T_{1} and T_{2}. In

In a real environment mosquitoes are exposed to seasonal temperature variations. The population dynamics of mosquitoes as well as the transmission rate of Dengue fever fluctuate in different seasons significantly. The winter temperature is a decisive factor determining the permanent breeding zones of Aedes aegypti and Aedes albopictus in a region. Unlike Aedes albopictus, Aedes aegypti cannot produce winter diapausing eggs, hence its main distribution regions are limited in tropical and subtropical regions since it is not able to resist the cold winter temperatures in temperate regions. Aedes albopictus however, can survive the winter temperature in many temperate regions in the form of diapausing eggs. In Europe, it has already established itself stably in many regions of southern Europe, and poses recently a great threat to central Europe. Therefore, the seasonality is introduced to the model system. At the mean time the mechanism of winter diapause of Aedes albopictus will be modeled as the primary difference to Aedes aegypti.

To describe the seasonal temperature variation a simple sine-function (Equation (16)) is applied, where T_{m} and T_{r} are annual mean temperature and daily mean temperature range, respectively (see _{0} determines the phase of the sinusoid. If t_{0} is set as 107, then the annual minimum temperature comes on day 16 (January 16), whereas the highest temperature occurs on day 198 (July 16).

T ( t ) = T m + T r sin ( 2 π t − t 0 365 ) (16)

Aedes aegypti

_{0} (day^{−1}). The temperature variation is shown in

mean temperature range of 6˚C. In

Aedes albopictus

Winter diapause of Aedes albopictus is mainly governed by temperature and photoperiod. According to the study of the seasonal dynamics of hatching and oviposition rate of Aedes albopictus in Rome [

W P ( T ) = 1 1 + e − k 0 ( T − T 0 ) (17)

∂ A ∂ t = f ( T ) ϕ ( 1 − A C ( P ) ) M − W P ( T ) ( γ ( T ) + μ A ( T ) ) A − β A A + K s f ( T ) W P ( T ) ∂ M ∂ t = ∇ ⋅ ( D ∇ M − v M ) + W P ( T ) r γ ( T ) A − μ M ( T ) M (18)

A compare of the observation data and simulation results is made in order to evaluate the plausibility of the model. According to the study of [

To simulate the population dynamics of Aedes albopictus in Rome, a temperature variation curve was obtained by fitting the temperature function i.e. Equation (16) to the temperature data provided by the Ufficio Centrale di Ecologia Agraria [_{0} (day^{−1}) leads to the dying out of this species. Hence a relative low predation pressure of 0.1 C_{0} (day^{−1}) was given. This is reasonable, since in a densely populated city like Rome the predation pressure is relative low.

According to the simulation results, the winged phase is active since around day 100 (early April), and its density peaks in week 34. During the winter, a small population of winged phase stays active until the beginning of January. The simulation results described above fit well with the observation data of [

Based on the simulations in one dimension, the dispersal of both mosquito species are simulated at a landscape scale. Since temporary breeders like Aedes aegypti and Aedes albopictus can adapt well to the peridomestic environment, where predator pressure are much lower than that in a natural habitat, Equation (19) is applied to distinguish the significantly different predator pressures between urban areas and natural regions [

β ( x , y ) = β max exp ( − ( P ( x , y ) P t r ) n ) (19)

By using the parameters listed in

In

Parameter | Standard value | Dimension |
---|---|---|

β_{max} | 700 | 1/(km^{2}・day) |

Ks | 10 | 1/km^{2} |

P_{tr} | 500 | 1/km^{2} |

n | 4 | 1 |

D_{0} | 0.083 | km^{2}/day |

C_{0} | 1000 | 1/km^{2} |

regions. The most abundant population densities of both phases are found in middle August (see

Similar as adults Aedes aegypti, adults Aedes albopictus have almost died out since the middle of November (see

In

Belonging to the top 100 invasive species of the world, Aedes albopictus has successfully invaded the southern Europe during the last decades. The distribution of this species in Europe may be altered dramatically in the future as a result of the climate change. Reliable regional climate change projections for Europe are available in the IPCC’s Fourth Assessment Report (AR4), which are based on variety of simulations under different emissions scenarios. Under the A1B scenario, an annual mean warming for 2080 to 2090 (with respect to 1980 to 1990) varies from 2.3˚C to 5.3˚C in the northern Europe and 2.2˚C to 5.1˚C in the southern Europe and Mediterranean region [

In the following scenarios the range expansion of Aedes albopictus will be simulated under current temperature as well as a mean temperature rise of 2˚C. The population is introduced in Frankfurt during middle April. A totally dispersal of 20 years of this species was simulated. The range expansion of Aedes albopictus under current temperature and under a temperature increase of two degrees were demonstrated in

Comparing these two figures one can find that under a temperature rise of 2˚C Aedes albopictus can spread much faster. For example, 10 years after introduction the expansion range of this species under a temperature increase of two degrees is considerably larger than that of the species under the current temperature (

As shown in

Under a higher temperature Aedes albopictus can establish itself more abundantly. Additionally, the increase of temperature can reduce the minimum viable population density for this species. As a consequence, the regions, which are not favorable for this species under the current temperature, can become suitable areas as the mean temperature rises.

In the present work, the spatial-temporal population dynamics of two invasive mosquito species, Aedes aegypti and Aedes albopictus, were qualitatively and quantitatively analyzed by using compartment models based on biology and habitat preference of both species. As specialties, the Allee effect, seasonal variation of ambient temperature, dependency of predation pressure on the human population density as well as winter diapause as the overwintering strategy of

Aedes albopictus were considered in the simulation. Moreover, the range expansions of both mosquito species in central Europe were simulated at a landscape scale. The effect of temperature rise relating to global warming on the range expansion of Aedes albopictus was also simulated.

The main conclusions of the present study are listed as follows.

1) Since both Aedes agypti and Aedes albopictus are highly domestic, human population density can affect the invasion speed as well as expansion range of both species. Densely populated urban areas provide not only abundant breeding sites, but also safe habitats with low predator pressure for both mosquito species.

2) Introduction of seasonal temperature variation is vital for the simulation of permanent establishment of both mosquito species in a region. Minimum winter temperature and predation pressure are two primary limiting factors for the permanent breeding of Aedes aegypti in temperate regions, whereas predation pressure and duration of unfavorable temperature after winter diapause determine the successful overwintering of Aedes albopictus in temperate regions.

3) Central Europe is not likely to become a permanent breeding region for Aedes aegypti, since this species is not able to survive the winter temperatures in this region. However, some densely populated urban areas like Frankfurt may become its “temporary regions”, when it is introduced during months with favorable temperature.

4) Under the current temperature, southwestern Germany, especially the areas along the Upper Rhine Valley, can provide suitable habitats for the permanent establishment of Aedes albopictus. It can also survive the winter temperatures in this region in the form of diapausing eggs. Hence persistent surveillance and control of Aedes albopictus in this area are necessary.

5) Ambient temperature exerts a significant influence on the propagation of Aedes albopictus in central Europe [

The model employed in the present work is based on the biology as well as ecological niche of both mosquito species. However, related to both aspects there are some limitations in the model as a result of the shortage of available data and insufficient research.

One of the major objectives of this work was to simulate the invasion of Aedes aegypti and Aedes albopictus in central Europe. Since there are abundant variations in genes, morphology and ecology of both species around the world [

Besides studies focusing on entomological parameters, more quantitative studies or surveillance programs on habitat preferences and biological invasions of both mosquito species should be performed. Some parameters in the model system like environmental capacity, predation pressure or diffusion coefficient are dependent on many factors, which makes direct measurements very difficult. So it would be more practical if these parameters were indirectly estimated through model calibration. For example it would be quite meaningful if the present model was first applied to simulate the range expansion of Aedes albopictus in Italy. This species has invaded Italy since 1990, and has now spread to most areas of the country with an altitude below 600 m [

In the present work the influence of microclimate on the invasion and permanent establishment of both species was not considered in the simulation. It is very likely that Aedes aegypti will be able to find necessary protections against winter weather (for example, through breeding in artificial containers) in some cities in North America [

We thank Prof. Dr. Hyun Mo Yang for his kind permission to use the entomological data.

He, W. and Richter, O. (2018) Modelling Large Scale Invasion of Aedes aegypti and Aedes albopictus Mosquitoes. Advances in Pure Mathematics, 8, 245-265. https://doi.org/10.4236/apm.2018.83013