Can a patchy model describe the potential spread of West Nile virus in Germany?
Suman Bhowmick, Jörn Gethmann, Igor M. Sokolov, Franz J. Conraths, Hartmut H. K. Lentz
CContents1 Introduction 32 Model Formulation 6 R of the patchy model . . . 15 a r X i v : . [ phy s i c s . b i o - ph ] J a n an a patchy model describe the potential spread ofWest Nile virus in Germany? Suman Bhowmick b,1 , J¨orn Gethmann a , Igor M. Sokolov b,c , Franz J.Conraths a , Hartmut H. K. Lentz a a Friedrich-Loeffler-InstitutInstitute of EpidemiologyS¨udufer 10, 17493 Greifswald, Germany b Institute of Physics, Humboldt University of BerlinNewtonstraße 15, 12489 Berlin c IRIS Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany
Abstract
In 2018, West Nile Virus (WNV) was detected for the first time in Germany.Since the first detection, 36 human cases and 175 cases in horses and birdsare detected. The transmission cycle of West Nile Virus includes birds andmosquitoes and – as dead-end hosts – people and horses. Spatial disseminationof the disease is caused by the movements of birds and mosquitoes. While theactivity and movement of mosquitoes are depending mainly on temperature,in the birds there is a complex movement pattern caused by local birds andlong range dispersal birds. To this end, we have developed a metapopulationnetwork model framework to delineate the potential spatial distribution andspread of WNV across Germany as well as to evaluate the risk throughout ourproposed network model. Our model facilitates the interconnection amongst thevector, local birds and long range dispersal birds contact networks. We haveassumed different distance dispersal kernels models for the vector and avianpopulations with the intention to include short and long range dispersal. Themodel includes spatial variation of mosquito abundance and the movements toresemble the reality.
Keywords:
SEIR, Network, Metapopulation, WNV, Spatial Corresponding author
Preprint submitted to Journal of L A TEX Templates January 28, 2021 . Introduction
In August 2018, West Nile Virus (WNV), was detected for the first timein Germany [1]. Genetic characterisation indicates, that the German cases arein the same Central European subclade as cases found in the Czech Republicand Austria [1]. Since the first detection, 36 human cases and 175 cases inhorses and birds are detected. The disease pattern is mainly focussed on theeastern part of Germany. The disease is maintained by an enzootic transmissioncycle between birds and mosquitoes. Humans and horses can get infected butthey will not spread the disease [2]. The spread of mosquito-borne infectiousdiseases is a spatio-temporal dynamic process that is being affected by multipleagents such as vector and host movements, pathogen transmission heterogeneity,environmental factors etc [3, 4]. Previous studies have revealed that increase inthe temperature and the host-vector mobility facilitate the spread of mosquito-borne diseases [5, 6]. According to [1], the summer of 2018 have provided thefavourable climatic conditions for the potential geographical spread of zoonoticarthropod-borne WNV in Germany and possibly it have been introduced bythe wild birds as they can act as amplifying hosts. A previous study on theWNV outbreak in the United States [7] has listed the likely pathways by whichWNV has spread as migratory birds, dispersal of nonmigratory birds or longrange birds, movement of mosquitoes by flight or wind, and human transportof mosquitoes, birds, or other animals. While [8, 9] state out that migratory orthe long range dispersal birds may play the significant role as a spreader to newregions along their major flyways across the globe, [7] concludes that despitethe fact that many studies have carried out, there is still no evidence for thishypothesis.In general, there are two migrating seasons for the long dispersal birds, springand autumn [10]. In the spring migration season birds are migrating from thesouth northwards to Germany for breeding. Once the birds arrived in Germany,they will breed and stay for summer in the same local area. In autumn, they3eave Germany and migrate into the south. On their way from the south, theymight get infected and introduce the virus into Germany. While they are inGermany, the temperature might be suitable for disease transmission [11, 12].On the other hand, in autumn, birds migrating from the North and East tooverwinter in Germany. These birds play a minor or no role in WNV transmis-sion as temperature in winter is not suitable for transmission.Another potential route of introduction are local birds, or vectors, spreadingthe virus on a local area. In 2017, [13] shows that they detect WNV positivevectors in the Czech republic, other cases were found in Hungary, and Austria[14, 15, 16].Other studies [17, 18] have included the importance of host movement toanalyse the infection transmission through spatial host networks in the hetero-geneous environment. In [19], the authors quantify the importance of movementof livestock and the dispersal of vector in the disease transmission.After the first introduction of WNV to birds and equines in Germany, thecases of WNV has increased in the following season [20]. The activity of WNV isdetected in the Eastern part to the Northern zone of Germany. The combinationof phylogenetic analysis and the wide distribution of WNV in Germany fromnorth to the south reveals that WNV may have been introduced to Germanyfrom Czech Republic already before 2018 [1]. These findings demonstrate thatthere is a further risk of potential spatial transmission of WNV in Germany inpossible with some additional cases in the mammals and in the birds.In our current work we are interested in the description of the spatial spreadof WNV in Germany under the effect of host (bird) and vector (mosquito)movements, including seasonality. Once WNV is introduced into Germany, themain actors in the local and spatial transmission are local birds and vectors. Inthis study, we assume, that migratory or the long distance dispersal birds thatsettle down for breeding will have a similar behaviour as local birds.Geographical and population movements are essential in the context of spa-tial transmission. There are several approaches to model the spatial transmis-sion of geographical and population movements. Partial differential equation4PDE)[21, 22, 23] are one of the choice for that. However, while modellingthe geo-spatial dissemination of disease, usually there is a separation betweenthe diffusion and the dispersal models are being made. In diffusion model, thetransmission occurs immediately to the neighbouring zones but in patchy envi-ronment, this kind of modelling assumptions are not preferable.[24] describeda general approach, how to model the spatial transmission of WNV by a dif-fusion model. Moreover, given the extent of spatial spread of WNV acrossGermany possibly due to the migratory or long range dispersal birds, distancebased dispersal models seem to be more applicable. While accounting on thedispersal model, metapopulation model is a valuable modelling approach forsuch purposes. In [25, 26], the authors investigate the impact of the host dis-persal amongst the multiple patches in the disease dynamics. Spatio-temporalfeatures in the progress of infectious agents of different hosts are in includedin [27, 28] using metapopulation models. In [29, 30], the authors deal with themulti-species epidemic models on n patches with migration what can potentiallybe employed in the vector-hosts model.Most WNV spread models are mathematical deterministic compartmentalmodels [31, 32, 33]. However, these models are usually developed at a localscale that do not necessarily include the global information about the differentcomponents such as mobility patterns of the hosts or vector or both, temperatureor landscape data types. In a previous study, we develop a temperature drivenmodel [34] that analyses the local spread of different regions in Germany. In thisstudy, we extend this model to a metapopulation-network associated model.In contrast to the results of our previous model [34], where the most suitableregion for the establishment of the disease is in the South West, the cases inGermany are mainly observed in the East.The new model should help us to understand the WNV spatial transmissionin Germany, and to understand the key factors of spatio-temporal transmissionof WNV. Hence, we systematically examine the relevance of the variables inour model. Our current endeavour is similar in spirit to other several models[35, 36, 37] that also direct to bring necessary information in the spatial scales.5 . Model Formulation
Using our local model as a basis [34], we have constructed a metapopulationmodel. The principle functionality of a metapopulation model is shown in figure1. Our model system comprises of heterogeneous networks of subpopulations orpatches what are connected by migration. Each subpopulation represents thepopulation of vectors and hosts in the habitat patch. The respective migrationsof individuals from one subpopulation to another subpopulation is governed bythe migration paths of connections among the subpopulations. Individuals canmigrate from a subpopulation to another on the network of connections amongsubpopulations. Transmission within each patch is modelled by a vector-hostcompartment model. Nonlinear Ordinary Differential Equations are employed tomodel the local transmission and the transmission due to the migration amongstthe adjoining habitat sites as well the population dynamics.
Subpop j
0. It isassumed that population growth, birth and the disease transmission happen inthe respective patches only but not during the migration. Assuming Q i is thepopulation in patch i , then the dynamics associated with Q i ( t ) is given by for i = 1 , ..., n dQ i dt = Π Q i − µ Q i Q i + (cid:88) j r ij Q j − (cid:88) j r ji Q i (1)where Π Q i represents recruitment in the population Q i , µ Q i is the death rateat the patch i . Equation (1) can be expressed in a more compact form as . Q = Π Q − diag ( µ Q ) Q + RQ (2)where, Q = ( Q , Q , ..., Q n ) (cid:124) , Π Q = (Π Q , Π Q , ..., Π Q n ) (cid:124) , µ Q = ( µ Q , µ Q , ..., µ Q n ) (cid:124) ,and R is the movement matrix and the elements of R are defined as R ij = r ij for i (cid:54) = j . When combining the local epidemiological model [34] and the metapopula-tion model subsequently will lead to equation 3 representing the model for themosquitoes with m ij as the rate of migration movement between two arbitrarynodes j and i . dS M i dt = [ b M i N M i − m M i S M i ] [1 − S M i /K M i ] − S M i K B i ( c i I BC i + c i I BSC i )+ (cid:88) j m ij S M j − (cid:88) j m ji S M i dE M i dt = S M i K B i ( c i I BC i + c i I BSC i ) − γ M E M i − m M i E M i + (cid:88) j m ij E M j − (cid:88) j m ji E M i dI M i dt = γ M E M i − m M i I M i + (cid:88) j m ij I M j − (cid:88) j m ji I M i (3)Equation 4 shows the model for birds and p ij represents the migration be-tween two arbitrary nodes j & i of the birds.7 S B i dt = (cid:20) b B i − ( b B i − m B i ) N B i K B i (cid:21) − m B i S B i − ( β i + β i ) I M i S B i K B i + (cid:88) j p ij S B j − (cid:88) j p ji S B i dE BC i dt = β i I M i S B i K B i − m B i E BC i − γ BC i E BC i + (cid:88) j p ij E BC j − (cid:88) j p ji E BC i dE BSC i dt = β i I M i S B i K B i − m B i E BSC i − γ BSC i E BSC i + (cid:88) j p ij E BSC j − (cid:88) j p ji E BSC i dI BC i dt = γ BC i E BC i − m B i I BC i − α i I BC i − d BC i I BC i + (cid:88) j p ij I BC j − (cid:88) j p ji I BC i dI BSC i dt = γ BSC i E BSC i − m B i I BSC i − α i I BSC i + γ i R BSC i − d BSC i I BSC i + (cid:88) j p ij I BSC j − (cid:88) j p ji I BSC i dR BC i dt = α i I BC i − m B i R BC i + (cid:88) j p ij R BC j − (cid:88) j p ji R BC i dR BSC i dt = α i I BSC i − m B i R BSC i − γ R BSC i + (cid:88) j p ij R BC j − (cid:88) j p ji R BC i (4)Similar expressions stand for the long range bird population but we decidenot put that here. We have appended in the Supplementary Information. Theinitial conditions are S B i (0) , S M i (0) > E BC i (0) , E BSC i (0) , I BC i (0) , I BSC i (0) , R BC i (0) , R BSC i (0) , E M i (0) , I M i (0) ≥ λ i = β i I Mi K Bi , η i = β i I Mi K Bi , δ i = c i I BCi K Bi and µ i = c i I BSCi K Bi .Λ i = [ b M i N M i − m M i S M i ] [1 − S M i /K M i ] , Π i = (cid:104) b B i − ( b B i − m B i ) N Bi K Bi (cid:105) Adding up Eqn (3) & Eqn (4) give us equations for the total mosquito andbirds populations, respectively, in patch i = 1 , ..., n : while using the notations8ntroduced above, we get dN B i dt = Π i − m B i N B i − d BC i I BC i − d BSC i I BSC i + (cid:88) Z (cid:88) j p ij Z j − (cid:88) j p ji Z i (5) dN M i dt = Λ i − m M i N Mi + (cid:88) Y (cid:88) j m ij Y j − (cid:88) j m ji Y i (6)Here, Y = S M , E M , I M and Z = S B , E BC , E BSC , I BC , I BSC , R BC , R BSC . Letthe total bird and mosquito population be denoted as N B and N M , respec-tively. Then after adding the population over all the patches, we should get thefollowing dN B dt = (cid:88) i (Π i − m B i N B i − d BC i I BC i − d BSC i I BSC i )+ (cid:88) i (cid:88) Z (cid:88) j p ij Z j − (cid:88) j p ji Z i (7)Since I BC i < N B i & I BSC i < N B i , it follows that (cid:88) i Π i − (cid:88) i ( m B i + d BC i + d BSC i ) N B i ≤ N B i dt ≤ (cid:88) i Π i − (cid:88) i m B i N B i (8)Thus (cid:88) i Π i − max ≤ i ≤ n { m B i + d BC i + d BSC i } N B i ≤ N B i dt ≤ (cid:88) i Π i min ≤ i ≤ n { m B i } N B i (9)From here, we can claim that ∀ t ≥ (cid:20) (cid:80) i Π i max ≤ i ≤ n { m B i + d BC i + d BSC i } , N B (0) (cid:21) ≤ N B ( t ) ≤ max (cid:20) (cid:80) i Π i max ≤ i ≤ n { m B i } , N B (0) (cid:21) (10)Therefore, the total populations of local birds are bounded.We can reframe 3 & 4 into the matrix form as follow: . S M = Λ − diag ( δ + µ ) S M + M S M . E M = diag ( δ + µ ) S M − diag ( γ M + m M ) E M + M E M . I M = diag ( γ M ) E M − diag ( m M ) I M + M I M (11)9nd for the local birds: . S B = Π − diag ( λ + η + m B ) S B + P S B . E BC = diag ( η ) S B − diag ( γ BC + m B ) E BC + P E BC . E BSC = diag ( λ ) S B − diag ( γ BSC + m B ) E BSC + P E
BSC . I BC = diag ( γ BC ) E BC − diag ( α + m B + d BC ) I BC + P I BC . I BSC = diag ( γ BSC ) E BSC − diag ( α + m B + d BSC ) I BSC + diag ( γ ) R BSC + P I
BSC . R BC = diag ( α ) I BC − diag ( m B ) R BC + P R BC . R BSC = diag ( α ) I BSC − diag ( m B + γ ) R BSC + P R
BSC , (12)where M , P are the movement matrices of the vector and the local bird popu-lation, respectively. The matrix form of the equations associated with the longrange dispersal birds are included in the Supplementary Information.
3. Network Framework
In the last section we show the ODE model for WNV disease spread betweenpatches. In this section we describe the mobility networks of mosquitoes andbirds. These networks are necessary to define the movement matrices for thevector ( M ) and the local bird ( P ) populations. The importance of vector movement in the spread of WNV is controversial.While [21] conclude that mosquito movements do not play an important role but[38, 39, 40, 41] conclude that due to the opportunistic bites from the mosquitoesto the host, mosquito movements are important. Information on exact mobilitypathways for mosquitoes are scarce. However, there are some estimations on theflight range of Culex [42, 43, 44]. [43] estimate the flight range of
Culex pipiens ( average maximum distance = 9695 m , minimum of maximum distance =350 m , and maximum of maximum distance = 22 , m ) and the dispersalcapacity is Strong . Given their dispersal capacity, it is more realistic to include10he precise and daily movement of the mosquitos. Later on we show that themosquito movement matters in case of spreading the disease from one patch i tothe neighbouring patch j . Let the distance between two patches be ( i & j ) as D ij ,then according to [45], the dispersal rates between two sub-populations ( M i,j )are assumed to be negative-exponential distribution. But for the simplicity wedecided to follow the distribution proposed by [46]. Here, we have used thefact that Culex pipens have Strong dispersal capability, henceforth the D max used by [46] is different than what we have considered [43] but the dispersalprobability is a function of the linear decreasing distance as in [46]. The networkwith such linear dispersal kernel is calculated as follows: Algorithm 1
Mosquito movement network algorithm procedure MosNet ( i, j ) (cid:46) Routine to create link between i & j D ij ← Euclidean Distance ( i, j ) (cid:46) Euclidean distance between i & j D max ← M aximum Dist (cid:46)
Maximum interaction radius of mosquitoes p ( D ij ) ← D max − D ij D max (cid:46) Probability of link connection between i & j p rand ← rand (0 , (cid:46) Generate a random no between 0 & 1 if p rand < p ( D ij ) then Create an undirected link between i & j In figure 2 we show an example of a randomly generated mosquito network
Figure 2: One realisation of the stochastic network of Mosquito .2. Host Mobility Model Birds are the natural reservoir for WNV [47, 7, 48]. In general, there is atransmission cycle between birds and vectors. Furthermore, there are discus-sions about the importance of bird-to-bird transmissions [49] but in our currentendeavour we have not included such transmission routes. It is obvious, thatbird movement might spread the disease within the home range of birds. Thehome range of birds depends on the required habitat, as well as feed supply andbird density [50, 51, 52]. As home ranges differs between species and dependson habitat suitability, it is difficult to define a single home range for the hostmobility model. Hence, we include two movement patterns, one to cover smallhome ranges of breeding birds with a maximal dispersal ranges from 1500 to2164 meter and minimal dispersal ranges from 80 to 170 meter [53] and a secondone to cover large home ranges with a maximum flight distance of 500 Km afterfollowing [35]. In the Figure: 3, we show an example for the local bird flightrange.
But, given the above said reasons, we decide to follow the seed dispersalmodel [54, 55, 56, 57] for the local bird mobility model. We use the seeddispersal as the proxy for the movement network of the birds. The dispersalprobability is of Weibull distribution [58, 59, 60, 61, 62, 63, 64]. . . . . . . Distance from habitat patch (m) P r obab ili t y D en s i t y Patch Area< 10ha10 − 25ha25 − 50ha50 − 250ha> 250ha
Figure 3: Dispersal probability of the bird
12o construct the movement matrix ( P ) of the local birds, we have madeuse of the following routine just the way we have constructed the mosquitomovement network ( M ) in Algorithm: 1 . Algorithm 2
Local bird movement network algorithm procedure LocalBirdNet ( i, j ) (cid:46) Routine to create link between i & j D ij ← Haversine Dist ( i, j ) (cid:46) Haversine distance between i & j p ( D ij ) ← W eibull Distribution ( D ij ) (cid:46) Probability of link connectionbetween i & j p rand ← rand (0 , (cid:46) Generate a random no between 0 & 1 if p rand < p ( D ij ) then Create an undirected link between i & j Figure 4: This is one realisation of the stochastic network of local bird
An example of such generated local bird network using Algorithm: 2 is shownin Figure: 4 Shape and the scale parameters are taken from [53, 65, 66] and theranges for the shape and scale parameters are following [2 . , .
26] and [001 , . The information for large home ranges in birds are scarce. Hence, we followthe approach of [67, 35], using a power-law distribution to model incidental long-range disease transmission routes. To model the spatial dynamics of differentinfectious diseases, power-law transmission is used in several occasions [68]. Wehave constructed the movement matrix ( N ) of the long dispersal bird using the13ollowing routine as we have done for the mosquito movement network and thelocal bird movement network in Algorithm: 1 and Algorithm: 2. Algorithm 3
Long range dispersal bird movement network algorithm procedure MigBirdNet ( i, j ) (cid:46) Routine to create link between i & j D ij ← Haversine Dist ( i, j ) (cid:46) Haversine distance between i & j p ( D ij ) ← P owerlaw Distribution ( D i,j ) (cid:46) Probability of linkconnection between i & j p rand ← rand (0 , (cid:46) Generate a random no between 0 & 1 if p rand < p ( D ij ) then Create an undirected link between i & j An example of such generated long range dispersal bird network using Al-gorithm: 3 is shown in Figure: 5
Figure 5: One realisation of the stochastic network of long range dispersal bird
The value of the power-law parameter has been taken from [35] with its valueranging ∼ U (2 , To find the disease free equilibrium E , we consider the following linearsystem Λ − diag ( δ + µ ) S M + M S M = 0Π − diag ( λ + η + m B ) S B + P S B = 0 (13)14r, in the compact form as HS = Ω (14)where H = diag ( δ + µ ) − M diag ( λ + η + m B ) − P , S = S M S B Ω = ΛΠ Since all off-diagonal entries of H are nonpositive and the sum of the en-tries in each column of H is positive, H is a nonsingular M -matrix, H − ≥ S = ( S M , S M , ..., S M n , S B , S B , ..., S B n ) = H − Ω > ∀ i .The model system (11) and (12) can be put in a compact form as thefollowing dxdt = A − B x dydt = D x + E y (15)where x = [ S M , S B ] T , y = [ E M , I M , E BC , E BSC , I BC , I BSC , R BC , R BSC ] T , A = ΛΠ , B = diag ( δ + µ ) + M diag ( λ + η + m B ) + P The expressions of D and E are included in the Supplementary Information. R of the patchy model To compute the basic reproduction number, we use the
Next generationmethod [70]. Using the notation used in [70], we can decompose the modelsystem (3) & (4) as F ( I ) − V ( I ) . F ( I ) & V ( I ) represent as the flow of newinfections and the remaining transfers within and out of the infected classes,respectively. For the simplicity of our matrix calculation, we have consideredthe subclinical case as γ i = 0 will give the clinical case. FBSC ( I ) = (cid:34) c IBSC SM KB , , β IM SB KB , γ RBSC , . . . , c nIBSCnSMnKBn , , β nIMnSBnKBn , γ nRBSCn (cid:35) T FBSC ( I ) = (cid:104) µ SM , , λ SB , γ RBSC , . . . , µnSMn, , λnSBn, γ nRBSCn (cid:105) T VBSC ( I ) = − − γM EM − mM EM (cid:80) m jEMj − (cid:80) mj EM γM EM − mM IM (cid:80) m jIMj − (cid:80) mj IM − γBSC EBSC − mB EBSC (cid:80) p jEBSCj − (cid:80) pj EBSC γBSC EBSC − mB IBSC − α IBSC − dBSC IBSC (cid:80) p jIBSCj − (cid:80) pj IBSC − γM EMn − mMnEMn + (cid:80) mnjEMj − (cid:80) mjnEMnγM EMn − mMnIMn + (cid:80) mnjIMj − (cid:80) mjnIMn − γBSCnEBSCn − mBnEBSCn + (cid:80) pnjEBSCj − (cid:80) pjnEBSCnγBSCnEBSCn − mBnIBSCn − α nIBSCn − dBSCnIBSCn + (cid:80) pnjIBSCj − (cid:80) pjnIBSCn Letting F BSC = [ ∂ F BSC | ( S , ) ] and V BSC = [ ∂ V BSC | ( S , ) ] as the Jaco-bian matrices evaluated at the disease free equilibrium ( S , ). Following [70],the matrix N GM
BSC = F BSC V BSC − is the next generation matrix for thesubclinical birds and it is well defined.The elements of the Jacobians have the following forms ∂µ i ∂E M i = ∂µ i ∂I M i = ∂µ i ∂E BSC i = ∂λ i ∂E M i = ∂λ i ∂E BSC i = ∂λ i ∂I BSC i = 0 ∂µ i ∂E M j = ∂µ i ∂I M j = ∂µ i ∂E BSC j = ∂λ i ∂E M j = ∂λ i ∂E BSC j = ∂λ i ∂I BSC j = 0 , i (cid:54) = j∂R BSC i ∂E M i = ∂R BSC i ∂E I i = ∂R BSC i ∂E BSC i = ∂R BSC i ∂I BSC i = 0 ∂R BSC i ∂E M j = ∂R BSC i ∂E I j = ∂R BSC i ∂E BSC j = ∂R BSC i ∂I BSC j = 0 , i (cid:54) = j The partial derivatives are evaluated at ( S , ). Matrices F BSC and V BSC are4 n × n and we express F BSC as F BSC = diag [ F BSC ii ], with i = 1 , , . . . , n and F BSC ii = ∂λ i ∂I Mi S B
00 0 0 0 ∂µ i ∂I BSCi S M V BSC = [ V BSC ij ], where V BSC ij = diag (cid:104) − m ij − m ij − p ij − p ij (cid:105) ; i (cid:54) = j and V BSC ii = γ M + m M i + (cid:80) m ji − γ M m M i + (cid:80) m ij γ BSC i + m B i + (cid:80) p ji − γ BSC i m B i + α i + d BSC i + (cid:80) p ji Similarly the next generation matrix (
N GM BC ) associated with the clinicalbirds is F BC V BC − . After including clinical and subclinical birds in differentpatches, the next generation matrix ( N GM ) of the system 3 & 4 is following
N GM = (cid:104) N GM BC N GM
BSC (cid:105)
So, the basic reproduction number is R = ρ ( N GM ) (16)where ρ is the spectral radius of the matrix N GM . According to [70], thelocal stability of the disease free equilibrium E = ( S , ) is governed by R . If R <
1, then E is asymptotically unstable and unstable whenever R >
4. Simulation Results
Most of the literature in the area of vector-borne disease modelling centresaround the movement on the long distance and long duration travelling of thehost species only. In those models, we can overlook the vector mobility butin our current effort we include the short and small scale flights of the vectorsof WNV. The influence of the vectorial capacity and the movements of themosquitoes are important features to the potential spread and hence sustainingWNV [46, 71]. Therefore, we conceive that the daily mobility and the short17ourneys carried by the vector species can not be ignored. It is of our interest toacknowledge that smaller temporal scale of WNV transmission can potentiallyinclude the new aspects in the spread of WNV in Germany.
Time S ub C li n i ca l P opu l a ti on IBSCn1IBSC1 (a)
Time S ub C li n i ca l P opu l a ti on IBSCn1IBSC1 (b)Figure 6: Graphical representations of effect of mosquito mobility in the time evolution ofinfected subclinical bird population. (a) In this simulation we have chosen the range ofmosquito mobility as the average maximum distance. (b) In this simulation we have chosenthe range of mosquito mobility as the minimum of maximum distance as described in thesection 3.1.
The expression of R for the metapopulation model is too long and compli-cated enough to perform theoretical analysis to understand the influence of themovements of the mosquitoes. So, we decide to consider only the subclinicalbirds and examine the influence of the mosquito movements. Here, we simulatethe situation when an infected local subclinical bird is introduced to a com-pletely susceptible population with the different flight range movements of themosquitoes. We consider two cases for the simulations, one with the inclusionof local mosquito mobility and another one without the mosquito migration. Inboth cases, we keep the subclinical birds movement enabled. The importanceof local mosquito interactions is visible in the Figure: 6. It shows the number ofsubclinical infected birds versus time. I BSC stands for the infected subclinical18irds in the patch number 1 and I BSCn stands for the subclinical birds in thepatch number 1 but without mosquito movements. It is interesting to observethat in the scenario when we do not include the mosquito movements, the in-fection spreading process takes a bit longer time and the peak is relatively flatwhereas when we include the mosquito mobility, the infection spreads quickerthan previously considered case and the peak is sharp and concentrated withsubclinical birds. For the experimental purpose, we change the maximum flightranges of mosquitoes and the influence of the mosquito movements are clearlyvisible in the Figure: 6a and Figure: 6b. Higher range of mosquito movementsfacilitate the potential transmission of WNV in the local birds population. The shape of the epidemiology of WNV spread in Germany is potentiallygoverned by the movements of the interacting species especially the birds asa prime host. It is widely accepted that the movements of the birds likely tofacilitate the dissemination of WNV in Germany [20, 1]. In this section we ex-plore the influence of the birds (both local and long dispersal birds) movementson the potential spatial spread of WNV in Germany. With the intention tocomprehend the impact of bird movements, we set up some simulations undercertain assumptions. In the following sections we elaborate them.
In this section we explore the influence of the local bird movements on thebasic reproduction number of the patchy model system (3) & (4). To keepour findings simple, we just consider two-patch model consisting only clinicalbirds and the corresponding basic reproduction number R given in (16). Theanalytical and the symbolic computations related to the clinical and subclinicalbirds are the same. Given the complex and the long expression of R (Includedin the Supplementary Information) even for the two-patch model, it is ratherdifficult to quantify theoretically the impact of the local bird migration in R and in the dissemination of WNV from one habitat patch to another. So, we take19he refuge of mathematical simulations under different conditions to understandit after following [72]. p R (a) p R (b)Figure 7: Graphical representations of local bird movements on R for a two patch model.The symbolic computations have been performed in [73]. (a) Impact of local bird (subclinical)migration from the second patch to the first patch and (b) Impact of local bird (subclinical)migration from the first patch to the second patch. To test the influence on local bird movements, we run the model for twopatches using the following parameters: m B = m B = 0 . γ BSC = γ BSC =0 . α = α = 0 . K M = K M = 1000, K B = K B = 100, c = c =0 . d IBSC = d IBSC = 0 . m M = m M = 0 . R )of patch 1 is less than 1 and for the patch 2 it is ( R ) greater than 1i.e. R = 0 . R = 1 . p = 0 .
91 and let let p to vary. Then we keep p = 0 . p to observe the influence of the clinical birds on the magnitude of the basicreproduction number. From the Figure: 7a, it is evident that with the increaseof migration from the patch 2 to the patch 1, the basic reproduction numberreduces whereas we can witness the opposite of this phenomenon in the Figure:7b, where with the increase of p yields the increase in the basic reproductionnumber of the two patch model.These simulations possibly give us the glimpse of the complexity of the poten-tial spread of WNV from one endemic patch to another one. The immigrationof the birds can trigger a series of such expansion of WNV and this kind ofcomplexities can not be explained through a localised model [34]. The authors in [48] investigate that the dispersal of long range birds actuallyestablish the spatial spread of WNV. It is plausible that the acquisition of WNVcan happen during dispersal of long range birds [9]. In this simulation exercise,we include the movements of the local birds and mosquitoes but discard themovements of the long range dispersal birds to understand the importance ofthe movements of long range dispersal birds in introducing and sustaining thebite of WNV in the new places. 21
Cumulative Cases 0 [5, 15) [15, 25) (25, 35] (a)
NCumulative Cases 0 [5, 15) [15, 25) (25, 35] (b)
NCumulative Cases 0 [5, 15) [15, 25) (25, 35] (c)
NCumulative Cases 0 [5, 15) [15, 25) (25, 35] (d)Figure 8: (a) Choropleth maps of potential spatial spread of WNV in Germany after season 1without the movements of long range dispersal birds with 0 .
75 and 0 .
001 as shape and scaleparameters (b) Choropleth maps of potential spatial spread of WNV in Germany after season1 without the movements of long range dispersal birds with 0 .
95 and 0 .
01 as shape and scaleparameters (c) Choropleth maps of potential spatial spread of WNV in Germany after season1 without the movements of long range dispersal birds with 1 . . . .
22e introduce I Inf (= 1) number of infected long range dispersal bird in se-lected city of Halle as the first case of WNV was reported in that city [1]. Weemploy different dispersal kernels of the local birds to spatially map the potentialspread of WNV under the above mentioned assumptions to run the simulationswith the different parameter values as described in Section 3.2.1. It is notice-able that without the inclusion of movements of the long range dispersal birds,the spread of WNV is rather concentrated and the pace at which transmits iscomparatively slow. From this simulation experiment we can infer the potentrole of long range dispersal birds to introduce WNV into the new territories. Wehave included the graphical representations of Weibull distribution parametersin Supplementary Information. We compute structural similarity (SSIM) index[74] score as mentioned in the Section 6 for each epidemiological simulation andappend the findings in the Supplementary Information.
5. Vector Control
One of the main applications of models for the potential spread of WNVamong the birds is to design of possible control strategies and create bufferzones to prevent, or minimise the spread of WNV. For the demonstration pur-pose we consider only the movement between two patches of the clinical birds.Mosquito movement is also considered. We follow [75] to assess mathematicallythe feasible control strategies. Consider the model system (3) and (4) and wetake account of the clinical birds and their movements between two patchesonly. The next generation matrix for the clinical birds turns out to be in thefollowing form
N GM BC = a a a a a a a a The explicit mathematical forms of a i , i = 1 : 8 are included in the Supplemen-tary Information. 23 GM BC can be readily put into the block matrix form as following: N GM BC = A where A = a a a a A = a a a a Then according to [75, Theorem 8 (1)], targeting either the vector or the hostpopulation to control WNV should be the effective strategy while counting onthe cost of corresponding one-group target strategies. According to Theorem 8,it is better to apply all available resources to target only one population only.In order to lessen the burden of WNV, the mosquito population needs to bereduced below a certain threshold. In our case mosquito control is only optionto explore. According to [75], the type reproduction number ( T M ) to control themosquito population is given by: T M = A A − p , where p is the fraction of thevector population to be controlled. Therefore, more than the 1 − T M fraction ofmosquitoes should be targeted to eradicate the impact of WNV in case of twopatch populations. Using the parameters described in 4.2.1, more than 56 .
6. Spatial spread in Germany
The study area comprised of 11,054 German Gemeinden (Municipalities).We use the deterministic metapopulation 11 & 12 and the equations associatedwith the movement of birds (included in the Supplementary Information) to ap-prehend the WNV transmission in the local bird populations in each
Gemeinde .We are caught increasingly between the complexity of the simulations and thepossibility to decipher the potential spatial spread of WNV in Germany qualita-tively. Birds are grouped according to their health status as depicted in 1. Weexplicitly represent the vector population and the mobility. Unfortunately, we24o not have the population distribution of birds across the different Gemeindenlevel so we have assumed uniform number of local birds as in [34].At first, we keep all the local birds initially susceptible in all the Gemeinden.We introduce I Inf (= 1) number of infected long range dispersal bird in selectedGemeinden. For the first season simulation the infection is seeded in the cityHalle as the first case of WNV was reported in that [1].Between-Gemeinden movements of vectors and the hosts take place on threedifferent contact networks: (i) the vector network, representing the mosquitoflies (ii) the local birds network, represents the movements of the local birds ina habitat patch and (iii) the long distance movement birds network, representssporadic movements of birds ranging distances between 0 and 500 km pathways.The nodes are the centroid of Gemeinden and the links amongst them areformed after employing Algorithm: 1, Algorithm: 2 and Algorithm: 3 for thethree distinct networks. In Figure 9, we depict the long range dispersal birdspathway being constructed through the power-law distance based kernel. N Figure 9: Dispersal networks for power-law distance based kernel of long range dispersal birdsin Germany according to the Algorithm: 3
In the simulation, we take note of the source of infection of each newlyinfected Gemeinden. Gemeinden that are previously free of WNV might beinfected through different networks of concern. We then take account of cu-25ulative number of infected birds, both the local and the long range dispersalbirds in each season of WNV circulation per Gemeinden in Germany.
NCumulative Cases 0 [5, 15) [15, 25) (25, 35] (a)
NReported Cases 0 [5, 8) [8, 13) (13, 16] (b)
NCumulative Cases 0 [5, 25) [25, 45) (45, 65] (c)
NReported Cases 0 [9, 12) [12, 15) (15, 19] (d)Figure 10: (a) Choropleth maps of potential spatial spread of WNV in Germany after season 1(b) Choropleth maps of cumulative reported cases of WNV in Gemeinden, Germany in 2018,data from [76] (c) Choropleth maps of potential spatial spread of WNV in Germany afterseason 2 (d) Choropleth maps of cumulative reported cases of WNV in Gemeinden, Germanyin 2019, data from [76] φ and compare their distribution in Fig-ure: 11. For more information please see the Supplementary Information for adetailed description. Season 1Season 2 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Structural Similarity Index S ea s on Figure 11: Spatial analysis of model outcome and the reported cases of WNV in Germanyduring the 2018 − st quartile,median and 3 rd quartile and the maximum of computed SSIM values. It is noticeable therange of SSIM scores in the first season is comparatively smaller to that off in the next season. We can notice that in the initial phase of WNV spread in Germany, themodel performs well while approximating the observed spatial dissemination.In the following season though the performance of the model is slightly worsecompared to the results from the previous season in the backdrop of reportedcases. Visual representations of the spatial R vs cumulative number of infectedbirds per Gemeinden are included in the Supplementary Information.28 . Discussion & Conclusion To comprehend the potential eco-epidemiological parameters and the diseasemechanism yield to the preservation, appearance and ultimately lead to thepotential spatial spread of WNV are essential to implement control strategiesfor the containment. After the first case of WNV detected in Germany in2018 [1] and the favourable weather condition accentuated its possible spreadacross Germany especially not only confined to its first detected place in theEastern and Southeastern Germany. Following this sporadic cases of WNV, inthe next season extraordinary high temperatures has allowed to decrease theextrinsic incubation period (EIP) and consequently the cases of WNV increasedmultifold [20] across Germany.In this work, we endeavour a deterministic metapopulation-network modelof WNV transmission in Germany. We put an effort to analyse the transmis-sion of WNV locally and employ the contact networks to investigate the WNVtransmission to the disease-free zones. The contact networks incorporate differ-ent distinct types of vector and host movements: local bird movements in theirhabitat patch accordingly, long-distance movements of birds as well as vectormovements. In our simulations we have opted for to include and to consider thedifferent movement networks of the vector and the long range dispersal birdsto capture the transmission patterns of WNV in Germany. To procure the spa-tial prediction of WNV circulation across the whole Germany, we compare thespatial dissemination of WNV after including and afterwards ruling out the po-tential role of the long range dispersal bird’s role in the spread of pathogens. Itis interesting to note that in the first season of WNV spread in Germany, thespatial transmission of WNV is well caught in a qualitative manner althoughin the next season similar feat is not being achieved after the inclusion of longrange dispersal birds. One possible reason could be attributed to the fact thatthe reported cases of WNV rely solely on the reporting and there might bereporting bias which might have hindered into the quantitative similarities ofthe reported cases and the simulated cases of WNV in Germany. Another pos-29ible explanation could be the role long range dispersal birds what is benightedin the spread of WNV. One thing should be noted that the uncertainty asso-ciated with the number of overwintering infected mosquitoes can potentiallytrigger the outbreak. From our simulations, it is clear that the role of longrange dispersal birds to introduce the WNV in the different parts of Germanyis recognisable and afterwards, the WNV circulation amongst the mosquito andthe local birds help to localise the dissemination. The spatial projections of themodel outcomes can explain this.Several previous models [17, 18, 77, 78, 79, 37] include network models toanalyse the role of birds and vectors in spatial disease transmission. 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