1. Introduction
Europe is confronting a rising threat of outbreaks of arthropod vector borne (AVB) tropical diseases as rising temperatures linked to climate change create a conducive environment for arthropod development and dispersion [
1]. Mosquitos, among others, are vectors of many viruses and parasitic pathogens. They can carry diseases such as malaria or yellow fever and traditionally most of these pathogens are found in Africa, Asia and Latin America. However, the expansion of arthropods vectors (AVs) in temperate climates is now a direct consequence of the increased mobility of people in the era of globalization [
1,
2]. Most importantly, climate change has created conditions conducive to maintaining and developing vector mosquitoes in new areas. This leads to an increase in mosquito populations and the potential for virus transmission [
3,
4].
In addition to endemic mosquito species that are present in Europe and particularly Greece, such as those belonging to the genera
Culex,
Anopheles and
Aedes, climate change and particularly the increase in average temperatures is expected to bring about extension to wider geographical units of Europe of more arthropod vector related diseases [
5,
6,
7]. Over the last ten years, worldwide climatic conditions, including the Mediterranean and Greece, have encouraged the development of the mosquito population and particularly the transmission of West Nile virus (WNV) [
8,
9,
10]. The primary reservoir of the virus in nature is mainly wild birds, from which mosquitoes are infected, while humans do not further transfer the virus to other mosquitoes [
11,
12,
13]. The majority of people infected with the virus are asymptomatic, around 20% have mild symptoms of viral syndrome and less than 1% has more serious central nervous system manifestations, mainly encephalitis and meningitis. The most severe events usually occur in the elderly, immunocompromised patients and, in general, individuals with underlying chronic diseases [
14].
In fact, vector-borne diseases are becoming a major threat, impacting both human and animal health with severe consequences in the governance of health risks attributed to these emerging diseases in European countries. Primary strategies focus on preventing human exposure by effective insect control. This work considers in mosquito-borne diseases and mostly those transmitted by
Culex sp. and how they are affected by weather variables. One of the most effective ways to achieve this goal is based on prioritizing efforts in identifying what considerations should be taken into account to guide decisions such as variable weather in itself (e.g., drought, extreme temperatures) that could be detrimental to mosquito survival [
15]. Arthropods are very sensitive to weather and therefore ongoing climatic trends of warming and more variable weather threaten to increase burden of these diseases [
16]. In recent years, progressions in Earth observation satellites together with geographical information systems have contributed to weather monitoring. Mathematical models have become a valuable tool for predicting dynamics of populations under climatic scenarios and different temperature regimes improving our knowledge about contribution of environmental and biodiversity factors in vector-borne diseases and helping public health decision making for better allocation of resources in the fight against many pathogens [
17].
Arthropod vector control, which relies on the use of insecticides, is the principal means of mitigating the spread of related diseases [
18,
19]. However, for such a strategy to be effective, it is important to predict the temporal change in mosquito abundance and how it is affected by weather conditions [
20,
21]. This is particularly relevant for epidemiological studies with arthropod vectors since their development and population dynamic are strongly affected by climatic conditions, changing conditions in a context of climate change.
For instance, weather conditions have direct and indirect effects on growth and development of mosquitoes. Additionally, gonotrophic activity by
Culex sp. in early spring and during the season is affected by water resources and this establishes the subsequent phenology and determines the potential date of the earliest mosquito-borne encephalitis virus transmission [
21]. Traditionally, the effect temperature over insect growth rate is captured by empirical linear and/or parabolic functions [
22]. However, most often temperature-dependent population growth is determined by temporally fluctuating vital rates for which classical demographic theory often does not apply [
23]. Moreover, actual mosquito flight patterns might be related to the extent of which blood-feeding behavior are shaped by available feeding resources [
24]. As a result, in field conditions mosquito dynamics may be complex and characterized by abrupt outbreaks and overlapping generations. From an applied perspective, understanding adult mosquito flight patterns and how it is affected by weather is essential for predicting their activity to prevent the risk of transmitting related vector-borne pathogens in nature [
25]. Identification, of periods of high population activity can guide effective targeting of the species. Consequently, several mathematical models have been used to connect the biological processes of vector dynamics and climate [
26,
27,
28,
29]. To date, most epidemiological and insect population models, have a deterministic nature and rely on some basic assumptions to define the various parameters of vector and disease dynamics under study [
11,
30]. Often these parameters are unknown and need first to be estimated to parameterize the model. Moreover, because of the impacts of various internal and external factors, the temporal evolution of population processes is non-linear and characterized by random perturbations making it difficult to analyze and forecast population dynamics using only empirical-temperature dependent growth and mechanistic models [
31].
Stochastic models, which recognize that all variables are probabilistic in nature and are handled as such, could be employed to model non-linear ecological processes and advance our understanding of vector population dynamics for public health planning [
31]. Markov chain (MC) models belong to the class of stochastic models in probability theory that are based on the Markov property, which assumes that future states of a process that evolves in time depend only on the current state and not on the events that occurred before [
31]. Such a framework provides a coherent approach to solving and inferring practical issues of decision-making since it integrates multiple sources of uncertain information in a probabilistic way, and which often characterizes noisy stochastic processes [
31,
32,
33].
Markov chain models have proved suitable for describing randomly changing systems such as queuing [
34] and manufacturing systems [
35], market trend analysis [
36,
37] and insurance methods [
38,
39]. Recently, MCs were used in modeling biological processes and health systems, such carcinogenesis [
40,
41] and medical cost health problems [
42]. Applications of MCs in modeling categorical data sequences can also be found [
43,
44,
45,
46], including air population modeling [
47], and weather forecasting [
48,
49], although there are fewer examples analyzing ecological time series and population dynamics [
32,
50].
To date, in the field of theoretical population biology, the concept of MCs can be associated to any to the projection of the Leslie matrix that contains information about its genealogy, and which can be transformed exactly into a Markov chain [
51]. Additionally, developmental stage transition models have been formulated to estimate the transition from one stage to another and under different environmental conditions [
52,
53]. Most of the age-structured matrix projection models have been used in demographic studies as a central tool to quantify the asymptotic growth rates (i.e., the dominant eigenvalue) and the reproductive values (dominant right and left eigenvalues). They have been also used in the context of MCs to estimate the stable age distribution and how it is affected under certain environments [
53,
54].
However, very few studies have emphasized the application of MCs in studying mosquito dynamics and the effect of detrimental weather variables. Chaves et al. 2014 [
24], for instance, has used a two-stage (larvae and adults) recruiting matrix model to propose a mechanism for environmental signal canalization into demographic parameters of
Aedes aegypti that could explain delayed high temperature induced mosquito outbreaks. However, to our best knowledge, Markov chains have not been applied to solve the problem of predicting and simulating the probability of
Culex sp. ecological time series and especially with respect to exogenous stochastic factors such as weather variables and particularly temperature. Hence, it is assumed that a temperature-conditioned Markov chain (CMC) model could be developed and applied for the first time to predict the dynamics of
Culex sp. vectors of important medical diseases. The major advantage of the CMC over traditional Markov chain models is that via appropriate conditioning their primary Markov chain properties are mixed with that of relevant climate factors [
55]. Additionally, the mosquito dynamics process is considered as random, in which no prior information of the system properties is needed, and thus the resultant dynamic stochastic model is purely data driven. It could be beneficial to develop new stochastic population modeling approaches that take into account the effect of one exogenous stochastic variable over the other by terms of conditional probabilities.
In addition, due to their complicated life cycle and overlapping of generations, mosquito populations are characterized by abrupt dynamics and thus cannot be easily predicted by traditional insect population models. Moreover, considering that most of their attributes can change in respect to random climate events, stochastic models become a suitable alternative candidate for describing and predicting their abundance [
3,
4,
11,
17].
In previous works [
56] it was shown that the relationship between climate factors and mosquito abundance is not linear over the full data length, and that mosquito populations exhibit a high degree of non-linear behavior under field conditions. As a result, periods of mosquito growth and different population levels are interrupted by the presence of unfavorable temperature conditions in a random way. To date, despite Markov chain models being used extensively in turbulence and predictability studies, as well as disease dynamics, they have not been used to model the abundance of
Culex sp.
In this context, the major objective of the current work was the development of a weather driven Markov chain model for simulating and predicting the population dynamics of arthropod vector dynamics. Moreover, the model is applied and tested on Culex sp. Mosquitos, which is the main vector of WNV transmission and thus of high medical importance. Additionally, the aim is to contribute to a precise prediction of the adult mosquito dynamics through the application of a conditional Markov chains. This paper presents a general stochastic modeling approach enabling the multivariate analysis of Culex sp. population dynamics and related weather variables. Avoiding inherent relationship assumptions and parameter estimation in deterministic models; stochastic models provide a realistic data-based alternative to simulation of complex systems and robust predictions that could make informed and sound decisions.
In the next section, we show the model derivation and final formulation, while in
Section 3 we apply the model for predicting
Culex sp. first with real data and numerical simulations of the population dynamics of mosquitos, then conditioned by the most detrimental weather variable [
56]. Additionally, we made efforts to validate the model using empirical data and which have not been used to parameterize the models. In the end, we briefly discuss the modeling, prediction results of the current subject field, as well as over the pros and cons of the proposed mathematical modeling approach and how it can lead to our understanding of
Culex sp. dynamics and help to reach decisions along the various interventions that can be made needed for public health.
2. Materials and Methods
Markov chain model is initially proposed for addressing the problem of predicting the time evolution of mosquito population dynamics throughout the season in a temperate, Mediterranean climate. First, the probability of a population at different population levels, particularly high levels, is predicted; and then the effect of a climate variable, particularly temperature, is included.
2.1. Stochastic Process of Ecological Time Series
Let X(t) be the ecological variable (e.g., mosquito population, or climatic variable), which is considered as a stochastic process that evolves in time t and is defined in a probability space (Ω, F, P). Where Ω is a sample space, F is a set of outcomes in the sample space and P assigns each event of F a probability. If the number of F is not countable then the process is denoted by (X (t): t ≥ 0), or (Xt) t ≥ 0. In the first case, the process is called a chain in discrete time and in the second, in continuous time. Here the first case is considered, since data have been observed in specific time points and not continuously.
Let S be the space created by all possible process values X(t) in discrete time. If S = (0, 1,...) the study refers to a stochastic process with integer values or a discrete state process, e.g., a population threshold or class that corresponds to the number of mosquitos captured in a day, or a class of mean temperature values for that day, etc. Hence, S is considered to take real and finite n values and this contemplative process is called an n-dimensional stochastic process.
2.2. The Markov Chain Model
The above stochastic process consists of a Markov chain which is determined by its initial state distribution and a transition probability matrix
P of size
m is [
32]:
The simplest kind of discrete variables the transition matrix may have two stages
S = (1, 2), which is defined in their simplest form as a high or low level of the ecological variable (e.g., mosquito vector population, temperature) or occurrence or not occurrence (e.g., rain). A sequence of weekly observations constitutes time series of that discrete variable. For the first order Markov chain, the transition probability to future state depends only on its current state. Thus, knowing that at week
i the variable
X is either in state 1 (low population levels
X(
i) =
a), or state 2 (high population levels
X(
i) =
b) the related row stochastic transition matrix is:
By considering m states,
S = (1,…,
m) a higher dimension of the transition matrix is formulated as follows:
A state Sj is said accessible from state Si (written Si → Sj) if the defined transition system starting in state Si has a positive probability to reach the state Sj at a certain point, i.e., ∃n > 0: > 0. Two states are said to communicate if both Si → Sj and Sj → Si. Moreover, any state Si is considered periodic if any return to state Si occur in multiplies of ki steps and ki is the period and ki = GCD (n: Pr(Xn = si |X0 = si) > 0), where GCD is the greatest common divisor. Thus, for ki = 1 the state Si is aperiodic, else if ki > 1 the state Si is periodic with period ki. In other words, a state is periodic if after a fixed number of transitions, ki > 1, the state can only return it itself otherwise it is aperiodic.
2.3. Data Inferred Markov Chain Modeling
Because the knowledge over the time evolution of the current mosquito population process is based on trap captures and is thus limited to derive laws and construct parameterizations from first principles, a data-driven method is used to construct parameterizations by inferring from data. Moreover, the number of adults is considered captured in the CO2 traps as a proxy of both, the size of the population and the related mosquito activity levels and despite that different traps vary in their ability to catch certain species.
First, the data are classified into different scale states (e.g., population levels) and a matrix is estimated for each scale state. In particular, for
m states there have to be
m2 matrix entries to be estimated. The transition probability matrix entry
P(
i,
j) is estimated as follows (32):
where
counting for the transition from m to n observed states observed in a given data set and
is the maximum likelihood estimator of
p(
i,
j).
Thus, the Markov chains are “trained with” data from real observations with the aim of mimicking the observed behavior of the population process afterwards in which a finite state MC is inferred from data by estimating its transition probability matrix:
2.4. The Conditional Markov Chain Model
The conditional Markov chain (CMC) model is formulated for the case analyzing the occurrence and level of mosquito population depending on the physical state of climate conditions. Particularly, since mosquitos are arthropods and all arthropods are poikilothermic organisms, their development and occurrence of states are affected by temperatures and rain (i.e., favorable versus unfavorable climate). This means that if a Markov chain is used to mimic the process of mosquito population occurrence
X(
t), it can be improved by taking into account the condition of a second process
Y(
t) which is related to a climate variable (e.g., state transitions of temperature or rain levels). Under this assumption, probabilities take the following form [
31,
57]:
2.5. Data Inferred Conditional Markov Chain Modeling
If a finite number of states is considered (say five as presented later), then it is possible to construct a CMC model by estimating a transition probability matrix,
, for each possible state. This can be done by knowing the time evolution of the ecological time series for a finite number of states that is used as basis to train the chain model. The transition matrix is estimated as follows [
32]:
In which 1 is the indicator function: 1(A) = 1 if A is true and 1(A) = 0 if A is false, while t runs over time instances in the data set used to train the Markov chain model.
2.6. Data Encoding and Determination of Transition States
In order to work with finite state conditional Markov chains, vector population dynamics and climatic variables, must be discrete and coded to a finite number of states. If the data are uniformly distributed, this can be done using tree classification schemes based on pre-defined thresholds.
However, since ecological data are most often not uniformly distributed, choosing thresholds is difficult and could result in classes to which no data are assigned and classes to which almost all data are assigned. Moreover, to overcome the problem of subjective defining the different mosquito population levels, k-means clustering algorithm has been used for partitioning the sequence of different population levels in different states based on their centroids.
In particular, each set of mosquito population observations
x1 was considered,
x2,…,
xn as a
d-dimensional real vector and implemented a standard,
k-means clustering algorithm to partition the n observations into
k sets (
k ≤
n) that correspond to discrete population states:
S = (
S1,
S2, …,
Sk) so as to minimize the within-cluster sum of squares (WCSS):
where
μi is the mean of points in
Si.
Let
nij denote the number of individuals who were in state
i in period
t − 1 and are in state
j in period
t. The probability of a mosquito population being in state
j in period
t given that they were in state
i in period
t − 1, denoted by
pij, can be estimated using the following formula [
58]:
The probability of transition from any given state i is equal to the proportion of mosquitos that started in state i and ended in state j as a proportion of all individuals in that started in state i.
Thus, using the above scheme, the observed behavioral stream of population dynamics was first converted into a symbolic sequence of population states to be used later on the estimation of the transition matrices. It was possible to estimate a transition matrix for each case using mosquito count data.
2.7. Markov Chain Model Validation and Equilibrium Distribution
Field data on frequencies of successive mosquito population levels, which were not used for the data inferred MC modeling, were assembled to obtain an empirical intensity transition matrix. Then the empirical transition matrices were generated, and the observed frequencies were compared visually, as well as statistically, with those obtained from the MC models. Two methods were used to evaluate the equidistribution between the observed sequences and that of the MC models.
First, the homogeneity of the transition matrices was tested using a Chi-square (ChiSq) minimum discrimination statistic test. The
verifyHomogeneity function was applied using the R package makovchain [
59,
60]. Considering in 2.1. the time evolution of the mosquito population as a stochastic process that generates:
i = 1, 2, …,
n discrete time Markov chain samples and that the cardinality of the state space is S the Homogeneity function verifies whether
l chains belong to the same unknown one. The function shows that its test statistics follows a chi-square law and is estimated as follows (59):
If there are l realizations of a Markov chain of order 1 with S states, the null hypothesis, H0, is that the transition probability matrix is the same for all i and for every possible pairing of j and k and P(X > Chsq), which is less than or equal to the significant level, α = 0.05
In the current work, the case of l = 2 chains was considered as two realizations of the S = 5-state Markov chains (theoretical and empirical) that are tested for homogeneity. The frequency entries, are: and i = 1,2, j = 1,2,… and k = 1,2,..,5.
Secondly, the asymptotic distribution properties of the theoretical and the empirical intensity transition matrices were compared using an entropy-based divergence distance measure [
54]. In this case, the mosquito population stochastic process was considered as homogenous, and thus starting from an initial distribution of population states a limiting probability,
, exist:
This is called the normalizing condition. Entropy is further associated:
represents the average amount of uncertainty of the population system for moving one step ahead being initially in state
Si. We are now interested in estimating the average uncertainty of the chain for moving one step ahead of any other initial state, which is [
54]:
and consider the Markov chain process as ergodic (e.g., MC and CMC models, as well as the observed mosquito population process used for model validation), so that:
Initially, we define a distance measure by introducing the following norm:
where
the entropy associated with the initial probability distribution
representing the different mosquito population levels and
,the entropy of each time step ahead t = 2, 3…
The above scheme quantifies the rate of convergence from a starting non-equilibrium probability distribution towards equilibrium.
2.8. Data
We applied the conditional Markov chain model using free mosquito trap data available from the open European Union Data Portal (EU ODP) (
http://data.europa.eu, accessed on 3 May 2019), which provides access to data from the European Union (EU) institutions and other EU bodies, which can be reused for commercial or non-commercial purposes (European Commission Decision 2011/833/EU). In particular, we used adult mosquito trap data of
Culex sp. sampled from 11 locations in central Macedonia–Greece. Data were handled as vectors which consisted of close to weekly time intervals of the number of adult mosquitos captured in CO2 traps from mid-May to September. We used data during two successive observation years (2011 and 2012) for training the Markov chain models, whilst data of one additional year, 2013, were used for validating the model performance.
Because of slight differences between the time intervals of some of the trap counts, data were transformed to mosquito per trap per day (MTD) and thereby averaged over the 11 nearby sampling locations [
56]. The MTD thus estimates the average number of mosquitos captured on a day that the trap is exposed in the field.
Weather data and particularly mean air temperatures in Celsius and rain events in mm, were obtained by the national observatory of Athens through a meteorological station, which is located in Makrohori town, which in the same location and latitude and near to the mosquito observation area (
http://stratus.meteo.noa.gr/front, accessed on 2 April 2020).
4. Discussion
Existing and emerging vector-borne diseases are representing one of the most important challenges to public health policy. The novelty of the work consists of the current methodology to simulate, predict the population dynamics of medically important vectors, and the determination for the first time of transition matrices from mosquito field data. This work is first of its kind which apply conditional Markov chains to predict Culex sp. adults’ dynamics and considering that most models are of deterministic nature. This approach could be helpful to develop control programs for vector-borne arthropods.
Compared to other models the MCs are simple and thus preferred in modeling complex systems and without detailed knowledge on their function, in order to study their performance and dependability to exogenous factors. However, although the current models have high prediction accuracy, they face the limitation of not providing a strict phenomenological explanation of the system. As a result, a direct biological interpretation of the current model parameters (transitions), as in the case of the Leslie model, cannot be made.
Moreover, the transition values estimated here might be the final result of the environmental conditions that are affecting vital life events of the Culex sp. natural populations. In other words, probabilistic models generated are an accurate abstraction of the particular species population process observed in this study and characteristic of the dataset used for their generation. From an entomological standpoint, in order to predict the population patterns of other insect species, the current models should be retrained with population and weather data of the new species and location of interest.
Moreover, it important to clarify that we have used empirical data of
Culex sp. adults captured in CO
2 traps which are further considered as a proxy for both, the size of the population, as well as the inherent mosquito activity of the particular study region. However, it is known in entomological studies that due to different modes of actions different traps may vary in their ability to catch certain species [
61,
62,
63,
64], while sampling condition such as the location of the trap [
65], or the number of nights over which sampling occur [
66], as well as weather conditions may also influence trapping results. Nevertheless, despite these limitations, the empirical data used provide means to interpret
Culex sp. activity that is highly likely to transmit the virus. Actually, variations in trapping outcomes of different data sets may not limit the application and inference of the current models since they deal with probabilities of population level successions rather than forecasting abundances of mosquito individuals
per se.
Based on the results of this study we conclude that model performance can be improved when temperature is taken into account and this is because environmental fluctuations are translated into population fluctuations through temperature-dependent difference in intrinsic demographic parameters (i.e., survivorship, fecundity, feeding behavior, etc). However, it is important to clarify that in this article we are mainly interested with predictions rather than forecasts despite that we consider the temporal dimension (i.e., weeks). Forecasting, sensu stricto, may be considered as a subfield of prediction which is used on the basis of future time series data generation. This is justified by the fact that we fit the model to a training data set, which results in a model that estimates the outcomes for unseen state transitions in terms of probabilities, rather than estimates of the actual mosquito abundance values (i.e., time series). Moreover, a similar approach has been applied on the validation data set to generate a future sequence of mosquito population states to be compared with the model predictions. This was also the main reason why we have used the Square-test to compare transition matrices, as well as related information measures, such as Shannon entropy, to compare the time evolution of the models towards equilibrium, rather than a correlation analysis.
Summarizing the modeling approach and related simulations, we consider here to look at some of the features of the conditional Markov chain modelling method under consideration. Classical conditional Markov chain models (also known as linear-chain conditional random fields in the literature e.g., Lafferty et al. 1999 [
57]), were defined by Bielecki and Rutkowski in 2004 [
55], for applications in finance and insurance. Conditional Markov chains, as proposed in the current work, have been also used in atmospheric science [
32]. This is done in response to the need for modeling dependence between dynamic systems in cases when some conditional properties of a system are important and should be accounted for. Hence, conditional Markov chains are defined as a versatile class of discriminative models for the distribution of a sequence of hidden or latent states conditional on a sequence of observable variables.
However, it should be noted that although the concept of Markov chains has been used already in pioneering works of theoretical biology [
51,
52,
53], in this work it is applied under a different context. For example, classical works in the field of transition models deal with age-structured cohorts, where survivorship and net maternity is known and extend the already know Leslie model to describe the limiting behavior of population growth and its sensitivity to environmental perturbations. Here, the use of Markov chains differs conceptually compared to classical theoretical Leslie projection schemes, since we have no prior information of the initial stage specific population structure and its demographic characteristics, and the only available information is the adult abundances which were estimated by traps. Contrarily, this work aims first on the partition of data, using a classical clustering algorithm, and later predicting the transitions of different mosquito population levels forced by temperature. Thus, this empirical work emphasizes applied modeling of a populations data sequence, which is most often available in entomological surveys, rather than a theoretical study which focus on a complete characterization of the species life-cycle transitions as a result of births and deaths.
Based on the simulation results it is apparent the model performs better when it is conditioned on temperature. In accordance with other studies, this work shows that environmental changes have impacts on demographic parameters and are reflected in the species population. In the case of the observed and predicted Culex sp. dynamics, it can be argued that temperature alteration during the season induce changes in developmental rates, survivorship and net maternity could underlie the transitions between the different population states.
In that sense our model simulations might suggest that a persistence of increased temperature levels for longer periods could be linked with an increase in fecundity and survival of immature stages, suggesting an increased fitness which is reflected to a sequence of high Culex sp. population levels. In the same sense, one cannot exclude the opposite function when temperature levels are lower. Moreover, other factors, which have not considered here, such as wind, relative humidity, could also affect Culex sp. feeding behavior and related dynamics. All these factors may be essential for the eco-biology of the species and affect its observed dynamics.
Given the importance of climate conditions for mosquito development, especially temperature and rain, it is necessary to take into account these variables in predicting mosquito populations. Furthermore, the estimation of the transition matrix through the use of empirical data first to define the system states and later on for training and validating the Markov chain model is a principal step for the simulation of realistic vector population projections and without the need for defining differential equations and related state variables. The CMC model that is proposed in this study might prove very suitable in public health decision making and especially for predicting arthropod vector population dynamic and vector controls. Several mathematical models have been developed to clarify and predict the dynamics of mosquito populations and to understand the role of environmental factors [
67,
68,
69,
70]. In most studies mosquito population dynamics are treated as deterministic processes (among others [
71,
72]) despite populations being driven by climate factors which are considered to be probabilistic [
73,
74,
75,
76,
77]. Therefore, it is difficult to perform a direct comparison of our modeling approach and results, to other related studies, since most of them are based on deterministic-dynamic population models. However, multivariate ecological and epidemiological time series are characterized by complex non-linear relationships and most often explicit a random behavior [
78].
Hence, the proposed MC stochastic model provides a robust alternative to traditional models. For instance, deterministic models cannot capture population fluctuations which are dominated by environmental conditions, variability in the controlling parameters as well as the random nature of population events, which occur in a real system [
33]. Moreover, the fact that the projection of mosquito population dynamics is improved by incorporation of information on temperature levels in terms of conditional probabilities is in accordance with most studies that acknowledge the significance of climate factors and temperatures particularly in arthropod vector population dynamics and related disease epidemics [
78,
79,
80].
Based on the current cross correlation results, we conclude that temperature exert a higher impact on the Culex sp. adult phenology compared to rain events, despite mosquitos thriving in wet conditions as rain indirectly affects the mosquito population by increasing breeding grounds. Therefore, it was judged as necessary to include the most influential meteorological variables (e.g., only temperature) to improve the performance of a simple MC model through the use of a CMC model instead. Actually, it was found that Markov chain model of arthropod vector population dynamics, conditioned over temperatures, performed better than single MC stochastic modeling of vector population dynamics.
After the importance of the meteorological conditions was found, it became apparent that once the population reaches a high state during a week, there is a very high probability that it will remain in this state the following week and so on. Moreover, considering that time evolution of temperature states is quite analogous of that of the arthropod vector states, we can conclude that if there is a high probability of increased temperatures, we expect an increased probability to observe very high mosquito levels. Thus, a part of the CMC mode results, modeling only temperatures through MC model may be proved very useful in judging whether during the same period the mosquito population is also high. This is of major importance for public health management and vector eradication programs considering saving costs and time for the establishment of mosquito surveillance programs over different areas. Thus, this information becomes crucial for preventing the transmission of mosquito disease prioritizing resources for optimal responses in vector eradication programs and mosquito surveillance programs.
Considering model validation, the transition probabilities of both Markov chain models (e.g., the simple, as well as the conditional one) do not differ significantly to the empirical data. However, there should also be some wariness as the data set used for model validation, despite being representative for a mosquito activity season, was only from a year of observations. Nevertheless, overall, the MC model and the empirical realization follow a similar pattern although despite some expected deviations.
The presented results are promising, although we stress that they were obtained under certain assumptions, such as the stability of the particular study environment and the conditioning over only one climate variable. For instance, Markov models are generally inappropriate over sufficiently short sequence lengths and time intervals yielding in a process which is deterministically related to time rather than random to resolve this problem we decided to evaluate the effect of different sequence lengths on the informational content of different MC sample trajectories. Additionally, considering the data sampling intervals of mosquito abundances, we have decided to use normalized weekly counts, for both, model training as well as for the predictions, since these are most often used in entomological studies to capture the dynamics of ecological processes.
In reality, additional ecological factors which have not been taken into account in this study may affect mosquito population dynamics in a more complex manner. Among such factors is the possibility of a parallel influence of two or more climate variables on the mosquito population dynamics or even a more substantial influence of lagged values that suggested also testing the model performance of a higher order Markov chain model.
Moreover, to reach more realistic Markov chain projection schemes, they should probably be compared and trained with additional observational data. Nevertheless, the current work outlines how the Markov chain models can be applied in ecological time series and particularly in modeling arthropod vector dynamics. Furthermore, the current work contributes to recent tendencies in ecological modeling which focus on the integration of climate factors and related weather variables in functioning of population processes. Another future direction which may be worth verifying would be the calculation of a multivariate semi-Markov conditional model with different orders.