Spatial dynamics of dengue fever spreading for the coexistence of two serotypes with an application to the city of São Paulo, Brazil
Introduction
Dengue is responsible for 96 million symptomatic cases and an estimated 40,000 death per year globally; also, around 3.9 billion people in 129 countries lives in risk regions for dengue [6], [44], [46]. Despite the efforts to combat and control the disease, dengue is one of the oldest diseases that humankind has contact [6], [26], [45], [53].
Having the vector for transmitting the disease as the Aedes aegypti female mosquito, the dengue fever has a peculiar problem: sequential infections may evolve to dengue shock syndrome, increasing disease severity due to a process called antibody-dependent enhancement [12], [21], [22], [40]. This is a big concern for a disease with five antigenically distinct viruses: DENV-1, DENV-2, DENV-3, DENV-4 and DENV-5 [1], [20], given that an individual has life-long immunity to the cured case serotype and a temporary low cross-immunity of 3 to 9 months for other serotypes [1], [32], [47], [61]. In Brazil, there is a concern that the co-circulation of multiple dengue serotypes, called hyperendemicity, is becoming standard for an increasing number of cities and metropolitan regions [11], [17], [59]. This is a problem since the 1990 decade, and it was confirmed by a few studies which report the presence of individuals with antibodies to three serotypes in northeastern Brazil in 2005 [4], [8], [24], [41], and the four serotypes with simultaneous circulation in Manaus [3], [4], [50].
The Aedes aegypti females mosquitoes breed in clean still water and go in search of human blood, reaching distances of eight hundred meters in six days [27]. Therefore, it is a disease with spreading dynamics dependent on the spatial distribution of humans, vectors and places of standing water. For instance, the work reported in Freitas et al. [18] showed that cases of dengue, zika and chikungunya were not detected at the same location and time in Rio de Janeiro in 2015/2016, with the reasons raised by the authors related to the spatial dynamics of the diseases: competition between viruses for breeding resources, and change in behaviour of susceptible human population as a response to increasing cases in the area. Such cases clusters were also studied in Bhoomiboonchoo et al. [5], where the depletion of cases occurred after spatial and temporal targeted interventions. Finally, other studies considered the landscape and seasonality to describe and predict dengue cases [31], [49], [62].
Therefore, the objective of this paper is to explore the spatial distributions of breeding places for mosquitoes in a model based on probabilistic cellular automata [47] for the co-circulation of two strains of the dengue virus. The basic reproduction number () is derived from the ordinary differential equations (ODE) proposed as a mean-field approximation of the cellular automata. Moreover, we propose a method for estimating the spatial basic reproduction number () to understand the real impact of this spatial configuration on the disease outbreak. The following points summarize the novelty of the paper: consider two dengue serotypes for numerical spatial analysis, given that many regions in Brazil face such a problem [3], [11], [59]; explore spatial distributions of mosquitoes breeding places; obtain the basic reproduction number for a model which takes into account the severity of a second infection of dengue; propose a spatial numerical basic reproduction number based on similar processes of spatial analysis; use the model for an accurate population distribution to estimate the impact of mosquitoes breeding places in residences and public spaces.
The basic reproduction number is “an epidemiologic metric used to describe the contagiousness or transmissibility of infectious agents” [13]. It represents the number of secondary infections from a single infected individual in a population fully susceptible to the disease. When ordinary differential equations (ODE) are used for compartmental models, is also a bifurcation parameter, where its value comes from the stability analysis of the system [28], [33].
Regarding vector-borne diseases, can be interpreted as the number of secondary disease cases in humans from a single infected human individual in a population of humans and vectors completely susceptible [7]; which is a metric most often used due to the difficulty of estimating biological and environmental mosquitoes information [13], [28], [38], [43], [58]. The value of for dengue depends on many variables, such as population density, social organization, and weather seasonality [13], with global warming being a concern as it may increase the of dengue in temperate regions [34].
The next-generation matrix can be used to derive the basic reproduction number. The matrix is formed by the infectious subsystem of a compartmental model linearised about the disease-free equilibrium state. It represents the expected number of secondary infections caused by a single infected individual for a population entirely susceptible for each combination of row and columns of the matrix. This method was first described in Diekmann et al. [15] and further elaborated in Van Den Driessche and Watmough [56]. Here, the basic reproduction number is calculated from the ordinary differential equations using the next-generation matrix [60]. The ODE parameters are obtained from the probabilistic cellular automata simulations, in which multiple breeding places distributions are used.
Heterogeneous spatial distributions of human and vector populations require alternatives for modelling vector-borne diseases dynamics. Releases of female Aedes aegypti and Aedes albopictus to estimate their flight range is an example of an experimental process to evaluate the spatial infection spreading in Rio de Janeiro, Brazil [27]. In the same city, patches of the map representing neighbourhoods were used to detect spatio-temporal clustering of dengue, chikungunya and Zika, assessing the cases of the disease per patch [18]. Patches of maps have also been used for space-time scan statistical analyses of dengue in China [62], and to mapping the basic reproduction number for vector-borne diseases in Netherlands [25]. Still, in the Hartemink et al. study, the basic reproduction number is calculated for 107 spots in the map, representing the local situation in these regions. Mathematical models demonstrate that the spatial density of the winged female Aedes aegypti and the aquatic form of mosquitoes are distance-dependent to breeding sites [5], [16]. Therefore, studies consider the distance to a point in the map as a variable to determine epidemiological parameters, such as the basic reproduction number [39], mosquito eggs count in a map [49], and the probability of being bitten by a tick in forests [57].
A water supply crisis took place in the southeast Brazil region in 2014–2015. It was partly responsible for a peak in annual dengue cases since households started to store water in inappropriate compartments, which helped the multiplication of mosquito breeding sites. Only in the city of São Paulo, more than 100,000 cases have been reported in 2015 [55]. This is one of the reasons to apply the developed model and methodology for a neighbourhood in São Paulo. Two scenarios are tested to explore the dynamics of the spatial distributions of vector breeding sites. The first considers that these breeding sites are on residential areas, and the second considers public squares, parks and schools.
This paper is organized as follows: the next section contains the methodology and description of the models, the results of the simulations are in Section 3, and a final discussion is presented in Section 4.
Section snippets
Methods
The human population is based on a probabilistic cellular automaton with cells representing individuals in one of the disease’s states. The lattice is square with side and a total human population of , and the neighbourhood, i.e., the neighbour cells that individuals visit per time step consists of visits inside a Moore radius per time step. The neighbourhood is different at each time step, with the visits being randomly chosen according to the following procedure: the individual
Results
For the simulations, we consider a lattice with side , and human individuals. The vector constant population consists of vector individuals initially distributed over the 3600 breeding cells. Further details will be explained later. The parameters for PCA simulations are , , , , , and . The simulation runs for time steps starting with the initial conditions , , and
The model in a real map
The model is adapted to a real map for evaluating the impact of breeding cells over space. Therefore, we used a region from the east part of the São Paulo city called Vila Curuça, where there are some required characteristics on the map, such as residential neighbourhood, approximately 10,000 residents per square kilometre, and public areas consisting of squares, parks and children’s education schools. Therefore, the map and the layers for each feature of the map were downloaded from [42] using
Discussion
This paper proposed a model based on probabilistic cellular automata to study the dengue fever dynamics on a few spatial distributions of breeding places. Ordinary differential equations were used to derive the basic reproduction number from the next-generation matrix method. Simulations were set so that the population was in the steady state for one strain of the disease when another strain was added to the population. Five spatial distributions of breeding places were tested, and the cases
Statements of ethical approval
Ethics approval was not required for this study.
Funding
PHTS is supported by grants #440025/2020-6, #307194/2019-1 and #402874/2016-1 of Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). FMMP is supported by scholarship grant #88887.505459/2020-0 of Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
PHTS and FMPP would like to show gratitude to Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and São Paulo Research Foundation (FAPESP) due to the current and past funded projects.
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