Some features on methodology of dengue modelling linked to data

Many natural and social systems, formed by huge number of elements, with nonlinear interactions between them, may generate different dynamical scenarios. Concepts and techniques of Statistical Physics, Nonlinear Dynamics, Complex Networks are useful to study those systems. The mathematical modelling is a kind of “umbrella method” to investigate their dynamical behaviour; in particular, in the context of living systems, there is vast literature on modelling of infectious diseases both communicable and vector-borne ones. Among the vector-borne diseases, our focus is on mathematical modelling of Dengue, also making a comment on the recent systematic review paper [1].

As a general view of the methodology used for mathematical modelling of diseases, I highlight two classifications of models. For the first one, related to the way of presenting the interacting elements of the system, there are population-based and individual-based models; Aguiar and collaborators consider mainly population-based models in their review paper, for which populations of humans and mosquitoes are divided into different compartments, which is traditionally used in Mathematical Epidemiology [2]. The individual-based models, mostly used in the context of Statistical Physics from the tradition of lattice models [3], are applied to infectious diseases [4], including cellular automata and agent models as well as individual-based models in complex networks [5]; see, for instance, a cellular automata dengue model in [6]. An intermediate approach is adopted through the meta-population models with the aim of considering spatially distributed populations, which is traditionally used in the context of ecological systems [7]. The second classification - this one was considered by the authors in the review paper as a criterion of classifying the selected papers about dengue modelling - is related to the scale of modelling; there are inter-host models (host-to-host and vector-host models) and intra-host models (or within host models). Even more it is possible to consider other scales, such as protein/gene scale and construct multi-scale models. It is worth saying that there is a well-established literature for multi-scale modelling of cancer [8]; I call attention of the reader to reference [9], - developed by researchers who are experts in cancer modelling - taking into account the scales of viruses, cells and populations to construct a multi-scale framework for COVID-19 (also reference 180 of [1]).

Among the vector-borne diseases, the complex dynamics of dengue fever has several factors, treated through mathematical modelling, also pointed out on the review paper: the multi-strain character with 4 serotypes, with long-term immunity to the homologous virus and temporary immunity to the heterologous viruses, the antigen-dependent enhancement (ADE) that may be related to severe forms of the disease, the impact of multi-strain character on a safe tetravalent vaccine, the seasonal variations on vector dynamics, the limitations and environment effects of larvicide and adulticide as vector control procedures. They highlight two relevant points discussed in the literature based on multi-strain dengue models, for which their research group has contributed: the emergence of deterministic chaos in long term dynamics of dengue and the conditions for adoption of tetravalent vaccines. Concerning multi-strain models, I remind that the effect of cross-reactivity in Dengue dynamics was already considered in the literature since nineties (see, for instance, a multi-strain dengue model in [10]).

It is worth noting that, nowadays, dengue infection spreads around the world reaching countries away from the tropical regions since dengue viruses became adapted to lower temperature and climate changes also impact in the mosquitoes dynamics [11]. Even more, the absence of a safe tetravalent vaccine against dengue, up to now, that also contemplate seronegative humans [12], and the uncertainties related to different efficacy against dengue for each serotype make the techniques of vector control [13], [14] still relevant to be investigated, also from the modelling point of view. Due to that current scenario, host-to-host models – that do not model the mosquito dynamics explicitly - may mask some features that could help the understanding of the dynamics of dengue transmission. Although that approach is assumed in the literature, as listed in Table 1 in [1], and there are some justifications from the mathematical and physical points of view [15], in my own opinion, it is a huge simplification even if the focus of the modelling is neither vector control nor seasonal effects. In fact, that point may be contained in a wide discussion about the differences between mathematical modelling of living systems and physical systems: the degree of simplification in modelling living systems, and, also social systems, should be more restricted than for physical systems mainly because the former ones are more adaptative and their variability is much more significative.

For sure, the variability of scenarios for living systems is well-embraced by estimating of parameters values of the model, which is a special chapter of the “novel of mathematical modelling”, since it is the way within reach of validating models in order to gain confidence for using them to perform predictions and to design scenarios; as an example see the vaccination coverage based on previous dengue epidemics [16]. The estimation of parameters is also associated with fitted real epidemic data, that was also considered as an item of raised literature in the review paper. In this chapter, the main point is the balance between the quantity of parameters to be fitted according to real data and the parameters whose values are fixed within range of values based on information on the literature about the disease. The former parameters depend on the specific epidemics and on the characteristics of the city where it occurs such as seasonal parameters. There was advance in techniques for estimating model parameter values, minimizing errors between real and simulated data, based on Machine Learning [17]. Furthermore sensitivity and optimal control analyses of parameters are other techniques for analysing their role for the results of the model.

There are some other nuances in performing numerical simulations linking modelling and data, such as considering report rate due to high sub-notification rates of cases, which increases for notifications by serotype. There is also a gap in real data of mosquitoes; although there are many data collection of mosquitoes using traps for Aedes Aegypti, there is no long time series of captured female mosquitoes (see an except study in Foz do Iguacu - Brazil [18]). The essential measures of basic ($R_0$) and effective ($R(t)$) reproduction numbers, for which there is vast literature, also link model to data (see [19] for dengue and [20] for COVID-19]); we highlight a general method of calculating $R(t)$ [21], based on the next generation method [22], commonly used for calculating $R_0$. Besides the co-circulation of dengue serotypes, Aguiar and collaborators discussed about concomitant epidemics of Dengue and COVID-19; I also call the attention of the reader for some results about cross-protection [23] and co-circulation [24] of Dengue and Zika viruses, both of them transmitted by Aedes Aegypti, as it occurs in Brazil in 2015-2016, when Zika virus enters in the country.

Finally the art of mathematical modelling of diseases, in particular of Dengue fever, involves many features and choices according to the raised research question. The compromise between simplification of the model and its capacity to describe real situations is a challenge of mathematical modelling. In my point of view, due to the complexity of living systems, the results obtained from mathematical modelling constitute a kind of puzzle, whose fit lead to a better understanding of the dynamics. In case of Dengue modelling, many puzzle pieces are fitted, but other ones still have to be fitted; in that direction, the publication of the commented review paper is a very useful contribution to “jigsaw puzzle assembly”.

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.

Communicated by J. Fontanari