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Mathematical and Theoretical Biology

Our research group is currently working on:

COVID19:

In March 2020, a multidisciplinary task force (so-called Basque Modelling Task Force, BMTF) was created to assist the Basque Health managers and the Basque Government during the COVID-19 responses. The BMTF Objectives are:
- description of the epidemic in terms of disease spreading and control.
- projections on the national health system needs during the increased population demand on hospital admissions.
- monitoring the disease transmission when the country lockdown was gradually lifted.
- evaluating the different epidemic phases (wave) severity and the impact of the new restriction measures, from lockdown implementation up to lockdown lifting.

As the COVID-19 pandemic is unfolding, mathematical models to describe the impact of the COVID19 pandemic in the Basque Country and other European regions are developed and analyzed. To describe accurately the process of disease spreading we use the stochastic SHARUCD model (susceptible (S), hospitalized (H) including all severe cases, asymptomatic (A) that includes also sub-clinical or mild infections, recovered (R), patients admitted to the intensive care units (U) from the recorded cumulative positive cases (C) which includes all new positive cases for each class, and death (D)).

The SHARUCD model is able to describe the available empirical data on disease incidence and is tested on short term predictions every week. This framework is now used to monitor the effect of the control measures on the infectivity of the population in order assist the Basque Health Managers and the Basque government [16-19].

SHARUCD Model validation from March 4 to October 15, 2020 (incidences). Data are plotted as black line. The mean of 200 stochastic realizations are plotted as a blue line. The 95% confidence intervals are obtained empirically from 200 stochastic realizations and are plotted as purple shadow. 15 days predictions are shown. In a) hospitalizations and in b) ICU admission. In c) deceased cases and in d) detected PCR positive cases. For more details, please visit our modeling webpage here

Public Health Epidemiology:

Mathematical modeling is an important tool for the understanding of infectious disease epidemiology and dynamics, leading to great advances in providing tools for identifying possible approaches to control disease transmission and for assessing the potential impact of different public health intervention measures. Focused On basic and applied aspects of host, pathogen, and environmental factors that influence disease emergence, transmission and spread, epidemiological models are formulated to describe the transmission of the disease and to predict future outbreaks, addressing specific public health questions on disease epidemiology, prevention and control.

M. Aguiar (Research Line Leader, Ikerbasque Researcher) has large experience in modeling infectious diseases dynamics with focus in dengue fever epidemiology. Multi-strain dengue dynamics have been modeled with extended Susceptible-Infected-Recovered (SIR)-type models including immunological aspects of the disease such as the Antibody Dependent-Enhancement (ADE) phenomenology; Maira has investigated a minimalist twoinfection dengue model, with at least two different dengue serotypes to describe differences between primary and secondary infections [1,2]. Deterministic chaos was found in much wider parameter regions (not predicted by previous models), no longer needing to restrict the infectivity on a secondary infection to be much larger than the infectivity on primary infection. The minimalistic model successfully described large fluctuations observed in empirical outbreak data, estimating lower infection rate for secondary dengue infections than for primary infections[1-3], anticipating results published recently in Duong et al. PNAS 2015. Aguiar has also shown that the combination of immunological aspects of the disease such as temporary cross-immunity (TCI) and ADE are the most important features to drive the complex dynamics in the system, more than the detailed number of serotypes to be added in the model [3]. However, this work is focusing on the multi-strain aspect of the disease and its effects on the host population only, taking effects of the vector dynamics into account only by the model parameters. Regarding the newly licensed dengue vaccine, opposing the predictions made by other groups, Aguiar et al. have discussed the risks behind this vaccine recommendation [4], after analyzing an age-structured model [5]. Using the publicly available vaccine trial data, vaccine efficacy was estimated via the Bayesian approach, predicting a significant reduction of hospitalizations only when the vaccine is given to seropositive individuals [6-9]. This work is still ongoing and includes research on:

i) infectious disease transmission dynamics and prediction; ii) infectious disease control (including vector control and vaccination, with eventual industry collaboration), iii) epidemiological dynamics of multi-strain infectious diseases, iv) interplay between pathogen serotypes and cross-immunity, v) epidemiological dynamics of vaccine preventable diseases, vi) epidemiological dynamics of vector-pathogen-host interaction [10], vii) within-host dynamics focusing on immune responses, viii) complex dynamics, empirical data analysis and public health intervention measures.

Models are parameterized on disease incidence data considering the prevalence of infection in different endemic countries. A combination of techniques, including non-linear time series analysis, classical bifurcation diagrams and Lyapunov exponent spectra are used.

Compilation of different analyses of multi-strain complex systems. In a) Bifurcation diagram showing two chaotic windows in wide parameter regions for the minimalistic multi-strain dengue model. In b) state space analyses of attractors found during the bifurcation analysis performed in a). In c) 2D Lyapunov spectra used to evaluate the prediction horizon of a simple 2-infection dengue model. In d) Spatial temporal pattern of dengue disease spreading in Thailand.

V. Anam (PhD Student "Predoc. Severo Ochoa" 2020; within-host modeling.) Vizda`s PhD thesis is focused on modeling the host-immune components involved in the progression of disease. Within-host modeling approach will be used to understand the immunopathogenesis of severe disease as primary and as subsequent infections with the same pathogens and also for different pathogens, including animal-to-human spillover of new pathogens with pandemic potential. The common feature of different infectious diseases will be investigated. Laboratory data for antibodies mediating immune response will be used to validate the model and then, the possible imperfect and selective vaccine efficacy will be analyzed.

Population dynamics and methodological topics in natural sciences and mathematics:

i) Time-scale separation and center manifold analysis describing vector-borne disease dynamics

In vector-borne diseases, the human hosts epidemiology often acts on a much slower time scale than the one of the mosquitoes transmitting as a vector from human to human, due to their vastly different life cycles. We investigate in how far the fast time scale of the mosquito epidemiology can be slaved by the slower human epidemiology, so that for the understanding of human disease data mainly the dynamics of the human time scale of immunological status is essential and only slightly perturbed by the mosquito dynamics [11].

ii) Stochastic processes and spatially extended systems

We describe spreading of diseases in geographical space via super-diffusion. Nowadays people travel a lot over wide distances and therefore the spread of the infection happens not only locally, i.e., from one person to the neighbor, but also for large distances. Super-diffusion has been suggested to model this type of epidemiological spreading in space. We consider the analytically tractable case of a diffusion like process on the lattice which is used as a surrogate process of human contacts in epidemiology. A stochastic process for a population is then used where the notion of distance is given by power law decaying connectivities, in good agreement with the analytics.

For dengue fever, good empirical data are available for different endemic countries. We develop spatial extended systems to investigate disease transmission and the impact of disease control.

iii) Bio-statistics, model comparison and parameter estimation of chaotic systems

The parameter estimation framework for population biological dynamical systems are applied to calibrate the models developed. Initial studies have been performed, partly on analytically solvable simple models, now ready for application also to complex system as investigated here. For parameter estimation in complex dynamical multi-strain models, a novel and ambitious application of recently developed techniques for parameter estimation for chaotic systems, a method called maximum likelihood iterated filtering (MIF) is being implemented and refined by including dynamic noise in likelihood functions for multi-strain dynamics. Indeed, understanding how stochasticity interacts with the deterministic components of epidemiological models would have important benefits on the practical predictability of the dynamical systems. To be predictive and not include unnecessary components in the mathematical models used, parsimony is important. Statistical methods are complemented with dynamical insight into the processes, in order to obtain biologically relevant parameter regions. New tools of non-linear data analysis like Takens embedding are available, and allow to obtain topological information (fixed points, periodic orbits and the nature of chaotic attractors) about the whole multi-strain epidemiological system from time series of overall infected only.

Besides using maximum likelihood methods and Bayesian approaches for the estimation of parameters, especially the Bayesian framework which gives normalized probabilities not only for parameters conditioned on given data, we also work with probabilities for different models conditioned on the same data.

Compilation of different analyses. Using the Bayesian approach, in a) vaccine efficacy distribution by serostatus and age. In b) vaccine efficacy estimation changing over time, likely to be related to waining immunity. In c) a spatially extended stochastic simulation for reinfection models and in d) time scale separation of mosquito infection and human infection.

D. Knopoff (BCAM Post-doctoral fellow; population dynamics, complex living systems) is a mathematician and chemical engineer. Damian`s research is mainly focused on the mathematical modeling of complex living systems, with special emphasis on crowd dynamics, socio-economic problems and, more recently, epidemiological models for antimicrobial resistant infections.

iv) Horizontal transfer and bacterial antibiotic resistance (Keywords: horizontal transfer, cell conjugation, mutualism, antibiotic resistance):

This work is carried out in collaboration with a team of biologists mainly interested in the mutualism established between legumes and soil bacteria known as rhizobia. These bacteria from soil infect plant roots and reproduce inside root nodules where they fix atmospheric N2 for plant nutrition, receiving carbohydrates in exchange. Host-plant sanctions against non N2 fixing, cheating bacterial symbionts have been proposed to act in the legume-Rhizobium symbiosis, to preserve the mutualistic relationship [20]. Modelling and understanding this phenomena is important to analyse and improve soil fertility and agricultural production. The obtained results agree with increasing experimental evidence and theoretical work showing that mutualisms can persist in the presence of cheating partners.

A hallmark in this biological scenario is the horizontal transmission of symbiotic plasmids, which also acts in the acquisition of bacterial resistance to antibiotics. Indeed, last year the UN issued an urgent warning on the growing peril of drug-resistant infections, stating that it may be the main cause of death in the world by 2050. This fact poses a lot of questions regarding the use of antibiotics: how does it lead to resistance, how to avoid this problem and, ultimately, what can mathematics do to tackle this issue. Consistently with these questions, in collaboration with physicians from the Red Cross, we are trying to model and predict the evolution of these infections through kinetic and epidemiological models [21]. Using a simple extension of a classical SIR model (see transfer diagram in (a)), we were able to find a correlation between the outpatient use of antibiotics and the onset of bacterial resistance (b):

v) Crowd dynamics (Keywords: crowd dynamics, kinetic theory, active particles, evacuation, fundamental diagrams):

The study of crowd dynamics can contribute to tackle safety problems of interest for the well-being of our society, for instance, by supporting crisis management in critical situations such as sudden evacuation dynamics induced through complex venues by incidents. Simulations can support crisis managers to handle the competition of antagonist groups in a crowd. Modeling is really challenging, since it has to tackle several issues related to the geometry of the domain where the crowd moves, interaction with walls and obstacles, signaling, as well of emotional features of single pedestrians, like stress contagion or presence of leaders. All of these features influence the dynamics towards emerging behaviors and pattern formation. In addition, numerical methods for solving kinetic theory models efficiently are needed, including the Monte Carlo particle methods that turn out to be particularly suitable for mitigating the computational cost of simulations.

Some contributions in this line of research include the evacuation of a crowd from bounded domains [22], while the recent Lecture Notes [23] present a detailed study of this topic. Coupling epidemiological models with crowd dynamics is a very promising line for further research perspectives.

Case-study: Dependence of evacuation time on the size of the exit door from a square room of side length 10 m. (a) Initial distribution of about 46 people grouped into two clusters moving one against the other with opposite directions and exit size 2.6 m. (b-e) Evacuation progress for t = 0 s, 1.51 s, 6.06 s, 12.87 s, 17.42 s, respectively. (f) Evacuation times for different sizes of the exit door.

Physiological and Medical Systems:

(Keywords: non-local electrophysiology, brain tumor modeling, radio frequency ablation)

N. Cusimano (BCAM Post-doctoral fellow; non-local electrophysiology, Bayesian analysis, spatio-temporal models of complex systems). Nicole has a solid background in electrophysiology and one of the main foci of her research has been the use of fractional order differential operators to describe electrical signal propagation in complex biological media characterized by high levels of heterogeneity (e.g., cardiac tissue) [12]. More theoretical aspects of her research involve the study of numerical methods for the discretization of fractional powers of elliptic operators and the solution of the corresponding fractional partial differential equations [13]. Nicole is particularly interested in merging theoretical results and experimental data [14] and is currently involved in the analysis of RNA-Seq data extracted from breast cancer cells for the study of the mechanisms underlying resistance to drug therapy. In this context, she has been working on both simulation-based methodologies for forward uncertainty propagation and Bayesian approaches to inverse uncertainty quantification. Finally, Nicole is contributing to the epidemiological SHARUCD model dashboard and has recently started working on COVID-19 as well as Influenza data for the Basque Country provided by the Basque Health Service (Osakidetza).

In a) Starting from an initial pool of candidate parameter sets, a genetic algorithm uses its characteristic operations (selection, crossover, and mutation) to evolve the population over successive generations and explore the parameter space. The figure shows the evolution of G1 over five successive generations (left) and the fitness value assigned to all individuals in the considered five populations (right).

In b) The non-local diffusion term in the space-fractional model of cardiac electrophysiology can be seen as the sum of standard diffusion (corresponding to an ideally homogeneous tissue) and a hypothetical non-local current quantifying the effect of tissue heterogeneities on excitation dynamics. This figure shows the differences in the local and non-local case, at two different spatial locations, in the action potential of the Beeler-Reuter model (top row) and in the non-local current during activation (mid row) and recovery (bottom row).

M. Echeverria Ferrero (PhD Student "Predoc. Severo Ochoa" 2018; radio frequency ablation). Marinas PhD thesis will consist of developing a patient-specific mathematical model for the treatment of cardiac arrhythmias. As an extension of [Petras, A. (2019). A computational model of open irrigated radio frequency catheter ablation accounting for mechanical properties of the cardiac tissue. International journal for numerical methods in biomedical engineering], the model will include both real geometries from medical imaging and realistic features of a medical procedure called radio frequency catheter ablation (RFCA), e.g., efficiently simulate the effect of catheter movement during the ablation procedure at different ablation sites, through simulations of RFCA. The mathematical model addresses the physical phenomena regarding the interaction between the RF energy and the cardiac tissue by means of Partial Differential Equations, which are solved numerically in an open-source Finite Element platform.

In a) Full 3D computational geometry. Right: detail of the computational catheter tips: the assembled catheter tip (top left), the saline pipes (top right), the thermistor (bottom left) and the electrode (bottom right). Green color for hemispherical catheter tip, blue for cylindrical.

In b) Lateral slices of the created lesion for spherical (top row) and cylindrical (bottom row) simulations, for catheter insertion forces of 5g, 10g, 15g and 20g. The pops in the tissue are highlighted as gray spheres (bubbles) and the popping time (in seconds) is indicated.

Selected Publications:

[1] Aguiar, M., Ballesteros, S., Kooi, B.W. & Stollenwerk, N. (2011). The role of seasonality and import in a minimalistic multi-strain dengue model capturing differences between primary and secondary infections: complex dynamics and its implications for data analysis. Journal of Theoretical Biology, 289, 181-196.
[2] Aguiar, M., Stollenwerk, N. & Kooi, W.B. (2012). Scaling of stochasticity in dengue hemorrhagic fever epidemics. Mathematical Modelling of Natural Phenomena, 7, 1-11.
[3] Aguiar, M., Kooi, W.B., Rocha, F., Ghaffari, P. & Stollenwerk, N. (2013). How much complexity is needed to describe the fluctuations observed in dengue hemorrhagic fever incidence data? Ecological Complexity, 16, 31-40.
[4] Aguiar, M., Stollenwerk, N. & Halstead S.B. (2016). The risks behind Dengvaxia recommendation. The Lancet Infectious Diseases, 16, 882-883.
[5] Aguiar, M., Stollenwerk N. & Halstead, S.B. (2016). Modeling the impact of the newly licensed dengue vaccine in endemic countries, PLoS Neglected Tropical Diseases, 10(12), e0005179.
[6] Aguiar, M., Halstead, S.B. & Stollenwerk, N. (2017). Consider stopping dengvaxia administration without immunological screening. Expert Review of Vaccines, 16 (4), 301-302.
[7] Aguiar, M. (2018). Dengue vaccination: a more ethical approach is needed. The Lancet, 391(10132), 1769-1770.
[8] Aguiar, M. & Stollenwerk, N. (2018). Dengvaxia: age as surrogate for serostatus. The Lancet Infectious Diseases, 18(3), 245.
[9] Aguiar, M. & Stollenwerk, N. (2018). Dengvaxia efficacy dependency on serostatus: a closer look at more recent data. Clinical Infectious Diseases, 66(4), 641-642.
[10] Rashkov, P., Venturino, E., Aguiar, M., Stollenwerk, N. & Kooi, B.W. (2019). On the role of vector modeling in a minimalistic epidemic model. Mathematical Biosciences and Engineering,16(5):4314-4338
[11] Rocha, F., Aguiar, M., Souza, M. & Stollenwerk, N. (2013). Time scale separation and center manifold analysis describing vector borne diseases dynamics. International Journal of Computer Mathematics, 90, 2105-2125.
[12] Cusimano, N., Gizzi, A., Fenton, F.H., Filippi, S.,& Gerardo-Giorda, L. (2020). Key aspects for effective mathematical modelling of fractional diffusion in cardiac electrophysiology: A quantitative study. Communications in Nonlinear Science and Numerical Simulation, 84, 105152.
[13] Cusimano, N., del Teso, F. & Gerardo-Giorda, L. (forthcoming). Numerical approximations for fractional elliptic equations via the method of semigroups. ESAIM: Mathematical Modeling and Numerical Analysis.
[14] Drovandi, C.C., Cusimano, N., Psaltis, S., Lawson, B.A.J., Pettitt, A.N., Burrage, P. & Burrage, K. (2016). Sampling methods for exploring between-subject variability in cardiac electrophysiology experiments. Journal of the Royal Society Interface, 13, 20160214.
[15] Conte, M., Gerardo-Giorda, L. & Groppi, M. (2020). Glioma invasion and its interplay with nervous tissue and therapy: A multiscale model. Journal of theoretical biology, 486, 110088.
[16] Aguiar, M., Ortuondo, E.M., Bidaurrazaga Van-Dierdonck, J. et al. Modelling COVID 19 in the Basque Country from introduction to control measure response. Sci Rep 10, 17306 (2020). Link
[17] Aguiar, M., Van-Dierdonck, J. B. & Stollenwerk, N. Reproduction ratio and growth rates: measures for an unfolding pandemic. PLoS ONE 15(7), e0236620 (2020). Link
[18] Aguiar, M. & Stollenwerk, N. Condition-specific mortality risk can explain differences in COVID-19 case fatality ratios around the globe. J. Public Health. Link
[19] Aguiar, M. & Stollenwerk, N. SHAR and effective SIR models: from dengue fever toy models to a COVID-19 fully parametrized SHARUCD framework. Commun. Biomath. Sci. 3(1), 60 - 89 (2020). Link
[20] Moyano, G., Marco, D., Knopoff, D., Torres, G., & Turner, C. (2017). Explaining coexistence of nitrogen fixing and non-fixing rhizobia in legume-rhizobia mutualism using mathematical modeling. Mathematical Biosciences, 292, 30-35. Link
[21] Knopoff, D. & Trucco, F. (2020). A compartmental model for antibiotic resistant bacterial infections over networks. International Journal of Biomathematics, 13(1), 2050001. Link
[22] Agnelli, J.P., Colasuonno, F. & Knopoff, D. (2015). A kinetic theory approach to the dynamics of crowd evacuation from bounded domains. Mathematical Models and Methods in Applied Sciences, 25(1), 109-129. Link
[23] Aylaj, B., Bellomo, N., Gibelli, L., & Knopoff, D. (2020). Crowd dynamics by kinetic theory modeling complexity, modeling, simulations, and safety. Synthesis Lectures on Mathematics and Statistics, 12(4), 1-98. Link

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Eusko Jaurlaritza - Gobierno Vasco ikerbasque - Basque Foundation for Science Bizkaia xede. Bizkaiko Foru Aldundia innobasque - Agencia vasca de la innovación Universidad del PaÌs Vasco (UPV/EHU)