Spatially extended SHAR epidemiological framework of infectious disease transmission

Mathematical models play an important role in epidemiology. The inclusion of a spatial component in epidemiological models is especially important to understand and address many relevant ecological and public health questions, e.g., when wanting to differentiate transmission patterns across geographical regions or when considering spatially heterogeneous intervention measures. However, the introduction of spatial effects can have significant consequences on the observed model dynamics and hence must be carefully analyzed and interpreted. Cellular automata epidemiological models typically rely on simplified computational grids but can provide valuable insight into the spatial dynamics of transmission within a population by suitably accounting for the connections between individuals in the considered community. In this paper, we describe a stochastic cellular automata disease model based on an extension of the traditional Susceptible-Infected-Recovered (SIR) compartmentalization of the population, namely, the Susceptible-Hospitalized-Asymptomatic-Recovered (SHAR) formulation, in which infected individuals either present a severe form of the disease, thus requiring hospitalization, or belong to the so-called mild/asymptomatic class. The critical transmission threshold is derived analytically in the nonspatial SHAR formulation, and this generalizes previously obtained theoretical results for the SIR model. We present simulation results discussing the effect of key model parameters and of spatial correlations on model outputs and propose an algorithm for tracking the evolution of infection clusters within the considered population. Focusing on the role of import and criticality on the overall dynamics, we conclude that the current spatial setting increases the critical transmission threshold in comparison to the nonspatial model.Fil: Knopoff, Damián Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: Cusimano, Nicole. Basque Center For Applied Mathematics; EspañaFil: Stollenwerk, Nico. Basque Center For Applied Mathematics; EspañaFil: Aguiar, Maíra. Basque Center For Applied Mathematics; Españ

Fractional models in space for diffusive processes in heterogeneous media with applications in cell motility and electrical signal propagation

This work addresses fundamental issues in the mathematical modelling of the diffusive motion of particles in biological and physiological settings. New mathematical results are proved and implemented in computer models for the colonisation of the embryonic gut by neural cells and the propagation of electrical waves in the heart, offering new insights into the relationships between structure and function. In particular, the thesis focuses on the use of non-local differential operators of non-integer order to capture the main features of diffusion processes occurring in complex spatial structures characterised by high levels of heterogeneity

On the order of the fractional Laplacian in determining the spatio-temporal evolution of a space-fractional model of cardiac electrophysiology

Space-fractional operators have been used with success in a variety of practical applications to describe transport processes in media characterised by spatial connectivity properties and high structural heterogeneity altering the classical laws of diffusion. This study provides a systematic investigation of the spatio-temporal effects of a space-fractional model in cardiac electrophysiology. We consider a simplified model of electrical pulse propagation through cardiac tissue, namely the monodomain formulation of the Beeler-Reuter cell model on insulated tissue fibres, and obtain a space-fractional modification of the model by using the spectral definition of the one-dimensional continuous fractional Laplacian. The spectral decomposition of the fractional operator allows us to develop an efficient numerical method for the space-fractional problem. Particular attention is paid to the role played by the fractional operator in determining the solution behaviour and to the identification of crucial differences between the non-fractional and the fractional cases. We find a positive linear dependence of the depolarization peak height and a power law decay of notch and dome peak amplitudes for decreasing orders of the fractional operator. Furthermore, we establish a quadratic relationship in conduction velocity, and quantify the increasingly wider action potential foot and more pronounced dispersion of action potential duration, as the fractional order is decreased. A discussion of the physiological interpretation of the presented findings is made

Fractional models for the migration of biological cells in complex spatial domains

Travelling wave phenomena are observed in many biological applications. Mathematical theory of standard reaction-diffusion problems shows that simple partial differential equations exhibit travelling wave solutions with constant wavespeed and such models are used to describe, for example, waves of chemical concentrations, electrical signals, cell migration, waves of epidemics and population dynamics. However, as in the study of cell motion in complex spatial geometries, experimental data are often not consistent with constant wavespeed. Non-local spatial models are successfully used to model anomalous diffusion and spatial heterogeneity in different physical contexts. We develop a fractional model based on the Fisher--Kolmogoroff equation, analyse it for its wavespeed properties, and relate the numerical results obtained from our simulations to experimental data describing enteric neural crest-derived cells migrating along the intact gut of mouse embryos. The model proposed essentially combines fractional and standard diffusion in different regions of the spatial domain and qualitatively reproduces the behaviour of neural crest-derived cells observed in the caecum and the hindgut of mouse embryos during in vivo experiments. References R. J. Adler, R. E. Feldman and M. S. Taqqu. A practical guide to heavy tails: Statistical techniques and applications. Birkauser, 1998. I. J. Allan and D. F. Newgreen. The origin and differentiation of enteric neurons of the intestine of the fowl embryo. The American Journal of Anatomy, 157, 137--154, 1980. doi:10.1002/aja.1001570203 B. J. Binder, K. A. Landman, M. J. Simpson, M. Mariani and D. F. Newgreen. Modeling proliferative tissue growth: A general approach and an avian case study. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 78(3), 1--13, 2008. doi:10.1103/PhysRevE.78.031912 M. A. Breau, A. Dahmani, F. Broders--Bondon, J. P. Thiery and S. Dufour. $\beta 1$ integrins are required for the invasion of the caecum and proximal hindgut by enteric neural crest cells. Development, 136, 2791--2801, 2009. doi:10.1242/dev.031419 K. Burrage, N. Hale and D. Kay. An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM Journal on Scientific Computing, 34(4), A2145--A2172. doi:10.1137/110847007 N. R. Druckenbrod and M. L. Epstein. The patterns of neural crest advance in the cecum and colon. Developmental Biology, 287, 125--133, 2005. doi:10.1016/j.ydbio.2005.08.040 H. Engler. On the speed of spread for fractional reaction-diffusion equations. International Journal of Differential Equations, 2010, Article ID 315421, 2010. doi:10.1155/2010/315421 M. Ilic, F. Liu, I. Turner and V. Anh. Numerical approximation of a fractional-in-space diffusion equation (II)--with nonhomogeneous boundary conditions. Fractional Calculus and Applied Analysis, 9, 333--349, 2006. http://eprints.qut.edu.au/23835/ P. K. Maini, D. L. S. McElwain and D. Leavesley. Travelling waves in a wound healing assay. Applied Mathematics Letters, 17, 575--580, 2004. doi:10.1016/S0893-9659(04)90128-0 R. McLennan, L. Dyson, K. W. Prather, J. A. Morrison, R. E. Baker, P. K. Maini and P. M. Kulesa. Multiscale mechanisms of cell migration during development: theory and experiment. Development, 139, 2935--2944, 2012. doi:10.1242/dev.081471 J. D. Murray. Mathematical Biology I and II. Springer Verlag, 2003. M. J. Simpson, D. C. Zhang, M. Mariani, K. A. Landman and D. F. Newgreen. Cell proliferation drives neural crest cell invasion of the intestine. Developmental Biology, 302, 553--568, 2007. doi:10.1016/j.ydbio.2006.10.017 H. M. Young, A. J. Bergner, R. B. Anderson, H. Enomoto, J. Milbrandt, D. F. Newgreen and P. M. Whitington. Dynamics of neural crest-derived cell migration in the embryonic mouse gut. Developmental Biology, 270, 455--473, 2004. doi:10.1016/j.ydbio.2004.03.015 P. Zhuang, F. Liu, V. Anh and I. Turner. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM Journal on Numerical Analysis, 47(3), 1760--1781, 2009. doi:10.1137/08073059

Fractional models for the migration of biological cells in complex spatial domains

Travelling wave phenomena are observed in many biological applications. Mathematical theory of standard reaction-diffusion problems shows that simple partial differential equations exhibit travelling wave solutions with constant wavespeed and such models are used to describe, for example, waves of chemical concentrations, electrical signals, cell migration, waves of epidemics and population dynamics. However, as in the study of cell motion in complex spatial geometries, experimental data are often not consistent with constant wavespeed. Non-local spatial models have successfully been used to model anomalous diffusion and spatial heterogeneity in different physical contexts. In this paper, we develop a fractional model based on the Fisher-Kolmogoroff equation and analyse it for its wavespeed properties, attempting to relate the numerical results obtained from our simulations to experimental data describing enteric neural crest-derived cells migrating along the intact gut of mouse embryos. The model proposed essentially combines fractional and standard diffusion in different regions of the spatial domain and qualitatively reproduces the behaviour of neural crest-derived cells observed in the caecum and the hindgut of mouse embryos during in vivo experiments